Handbook of Nonlinear Partial Differential Equations, Second Edition

HANDBOOK OF

NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS SECOND EDITION

C8723FM.indd 1

11/14/11 12:25:50 PM

Handbooks of Mathematical Equations

Handbook of Linear Partial Differential Equations for Engineers and Scientists A. D. Polyanin, 2002 Handbook of First Order Partial Differential Equations A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, 2002 Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition A. D. Polyanin and V. F. Zaitsev, 2003 Handbook of Nonlinear Partial Differential Equations A. D. Polyanin and V. F. Zaitsev, 2004 Handbook of Integral Equations, 2nd Edition A. D. Polyanin and A. V. Manzhirov, 2008 See also: Handbook of Mathematics for Engineers and Scientists A. D. Polyanin and A. V. Manzhirov, 2007

C8723FM.indd 2

11/14/11 12:25:50 PM

HANDBOOK OF

NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS SECOND EDITION

Andrei D. Polyanin Valentin F. Zaitsev

C8723FM.indd 3

11/14/11 12:25:50 PM

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2012 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20111107 International Standard Book Number-13: 978-1-4200-8724-6 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page v

CONTENTS Preface to the new edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii Preface to the first edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviii Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi Some notations and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii

Part I. Exact Solutions of Nonlinear Partial Differential Equations 1. First-Order Quasilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Equations with Two Independent Variables Containing Arbitrary Parameters . . 1.1.1. Coefficients of Equations Contain Power-Law Functions . . . . . . . . . . . . . 1.1.2. Coefficients of Equations Contain Exponential Functions . . . . . . . . . . . . . 1.1.3. Coefficients of Equations Contain Hyperbolic Functions . . . . . . . . . . . . . . 1.1.4. Coefficients of Equations Contain Logarithmic Functions . . . . . . . . . . . . . 1.1.5. Coefficients of Equations Contain Trigonometric Functions . . . . . . . . . . . 1.2. Equations with Two Independent Variables Containing Arbitrary Functions . . . 1.2.1. Equations Contain Arbitrary Functions of One Variable . . . . . . . . . . . . . . 1.2.2. Equations Contain Arbitrary Functions of Two Variables . . . . . . . . . . . . . 1.3. Other Quasilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1. Equations with Three Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2. Equations with Arbitrary Number of Independent Variables . . . . . . . . . . . 2. First-Order Equations with Two Independent Variables Quadratic in Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Equations Containing Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂w 2.1.1. Equations of the Form ∂w ∂x ∂y = f (x, y, w) . . . . . . . . . . . . . . . . . . . . . . . . . . ∂w ∂w 2.1.2. Equations of the Form f (x, y, w) ∂w ∂x ∂y + g(x, y, w) ∂x = h(x, y, w) . . . . ∂w ∂w ∂w 2.1.3. Equations of the Form f (x, y, w) ∂w ∂x ∂y + g(x, y, w) ∂x + h(x, y, w) ∂y = s(x, y, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
∂w 2 2.1.4. Equations of the Form ∂w = g(x, y, w) . . . . . . . . . . . . ∂x + f (x, y, w) ∂y
2 ∂w 2.1.5. Equations of the Form ∂w + g(x, y, w) ∂w = h(x, y, w) ∂x + f (x, y, w) ∂x

∂y ∂w 2 ∂w 2 2.1.6. Equations of the Form f (x, y, w) ∂x + g(x, y, w) ∂y = h(x, y, w) .
2 ∂w 2.1.7. Equations of the Form f (x, y) ∂w + g(x, y) ∂w ∂x ∂x ∂y = h(x, y, w) . . . . . . 2.1.8. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂w 2.2.1. Equations of the Form ∂w ∂x ∂y = f (x, y, w) . . . . . . . . . . . . . . . . . . . . . . . . . .
∂w 2 2.2.2. Equations of the Form f (x, y) ∂w = h(x, y, w) . . . . . . ∂x + g(x, y, w) ∂y
2 ∂w 2.2.3. Equations of the Form ∂w + g(x, y, w) ∂w = h(x, y, w) ∂x + f (x, y, w) ∂x

∂y ∂w 2 ∂w 2 2.2.4. Equations of the Form f (x, y, w) ∂x + g(x, y, w) ∂y = h(x, y, w) .
2 ∂w 2.2.5. Equations of the Form ∂w + f (x, y, w) ∂w ∂x ∂x ∂y = g(x, y, w) . . . . . . . . . 2.2.6. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 3 3 11 14 16 17 19 19 30 35 35 39 43 43 43 45 47 51 59 63 69 73 77 77 79 86 89 92 95

v

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page v

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

vi

Page vi

CONTENTS

3. First-Order Nonlinear Equations with Two Independent Variables of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

3.1. Nonlinear Equations Containing Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . 99 3.1.1. Equations Contain the Fourth Powers of Derivatives . . . . . . . . . . . . . . . . . 99 3.1.2. Equations Contain Derivatives in Radicands . . . . . . . . . . . . . . . . . . . . . . . . 101 3.1.3. Equations Contain Arbitrary Powers of Derivatives . . . . . . . . . . . . . . . . . . 102 3.1.4. More Complicated Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2. Equations Containing Arbitrary Functions of Independent Variables . . . . . . . . . . 107 3.2.1. Equations Contain One Arbitrary Power of Derivative . . . . . . . . . . . . . . . . 107 3.2.2. Equations Contain Two or Three Arbitrary Powers of Derivatives . . . . . . 111 3.3. Equations Containing Arbitrary Functions of Derivatives . . . . . . . . . . . . . . . . . . . 3.3.1. Equations Contain Arbitrary Functions of One Variable . . . . . . . . . . . . . . 3.3.2. Equations Contain Arbitrary Functions of Two Variables . . . . . . . . . . . . . 3.3.3. Equations Contain Arbitrary Functions of Three Variables . . . . . . . . . . . . 3.3.4. Equations Contain Arbitrary Functions of Four Variables . . . . . . . . . . . . .

113 113 116 120 123

4. First-Order Nonlinear Equations with Three or More Independent Variables 125 4.1. Nonlinear Equations with Three Variables Quadratic in Derivatives . . . . . . . . . . 4.1.1. Equations Contain Squares of One or Two Derivatives . . . . . . . . . . . . . . . 4.1.2. Equations Contain Squares of Three Derivatives . . . . . . . . . . . . . . . . . . . . . 4.1.3. Equations Contain Products of Derivatives with Respect to Different Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4. Equations Contain Squares and Products of Derivatives . . . . . . . . . . . . . .

125 125 130 131 133

4.2. Other Nonlinear Equations with Three Variables Containing Parameters . . . . . . 4.2.1. Equations Cubic in Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Equations Contain Roots and Moduli of Derivatives . . . . . . . . . . . . . . . . . . 4.2.3. Equations Contain Arbitrary Powers of Derivatives . . . . . . . . . . . . . . . . . .

134 134 136 136

4.3. Nonlinear Equations with Three Variables Containing Arbitrary Functions . . . . 4.3.1. Equations Quadratic in Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Equations with Power Nonlinearity in Derivatives . . . . . . . . . . . . . . . . . . . 4.3.3. Equations with Arbitrary Dependence on Derivatives . . . . . . . . . . . . . . . . . 4.3.4. Nonlinear Equations of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 139 146 148 150

4.4. Nonlinear Equations with Four Independent Variables . . . . . . . . . . . . . . . . . . . . . . 154 4.4.1. Equations Quadratic in Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.4.2. Equations Contain Power-Law Functions of Derivatives . . . . . . . . . . . . . . 156 4.5. Nonlinear Equations with Arbitrary Number of Variables Containing Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.5.1. Equations Quadratic in Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.5.2. Equations with Power-Law Nonlinearity in Derivatives . . . . . . . . . . . . . . . 161 4.6. Nonlinear Equations with Arbitrary Number of Variables Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1. Equations Quadratic in Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2. Equations with Power-Law Nonlinearity in Derivatives . . . . . . . . . . . . . . . 4.6.3. Equations Contain Arbitrary Functions of Two Variables . . . . . . . . . . . . . 4.6.4. Nonlinear Equations of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

162 162 167 168 169

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page vi

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

CONTENTS

Page vii

vii

5. Second-Order Parabolic Equations with One Space Variable . . . . . . . . . . . . . . 175 5.1. Equations with Power-Law Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 ∂2w 2 . . . . . . . . . . . . . . . . . . . . . . 175 5.1.1. Equations of the Form ∂w ∂t = a ∂x2 + bw + cw ∂w ∂2w 2 3 ∂t = a ∂x2 + b0 + b1 w + b2 w + b3 w . . . . . . . . . . 2 ∂ w k m n .............. 5.1.3. Equations of the Form ∂w ∂t = a ∂x2 + bw + cw + sw 2 ∂ w 5.1.4. Equations of the Form ∂w ∂t = a ∂x2 + f (x, t, w) . . . . . . . . . . . . . . . . . . . . . . ∂2w k ∂w 5.1.5. Equations of the Form ∂w ∂t = a ∂x2 + bw ∂x + f (w) . . . . . . . . . . . . . . . . . ∂2w ∂w 5.1.6. Equations of the Form ∂w ∂t = a ∂x2 + f (x, t, w) ∂x + g(x, t, w) . . . . . . . . .
2 ∂ w ∂w 2 5.1.7. Equations of the Form ∂w + f (x, t, w) . . . . . . . . . . . . . ∂t = a ∂x2 + b ∂x
∂w ∂2w 5.1.8. Equations of the Form ∂t = a ∂x2 + f x, t, w, ∂w .................. ∂x
2w ∂w ∂ ∂w 5.1.9. Equations of the Form ∂t = awk ∂x2 + f x, t, w, ∂x . . . . . . . . . . . . . . .
∂ m ∂w 5.1.10. Equations of the Form ∂w ∂t = a ∂x w ∂x
. . . . . . . . . . . . . . . . . . . . . . . . ∂ m ∂w k 5.1.11. Equations of the Form ∂w ∂t = a ∂x w ∂x
+ bw . . . . . . . . . . . . . . . . . . . ∂ m ∂w k1 k2 k3 5.1.12. Equations of the Form ∂w ∂t = a ∂x w ∂x + bw + c1 w + c2 w + c3 w ∂ ∂w 5.1.13. Equations of the Form ∂w ∂t = ∂x f (w) ∂x
+ g(w) . . . . . . . . . . . . . . . . . . ∂w ∂ m 5.1.14. Equations of the Form ∂t = a ∂x w ∂w ∂x  + f (x, t, w) . . . . . . . . . . . . . .  ∂w ∂ ∂w 5.1.15. Equations of the Form ∂t = ∂x f (w) ∂x + g(w) ∂w ∂x . . .
. . . . . . . . . . . .   ∂w ∂ ∂w 5.1.16. Equations of the Form ∂t = ∂x f (w) ∂x + g x, t, w, ∂w .......... ∂x

5.1.2. Equations of the Form

5.2.

5.3.

5.4.

5.5.

5.6.

5.1.17. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equations with Exponential Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂2w λw + b e2λw . . . . . . . . . . . . . 5.2.1. Equations of the Form ∂w 1e 2 ∂t = a ∂x2 + b0 + b
∂ λw ∂w + f (w) . . . . . . . . . . . . . . . . . . . 5.2.2. Equations of the Form ∂w = a e ∂t ∂x ∂x   ∂ ∂w 5.2.3. Equations of the Form ∂w = f (w) ∂t ∂x ∂x + g(w) . . . . . . . . . . . . . . . . . . . 5.2.4. Other Equations Explicitly Independent of x and t . . . . . . . . . . . . . . . . . . . 5.2.5. Equations Explicitly Dependent on x and/or t . . . . . . . . . . . . . . . . . . . . . . . Equations with Hyperbolic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Equations Involving Hyperbolic Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. Equations Involving Hyperbolic Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3. Equations Involving Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4. Equations Involving Hyperbolic Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . Equations with Logarithmic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂2w 5.4.1. Equations of the Form ∂w ∂t = a ∂x2 + f (x, t, w) . . . . . . . . . . . . . . . . . . . . . . 5.4.2. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equations with Trigonometric Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1. Equations Involving Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2. Equations Involving Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3. Equations Involving Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4. Equations Involving Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5. Equations Involving Inverse Trigonometric Functions . . . . . . . . . . . . . . . . Equations Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂2w 5.6.1. Equations of the Form ∂w ∂t = a ∂x2 + f (x, t, w) . . . . . . . . . . . . . . . . . . . . . . 5.6.2. Equations of the Form

∂w ∂t

176 182 185 186 190 193 195 197 210 219 228 233 236 237 242 245 254 254 256 259 262 266 268 268 269 270 270 271 271 273 276 276 277 278 278 279 279 279

2

= a ∂∂xw2 + f (x, t) ∂w ∂x + g(x, t, w) . . . . . . . . . . . 284

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page vii

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

viii

Page viii

CONTENTS

5.6.4. 5.6.5. 5.6.6. 5.6.7. 5.6.8.

∂w ∂t ∂w ∂t ∂w ∂t ∂w ∂t

2

∂w ∂t

2 aw ∂∂xw2

= a ∂∂xw2 + f (x, t, w) ∂w + g(x, t, w) . . . . . . . . .
∂x ∂2w ∂w 2 Equations of the Form = a ∂x2 + b ∂x + f (x, t, w) . . . . . . . . . . . . .
2 2 Equations of the Form = a ∂∂xw2 + b ∂w + f (x, t, w) ∂w ∂x ∂x + g(x, t, w)
2 ∂ w ∂w 2 Equations of the Form = a ∂x2 + f (x, t, w) ∂x + g(x, t, w) ∂w ∂x + h(x, t, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
∂2w ∂w .................. Equations of the Form ∂w ∂t = a ∂x2 + f x, t, w, ∂x
2 ∂ w ∂w Equations of the Form ∂w ............. = f (x, t) + g x, t, w, 2 ∂t ∂x ∂x

5.6.3. Equations of the Form

5.6.9. Equations of the Form

=

+

∂w ∂t

f (x, t, w) ∂w ∂x + g(x, t, w)
2 ∂2 w b) ∂x2 + f (x, t, w) ∂w ∂x

286 290 293 294 298 299

. . . . . . . 303

5.6.10. Equations of the Form = (aw + + + h(x, t, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 g(x, t, w) ∂w ∂x 5.6.11. 5.6.12. 5.6.13. 5.6.14. 5.6.15.

Equations of the Form Equations of the Form Equations of the Form Equations of the Form Equations of the Form

∂w ∂t ∂w ∂t ∂w ∂t ∂w ∂t ∂w ∂t ∂w ∂t ∂w ∂t ∂w ∂t

2

= awm ∂∂xw2 + f (x, t) ∂w ∂x + g(x, t, w) . . . . . . .
∂ ∂w = a ∂x w ∂x + f (x, t) ∂w ∂x + g(x, t, w) . . . . .
∂ ∂w = a ∂x wm ∂w + f (x, t) ∂x
∂x + g(x, t, w) . . . ∂w ∂ λw = a ∂x e ∂x + f (x, t, w) . . . . . . . . . . . . . .  
∂ ∂w = ∂x f (w) ∂w .......... ∂x + g x, t, w, ∂x 2

= f (x, w) ∂∂xw2 . . . . . . . . . . . . . . . . . . . . . . . . . .
2 5.6.17. Equations of the Form = f (x, t, w) ∂∂xw2 + g x, t, w, ∂w .......... ∂x

∂w ∂ 2 w ∂w 5.6.18. Equations of the Form = f x, w, ∂x ∂x2 + g x, t, w, ∂x . . . . . . . 5.6.19. Evolution Equations Nonlinear in the Second Derivative . . . . . . . . . . . . . 5.6.20. Nonlinear Equations of the Thermal (Diffusion) Boundary Layer . . . . . 5.7. Nonlinear Schr¨odinger Equations and Related Equations . . . . . . . . . . . . . . . . . . . ∂2w 5.7.1. Equations of the Form i ∂w ∂t + ∂x2 + f (|w|)w = 0 Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 1 ∂ n ∂w 5.7.2. Equations of the Form i ∂w ∂t + xn ∂x x ∂x + f (|w|)w = 0 Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.3. Other Equations Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . 5.7.4. Equations with Cubic Nonlinearities Involving Arbitrary Functions . . . . 5.7.5. Equations of General Form Involving Arbitrary Functions of a Single Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.6. Equations of General Form Involving Arbitrary Functions of Two Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.16. Equations of the Form

6. Second-Order Parabolic Equations with Two or More Space Variables . . . . . 6.1. Equations with Two Space Variables Involving Power-Law Nonlinearities ....   ∂  ∂ ∂w ∂w p ....... 6.1.1. Equations of the Form ∂w = f (x) + g(y) + aw ∂t ∂x ∂x ∂y
∂y
∂ n ∂w + b ∂ wk ∂w . . . . . . . . . . . . . 6.1.2. Equations of the Form ∂w = a w ∂t ∂x ∂x ∂y ∂y   ∂   ∂ ∂w ∂w 6.1.3. Equations of the Form ∂w = f (w) + g(w) ∂t ∂x ∂x ∂y ∂y + h(w) . . . . . 6.1.4. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Equations with Two Space Variables Involving Exponential Nonlinearities ....   ∂  ∂w ∂ ∂w ∂w λw 6.2.1. Equations of the Form ∂t = ∂x f (x) ∂x + ∂y g(y) ∂y + ae ......

∂w ∂ ∂w ∂ ∂w βw λw 6.2.2. Equations of the Form ∂t = a ∂x e ∂x + b ∂y e ∂y + f (w) . . . . .

308 311 312 316 318 327 329 339 346 346 348 348 352 354 355 358 362 367 367 367 368 374 377 384 384 385

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page viii

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

CONTENTS

6.3. Other Equations with Two Space Variables Involving Arbitrary Parameters . . . . 6.3.1. Equations with Logarithmic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2. Equations with Trigonometrical Nonlinearities . . . . . . . . . . . . . . . . . . . . . . 6.4. Equations Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1. Heat and Mass Transfer Equations in Quiescent or Moving Media with Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . ∂ ∂w ∂ ∂w 6.4.2. Equations of the Form ∂w ∂t = ∂x f (x) ∂x  + ∂y g(y) ∂y + h(w) . . . . . . ∂ ∂w ∂ ∂w 6.4.3. Equations of the Form ∂w ∂t = ∂x f (w) ∂x + ∂y g(w) ∂y + h(t, w) . . . . 6.4.4. Other Equations Linear in the Highest Derivatives . . . . . . . . . . . . . . . . . . . 6.4.5. Nonlinear Diffusion Boundary Layer Equations . . . . . . . . . . . . . . . . . . . . . 6.4.6. Equations Nonlinear in the Highest Derivatives . . . . . . . . . . . . . . . . . . . . . . 6.5. Equations with Three or More Space Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1. Equations of Mass Transfer in Quiescent or Moving Media with Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2. Heat Equations with Power-Law or Exponential Temperature-Dependent Thermal Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3. Equations of Heat and Mass Transfer in Anisotropic Media . . . . . . . . . . . 6.5.4. Other Equations with Three Space Variables . . . . . . . . . . . . . . . . . . . . . . . . 6.5.5. Equations with n Space Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6. Nonlinear Schr¨odinger Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1. Two-Dimensional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2. Three and n-Dimensional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Page ix

ix 388 388 389 390 390 392 394 398 402 404 406 406 409 412 414 417 425 425 428

7. Second-Order Hyperbolic Equations with One Space Variable . . . . . . . . . . . . 433 7.1. Equations with Power-Law Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 2 2 7.1.1. Equations of the Form ∂∂tw2 = ∂∂xw2 + aw + bwn + cw2n–1 . . . . . . . . . . . . . . 433 7.1.2. Equations of the Form 7.1.3. Equations of the Form 7.1.4. Equations of the Form

∂2w ∂t2 ∂2w ∂t2 ∂2w ∂t2 ∂2w ∂t2 ∂2w ∂t2 ∂2w ∂t2 ∂2w ∂t2

2

= a ∂∂xw2 + f (x, t, w) . . . . . . . . . . . . . . . . . . . . . 436
2 = a ∂∂xw2 + f x, t, w, ∂w . . . . . . . . . . . . . . . . . 439 ∂x
∂2w ∂w = f (x) ∂x2 + g x, t, w, ∂x . . . . . . . . . . . . . . 442 2

= awn ∂∂xw2 + f (x, w) . . . . . . . . . . . . . . . . . . . .
∂ 7.1.6. Equations of the Form = a ∂x wn ∂w ......................... ∂x
∂w ∂ n 7.1.7. Equations of the Form = a ∂x w ∂x + bwk . . . . . . . . . . . . . . . . . . .
∂ k1 k2 k3 . . . 7.1.8. Equations of the Form = a ∂x wn ∂w ∂x + b1 w + b2 w + b3 w 7.1.9. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Equations with Exponential Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 7.2.1. Equations of the Form ∂∂tw2 = a ∂∂xw2 + beβw + ceγw . . . . . . . . . . . . . . . . . . . 7.1.5. Equations of the Form

∂2w ∂t2 ∂2w ∂t2

2

= a ∂∂xw2 + f (x, t, w) . . . . . . . . . . . . . . . . . . . . .
2 .............. 7.2.3. Equations of the Form = f (x) ∂∂xw2 + g x, t, w, ∂w ∂x 7.2.4. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Other Equations Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1. Equations with Hyperbolic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2. Equations with Logarithmic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3. Sine-Gordon Equation and Other Equations with Trigonometric Nonlinearities . . . . . . . . .2 . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . ∂ ∂w 7.3.4. Equations of the Form ∂∂tw2 + a ∂w ∂t = ∂x f (w) ∂x . . . . . . . . . . . . . . . . . . .   2 ∂ ∂w 7.3.5. Equations of the Form ∂∂tw2 + f (w) ∂w ∂t = ∂x g(w) ∂x . . . . . . . . . . . . . . . . 7.2.2. Equations of the Form

447

450 454 458 460 469 469 472 475 480 485 485 486 490 494 496

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page ix

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

x

Page x

CONTENTS

7.4. Equations Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 7.4.1. Equations of the Form ∂∂tw2 = a ∂∂xw2 + f (x, t, w) . . . . . . . . . . . . . . . . . . . . .
2 2 7.4.2. Equations of the Form ∂∂tw2 = a ∂∂xw2 + f x, t, w, ∂w ................. ∂x
∂2w ∂2w ∂w 7.4.3. Equations of the Form ∂t2 = f (x) ∂x2 + g x, t, w, ∂x . . . . . . . . . . . . . .
2 2 7.4.4. Equations of the Form ∂∂tw2 = f (w) ∂∂xw2 + g x, t, w, ∂w .............. ∂x
∂2w ∂2w ∂w 7.4.5. Equations of the Form ∂t2 = f (x, w) ∂x2 + g x, t, w, ∂x . . . . . . . . . . . .
2 2 7.4.6. Equations of the Form ∂∂tw2 = f (t, w) ∂∂xw2 + g x, t, w, ∂w ............ ∂x 7.4.7. Other Equations Linear in the Highest Derivatives . . . . . . . . . . . . . . . . . . .
∂2w ∂w = F x, y, w, ∂w 7.5. Equations of the Form ∂x∂y ......................... ∂x , ∂y

499 499 505 511 517 526 529 530 540

∂2w

7.5.1. Equations Involving Arbitrary Parameters of the Form ∂x∂y = f (w) . . . . 540 7.5.2. Other Equations Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . 544 7.5.3. Equations Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

8. Second-Order Hyperbolic Equations with Two or More Space Variables . . . 8.1. Equations with Two Space Variables Involving Power-Law Nonlinearities . . . .   ∂  2 ∂ ∂w p ...... f (x) ∂w 8.1.1. Equations of the Form ∂∂tw2 = ∂x ∂x + ∂y g(y) ∂y + aw

2 ∂ ∂ k ∂w 8.1.2. Equations of the Form ∂∂tw2 = a ∂x wn ∂w ............ ∂x + b ∂y w ∂y     2 ∂ ∂ ∂w 8.1.3. Equations of the Form ∂∂tw2 = ∂x f (w) ∂w ........... ∂x + ∂y g(w) ∂y 8.1.4. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Equations with Two Space Variables Involving Exponential Nonlinearities . . . .   ∂  2 ∂ ∂w λw . . . . . 8.2.1. Equations of the Form ∂∂tw2 = ∂x f (x) ∂w ∂x + ∂y g(y) ∂y + ae

2 ∂ ∂ λw ∂w . . . . . . . . . . . 8.2.2. Equations of the Form ∂∂tw2 = a ∂x eβw ∂w ∂x + b ∂y e ∂y 8.2.3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Nonlinear Telegraph Equations with Two Space Variables . . . . . . . . . . . . . . . . . . 8.3.1. Equations Involving Power-Law Nonlinearities . . . . . . . . . . . . . . . . . . . . . . 8.3.2. Equations Involving Exponential Nonlinearities . . . . . . . . . . . . . . . . . . . . . 8.4. Equations with Two Space Variables Involving Arbitrary Functions . . . . . . . . . .   ∂  2 ∂ ∂w f (x) ∂w 8.4.1. Equations of the Form ∂∂tw2 = ∂x ∂x + ∂y g(y) ∂y + h(w) . . . . . .   ∂   2 ∂ ∂w 8.4.2. Equations of the Form ∂∂tw2 = ∂x f (w) ∂w ∂x + ∂y g(w) ∂y + h(w) . . . . . 8.4.3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5. Equations with Three Space Variables Involving Arbitrary Parameters . . . . . . . .   ∂  ∂  2 ∂ ∂w ∂w f (x) ∂w 8.5.1. Equations of the Form ∂∂tw2 = ∂x ∂x + ∂y g(y) ∂y + ∂z h(z) ∂z + awp . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . ∂ ∂ ∂w ∂ ∂w 8.5.2. Equations of the Form ∂∂tw2 = ∂x f (x) ∂w ∂x + ∂y g(y) ∂y + ∂z h(z) ∂z + aeλw . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . ∂ ∂ ∂ m ∂w k ∂w wn ∂w 8.5.3. Equations of the Form ∂∂tw2 = a ∂x ∂x +b ∂y w ∂y +c ∂z w ∂z + swp . . . . . . . . . . . . . . . . . .2. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . ∂ ∂ λ2 w ∂w + 8.5.4. Equations of the Form ∂∂tw2 = a ∂x eλ1 w ∂w ∂x + b ∂y e ∂y
∂ βw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c ∂z eλ3 w ∂w + se ∂z 8.6. Equations with Three or More Space Variables Involving Arbitrary Functions .     2 ∂ ∂ 8.6.1. Equations of the Form ∂∂tw2 = ∂x f1 (x) ∂w + ∂y f2 (y) ∂w + ∂x ∂y   ∂ ∂w ∂z f3 (z) ∂z + g(w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

553 553 553 555 565 570 574 574 577 582 583 583 587 589 589 593 599 604 604 606 608 615 624 624

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page x

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

CONTENTS

Page xi

xi

    2 ∂ ∂ f1 (w) ∂w + ∂y f2 (w) ∂w + 8.6.2. Equations of the Form ∂∂tw2 = ∂x ∂x ∂y   ∂ ∂w ∂z f3 (w) ∂z + g(w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 8.6.3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

9. Second-Order Elliptic Equations with Two Space Variables . . . . . . . . . . . . . . . 641 9.1. Equations with Power-Law Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 2 2 9.1.1. Equations of the Form ∂∂xw2 + ∂∂yw2 = aw + bwn + cw2n–1 . . . . . . . . . . . . . . 641 9.1.2. Equations of the Form 9.1.3. Equations of the Form

2 ∂2w + ∂∂yw2 = f (x, y, w) . . . . . . . . . . . . . . . . . . . . . . ∂x2
2 ∂2w ∂w + a ∂∂yw2 = F x, y, w, ∂w ............ ∂x , ∂y ∂x2  ∂   ∂ ∂w ∂w + f2 (x, y) ∂y = g(w) . . . . . . ∂x f1 (x, y) ∂x  ∂ ∂y   ∂ ∂w ∂w ∂x f1 (w) ∂x + ∂y f2 (w) ∂y = g(w) . . . . . . . . .

9.1.4. Equations of the Form 9.1.5. Equations of the Form 9.1.6. Other Equations Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . 9.2. Equations with Exponential Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 9.2.1. Equations of the Form ∂∂xw2 + ∂∂yw2 = a + beβw + ceγw . . . . . . . . . . . . . . . . . 9.2.2. Equations of the Form

2 ∂2w + ∂∂yw2 = f (x, y, w) . . . . . . . . . . . . . . . . . . . . . . ∂x2  ∂   ∂ ∂w + ∂y f2 (x, y) ∂w = g(w) . . . . . . ∂x f1 (x, y) ∂x ∂y  ∂  ∂ ∂w ∂w ∂x f1 (w) ∂x + ∂y f2 (w) ∂y = g(w) . . . . . . . . .

9.2.3. Equations of the Form 9.2.4. Equations of the Form 9.2.5. Other Equations Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . 9.3. Equations Involving Other Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1. Equations with Hyperbolic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2. Equations with Logarithmic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3. Equations with Trigonometric Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . 9.4. Equations Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 9.4.1. Equations of the Form ∂∂xw2 + ∂∂yw2 = F (x, y, w) . . . . . . . . . . . . . . . . . . . . . .
2 2 ∂w 9.4.2. Equations of the Form a ∂∂xw2 + b ∂∂yw2 = F x, y, w, ∂w ∂x , ∂y . . . . . . . . . . . .   ∂ f (x) ∂w + 9.4.3. Heat and Mass Transfer Equations of the Form ∂x ∂x   ∂ ∂w ............ ∂y g(y) ∂y = h(w) . . . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .  ∂ ∂w ∂ ∂w 9.4.4. Equations of the Form ∂x f (x, y, w) ∂x + ∂y g(x, y, w) ∂y = h(x, y, w) 9.4.5. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Second-Order Elliptic Equations with Three or More Space Variables . . . . 10.1. Equations with Three Space Variables Involving ..   ∂ Power-Law  ∂Nonlinearities   ∂ ∂w ∂w p 10.1.1. Equations of the Form ∂x f (x) ∂w + g(y) + h(z) = aw ∂x  ∂y  ∂y  ∂z  ∂z   ∂ ∂ ∂w ∂ ∂w 10.1.2. Equations of the Form ∂x f (w) ∂w ∂x + ∂y g(w) ∂y + ∂z g(w) ∂z = 0

644 645 646 648 654 662 662 664 665 668 671 675 675 677 680 682 682 690 697 699 707 713 713 713 716

10.2. Equations with Three Space Variables Involving . . 722   ∂  Exponential  ∂ Nonlinearities  ∂ ∂w ∂w ∂w λw 10.2.1. Equations of the Form ∂x f (x) ∂x + ∂y g(y) ∂y + ∂z h(z) ∂z = ae 722

∂ ∂w ∂ ∂w λ w λ w 1 2 10.2.2. Equations of the Form a1 ∂x e ∂x + a2 ∂y e ∂y +
∂ ∂w λ w βw 2 a3 ∂y e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 ∂y = be 10.3. Three-Dimensional Equations Involving Arbitrary Functions . . . . . . . .. . . . . . . 730 ∂ 10.3.1. Heat and Mass Transfer Equations of the Form ∂x f1 (x) ∂w ∂x +     ∂ ∂w ∂ ∂w ∂y f2 (y) ∂y + ∂z f3 (z) ∂z = g(w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730 10.3.2. Heat and Mass Transfer Equations with Complicating Factors . . . . . . . 734

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xi

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

xii

Page xii

CONTENTS

10.3.3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 10.4. Equations with n Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739     ∂ ∂w ∂w 10.4.1. Equations of the Form ∂x f1 (x1 ) ∂x + · · · + ∂x∂ n fn (xn ) ∂x = n 1 1 g(x1 , . . . , xn , w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 10.4.2. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 11. Second-Order Equations Involving Mixed Derivatives and Some Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 11.1. Equations Linear in the Mixed Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1. Calogero Equation and Related Equations . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2. Khokhlov–Zabolotskaya and Related Equations . . . . . . . . . . . . . . . . . . . 11.1.3. Equation of Unsteady Transonic Gas Flows . . . . . . . . . . . . . . . . . . . . . . .
∂w ∂ 2 w ∂w ∂w ∂2 w 11.1.4. Equations of the Form ∂w ........ ∂y ∂x∂y – ∂x ∂y 2 = F x, y, ∂x , ∂y 11.1.5. Other Equations with Two Independent Variables . . . . . . . . . . . . . . . . . . 11.1.6. Other Equations with Three and More Independent Variables . . . . . . .

745 745 749 756 760 762 770

11.2. Equations Quadratic in the Highest Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 11.2.1. Equations of the Form ∂∂xw2 ∂∂yw2 = F (x, y) . . . . . . . . . . . . . . . . . . . . . . . . .
2 2 ∂2 w 2 11.2.2. Monge–Amp`ere Equation ∂x∂y – ∂∂xw2 ∂∂yw2 = F (x, y) . . . . . . . . . . . .
∂2 w ∂2w
∂2w 2 ∂w .... 11.2.3. Equations of the Form ∂x∂y – ∂x2 ∂y2 = F x, y, w, ∂w ∂x , ∂y
∂2w 2 ∂2w ∂2w 11.2.4. Equations of the Form ∂x∂y = f (x, y) ∂x2 ∂y2 + g(x, y) . . . . . . . . . 11.2.5. Other Equations with Two Independent Variables . . . . . . . . . . . . . . . . . . 11.2.6. Pleba´nski Heavenly Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

772 772 774 787 794 798 802

11.3. Bellman Type Equations and Related Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 805 11.3.1. Equations with Quadratic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . 805 11.3.2. Equations with Power-Law Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . 808 12. Second-Order Equations of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811 12.1. Equations Involving the First Derivative in t . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
∂w ∂ 2 w 12.1.1. Equations of the Form ∂w ...................... ∂t = F w, ∂x , ∂x2
2 ∂w ∂ w 12.1.2. Equations of the Form ∂w .................... ∂t = F t, w, ∂x , ∂x2
∂w ∂w ∂ 2 w 12.1.3. Equations of the Form ∂t = F x, w, ∂x , ∂x2 . . . . . . . . . . . . . . . . . . . .
∂w ∂ 2 w 12.1.4. Equations of the Form ∂w .................. ∂t = F x, t, w, ∂x , ∂x2
∂w ∂w ∂ 2 w 12.1.5. Equations of the Form F x, t, w, ∂t , ∂x , ∂x2 = 0 . . . . . . . . . . . . . . . . 12.1.6. Equations with Three Independent Variables . . . . . . . . . . . . . . . . . . . . . .

12.2. Equations Involving Two or More Second Derivatives . . . . . . . . . . . . . . . . . . . . .
2 ∂2 w 12.2.1. Equations of the Form ∂∂tw2 = F w, ∂w ..................... ∂x , ∂x2
∂2w ∂w ∂w ∂ 2 w 12.2.2. Equations of the Form ∂t2 = F x, t, w, ∂x , ∂t , ∂x2 . . . . . . . . . . . . . 12.2.3. Equations Linear in the Mixed Derivative . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.4. Equations with Two Independent Variables, Nonlinear in Two or More Highest Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.5. Equations with n Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . .

811 811 820 825 830 836 838 839 839 843 847 849 853

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xii

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

CONTENTS

Page xiii

xiii

13. Third-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1. Equations Involving the First Derivative in t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂3w ∂w 13.1.1. Korteweg–de Vries Equation ∂w ∂t + a ∂x3 + bw ∂x = 0 . . . . . . . . . . . . . . 13.1.2. Cylindrical, Spherical, and Modified Korteweg–de Vries Equations . . ∂3w ∂w 13.1.3. Generalized Korteweg–de Vries Equation ∂w ∂t + a ∂x3 + f (w) ∂x = 0 . 13.1.4. Equations Reducible to the Korteweg–de Vries Equation . . . . . . . . . . .
∂3w ∂w 13.1.5. Equations of the Form ∂w ∂t + a ∂x3 + F w, ∂x = 0 . . . . . . . . . . . . . . . .
∂3w ∂w 13.1.6. Equations of the Form ∂w ∂t + a ∂x3 + F x, t, w, ∂x = 0 . . . . . . . . . . . .
∂w ∂ 2 w ∂ 3 w ............... 13.1.7. Equations of the Form ∂w ∂t = F x, w, ∂x , ∂x2 , ∂x3 13.2. Equations Involving the Second Derivative in t . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1. Equations with Quadratic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3. Hydrodynamic Boundary Layer Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1. Steady Hydrodynamic Boundary Layer Equations for a Newtonian Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2. Steady Boundary Layer Equations for Non-Newtonian Fluids . . . . . . . 13.3.3. Unsteady Boundary Layer Equations for a Newtonian Fluid . . . . . . . . . 13.3.4. Unsteady Boundary Layer Equations for Non-Newtonian Fluids . . . . . 13.3.5. Related Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4. Equations of Motion of Ideal Fluid (Euler Equations) . . . . . . . . . . . . . . . . . . . . . 13.4.1. Stationary Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2. Nonstationary Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5. Other Third-Order Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.1. Equations Involving Second-Order Mixed Derivatives . . . . . . . . . . . . . . 13.5.2. Equations Involving Third-Order Mixed Derivatives . . . . . . . . . . . . . . . 13.5.3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

857 857 857 866 871 875 879

14. Fourth-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1. Equations Involving the First Derivative in t . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
∂4w ∂w 14.1.1. Equations of the Form ∂w ............... ∂t = a ∂x4 + F x, t, w, ∂x 14.1.2. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2. Equations Involving the Second Derivative in t . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1. Boussinesq Equation and Its Modifications . . . . . . . . . . . . . . . . . . . . . . . 14.2.2. Other Equations with Quadratic Nonlinearities . . . . . . . . . . . . . . . . . . . . 14.2.3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3. Equations Involving Mixed Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1. Kadomtsev–Petviashvili Equation and Related Equations . . . . . . . . . . . 14.3.2. Stationary Hydrodynamic Equations (Navier–Stokes Equations) . . . . . 14.3.3. Nonstationary Hydrodynamic Equations (Navier–Stokes Equations) . 14.3.4. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

977 977 977 982 987 987 993 998 1000 1000 1003 1013 1028

15. Equations of Higher Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1. Equations Involving the First Derivative in t and Linear in the Highest Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1. Fifth-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.2. Some Equations with Sixth- to Ninth-Order . . . . . . . . . . . . . . . . . . . . . . . ∂nw 15.1.3. Equations of the Form ∂w n + f (x, t, w) . . . . . . . . . . . . . . . . . . . . ∂t = a ∂x ∂nw ∂w 15.1.4. Equations of the Form ∂w = a ∂t ∂xn + f (w) ∂x . . . . . . . . . . . . . . . . . . . . .

1031

882 884 896 896 900 903 903 911 917 930 935 938 938 942 949 949 958 973

1031 1031 1039 1043 1044

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xiii

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

xiv

Page xiv

CONTENTS ∂w ∂t ∂w ∂t ∂w ∂t ∂w ∂t

n

∂w = a ∂∂xw n + f (x, t, w) ∂x + g(x, t, w) . . . . . . .
n ∂w = a ∂∂xw ............... n + F x, t, w, ∂x
n ∂w ∂ n–1 w 15.1.7. Equations of the Form = a ∂∂xw ..... n + F x, t, w, ∂x , . . . , n–1 ∂x ∂nw ∂w = aw ∂xn + f (x, t, w) ∂x + g(x, t, w) . . . . . 15.1.8. Equations of the Form 15.1.9. Other Equations Involving Arbitrary Parameters and/or Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.10. Nonlinear Equations Involving Arbitrary Linear Differential Operators 15.1.11. Equations of the Burgers and the Korteweg–de Vries Hierarchies and Related Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15.1.5. Equations of the Form 15.1.6. Equations of the Form

15.2. General Form Equations Involving the First Derivative in
t . . . . . . . . . . . . . . . . . ∂w ∂nw 15.2.1. Equations of the Form ∂w .................. ∂t = F w, ∂x , . . . , ∂xn
∂w ∂w ∂nw 15.2.2. Equations of the Form ∂t = F t, w, ∂x , . . . , ∂xn . . . . . . . . . . . . . . . .
∂w ∂nw 15.2.3. Equations of the Form ∂w ∂t = F x, w, ∂x , . . . , ∂xn
. . . . . . . . . . . . . . . ∂w ∂nw .............. 15.2.4. Equations of the Form ∂w ∂t = F x, t, w, ∂x , . . . , ∂xn

15.3. Equations Involving the Second Derivative in t . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 n 15.3.1. Equations of the Form ∂∂tw2 = a ∂∂xw n + f (x, t, w) . . . . . . . . . . . . . . . . . . .
2 n ∂w 15.3.2. Equations of the Form ∂∂tw2 = a ∂∂xw .............. n + F x, t, w, ∂x n–1
∂2w ∂nw ∂w 15.3.3. Equations of the Form ∂t2 = a ∂xn + F x, t, w, ∂x , . . . , ∂∂xn–1w . . . . ∂2w

1047 1051 1057 1059 1062 1065 1067 1070 1070 1077 1079 1084 1088 1088 1089 1094

∂nw ∂xn

15.3.4. Equations of the Form ∂t2 = aw + f (x, t, w) ∂w ∂x + g(x, t, w) . . . . 1098 15.3.5. Other Equations Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . 1099 15.3.6. Equations Involving Arbitrary Differential Operators . . . . . . . . . . . . . . . 1101

15.4. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103 15.4.1. Equations Involving Mixed Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103 n ∂mw 15.4.2. Equations Involving ∂∂xw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110 n and ∂y m 16. Systems of Two First-Order Partial Differential Equations . . . . . . . . . . . . . . . 1115 ∂w 16.1. Systems of the Form ∂u ∂x = F (u, w), ∂t = G(u, w) . . . . . . . . . . . . . . . . . . . . . . . . 1115 16.1.1. Systems Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 1115 16.1.2. Systems Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117

16.2. Other Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122 16.2.1. Gas Dynamic Type Systems Linearizable with the Hodograph Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122 16.2.2. Other Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1130 17. Systems of Two Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133 2

2

∂ u ∂w ∂ w 17.1. Systems of the Form ∂u ∂t = a ∂x2 + F (u, w), ∂t = b ∂x2 + G(u, w) . . . . . . . . . . 17.1.1. Arbitrary Functions Depend on a Linear Combination of the Unknowns 17.1.2. Arbitrary Functions Depend on the Ratio of the Unknowns . . . . . . . . . 17.1.3. Arbitrary Functions Depend on the Product of Powers of the Unknowns 17.1.4. Arbitrary Functions Depend on Sum or Difference of Squares of the Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.5. Arbitrary Functions Depend on the Unknowns in a Complex Way . . . 17.1.6. Some Systems Depending on Arbitrary Parameters . . . . . . . . . . . . . . . .

1133 1133 1137 1145 1146 1148 1150

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xiv

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xv

xv

CONTENTS



a ∂ ∂w b ∂ n ∂u n ∂w 17.2. Systems of the Form ∂u ∂t = xn ∂x x ∂x + F (u, w), ∂t = xn ∂x x ∂x + G(u, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1. Arbitrary Functions Depend on a Linear Combination of the Unknowns 17.2.2. Arbitrary Functions Depend on the Ratio of the Unknowns . . . . . . . . . 17.2.3. Arbitrary Functions Depend on the Product of Powers of the Unknowns 17.2.4. Arbitrary Functions Depend on Sum or Difference of Squares of the Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.5. Arbitrary Functions Have Different Arguments . . . . . . . . . . . . . . . . . . . .

1163 1164

17.3. Other Systems of Two Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1. Second-Order Equations Involving Real Functions of Real Variables . 17.3.2. Second-Order Nonlinear Equations of Laser Systems . . . . . . . . . . . . . . 17.3.3. Systems Involving Third-Order Evolution Equations . . . . . . . . . . . . . . .

1165 1165 1170 1172

1156 1157 1159 1162

18. Systems of Two Second-Order Klein–Gordon Type Hyperbolic Equations . 1173 2

2

2

2

18.1. Systems of the Form ∂∂tu2 = a ∂∂xu2 + F (u, w), ∂∂tw2 = b ∂∂xw2 + G(u, w) . . . . . . . . . 18.1.1. Arbitrary Functions Depend on a Linear Combination of the Unknowns 18.1.2. Arbitrary Functions Depend on the Ratio of the Unknowns . . . . . . . . . 18.1.3. Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 ∂ ∂2 w b ∂ n ∂w 18.2. Systems of the Form ∂∂tu2 = xan ∂x xn ∂u ∂x + F (u, w), ∂t2 = xn ∂x x ∂x + G(u, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1. Arbitrary Functions Depend on a Linear Combination of the Unknowns 18.2.2. Arbitrary Functions Depend on the Ratio of the Unknowns . . . . . . . . . 18.2.3. Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1173 1173 1175 1177 1178 1178 1181 1183

19. Systems of Two Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185 19.1. Systems of the Form ∆u = F (u, w), ∆w = G(u, w) . . . . . . . . . . . . . . . . . . . . . . . 19.1.1. Arbitrary Functions Depend on a Linear Combination of the Unknowns 19.1.2. Arbitrary Functions Depend on the Ratio of the Unknowns . . . . . . . . . 19.1.3. Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1185 1185 1187 1189

19.2. Other Systems of Two Second-Order Elliptic Equations . . . . . . . . . . . . . . . . . . . 1191 19.3. Von K´arm´an Equations (Fourth-Order Elliptic Equations) . . . . . . . . . . . . . . . . . . 1192 20. First-Order Hydrodynamic and Other Systems Involving Three or More Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197 20.1. Equations of Motion of Ideal Fluid (Euler Equations) . . . . . . . . . . . . . . . . . . . . . 20.1.1. Euler Equations in Various Coordinate Systems . . . . . . . . . . . . . . . . . . . 20.1.2. Two-Dimensional Euler Equations for Incompressible Ideal Fluid (Plane Flows) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1.3. Other Solutions with Two Nonzero Components of the Fluid Velocity 20.1.4. Rotationally Symmetric Motions of Fluid . . . . . . . . . . . . . . . . . . . . . . . . . 20.1.5. Euler Equations for Barotropic Gas Flow . . . . . . . . . . . . . . . . . . . . . . . . .

1197 1197 1198 1199 1200 1201

20.2. Adiabatic Gas Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.2. One-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.3. Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.4. Three-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1204 1204 1205 1215 1219

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xv

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

xvi

Page xvi

CONTENTS

20.3. Systems Describing Fluid Flows in the Atmosphere, Seas, and Oceans . . . . . . 20.3.1. Equations of Breezes and Monsoons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.2. Equations of Atmospheric Circulation in the Equatorial Region . . . . . . 20.3.3. Equations of Dynamic Convection in the Sea . . . . . . . . . . . . . . . . . . . . . . 20.3.4. Equations of Flows in the Baroclinic Layer of the Ocean . . . . . . . . . . . 20.4. Chromatography Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.1. Langmuir Isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.2. Generalized Langmuir Isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.3. Power Isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.4. Exponential Isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5. Other Hydrodynamic-Type Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5.1. Hydrodynamic-Type Systems of Diagonal Form . . . . . . . . . . . . . . . . . . . 20.5.2. Hydrodynamic-Type Systems of Nondiagonal Form . . . . . . . . . . . . . . . 20.6. Ideal Plasticity with the von Mises Yield Criterion . . . . . . . . . . . . . . . . . . . . . . . . 20.6.1. Two-Dimensional Equations. Plane Case . . . . . . . . . . . . . . . . . . . . . . . . . 20.6.2. Axisymmetric Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6.3. Three-Dimensional Equations. Steady-State Case . . . . . . . . . . . . . . . . . 20.6.4. Dynamic Case. Two-Dimensional Equations . . . . . . . . . . . . . . . . . . . . . .

1223 1223 1225 1227 1229 1231 1231 1233 1234 1235 1236 1236 1236 1238 1238 1239 1240 1244

21. Navier–Stokes and Related Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247 21.1. Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247 21.1.1. Navier–Stokes Equations in Various Coordinate Systems . . . . . . . . . . . 1247 21.1.2. General Properties of the Navier–Stokes Equations . . . . . . . . . . . . . . . . 1249 21.2. Solutions with One Nonzero Component of the Fluid Velocity . . . . . . . . . . . . . . 1251 21.2.1. Unidirectional Plane Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1251 21.2.2. Unidirectional Flows in Tubes of Various Cross-Sections. External Flow Around a Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253 21.2.3. One-Dimensional Rotation Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 1257 21.2.4. Purely Radial Fluid Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1260 21.3. Solutions with Two Nonzero Components of the Fluid Velocity . . . . . . . . . . . . . 1263 21.3.1. Two-Dimensional Solutions in the Rectangular Cartesian Coordinates (Plane Flows) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263 21.3.2. Two-Dimensional Solutions in the Cylindrical Coordinates (Plane Flows) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1270 21.3.3. Axisymmetric Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274 21.3.4. Other Fluid Flows with Two-Nonzero Velocity Components . . . . . . . . 1282 21.4. Solutions with Three Nonzero Fluid Velocity Components Dependent on Two Space Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285 21.4.1. Quasi-plane Flows (with the Fluid Velocity Components Independent of z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285 21.4.2. Cylindrical and Conical Vortex Flows. Von K´arm´an-Type Rotationally Symmetric Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1290 21.4.3. Rotationally Symmetric Motions of General Form . . . . . . . . . . . . . . . . . 1297 21.5. Solutions with Three Nonzero Fluid Velocity Components Dependent on Three Space Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1302 21.5.1. Three-Dimensional Stagnation-Point Type Flows . . . . . . . . . . . . . . . . . . 1302 21.5.2. Solutions with Linear Dependence of the Velocity Components on Two Space Variables. Axial Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1303 21.5.3. Solutions with Linear Dependence of the Velocity Components on Two Space Variables. General Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xvi

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

CONTENTS

Page xvii

xvii

21.5.4. Solutions with the Linear Dependence of the Velocity Components on One Space Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323 21.5.5. Other Three-Dimensional Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327 21.6. Convective Fluid Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6.1. Equations for Convective Fluid Motions . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6.2. Steady-State Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6.3. Unsteady Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1329 1329 1329 1331

21.7. Boundary Layer Equations (Prandtl Equations) . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7.1. Equations and Boundary Conditions. Stream Function . . . . . . . . . . . . . 21.7.2. Self-Similar Solutions of Some Boundary Layer Problems . . . . . . . . . . 21.7.3. Other Solutions of the Boundary Layer Equations . . . . . . . . . . . . . . . . .

1333 1333 1333 1336

22. Systems of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337 22.1. Nonlinear Systems of Two Equations Involving the First Derivatives with Respect to t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337 22.2. Nonlinear Systems of Two Equations Involving the Second Derivatives with Respect to t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344 22.3. Other Nonlinear Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347 22.4. Nonlinear Systems of Many Equations Involving the First Derivatives with Respect to t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1348

Part II. Exact Methods for Nonlinear Partial Differential Equations

1353

23. Methods for Solving First-Order Quasilinear Equations . . . . . . . . . . . . . . . . . 1355 23.1. Characteristic System. General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355 23.1.1. Equations with Two Independent Variables. General Solution . . . . . . . 1355 23.1.2. Quasilinear Equations with n Independent Variables . . . . . . . . . . . . . . . 1356 23.2. Cauchy Problem. Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . 23.2.1. Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2.2. Procedure of Solving the Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . 23.2.3. Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1357 1357 1358 1359

23.3. Qualitative Features and Discontinuous Solutions of Quasilinear Equations . . 23.3.1. Model Equation of Gas Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3.2. Solution of the Cauchy Problem. Rarefaction Wave. Wave “Overturn” 23.3.3. Shock Waves. Jump Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3.4. Utilization of Integral Relations for Determining Generalized Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3.5. Conservation Laws. Viscosity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3.6. Hopf’s Formula for the Generalized Solution . . . . . . . . . . . . . . . . . . . . . . 23.3.7. Problem of Propagation of a Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1360 1360 1360 1362 1364 1365 1366 1367

23.4. Quasilinear Equations of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4.1. Quasilinear Equations in Conservative Form . . . . . . . . . . . . . . . . . . . . . . 23.4.2. Generalized Solution. Jump Condition and Stability Condition . . . . . . 23.4.3. Method for Constructing Stable Generalized Solutions . . . . . . . . . . . . .

1368 1368 1369 1370

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xvii

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

xviii

Page xviii

CONTENTS

24. Methods for Solving First-Order Nonlinear Equations . . . . . . . . . . . . . . . . . . . 24.1. Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.1.1. Complete, General, and Singular Integrals . . . . . . . . . . . . . . . . . . . . . . . . 24.1.2. Method of Separation of Variables. Equations of Special Form . . . . . . 24.1.3. Lagrange–Charpit Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.1.4. Construction of a Complete Integral with the Aid of Two First Integrals 24.1.5. Case where the Equation Does Not Depend on w Explicitly . . . . . . . . . 24.1.6. Hamilton–Jacobi Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2. Cauchy Problem. Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . 24.2.1. Statement of the Problem. Solution Procedure . . . . . . . . . . . . . . . . . . . . 24.2.2. Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2.3. Cauchy Problem for the Hamilton–Jacobi Equation . . . . . . . . . . . . . . . . 24.2.4. Examples of Solving the Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . 24.3. Generalized Viscosity Solutions and Their Applications . . . . . . . . . . . . . . . . . . . 24.3.1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3.2. Viscosity Solutions Based on the Use of a Parabolic Equation . . . . . . . 24.3.3. Viscosity Solutions Based on Test Functions and Differential Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3.4. Local Structure of Generalized Viscosity Solutions . . . . . . . . . . . . . . . . 24.3.5. Generalized Classical Method of Characteristics . . . . . . . . . . . . . . . . . . . 24.3.6. Examples of Viscosity (Nonsmooth) Solutions . . . . . . . . . . . . . . . . . . . .

1373 1373 1373 1374 1376 1377 1378 1379 1379 1379 1380 1380 1381 1382 1382 1382

25. Classification of Second-Order Nonlinear Equations . . . . . . . . . . . . . . . . . . . . 25.1. Semilinear Equations in Two Independent Variables . . . . . . . . . . . . . . . . . . . . . . 25.1.1. Types of Equations. Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . 25.1.2. Canonical Form of Parabolic Equations (Case b2 – ac = 0) . . . . . . . . . . 25.1.3. Canonical Form of Hyperbolic Equations (Case b2 – ac > 0) . . . . . . . . 25.1.4. Canonical Form of Elliptic Equations (Case b2 – ac < 0) . . . . . . . . . . . . 25.2. Nonlinear Equations in Two Independent Variables . . . . . . . . . . . . . . . . . . . . . . . 25.2.1. Nonlinear Equations of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.2. Quasilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1389 1389 1389 1390 1390 1392 1392 1392 1393

1383 1383 1385 1386

26. Transformations of Equations of Mathematical Physics . . . . . . . . . . . . . . . . . . 1395 26.1. Point Transformations: Overview and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 1395 26.1.1. General Form of Point Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 1395 26.1.2. Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396 26.1.3. Simple Nonlinear Point Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 1396 26.2. Hodograph Transformations (Special Point Transformations) . . . . . . . . . . . . . . . 1397 26.2.1. One PDE: One of the Independent Variables Is Taken to Be the Dependent One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397 26.2.2. One PDE: Method of Conversion to an Equivalent System of Equations 1398 26.2.3. System of Two PDEs: One of the Independent Variables Is Taken to Be the Dependent One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1402 26.2.4. System of Two PDEs: Both of the Independent Variables Are Taken to Be the Dependent Ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1402 26.3. Contact Transformations. Legendre and Euler Transformations . . . . . . . . . . . . . 1403 26.3.1. Preliminary Remarks. Contact Transformations for Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403 26.3.2. General Form of Contact Transformations for Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xviii

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

CONTENTS

Page xix

xix

26.3.3. Legendre Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406 26.3.4. Euler Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407 26.3.5. Legendre Transformation with Many Variables . . . . . . . . . . . . . . . . . . . . 1408 26.4. Differential Substitutions. Von Mises Transformation . . . . . . . . . . . . . . . . . . . . . 1409 26.4.1. Differential Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409 26.4.2. Von Mises Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410 26.5. B¨acklund Transformations. RF Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413 26.5.1. B¨acklund Transformations for Second-Order Equations . . . . . . . . . . . . 1413 26.5.2. RF Pairs and Their Use for Constructing B¨acklund Transformations . . 1415 26.6. Some Other Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1422 26.6.1. Crocco Transformation. Order Reduction of Hydrodynamic Type Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1422 26.6.2. Transformations Based on Conservation Laws . . . . . . . . . . . . . . . . . . . . 1425 27. Traveling-Wave Solutions and Self-Similar Solutions . . . . . . . . . . . . . . . . . . . . 1429 27.1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1429 27.2. Traveling-Wave Solutions. Invariance of Equations under Translations . . . . . . 27.2.1. General Form of Traveling-Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . 27.2.2. Invariance of Solutions and Equations under Translation Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2.3. Functional Equation Describing Traveling-Wave Solutions . . . . . . . . . . 27.3. Self-Similar Solutions. Invariance of Equations Under Scaling Transformations 27.3.1. General Form of Self-Similar Solutions. Similarity Method . . . . . . . . . 27.3.2. Examples of Self-Similar Solutions to Mathematical Physics Equations 27.3.3. More General Approach Based on Solving a Functional Equation. Some Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.3.4. Generalized Self-Similar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1429 1429 1430 1431 1431 1431 1432 1434 1436

28. Elementary Theory of Using Invariants for Solving Equations . . . . . . . . . . . . 1439 28.1. Introduction. Symmetries. General Scheme of Using Invariants for Solving Mathematical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1439 28.1.1. Symmetries. Transformations Preserving the Form of Equations. Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1439 28.1.2. General Scheme of Using Invariants for Solving Mathematical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1440 28.2. Algebraic Equations and Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.1. Algebraic Equations with Even Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.2. Reciprocal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.3. Systems of Algebraic Equations Symmetric with Respect to Permutation of Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3. Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3.1. Transformations Preserving the Form of Equations. Invariants . . . . . . . 28.3.2. Order Reduction Procedure for Equations with n ≥ 2 (Reduction to Solvable Form with n = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3.3. Simple Transformations. Invariant Determination Procedure . . . . . . . . 28.3.4. Analysis of Some Ordinary Differential Equations. Useful Remarks .

1441 1441 1442 1444 1445 1445 1446 1446 1447

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xix

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

xx

Page xx

CONTENTS

28.4. Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.1. Transformations Preserving the Form of Equations. Invariants . . . . . . . 28.4.2. Procedure for Constructing Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . 28.4.3. Examples of Constructing Invariant Solutions to Nonlinear Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.4. Simple Inverse Problems (Determination of the Form of Equations from Their Properties) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.5. General Conclusions and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1450 1450 1450 1451 1455 1457

29. Method of Generalized Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . 1459 29.1. Exact Solutions with Simple Separation of Variables . . . . . . . . . . . . . . . . . . . . . . 1459 29.1.1. Multiplicative and Additive Separable Solutions . . . . . . . . . . . . . . . . . . . 1459 29.1.2. Simple Separation of Variables in Nonlinear Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1459 29.1.3. Complex Separation of Variables in Nonlinear Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1462 29.2. Structure of Generalized Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1464 29.2.1. General Form of Solutions. Classes of Nonlinear Equations Considered 1464 29.2.2. General Form of Functional Differential Equations . . . . . . . . . . . . . . . . 1464 29.3. Simplified Scheme for Constructing Generalized Separable Solutions . . . . . . . 1465 29.3.1. Description of the Simplified Scheme for Constructing Solutions Based on Presetting One System of Coordinate Functions . . . . . . . . . . . . . . . . 1465 29.3.2. Examples of Finding Exact Solutions of Second- and Third-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465 29.4. Solution of Functional Differential Equations by Differentiation . . . . . . . . . . . . 1467 29.4.1. Description of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467 29.4.2. Examples of Constructing Exact Generalized Separable Solutions . . . 1467 29.5. Solution of Functional-Differential Equations by Splitting . . . . . . . . . . . . . . . . . 1471 29.5.1. Preliminary Remarks. Description of the Method . . . . . . . . . . . . . . . . . . 1471 29.5.2. Solutions of Simple Functional Equations and Their Application . . . . 1472 29.6. Titov–Galaktionov Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1477 29.6.1. Method Description. Linear Subspaces Invariant under a Nonlinear Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1477 29.6.2. Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1479 29.6.3. Finding Linear Subspaces Invariant Under a Given Nonlinear Operator 1480 29.6.4. Generalizations to Pseudo-Differential Equations . . . . . . . . . . . . . . . . . . 1483 30. Method of Functional Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 30.1. Structure of Functional Separable Solutions. Solution by Reduction to Equations with Quadratic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.1.1. Structure of Functional Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . 30.1.2. Solution by Reduction to Equations with Quadratic (or Power) Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.2. Special Functional Separable Solutions. Generalized Traveling-Wave Solutions 30.2.1. Special Functional Separable and Generalized Traveling-Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.2.2. General Scheme for Constructing Generalized Traveling-Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1487 1487 1487 1487 1487 1487 1489

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xx

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

CONTENTS

30.3. Differentiation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3.1. Basic Ideas of the Method. Reduction to a Standard Equation . . . . . . . 30.3.2. Examples of Constructing Functional Separable Solutions . . . . . . . . . . 30.4. Splitting Method. Solutions of Some Nonlinear Functional Equations and Their Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.4.1. Splitting Method. Reduction to a Standard Functional Equation . . . . . 30.4.2. Three-Argument Functional Equations of Special Form . . . . . . . . . . . . 30.4.3. Functional Equation f (t) + g(x) = Q(z), with z = ϕ(x) + ψ(t) . . . . . . . 30.4.4. Functional Equation f (t) + g(x) + h(x)Q(z) + R(z) = 0, with z = ϕ(x) + ψ(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.4.5. Functional Equation f (t)+g(x)Q(z)+h(x)R(z) = 0, with z = ϕ(x)+ψ(t) 30.4.6. Equation f1 (x) + f2 (y) + g1 (x)P (z) + g2 (y)Q(z) + R(z) = 0, z = ϕ(x) + ψ(y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Page xxi

xxi 1492 1492 1492 1496 1496 1497 1498 1498 1500 1501

31. Direct Method of Symmetry Reductions of Nonlinear Equations . . . . . . . . . 31.1. Clarkson–Kruskal Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1. Simplified Scheme. Examples of Constructing Exact Solutions . . . . . . 31.1.2. Description of the Method: A Special Form for Symmetry Reduction 31.1.3. Description of the Method: the General Form for Symmetry Reduction 31.2. Some Modifications and Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1. Symmetry Reductions Based on the Generalized Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2. Similarity Reductions in Equations with Three or More Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1503 1503 1503 1505 1506 1507

32. Classical Method of Symmetry Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1. One-Parameter Transformations and Their Local Properties . . . . . . . . . . . . . . . . 32.1.1. One-Parameter Transformations. Infinitesimal Operator . . . . . . . . . . . . 32.1.2. Invariant of an Infinitesimal Operator. Transformations in the Plane . . 32.1.3. Formulas for Derivatives. Coordinates of the First and Second Prolongations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2. Symmetries of Nonlinear Second-Order Equations. Invariance Condition . . . . 32.2.1. Invariance Condition. Splitting in Derivatives . . . . . . . . . . . . . . . . . . . . . 32.2.2. Examples of Finding Symmetries of Nonlinear Equations . . . . . . . . . . . 32.3. Using Symmetries of Equations for Finding Exact Solutions. Invariant Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3.1. Using Symmetries of Equations for Constructing One-Parameter Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3.2. Procedure for Constructing Invariant Solutions . . . . . . . . . . . . . . . . . . . . 32.3.3. Examples of Constructing Invariant Solutions to Nonlinear Equations 32.3.4. Solutions Induced by Linear Combinations of Admissible Operators . 32.4. Some Generalizations. Higher-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4.1. One-Parameter Lie Groups of Point Transformations. Group Generator 32.4.2. Group Invariants. Local Transformations of Derivatives . . . . . . . . . . . . 32.4.3. Invariant Condition. Splitting Procedure. Invariant Solutions . . . . . . . . 32.5. Symmetries of Systems of Equations of Mathematical Physics . . . . . . . . . . . . . 32.5.1. Basic Relations Used in Symmetry Analysis of Systems of Equations 32.5.2. Symmetries of Equations of Steady Hydrodynamic Boundary Layer .

1513 1513 1513 1514

1507 1510

1515 1516 1516 1516 1520 1520 1520 1522 1524 1526 1526 1526 1527 1527 1527 1528

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxi

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

xxii

Page xxii

CONTENTS

33. Nonclassical Method of Symmetry Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . 33.1. General Description of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.1.1. Description of the Method. Invariant Surface Condition . . . . . . . . . . . . 33.1.2. Scheme for Constructing Exact Solutions by the Nonclassical Method 33.2. Examples of Constructing Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2.1. Newell–Whitehead Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2.2. Nonlinear Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1533 1533 1533 1533 1534 1534 1536

34. Method of Differential Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.1. Preliminary Remarks. Method of Differential Constraints for Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.1.1. Description of the Method. First-Order Differential Constraints . . . . . 34.1.2. Differential Constraints of Arbitrary Order. General Consistency Method for Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.1.3. Some Generalizations. The Case of Several Differential Constraints . 34.2. Description of the Method for Partial Differential Equations . . . . . . . . . . . . . . . 34.2.1. Preliminary Remarks. A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . 34.2.2. General Description of the Method of Differential Constraints . . . . . . . 34.3. First-Order Differential Constraints for PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.3.1. Second-Order Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.3.2. Second-Order Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.3.3. Second-Order Equations of General Form . . . . . . . . . . . . . . . . . . . . . . . . 34.4. Second-Order Differential Constraints for PDEs. Some Generalized . . . . . . . . 34.4.1. Second-Order Differential Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.4.2. Higher-Order Differential Constraints. Determining Equations . . . . . . 34.4.3. Usage of Several Differential Constraints. Systems of Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.5. Connection Between the Method of Differential Constraints and Other Methods 34.5.1. Generalized/Functional Separation of Variables vs. Differential Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.5.2. Direct Method of Symmetry Reductions and Differential Constraints . 34.5.3. Nonclassical Method of Symmetry Reductions and Differential Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1539

35. Painlev´e Test for Nonlinear Equations of Mathematical Physics . . . . . . . . . . 35.1. Movable Singularities of Solutions of Ordinary Differential Equations . . . . . . . 35.1.1. Examples of Solutions Having Movable Singularities . . . . . . . . . . . . . . 35.1.2. Classification Results for Nonlinear First- and Second-Order Equations 35.1.3. Painlev´e Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.1.4. Painlev´e Test for Ordinary Differential Equations . . . . . . . . . . . . . . . . . . 35.1.5. Remarks on the Painlev´e Test. Fuchs Indices. Examples . . . . . . . . . . . . 35.1.6. The Painlev´e Test for Systems of Ordinary Differential Equations . . . . 35.2. Solutions of Partial Differential Equations with a Movable Pole. Method Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.2.1. Simple Scheme for Studying Nonlinear Partial Differential Equations 35.2.2. General Scheme for Analysis of Nonlinear Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.2.3. Basic Steps of the Painlev´e Test for Nonlinear Equations . . . . . . . . . . . 35.2.4. Some Remarks. Truncated Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . .

1565 1565 1565 1565 1566 1566 1567 1569

1539 1539 1542 1543 1545 1545 1546 1547 1547 1551 1553 1553 1553 1555 1557 1561 1561 1562 1562

1569 1570 1570 1570 1571

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxii

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

CONTENTS

Page xxiii

xxiii

35.3. Performing the Painlev´e Test and Truncated Expansions for Studying Some Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1572 35.3.1. Equations Passing the Painlev´e Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1572 35.3.2. Checking Whether Nonlinear Systems of Equations of Mathematical Physics Pass the Painlev´e Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576 35.3.3. Construction of Solutions of Nonlinear Equations That Fail the Painlev´e Test, Using Truncated Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1577 36. Methods of the Inverse Scattering Problem (Soliton Theory) . . . . . . . . . . . . . 36.1. Method Based on Using Lax Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.1.1. Method Description. Consistency Condition. Lax Pairs . . . . . . . . . . . . 36.1.2. Examples of Lax Pairs for Nonlinear Equations of Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.2. Method Based on a Compatibility Condition for Systems of Linear Equations 36.2.1. General Scheme. Compatibility Condition. Systems of Two Equations 36.2.2. Solution of the Determining Equations in the Form of Polynomials in λ 36.3. Method Based on Linear Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.3.1. Description of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.3.2. Specific Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.4. Solution of the Cauchy Problem by the Inverse Scattering Problem Method . . 36.4.1. Preliminary Remarks. Direct and Inverse Scattering Problems . . . . . . . 36.4.2. Solution of the Cauchy Problem for Nonlinear Equations by the Inverse Scattering Problem Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.4.3. N -Soliton Solution of the Korteweg–de Vries Equation . . . . . . . . . . . .

1579 1579 1579

37. Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.1. Basic Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.1.1. General Form of Conservation Laws. Integrals of Motion . . . . . . . . . . . 37.1.2. Conservation Laws for Some Nonlinear Equations of Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.2. Equations Admitting Variational Form. Noetherian Symmetries . . . . . . . . . . . . 37.2.1. Lagrangian. Euler–Lagrange Equation. Noetherian Symmetries . . . . . 37.2.2. Examples of Constructing Conservation Laws Using Noetherian Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1593 1593 1593

38. Nonlinear Systems of Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . 38.1. Overdetermined Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.1.1. Overdetermined Systems of First-Order Equations in One Unknown . 38.1.2. Other Overdetermined Systems of Equations in One Unknown . . . . . . 38.2. Pfaffian Equations and Their Solutions. Connection with Overdetermined Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.2.1. Pfaffian Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.2.2. Condition for Integrability of the Pfaffian Equation by a Single Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.2.3. Pfaffian Equations Not Satisfying the Integrability Condition . . . . . . . . 38.3. Systems of First-Order Equations Describing Convective Mass Transfer with Volume Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.3.1. Traveling-Wave Solutions and Some Other Invariant Solutions . . . . . . 38.3.2. Systems Reducible to an Ordinary Differential Equation . . . . . . . . . . . . 38.3.3. Some Nonlinear Problems of Suspension Transport in Porous Media . 38.3.4. Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1599 1599 1599 1600

1580 1581 1581 1582 1584 1584 1585 1587 1587 1589 1591

1593 1595 1595 1596

1600 1600 1601 1602 1602 1602 1603 1604 1606

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxiii

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

xxiv

Page xxiv

CONTENTS

38.4. First-Order Hyperbolic Systems of Quasilinear Equations. Systems of Conservation Laws of Gas Dynamic Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.4.1. Systems of Two Quasilinear Equations. Systems in the Form of Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.4.2. Self-Similar Continuous Solutions. Hyperbolic Systems . . . . . . . . . . . . 38.4.3. Simple Riemann Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.4.4. Linearization of Systems of Gas Dynamic Type by the Hodograph Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.4.5. Cauchy and Riemann Problems. Qualitative Features of Solutions . . . 38.4.6. Reduction of Systems to the Canonical Form. Riemann Invariants . . . 38.4.7. Hyperbolic n × n Systems of Conservation Laws. Exact Solutions . . . 38.4.8. Shock Waves. Rankine–Hugoniot Jump Conditions . . . . . . . . . . . . . . . . 38.4.9. Shock Waves. Evolutionary Conditions. Lax Condition . . . . . . . . . . . . 38.4.10. Solutions for the Riemann Problem. Solutions Describing Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.5. Systems of Second-Order Equations of Reaction-Diffusion Type . . . . . . . . . . . 38.5.1. Traveling-Wave Solutions and Some Other Invariant Solutions . . . . . . 38.5.2. Generalized Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1618 1620 1620 1620

Part III. Symbolic and Numerical Solutions of Nonlinear PDEs with Maple, Mathematica, and MATLAB

1623

39. Nonlinear Partial Differential Equations with Maple . . . . . . . . . . . . . . . . . . . . 39.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.2. Brief Introduction to Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.2.1. Maple Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.3. Analytical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.3.1. Constructing Analytical Solutions in Terms of Predefined Functions . 39.3.2. Constructing Solutions via Transformations . . . . . . . . . . . . . . . . . . . . . . . 39.3.3. Constructing Traveling-Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 39.3.4. Ansatz Methods (Tanh-Coth Method, Sine-Cosine Method, and Exp-Function Method) for Constructing Traveling-Wave Solutions . . . 39.3.5. Constructing Self-Similar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.3.6. Constructing Solutions along Characteristics . . . . . . . . . . . . . . . . . . . . . . 39.3.7. Constructing Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.4. Analytical Solutions of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.4.1. Constructing Traveling-Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 39.4.2. Constructing Generalized Separable Solutions . . . . . . . . . . . . . . . . . . . . . 39.5. Constructing Exact Solutions Using Symbolic Computation. What Can Go Wrong? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.5.1. Constructing New Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.5.2. Removing Redundant Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.6. Some Errors That People Commonly Do When Constructing Exact Solutions with the Use of Symbolic Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.6.1. General Description of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.6.2. Examples in Which “New Solutions” Are Obtained . . . . . . . . . . . . . . . . 39.7. Numerical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.7.1. Constructing Numerical Solutions in Terms of Predefined Functions . 39.7.2. Constructing Finite Difference Approximations . . . . . . . . . . . . . . . . . . .

1607 1607 1607 1609 1610 1611 1611 1614 1615 1616

1625 1625 1625 1627 1629 1629 1635 1637 1641 1645 1647 1651 1659 1659 1661 1662 1662 1665 1668 1668 1669 1672 1672 1677

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxiv

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

CONTENTS

Page xxv

xxv

39.8. Analytical-Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.8.1. Analytical Derivation of Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 39.8.2. Constructing Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.8.3. Comparison of Asymptotic and Numerical Solutions . . . . . . . . . . . . . . .

1680 1680 1683 1684

40. Nonlinear Partial Differential Equations with Mathematica . . . . . . . . . . . . . . 40.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.2. Brief Introduction to Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.2.1. Mathematica Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.3. Analytical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.3.1. Constructing Solutions Using Predefined Functions . . . . . . . . . . . . . . . . 40.3.2. Constructing Solutions via Transformations . . . . . . . . . . . . . . . . . . . . . . . 40.3.3. Constructing Traveling-Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 40.3.4. Ansatz Methods (Tanh-Coth Method, Sine-Cosine Method, and Exp-Function Method) for Constructing Traveling-Wave Solutions . . . 40.3.5. Constructing Self-Similar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.3.6. Constructing Solutions Along Characteristics . . . . . . . . . . . . . . . . . . . . . 40.3.7. Constructing Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.4. Analytical Solutions of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.4.1. Constructing Traveling-Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 40.4.2. Constructing Generalized Separable Solutions . . . . . . . . . . . . . . . . . . . . . 40.5. Numerical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.5.1. Constructing Numerical Solutions in Terms of Predefined Functions . 40.5.2. Constructing Finite-Difference Approximations . . . . . . . . . . . . . . . . . . . 40.6. Analytical-Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.6.1. Analytical Derivation of Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 40.6.2. Constructing Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.6.3. Comparison of Asymptotic and Numerical Solutions . . . . . . . . . . . . . . .

1687 1687 1687 1689 1691 1692 1694 1697

41. Nonlinear Partial Differential Equations with MATLAB . . . . . . . . . . . . . . . . . 41.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2. Brief Introduction to MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1. MATLAB Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3. Numerical Solutions Via Predefined Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.1. Scalar Nonlinear PDEs in One Space Dimension . . . . . . . . . . . . . . . . . . 41.3.2. Systems of Nonlinear PDEs in One Space Dimension . . . . . . . . . . . . . . 41.3.3. Nonlinear Elliptic PDEs in Two Space Dimensions . . . . . . . . . . . . . . . . 41.3.4. Systems of Nonlinear Elliptic PDEs in Two Space Dimensions . . . . . . 41.3.5. Nonlinear Elliptic PDEs. Geometrical Models . . . . . . . . . . . . . . . . . . . . 41.4. Solving Cauchy Problems. Method of Characteristics . . . . . . . . . . . . . . . . . . . . . 41.5. Constructing Finite-Difference Approximations . . . . . . . . . . . . . . . . . . . . . . . . . .

1735 1735 1735 1738 1741 1742 1745 1748 1752 1754 1756 1760

Supplements. Painlev´e Transcendents. Functional Equations

1767

42. Painlev´e Transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1. Preliminary Remarks. Singular Points of Solutions . . . . . . . . . . . . . . . . . . . . . . . 42.2. First Painlev´e Transcendent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3. Second Painlev´e Transcendent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4. Third Painlev´e Transcendent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1769 1769 1770 1771 1772

1700 1705 1706 1710 1719 1719 1721 1722 1722 1725 1728 1729 1731 1733

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxv

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

xxvi

Page xxvi

CONTENTS

42.5. Fourth Painlev´e Transcendent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.6. Fifth Painlev´e Transcendent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.7. Sixth Painlev´e Transcendent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.8. Examples of Solutions to Nonlinear Equations in Terms of Painlev´e Transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1775 1776 1777 1778

43. Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.1. Method of Differentiation in a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.1.1. Classes of Equations. Description of the Method . . . . . . . . . . . . . . . . . . 43.1.2. Examples of Solutions of Some Specific Functional Equations . . . . . . 43.2. Method of Differentiation in Independent Variables . . . . . . . . . . . . . . . . . . . . . . . 43.2.1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2.2. Examples of Solutions of Some Specific Functional Equations . . . . . . 43.3. Method of Argument Elimination by Test Functions . . . . . . . . . . . . . . . . . . . . . . 43.3.1. Classes of Equations. Description of the Method . . . . . . . . . . . . . . . . . . 43.3.2. Examples of Solutions of Specific Functional Equations . . . . . . . . . . . . 43.4. Nonlinear Functional Equations Reducible to Bilinear Equations . . . . . . . . . . . 43.4.1. Bilinear Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.4.2. Functional-Differential Equations Reducible to a Bilinear Equation . . 43.4.3. Nonlinear Functional Equations Containing the Complex Argument . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1783 1783 1783 1784 1785 1785 1785 1786 1786 1787 1789 1789 1790 1790 1795

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxvi

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxvii

PREFACE TO THE NEW EDITION The Handbook of Nonlinear Partial Differential Equations, a unique reference for scientists and engineers, contains over 3,000 nonlinear partial differential equations with solutions, as well as exact, symbolic, and numerical methods for solving nonlinear equations. First-, second-, third-, fourth-, and higher-order nonlinear equations and systems of equations are considered. Equations of parabolic, hyperbolic, elliptic, mixed, and general types are discussed. A large number of new exact solutions to nonlinear equations are described. In total, the handbook contains several times more nonlinear PDEs and exact solutions than any other book currently available. In selecting the material, the authors gave the highest priority to the following five major types of equations: • Equations that arise in various applications (heat and mass transfer theory, wave theory, nonlinear mechanics, hydrodynamics, gas dynamics, plasticity theory, nonlinear acoustics, combustion theory, nonlinear optics, theoretical physics, differential geometry, control theory, chemical engineering sciences, biology, and others). • Equations of general form that depend on arbitrary functions; exact solutions of such equations are of principal value for testing numerical and approximate methods. • Equations for which the general solution or solutions of quite general form, with arbitrary functions, could be obtained. • Equations that involve many free parameters. • Equations whose solutions are reduced to solving linear partial differential equations or linear integral equations. The second edition has been substantially updated, revised, and expanded. More than 1,500 new equations with exact solutions, as well some methods and many examples, have been added. The new edition has been increased by a total of over 1,000 pages. New to the second edition: • • • • •

First-order nonlinear partial differential equations with solutions. Some second-, third-, fourth-, and higher-order nonlinear equations with solutions. Parabolic, hyperbolic, elliptic, and other systems of equations with solutions. Some exact methods and transformations. Symbolic and numerical methods for solving nonlinear PDEs with Maple, Mathematica, and MATLAB. • Many new examples and tables included for illustrative purposes. • A large list of references consisting of over 1,300 sources. • Extensive table of contents and detailed index. Note that the first part of the book can be used as a database of test problems for numerical and approximate methods for solving nonlinear partial differential equations. We would like to express our deep gratitude to Alexei Zhurov and Alexander Aksenov for fruitful discussions and valuable remarks. We are very thankful to Inna Shingareva and Carlos Liz´arraga-Celaya who wrote three chapters (39–41) of the book at our request. xxvii

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxvii

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

xxviii

Page xxviii

PREFACE

The authors hope that the handbook will prove helpful for a wide audience of researchers, university and college teachers, engineers, and students in various fields of applied mathematics, mechanics, physics, chemistry, biology, economics, and engineering sciences. Andrei D. Polyanin Valentin F. Zaitsev September 2011

PREFACE TO THE FIRST EDITION Nonlinear partial differential equations are encountered in various fields of mathematics, physics, chemistry, and biology, and numerous applications. Exact (closed-form) solutions of differential equations play an important role in the proper understanding of qualitative features of many phenomena and processes in various areas of natural science. Exact solutions of nonlinear equations graphically demonstrate and allow unraveling the mechanisms of many complex nonlinear phenomena such as spatial localization of transfer processes, multiplicity or absence of steady states under various conditions, existence of peaking regimes, and many others. Furthermore, simple solutions are often used in teaching many courses as specific examples illustrating basic tenets of a theory that admit mathematical formulation. Even those special exact solutions that do not have a clear physical meaning can be used as “test problems” to verify the consistency and estimate errors of various numerical, asymptotic, and approximate analytical methods. Exact solutions can serve as a basis for perfecting and testing computer algebra software packages for solving differential equations. It is significant that many equations of physics, chemistry, and biology contain empirical parameters or empirical functions. Exact solutions allow researchers to design and run experiments, by creating appropriate natural conditions, to determine these parameters or functions. This book contains more than 1,600 nonlinear mathematical physics equations and nonlinear partial differential equations and their solutions. A large number of new exact solutions to nonlinear equations are described. Equations of parabolic, hyperbolic, elliptic, mixed, and general types are discussed. Second-, third-, fourth-, and higher-order nonlinear equations are considered. The book presents exact solutions to equations of heat and mass transfer, wave theory, nonlinear mechanics, hydrodynamics, gas dynamics, plasticity theory, nonlinear acoustics, combustion theory, nonlinear optics, theoretical physics, differential geometry, control theory, chemical engineering sciences, biology, and other fields. Special attention is paid to general-form equations that depend on arbitrary functions; exact solutions of such equations are of principal value for testing numerical and approximate methods. Almost all other equations contain one or more arbitrary parameters (in fact, this book deals with whole families of partial differential equations), which can be fixed by the reader at will. In total, the handbook contains significantly more nonlinear PDEs and exact solutions than any other book currently available. The supplement of the book presents exact analytical methods for solving nonlinear mathematical physics equations. When selecting the material, the authors have given a pronounced preference to practical aspects of the matter; that is, to methods that allow effectively “constructing” exact solutions. Apart from the classical methods, the book also describes wide-range methods that have been greatly developed over the last decade (the nonclassical and direct methods for symmetry reductions, the differential constraints

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxviii

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

PREFACE

Page xxix

xxix

method, the method of generalized separation of variables, and others). For the reader’s better understanding of the methods, numerous examples of solving specific differential equations and systems of differential equations are given throughout the book. For the convenience of a wide audience with different mathematical backgrounds, the authors tried to do their best, wherever possible, to avoid special terminology. Therefore, some of the methods are outlined in a schematic and somewhat simplified manner, with necessary references made to books where these methods are considered in more detail. Many sections were written so that they could be read independently from each other. This allows the reader to quickly get to the heart of the matter. The handbook consists of chapters, sections, and subsections. Equations and formulas are numbered separately in each subsection. The equations within subsections are arranged in increasing order of complexity. The extensive table of contents provides rapid access to the desired equations. Separate parts of the book may be used by lecturers of universities and colleges for practical courses and lectures on nonlinear mathematical physics equations for graduate and postgraduate students. Furthermore, the books may be used as a database of test problems for numerical and approximate methods for solving nonlinear partial differential equations. We would like to express our deep gratitude to Alexei Zhurov for fruitful discussions and valuable remarks. The authors hope that this book will be helpful for a wide range of scientists, university teachers, engineers, and students engaged in the fields of mathematics, physics, mechanics, control, chemistry, and engineering sciences. Andrei D. Polyanin Valentin F. Zaitsev September 2003

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxix

This page intentionally left blank

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxxi

AUTHORS Andrei D. Polyanin, D.Sc., Ph.D., is a well-known scientist of broad interests and is active in various areas of mathematics, mechanics, and chemical engineering sciences. He is one of the most prominent authors in the field of reference literature on mathematics. Professor Polyanin graduated with honors from the Department of Mechanics and Mathematics at Moscow State University in 1974. He received his Ph.D. in 1981 and D.Sc. in 1986 at the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences. Since 1975, Professor Polyanin has been working at the Institute for Problems in Mechanics of the Russian Academy of Sciences; he is also professor of mathematics at Bauman Moscow State Technical University. He is a member of the Russian National Committee on Theoretical and Applied Mechanics and the Mathematics and Mechanics Expert Council of the Higher Certification Committee of the Russian Federation. Professor Polyanin has made important contributions to the theory of differential and integral equations, mathematical physics, engineering mathematics, theory of heat and mass transfer, and chemical hydrodynamics. He has obtained exact solutions for several thousand ordinary differential, partial differential, and integral equations. Professor Polyanin is the author of more than 30 books in English, Russian, German, and Bulgarian as well as over 150 research papers and three patents. He has written a number of fundamental handbooks, including: A. D. Polyanin and V. F. Zaitsev’s Handbook of Exact Solutions for Ordinary Differential Equations (CRC Press, 1995 and 2003); A. D. Polyanin and A. V. Manzhirov’s Handbook of Integral Equations (CRC Press, 1998 and 2008); A. D. Polyanin’s Handbook of Linear Partial Differential Equations for Engineers and Scientists (Chapman & Hall/CRC Press, 2002); A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux’s Handbook of First Order Partial Differential Equations (Taylor & Francis, 2002); A. D. Polyanin and V. F. Zaitsev’s Handbook of Nonlinear Partial Differential Equations (Chapman & Hall/CRC Press, 2004); A. D. Polyanin and A. V. Manzhirov’s Handbook of Mathematics for Engineers and Scientists (Chapman & Hall/CRC Press, 2007), and A. D. Polyanin and A. I. Chernoutsan’s (Eds.) A Concise Handbook of Mathematics, Physics, and Engineering Sciences (Chapman & Hall/CRC Press, 2010). Professor Polyanin is editor-in-chief of the international scientific educational website EqWorld — The World of Mathematical Equations (http://eqworld.ipmnet.ru), which is visited by several thousand users a day worldwide. He is also editor of the book series Differential and Integral Equations and Their Applications (Chapman & Hall/CRC Press, London/Boca Raton). Professor Polyanin is a member of the editorial board of the journal Theoretical Foundations of Chemical Engineering. In 1991, Professor Polyanin was awarded a Chaplygin Prize of the Russian Academy of Sciences for his research in mechanics. In 2001, he received an award from the Ministry of Education of the Russian Federation. Address: Institute for Problems in Mechanics, Vernadsky Ave. 101 Bldg 1, 119526 Moscow, Russia Home page: http://eqworld.ipmnet.ru/polyanin-ew.htm

xxxi

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxxi

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

xxxii

Page xxxii

AUTHORS

Valentin F. Zaitsev, Ph.D., D.Sc., is a noted scientist in the fields of ordinary differential equations, mathematical physics, and nonlinear mechanics. Dr. Zaitsev graduated from the Radio Electronics Faculty of the Leningrad Polytechnical Institute (now St. Petersburg Technical University) in 1969 and received his Ph.D. degree in 1983 at the Leningrad State University. His Ph.D. thesis was devoted to the group approach of the study of some classes of ordinary differential equations. In 1992, Professor Zaitsev received his D.Sc. and his thesis was dedicated to the discrete group analysis of ordinary differential equations. From 1971–1996, Dr. Zaitsev worked in the Research Institute for Computational Mathematics and Control Processes of the St. Petersburg State University. Since 1996, Professor Zaitsev has been a member of the staff of the Russian State Pedagogical University (St. Petersburg). Professor Zaitsev has made important contributions to new methods in the theory of ordinary and partial differential equations. He is the author of more than 150 scientific publications, including 23 books, and 1 patent. Address: Russian State Pedagogical University, 48 Naberezhnaya reki Moiki, 191186 Saint-Petersburg, Russia Home page: http://eqworld.ipmnet.ru/en/board/zaitsev.htm

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxxii

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxxiii

SOME NOTATIONS AND REMARKS Latin Characters C1 , C2 , . . . curl u div u grad a r, ϕ, z r, θ, ϕ t w x, y, z x1 , . . . , xn

are arbitrary constants; curl of a vector u, sometimes also denoted rot u; ∂u2 1 divergence of a vector u; div u = ∂u ∂x + ∂y in two-dimensional case u = (u1 , u2 ); gradient of a scalar a, also denoted by ∇a, with ∇ being the nabla vector differential operator; p cylindrical coordinates, r =p x2 + y 2 and x = r cos ϕ, y = r sin ϕ; spherical coordinates, r = x2 + y 2 + z 2 and x = r sin θ cos ϕ, y = sin θ sin ϕ, z = r cos θ; time (t ≥ 0); unknown function (dependent variable); space (Cartesian) coordinates; Cartesian coordinates in n-dimensional space.

Greek Characters ∆ Laplace operator: ∂2 ∂2 ∆ = ∂x 2 + ∂y 2 , in two-dimensional case; ∆= ∆=

∂2 ∂x2 n P

k=1

+

∂2 ∂y 2

∂2 , ∂x2k

+

∂2 , ∂z 2

in three-dimensional case;

in n-dimensional case;

∆∆ biharmonic operator; in two-dimensional case, ∆∆ =

∂4 ∂x4

4

+ 2 ∂x∂2 ∂y2 +

∂4 . ∂y 4

Brief Notation for Derivatives Partial derivatives: ∂w ∂w ∂2w wx = , wt = , wxx = , ∂x ∂t ∂x2 Ordinary derivatives for f = f (x): fx′ =

df , dx

′′ fxx =

d2 f , dx2

′′′ fxxx =

wxt =

d3 f , dx3

∂2w , ∂x∂t

′′′′ fxxxx =

wtt =

d4 f , dx4

∂2w , ∂t2

fx(n) =

wxxx = dn f dxn

∂3w , ∂x3

with

...

n ≥ 5.

Brief Notation for Differential Operators ∂ ∂ ∂ ∂ , ∂y = , ∂t = , ∂w = (differential operators in x, y, t, and w); ∂x ∂y ∂t ∂w ∂ ∂ ∂ ∂ Dx = + wx + wxx + wxt + · · · (total differential operator in x); ∂x ∂w ∂wx ∂wt ∂ ∂ ∂ ∂ Dt = + wt + wxt + wtt +··· (total differential operator in t). ∂t ∂w ∂wx ∂wt In the last two relations, w is assumed to be dependent on x and t, w = w(x, t). ∂x =

xxxiii

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxxiii

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

xxxiv

Page xxxiv

SOME NOTATIONS AND REMARKS

Remarks 1. The book presents solutions of the following types: (a) expressible in terms of elementary functions explicitly, implicitly, or parametrically; (b) expressible in terms of elementary functions and integrals of elementary functions; (c) expressible in terms of elementary functions, functions involved in the equation (if the equation contains arbitrary functions), and integrals of the equation functions and/or other elementary functions; (d) expressible in terms of ordinary differential equations or finite systems of ordinary differential equations; (e) expressible in terms of solutions to linear partial differential equations or linear integral equations. 2. The book also deals with solutions described by equations with fewer new variables than those in the original equations. An expression that solves an equations in three independent variables and is determined by an equation in two independent variables will be called a two-dimensional solution. 3. As a rule, the book does not present simple solutions that depend on only one of the variables involved in the original equation. 4. Equations are numbered separately within each subsection. In Part I, when referencing a particular equation, we use a notation like 3.2.1.5, which implies equation 5 from Subsection 3.2.1. In Parts II and III, equation numbers are enclosed in parentheses; for example, (27.2.1.4) means equation 4 from Subsection 27.2.1. f (x) 5. If a formula or a solution contains an expression like , it is often not stated a–2 that the assumption a ≠ 2 is implied. 6. If a formula or a solution contains derivatives of some functions, then the functions are assumed to be differentiable. 7. If a formula or a solution contains definite integrals, then the integrals are supposed to be convergent. 8. Though incomplete, very simple and graphical classification of solutions by their appearance is used in the book. For equations in two independent variables, x and t, and one unknown, w, the solution name and structure are as follows (x and t in the solutions below can be swapped): No. 1 2 3 4 5 6 7 8

Solution name Traveling-wave solution∗ Additive separable solution Multiplicative separable solution Self-similar solution∗∗ Generalized self-similar solution Generalized traveling-wave solution Generalized separable solution Functional separable solution

Solution structure w = F (z), z = αx + βt, αβ ≠ 0 w = ϕ(x) + ψ(t) w = ϕ(x)ψ(t) w = tα F (z), z = xtβ w = ϕ(t)F (z), z = xψ(t) w = F (z), z = ϕ(t)x + ψ(t) w = ϕ1 (x)ψ1 (t) + · · · + ϕn (x)ψn (t) w = F (z), z = ϕ1 (x)ψ1 (t) + · · · + ϕn (x)ψn (t)

∗ For uniformity of presentation, we also use this term in the cases where the variable t plays the role of a spatial coordinate. ∗∗ Sometimes, a solution of the form w = ¯ tα F (z) with z = x¯¯ tβ , where x¯ = x + C1 and ¯ t = t + C2 , will also be called a self-similar solution.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxxiv

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

SOME NOTATIONS AND REMARKS

Page xxxv

xxxv

9. In this book, the term integrable equation applies to nonlinear partial differential equations that admit solution in terms of integrals or can be linearized, which means that their solutions can be expressed in terms of solutions to linear differential equations or linear integral equations. 10. The previous handbooks by Zaitsev and Polyanin (1996, 2003), Polyanin and Zaitsev (2002, 2004), and Polyanin, Zaitsev, and Moussiaux (2002) were extensively used in compiling this book; references to these sources are often omitted. 11. The ODE and PDE are conventional abbreviations for ordinary differential equation and partial differential equation, respectively. This symbol indicates references to literature sources whenever: (a) at least one of the solutions was obtained in the cited source (even though the solution contained “correctable” misprints in signs or coefficients); (b) the cited source provides further information on the equation in question and their solutions.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page xxxv

This page intentionally left blank

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 1

Part I

Exact Solutions of Nonlinear Partial Differential Equations

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 1

This page intentionally left blank

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 3

Chapter 1

First-Order Quasilinear Equations Preliminary remarks. The equations in each subsection are arranged in order of complexity of their form. The functions Φ(u1 , . . . , un ) appearing below in general solutions are arbitrary functions of their arguments uk , and the uk are some functions the independent and dependent variables.

1.1. Equations with Two Independent Variables Containing Arbitrary Parameters 1.1.1. Coefficients of Equations Contain Power-Law Functions ◮ Equations of the form F (x, y) ∂w + G(x, y) ∂w = H(x, y, w). ∂x ∂y ∂w ∂w +b = cw n + sw m. ∂x ∂y Z dw General solution: a = x + Φ(bx – ay). cwn + swm

1.

2.

a

∂w

+k

∂w

= (ax + by + cw)n + s.

∂x ∂y The substitution v(x, y) = ax + by + cw(x, y) leads to an equation of the form 1.1.1.1 with m = 0: ∂v ∂v +k = cv n + a + bk + cs. ∂x ∂y ∂w

∂w

= bxnw + cxmw k. ∂x ∂y General solution:  Z m  1 x dx 1–k w = F (x)Φ(y – ax) + c(1 – k)F (x) , F (x)

3.

+a



 b(1 – k) n+1 F (x) = exp x . n+1

∂w ∂w +b = cw + (βxn + λy m)w k. ∂x ∂y This is a special case of equation 11.2.1.3 with f (x) = βxn and g(y) = λy m . 4.

a

5.

∂w ∂w +a = bxny mw k. ∂x ∂y

This is a special case of equation 1.2.1.8 with f (x) = bxn , g(y) = y m , and h(w) = wk . 3

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 3

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4 6.

Page 4

FIRST-ORDER QUASILINEAR EQUATIONS ax

∂w ∂w + bx = cw k. ∂x ∂y

General solution: c ln |x| + Φ(bx – ay) = a 7.

ax

∂w ∂x

+ by

∂w



1 1–k 1–k w

if k ≠ 1, if k = 1.

ln |w|

= cw k.

∂y

General solution:
c ln |x| + Φ |x|b |y|–a = a 8.

ay



1 1–k 1–k w

ln |w|

if k ≠ 1, if k = 1.

∂w ∂w + bx = cw k. ∂x ∂y

General solution for ab > 0:

9.

x

∂w ∂x

√
c √ ln ab x + ay + Φ ay 2 – bx2 = ab +y

∂w ∂y

=w–a

General solution: w +

q

q



1 1–k 1–k w

ln |w|

if k ≠ 1, if k = 1.

w 2 – x2 – y 2 .

w2 – x2 – y 2 =

y Φ . x xa–1 1

∂w ∂w + by 2 = cw 2 . ∂x ∂y   –1 1 1 1 1 General solution: w = +Φ – . c ax by ax 10.

ax2

11.

ax2

∂w ∂w – bx2 = (w – bx – ay)2 . ∂x ∂y

General solution: w =

12.

[(a + b)x + ay]Φ(bx + ay) – ax(bx + ay) . Φ(bx + ay) – ax

  2 2 ∂w m(y 2 ± a2 ) 2 2 ∂w n m n(x ± a ) (x ± a ) + (y ± a ) = bx y + w2. ∂x ∂y x y 2

2

General solution:

w=

1 , Φ(v) – bxn y m

where

 a(x – y)    2 a + xy v= (x – a)(y + a)    (x + a)(y – a)

for the upper sign, for the lower sign.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 4

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.1. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY PARAMETERS



2

∂w

∂w

Page 5

5



–y = a(x2 + y 2 )w 2 . ∂x ∂y  –1 a(y 2 – x2 ) General solution: w = + Φ(xy) . 2(xy + a2 ) 13.

(xy + a ) x

14.

axn

∂w ∂x

+ by k

∂w

= cw m + s.

∂y

This is a special case of equation 1.2.1.11 with f (w) = cwm + s. 15.

ay n

∂w ∂x

+ bxk

∂w

= cw m + s.

∂y

This is a special case of equation 1.2.1.12 with f (w) = cwm + s. 16.

y

2a



ax

∂w ∂x

+ by

General solution: Φ

17.

b1 xn1

∂w ∂x



xb ya

+ b2 y n2

∂w ∂y ,y



c/b

∂w ∂y

= c(y aw – xb)2 .  ya exp a = 0. y w – xb 

= aw + (c1 xm1 + c2 y m2 )w k.

This is a special case of equation 1.2.1.17 with f1 (x) = b1 xn1 , f2 (y) = b2 y n2 , g1 (x) = c1 xm1 , and g2 (y) = c2 y m2 . ◮ Equations of the form F (x, y, w) ∂w + G(x, y, w) ∂w = H(x, y, w). ∂x ∂y 18.

∂w ∂x

+ aw

∂w

= 0.

∂y

Hopf’s equation. It is used as a model equation of the nonlinear wave theory and gas dynamics, where the independent variables x and y play the role of time and the spatial coordinate, respectively. 1◦ . General solution: Φ(axw – y, w) = 0

or

y = axw + Φ(w),

where Φ and Φ are arbitrary functions. 2◦ . The solution of the Cauchy problem with the initial condition w = ϕ(y)

at x = 0

can be represented in parametric form as y = ξ + aϕ(ξ)x,

w = ϕ(ξ).

For a > 0 and ϕ′ (ξ) > 0, these relations describe the classical single-valued solution.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 5

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

6

Page 6

FIRST-ORDER QUASILINEAR EQUATIONS

3◦ . Consider the Cauchy problem with the discontinuous initial condition w(0, y) =

n

w1 w2

for y < 0, for y > 0.

It is assumed that a > 0, w1 > 0, and w2 > 0. For w1 < w2 , the generalized solution has the form w(x, y) =

(

w1 y/(ax) w2

for y/x < V1 , for V1 ≤ y/x ≤ V2 , for y/x > V2 ,

where

V1 = aw1 , V2 = aw2 .

This solution is continuous in the half-plane x > 0 and describes a “rarefaction wave.” For w1 > w2 , the generalized solution has the form  w1 for y/x < V , w(x, y) = where V = 12 a(w1 + w2 ). w2 for y/x > V , This solution experiences a discontinuity along the line y = V x and describes a “shock wave.” 4◦ . In Subsection 23.3, examples are given which illustrate qualitative features of solutions to Hopf’s equations, including the wave breaking effect and shock waves. These subsections also present general formulas that permit one to construct discontinuous solutions for arbitrary initial conditions. The references listed below contain a lot of solutions to Cauchy problems which describe the merging and decay of discontinuities, periodic waves, and other nonlinear physical effects. References for equation 1.1.1.18: E. Hopf (1950), P. D. Lax (1954), O. A. Oleinik (1954, 1957, 1959), R. Courant (1962), G. B. Whitham (1974), J. Lighthill (1978), B. L. Rozhdestvenskii and N. N. Yanenko (1983), R. J. LeVeque (1992), A. G. Kulikovskii and E. I. Sveshnikova (1995).

19.

∂w ∂x

+ aw

∂w ∂y

= b.


General solution: Φ w – bx, aw2 – 2by = 0. 20.

∂w ∂x

+ aw

∂w ∂y

= bx.


General solution: y = axw – 13 abx3 + Φ w – 12 bx2 . ∂w ∂w + aw = by. ∂x ∂y Z y General solution: x = p

21.

y0

dt ab(t2 – y 2 ) + a2 w2

arbitrary. 22.

∂w ∂x

+ aw

∂w ∂y


+ Φ aw2 – by 2 , where y0 may be taken

+ bw = 0.

This is a model equation of nonlinear waves with damping (a, b > 0).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 6

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.1. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY PARAMETERS

Page 7

7


1◦ . General solution: Φ aw + by, webx = 0.

2◦ . The solution of the Cauchy problem with the initial condition w(0, y) = f (y) can be represented in parametric form as
a y=ξ+ 1 – e–bx f (ξ), w = e–bx f (ξ). b Reference: G. B. Whitham (1974).

23.

∂w ∂x

+ (aw + bx)

∂w ∂y

= 0.

General solution: y = axw + 12 bx2 + Φ(w). 24.

∂w ∂x

+ (aw + by)

General solution: x = 25.

∂w ∂x

∂w ∂y

= 0.

1 ln |aw + by| + Φ(w). b

+ (w + ax + by + c)

∂w ∂y

= 0.

General solution: bx = ln a + b(w + ax + by + c) + Φ(w).

26.

∂w

+ (aw + bxk + c)

∂x General solution:

y=

27.

∂w ∂x

+ axw

∂w ∂y

(

∂w ∂y

= 0.

b xk+1 + cx + Φ(w) k+1 axw + b ln |x| + cx + Φ(w) axw +

if k ≠ –1, if k = –1.

= bx.


General solution: Φ aw2 – 2by, 2w – bx2 = 0. 28.

(a – w)

∂w ∂w + (a + w) = 0. ∂x ∂y


General solution: Φ w, x(w + a) + y(w – a) = 0. ∂w ∂w + (b1 + b2 w) = c. ∂x ∂y 1◦ . General solution for c = 0: 29.

(a1 + a2 w)


Φ w, x(b1 + b2 w) – y(a1 + a2 w) = 0.

2◦ . General solution for c ≠ 0:


Φ a2 w2 + 2a1 w – 2cx, b2 w2 + 2b1 w – 2cy = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 7

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

8 30.

Page 8

FIRST-ORDER QUASILINEAR EQUATIONS (w – ay)

∂w ∂w + by = cw. ∂x ∂y


General solution: c(bx + ay) – bw = Φ wb y –c . 31.

[m(w – c) – n(y – b)]

∂w ∂w + [n(x – a) – l(w – c)] = l(y – b) – m(x – a). ∂x ∂y

This is an equation of a helical surface and the surface of bodies of
revolution. General solution: Φ (x – a)2 + (y – b)2 + (w – c)2 , lx + my + nw = 0. 32.

[b(x + y) – c(x + w)]

∂w ∂w + [c(y + w) – a(x + y)] = a(x + w) – b(y + w). ∂x ∂y

General solution: Φ(ax + by + cw, xy + yw + xw) = 0. ∂w ∂w + by = cw. ∂x ∂y  c  General solution: Φ wb y –c , aw + cx – y 2 = 0. 2b 33.

(y 2 – aw)

∂w ∂w + [1 – (y + a)2 ] = (y + a)w – x. ∂x ∂y
General solution: Φ x + (y + a)w, x(y + a) + w = 0.

34.

35.

[x(y + a) – w]

∂w ∂x

+ (aw 2 + bw + c)

∂w ∂y

= s.


General solution: Φ w – sx, 13 aw3 + 12 bw2 + cw – sy = 0. 36.

∂w ∂w + (aw 2 + bw + cx + s) = 0. ∂x ∂y

General solution: y = x(aw2 + bw + s) + 12 cx2 + Φ(w). 37.

∂w ∂w + (aw 2 + bw + cy + s) = 0. ∂x ∂y

General solution: x =

38.

∂w ∂x

1 ln |aw2 + bw + cy + s| + Φ(w). c

+ (a2 w 2 + a1 w + b2 x2 + b1 x + b0 )

∂w ∂y

= 0.

General solution: y = x(a2 w2 + a1 w) + 13 b2 x3 + 12 b1 x2 + b0 x.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 8

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.1. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY PARAMETERS

39.

(a + bx2 )

Page 9

9

∂w ∂w + (α + βw 2 ) = γ. ∂x ∂y

1◦ . General solution for ab > 0: √  p
Φ ab w – γ arctan b/a x , αw + 13 βw3 – γy = 0.

2◦ . General solution for ab < 0: √   √ bx – –ab 3 1 , αw + 3 βw – γy = 0. Φ 2 –ab w – γ ln √ bx + –ab ∂w

∂w

= w 2 – a2 xy. ∂x ∂y   w w + a ln |y|, + a ln |x| = 0. General solution: Φ x y

40.

x(w + ax)

41.

ax2 (by – cw)

42.

b(by + cw)2

+ y(w + ay)

∂w

+ by 2 (cw – ax)

∂w

= cw 2 (ax – by). ∂x ∂y General solution: Φ(xyw, abxy + bcyw + acxw) = 0. ∂w ∂w – a2 x(by + 2cw) = a2 xw. ∂x ∂y


General solution: Φ wb (by + y + 2cw), a2 x2 + b2 y 2 – bc2 w2 = 0.

∂w ∂w + 2axy = 2axw. ∂x ∂y  2  ax + by 2 + cw2 w General solution: Φ , = 0. y y 43.

(ax2 – by 2 – cw 2 )

44.

by(3ax2 + by 2 + cw 2 )

∂w



∂x

– 2ax(ax2 + cw 2 )

ax2 + by 2 + cw2 2ax2 + by 2 General solution: Φ , w w2 45.

b(a2 xy – abyw – b2 w 2 )

∂w ∂x



∂w ∂y

= 2abxyw.

= 0.

+ ab(bxw – axy – ay 2 )

∂w

∂y = a (axy + bxw + byw + ay 2 – ax2 ). 2

General solution: Φ(ax2 + ay 2 + 2byw, a2 x2 + b2 w2 + 2a2 xy) = 0. 46.

xy(a2 xy + 2b2 w 2 ) 

∂w ∂x

+ byw(byw – ax2 )

w b2 w2 bxw , + + ay General solution: Φ y ax y



∂w ∂y

= bw 2 (byw – ax2 ).

= 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 9

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

10

Page 10

FIRST-ORDER QUASILINEAR EQUATIONS

∂w ∂w + (aw k + b) = c. ∂x ∂y General solution:
Φ w, y – awk x – bx = 0   awk+1 + bw – cy = 0 Φ w – cx, k+1

47.

48.

∂w ∂x

+ (aw k + bx)

∂w ∂y

if c = 0, if c ≠ 0.

= 0.

General solution: y = axwk + 12 bx2 + Φ(w). 49.

∂w ∂x

+ (aw k + by)

General solution: x = 50.

∂w ∂y

= 0.

1 ln |awk + by| + Φ(w). b

∂w ∂w + (aw k + bxm) = cxn. ∂x ∂y

This is a special case of equation 1.2.1.45 with f (w) = awk , g(x) = bxm , and h(x) = cxn . 51.

∂w ∂x

+ (aw k + bxm)

∂w ∂y

= cw n.

This is a special case of equation 1.2.1.46 with f (w) = awk , g(x) = bxm , and h(w) = cwn . ∂w ∂w + axny mw k = 0. ∂x ∂y General solution: a 1 y 1–m – xn+1 wk = Φ(w) 1–m n+1 a ln |y| – xn+1 wk = Φ(w) n+1 1 y 1–m – awk ln |x| = Φ(w) 1–m ln |y| – awk ln |x| = Φ(w)   ∂w n ∂w +k = 1. 53. (ax + by + cw) ∂x ∂y Z du General solution: = x + Φ(y – kx), –n cu + a + bk 52.

54.

a+

p

w – αx – βy


∂w ∂x

+b

∂w ∂y

if n ≠ –1, m ≠ 1; if n ≠ –1, m = 1; if n = –1, m ≠ 1; if n = –1, m = 1.

where

u = ax + by + cw.

where

c = αa + βb.

= αa + βb.

General solution: p
Φ cy – bw, αy + 2b w – αx – βy = 0,

⊲ See also equations of the general form containing arbitrary functions in Section 1.2.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 10

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.1. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY PARAMETERS

Page 11

11

1.1.2. Coefficients of Equations Contain Exponential Functions ◮ Equations of the form F (x, y) ∂w + G(x, y) ∂w = H(x, y, w). ∂x ∂y 1.

a

∂w ∂x

+b

∂w ∂y

= ecw .

General solution: w = –

2.

∂w ∂x

+a

∂w ∂y

= b + ceβx+λw .

General solution: w = –

3.

∂w ∂x

+a

∂w ∂y

1 h c i ln Φ(bx – ay) – x . c a

1 h –bλx cλ βx i ln e Φ(y – ax) – e . λ β + bλ

= beβx + ceγx+λw.

This is a special case of equation 1.2.1.2 with f (x) = beβx and g(x) = ceγx . 4.

a

∂w ∂w +b = c + (keβx + seγy )eλw . ∂x ∂y

This is a special case of equation 1.2.1.4 with f (x) = keβx and g(y) = seγy . 5.

a

∂w ∂w +b = ec(w+λx+βy) . ∂x ∂y

General solution:

6.

ax

∂w ∂x

 i 1 h 1  c(λx+βy)  – ln Φ(bx – ay) – e  c aλ + bβ w= h  β 1 c i   (bx – ay) – ln Φ(bx – ay) – x a c a + by

∂w ∂y

if λ ≠ –bβ/a, if λ = –bβ/a.

= ecw .

General solution: w = –

i
c 1 h ln Φ |x|b |y|–a – ln |x| . c a

∂w ∂w + by = ec(w+λx+βy) . ∂x ∂y   Z 1 c exp(cλx + cβxb/a u–1/a ) General solution: w = – ln Φ(u) – dx , where u = xb y –a . c a x In the integration, u is treated as a constant. 7.

ax

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 11

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

12

Page 12

FIRST-ORDER QUASILINEAR EQUATIONS

∂w ∂w + bx = ec(w+λx+βy) . ∂x ∂y General solution:     Z
1 c cβ p 2 dx 2 2 w = – ln Φ bx – ay – 3/2 exp cλx + √ bx – u √ , c a a bx2 – u 8.

ay

where u = bx2 – ay 2 . In the integration, u is treated as a constant. 9.

axn

∂w ∂x

+ by k

∂w

= ceλw + s.

∂y

This is a special case of equation 1.2.1.11 with f (w) = ceλw + s. 10.

ay n

∂w ∂x

+ bxk

∂w ∂y

= ceλw + s.

This is a special case of equation 1.2.1.12 with f (w) = ceλw + s. 11.

aeλx

∂w ∂x

+b

∂w

= eγ(w+βy) .

∂y

General solution: w = – ∂w

13.

aeλy

14.

aeαx

∂x

∂w

+ beβy

∂w

= eγw. ∂y
γ –λx i 1 h General solution: w = – ln Φ bβe–λx – aλe–βy + e . γ aλ 12.

aeλx


1 γβy i 1 h ln Φ aλy + be–λx – e . γ bβ

∂w

= ceγw + k. ∂x ∂y This is a special case of equation 1.2.1.14 with f (w) = ceγw + k. ∂w ∂x

+ beβy

+ beβy

∂w ∂y

= ec(w+λx+µy) .

  Z
–cµ/β 1 (aα)cµ/β c (cλ–α)x –αx General solution: w = – ln Φ(u) – e bβe –u dx , where c a u = bβe–αx – aαe–βy . In the integration, u is treated as a constant. aeβy

∂w

+ beαx

∂w

= ec(w+λx+µy) . ∂x ∂y General solution:  Z
(cµ–β)/β cα   ecλx bβeαx – u dx  Φ(u) – cµ/β (aα) e–cw =    Φ(u) – 1 ecλx aλ 15.

if β ≠ cµ, if β = cµ,

where u = bβeαx – aαeβy . In the integration, u is treated as a constant.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 12

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.1. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY PARAMETERS

Page 13

13

∂w ∂w +b = ecw . ∂x ∂y
c i 1 h General solution: w = – ln Φ bβe–αx + aαeβy – y . c b 16.

aeαx+βy

17.

b1 eβ1 x

∂w ∂x

+ b2 eβ2 y

∂w ∂y

= a + (c1 eγ1 x + c2 eγ2 y )eλw.

This is a special case of equation 1.2.1.18 with f1 (x) = b1 eβ1 x , f2 (y) = b2 eβ2 y , g1 (x) = c1 eγ1 x , and g2 (y) = c2 eγ2 y . + G(x, y, w) ∂w = H(x, y, w). ◮ Equations of the form F (x, y, w) ∂w ∂x ∂y 18.

∂w ∂x

+ (aeλw + b)

∂w

= 0.

∂y

General solution: y = x(aeλw + b) + Φ(w). 19.

∂w ∂x

+ (aeλw + bx)

∂w ∂y

= 0.

General solution: y = axeλw + 12 bx2 + Φ(w). 20.

∂w ∂x

+ (aeλw + by)

General solution: x = 21.

∂w ∂x

∂w ∂y

1 ln |aeλw + by| + Φ(w). b

+ (aeλw + beβx + c)

General solution: y = axeλw + 22.

∂w ∂x

= 0.

+ (aeλw + bxm)

∂w ∂y

= 0.

b βx e + cx + Φ(w). β

∂w ∂y

= cxn.

This is a special case of equation 1.2.1.45 with f (w) = aeλw , g(x) = bxm , and h(x) = cxn . 23.

∂w ∂x

+ (aeλw + bxm)

∂w ∂y

= cw n.

This is a special case of equation 1.2.1.46 with f (w) = aeλw , g(x) = bxm , and h(w) = cwn . 24.

∂w ∂x

+ (aeλw + beβx)

∂w ∂y

= ceγx.

This is a special case of equation 1.2.1.45 with f (w) = aeλw , g(x) = beβx , and h(x) = ceγx . 25.

∂w ∂x

+ (aeλw + beβx)

∂w ∂y

= ceγw.

This is a special case of equation 1.2.1.46 with f (w) = aeλw , g(x) = beβx , and h(w) = ceγw . ⊲ See also equations of the general form containing arbitrary functions in Section 1.2.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 13

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

14

Page 14

FIRST-ORDER QUASILINEAR EQUATIONS

1.1.3. Coefficients of Equations Contain Hyperbolic Functions ◮ Equations of the form F (x, y) ∂w + G(x, y) ∂w = H(x, y, w). ∂x ∂y 1.

a

∂w ∂w +b = c coshk(λw) + s. ∂x ∂y

This is a special case of equation 1.2.1.5 with f (w) = c coshk (λw) + s. 2.

a

∂w ∂x

+b

∂w

= c sinhk(λw) + s.

∂y

This is a special case of equation 1.2.1.5 with f (w) = c sinhk (λw) + s. ∂w ∂w +b = c cosh(w + αx + βy). ∂x ∂y This is a special case of equation 1.2.1.6 with f (u) = c cosh u. 3.

a

4.

a

∂w ∂x

+b

∂w

= cxn coshk(λw).

∂y

This is a special case of equation 1.2.1.7 with f (x) = cxn , g(w) = coshk (λw). 5.

a

∂w ∂w +b = cxn sinhk(λw). ∂x ∂y

This is a special case of equation 1.2.1.7 with f (x) = cxn , g(w) = sinhk (λw). 6.

ax

∂w ∂x

+ by

∂w ∂y

= c tanhk(λw) + s.

This is a special case of equation 1.2.1.9 with f (w) = c tanhk (λw) + s. 7.

ay

∂w ∂x

+ bx

∂w ∂y

= c tanhk(λw) + s.

This is a special case of equation 1.2.1.10 with f (w) = c tanhk (λw) + s. 8.

a coshn(λx)

∂w

+ b coshm(βy)

∂w

= coshk(γw).

∂x ∂y This is a special case of equation 1.2.1.15 with f (x) = a coshn (λx), g(y) = b coshm (βy), and h(w) = coshk (γw). ◮ Equations of the form F (x, y, w) ∂w + G(x, y, w) ∂w = H(x, y, w). ∂x ∂y 9.

∂w

+ aw

∂w

∂x ∂y General solution:

= b cosh(λx).

y = axu +

ab cosh(λx) + Φ(u), λ2

where

u=w–

b sinh(λx). λ

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 14

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.1. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY PARAMETERS

Page 15

15

 ∂w ∂w  + a coshk(λw) + b = 0. ∂x ∂y   General solution: y = x a coshk (λw) + b + Φ(w). 10.

 ∂w ∂w  + a coshk(λw) + b coshn(βx) + c = 0. ∂x ∂y Z k General solution: y = ax cosh (λw) + b coshn (βx) dx + cx + Φ(w). 11.

12.

∂w ∂w + aw = b sinh(λx). ∂x ∂y

General solution: y = axu +

ab sinh(λx) + Φ(u), λ2

where

u=w–

b cosh(λx). λ

 ∂w ∂w  + a sinhk(λw) + b = 0. ∂x ∂y   General solution: y = x a sinhk (λw) + b + Φ(w).

13.

 ∂w ∂w  + a sinhk(λw) + b sinhn(βx) + c = 0. ∂x ∂y Z k General solution: y = ax sinh (λw) + b sinhn (βx) dx + cx + Φ(w). 14.

 ∂w ∂w  + a tanhk(λw) + b = 0. ∂x ∂y   General solution: y = x a tanhk (λw) + b + Φ(w). 15.

 ∂w ∂w  + a tanhk(λw) + b tanhn(βx) + c = 0. ∂x ∂y Z k General solution: y = ax tanh (λw) + b tanhn (βx) dx + cx + Φ(w). 16.

  ∂w + a cothk(λw) + b = 0. ∂x ∂y   General solution: y = x a cothk (λw) + b + Φ(w).

17.

∂w

  ∂w + a cothk(λw) + b cothn(βx) + c = 0. ∂x ∂y Z k General solution: y = ax coth (λw) + b cothn (βx) dx + cx + Φ(w).

18.

∂w

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 15

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

16

Page 16

FIRST-ORDER QUASILINEAR EQUATIONS

1.1.4. Coefficients of Equations Contain Logarithmic Functions ◮ Equations of the form F (x, y) ∂w + G(x, y) ∂w = H(x, y, w). ∂x ∂y 1.

a

∂w ∂x

+b

∂w ∂y

= c lnk(λw) + s.

This is a special case of equation 1.2.1.5 with f (w) = c lnk (λw) + s. ∂w ∂w +b = c ln(w + αx + βy). ∂x ∂y This is a special case of equation 1.2.1.6 with f (u) = c ln u. 2.

a

3.

a

∂w ∂w +b = cxn lnk(λw). ∂x ∂y

This is a special case of equation 1.2.1.7 with f (x) = cxn and g(w) = lnk (λw). 4.

ax

∂w ∂w + by = c lnk(λw) + s. ∂x ∂y

This is a special case of equation 1.2.1.9 with f (w) = c lnk (λw) + s. 5.

ay

∂w ∂x

+ bx

∂w ∂y

= c lnk(λw) + s.

This is a special case of equation 1.2.1.10 with f (w) = c lnk (λw) + s. a lnn(λx)

∂w

+ b lnm(βy)

∂w

= c lnk(γw). ∂x ∂y This is a special case of equation 1.2.1.15 with f (x) = a lnn (λx), g(y) = b lnm (βy), and h(w) = c lnk (γw). 6.

◮ Equations of the form F (x, y, w) ∂w + G(x, y, w) ∂w = H(x, y, w). ∂x ∂y ∂w ∂w + aw = b ln(λx). ∂x ∂y General solution: 7.

y = axu + 12 abx2 ln(λx) – 34 abx2 + Φ(u),

where

u = w + bx – bx ln(λx).

  ∂w + a lnk(λw) + b = 0. ∂x ∂y   General solution: y = x a lnk (λw) + b + Φ(w). 8.

9.

∂w

  ∂w + a lnk(λw) + bx = 0. ∂x ∂y

∂w

General solution: y = ax lnk (λw) + 12 bx2 + Φ(w).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 16

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.1. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY PARAMETERS

Page 17

17

 ∂w ∂w  + a lnk(λw) + by = 0. ∂x ∂y

10.

General solution: x =

1 ln |a lnk (λw) + by| + Φ(w). b

 ∂w ∂w  + a lnk(λw) + b lnn(βx) + c = 0. ∂x ∂y Z k General solution: y = ax ln (λw) + lnn (βx) dx + cx + Φ(w).

11.

1.1.5. Coefficients of Equations Contain Trigonometric Functions ◮ Equations of the form F (x, y, w) ∂w + G(x, y, w) ∂w = H(x, y, w). ∂x ∂y 1.

a

∂w ∂x

+b

∂w ∂y

= c cosk(λw) + s.

This is a special case of equation 1.2.1.5 with f (w) = c cosk (λw) + s. 2.

a

∂w ∂w +b = c sink(λw) + s. ∂x ∂y

This is a special case of equation 1.2.1.5 with f (w) = c sink (λw) + s. 3.

a

∂w ∂x

+b

∂w ∂y

= c cos(w + αx + βy).

This is a special case of equation 1.2.1.6 with f (u) = c cos u. 4.

a

∂w ∂w +b = cxn cosk(λw). ∂x ∂y

This is a special case of equation 1.2.1.7 with f (x) = cxn and g(w) = cosk (λw). 5.

a

∂w ∂x

+b

∂w ∂y

= cxn sink(λw).

This is a special case of equation 1.2.1.7 with f (x) = cxn and g(w) = sink (λw). 6.

ax

∂w ∂x

+ by

∂w ∂y

= c tank(λw) + s.

This is a special case of equation 1.2.1.9 with f (w) = c tank (λw) + s. 7.

ay

∂w ∂w + bx = c tank(λw) + s. ∂x ∂y

This is a special case of equation 1.2.1.10 with f (w) = c tank (λw) + s.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 17

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

18 8.

Page 18

FIRST-ORDER QUASILINEAR EQUATIONS a cosn(λx)

∂w ∂w + b cosm(βy) = cosk(γw) + c. ∂x ∂y

This is a special case of equation 1.2.1.15 with f (x) = a cosn (λx), g(y) = b cosm (βy), and h(w) = cosk (γw) + c. ◮ Equations of the form F (x, y, w) ∂w + G(x, y, w) ∂w = H(x, y, w). ∂x ∂y 9.

∂w ∂w + aw = b cos(λx). ∂x ∂y

General solution: y = axu –

ab cos(λx) + Φ(u), λ2

where

u=w–

b sin(λx). λ

  ∂w + a cosk(λw) + b = 0. ∂x ∂y   General solution: y = x a cosk (λw) + b + Φ(w).

10.

∂w

 ∂w ∂w  + a cosk(λw) + b cosn(βx) + c = 0. ∂x ∂y Z General solution: y = ax cosk (λw) + b cosn (βx) dx + cx + Φ(w). 11.

12.

∂w ∂x

+ aw

∂w ∂y

= b sin(λx).

General solution: y = axu –

ab sin(λx) + Φ(u), λ2

where

u=w+

b cos(λx). λ

  ∂w + a sink(λw) + b = 0. ∂x ∂y   General solution: y = x a sink (λw) + b + Φ(w). 13.

∂w

  ∂w + a sink(λw) + b sinn(βx) + c = 0. ∂x ∂y Z General solution: y = ax sink (λw) + b sinn (βx) dx + cx + Φ(w).

14.

∂w

 ∂w ∂w  + a tank(λw) + b = 0. ∂x ∂y   General solution: y = x a tank (λw) + b + Φ(w).

15.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 18

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.2. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY FUNCTIONS

Page 19

19

 ∂w ∂w  + a tank(λw) + b tann(βx) + c = 0. ∂x ∂y Z k General solution: y = ax tan (λw) + b tann (βx) dx + cx + Φ(w).

16.

  ∂w + a cotk(λw) + b = 0. ∂x ∂y   General solution: y = x a cotk (λw) + b + Φ(w). ∂w

17.

 ∂w ∂w  + a cotk(λw) + b cotn(βx) + c = 0. ∂x ∂y Z k General solution: y = ax cot (λw) + b cotn (βx) dx + cx + Φ(w). 18.

1.2. Equations with Two Independent Variables Containing Arbitrary Functions 1.2.1. Equations Contain Arbitrary Functions of One Variable ◮ Equations of the form F (x, y) ∂w + G(x, y) ∂w = H(x, y, w). ∂x ∂y 1.

∂w ∂x

+a

∂w ∂y

= f (x)w + g(x)w k.

General solution: w

2.

1–k

= F (x)Φ(y – ax) + (1 – k)F (x)

∂w ∂x

+a

∂w ∂y

Z

g(x) dx, F (x)

  Z F (x) = exp (1 – k) f (x) dx .

= f (x) + g(x)eλw.

General solution: –λw

e

3.

a

= F (x)Φ(y – ax) – λF (x)

Z

g(x) dx, F (x)

 Z  F (x) = exp –λ f (x) dx .

  ∂w ∂w +b = cw + f (x) + g(y) w k. ∂x ∂y

General solution:

Z Z ab 1–k f (x) dx g(y) dy w = E1 (x)Φ(bx – ay) + bE1 (x) + aE2 (y) , 1–k E1 (x) E2 (y)     c c where E1 (x) = exp (1 – k)x and E2 (y) = exp (1 – k)y . a b

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 19

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

20 4.

Page 20

FIRST-ORDER QUASILINEAR EQUATIONS a

  ∂w ∂w +b = c + f (x) + g(y) eλw . ∂x ∂y

General solution:

ab –λw = E1 (x)Φ(bx – ay) – bE1 (x) e λ

Z

f (x) dx – aE2 (y) E1 (x)

Z

g(y) dy , E2 (y)

 cλ   cλ  where E1 (x) = exp – x and E2 (y) = exp – y . a b ∂w ∂w +b = f (w). ∂x ∂y Z x dw General solution: = + Φ(bx – ay). f (w) a 5.

a

6.

a

∂w ∂x

+b

∂w

= f (w + αx + βy).

∂y

The substitution u = w + αx + βy leads to an equation of the form 1.2.1.5: a

7.

a

∂w ∂x

+b

∂w

= f (x)g(w).

∂y

General solution:

∂u ∂u +b = f (u) + aα + bβ. ∂x ∂y

Z

dw 1 = g(w) a

Z

f (x) dx + Φ(bx – ay).

∂w ∂w +a = f (x)g(y)h(w). ∂x ∂y Z Z x dw General solution: = f (t)g(y – ax + at) dt + Φ(y – ax), where x0 may be taken h(w) x0 arbitrary. 8.

9.

ax

∂w ∂x

+ by

General solution: ∂w

∂y Z

∂x

+ bx

= f (w).


dw 1 = ln |x| + Φ |x|b |y|–a . f (w) a

∂w

= f (w). ∂y Z √
dw 1 General solution: = √ ln ab x + ay + Φ ay 2 – bx2 , f (w) ab 10.

ay

∂w

ab > 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 20

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.2. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY FUNCTIONS

∂w ∂w + by k = f (w). ∂x ∂y General solution: Z dw 1 = x1–n + Φ(u), f (w) a(1 – n)

11.

21

axn

ay n

∂w

+ bxk

u=

1 1 x1–n – y 1–k . a(1 – n) b(1 – k)

∂w

= f (w). ∂x ∂y General solution: – n Z Z  n+1 dw b n + 1 k+1 a dx, = x –u f (w) a k+1 12.

Page 21

u=

b n + 1 k+1 x – y n+1 . a k+1

In the integration, u is considered a parameter.

∂w + beβy = f (w). ∂x ∂y Z dw 1 General solution: = – e–λx + Φ(u), where u = aλe–βy – bβe–λx . f (w) aλ 13.

aeλx

∂w

∂w + beβx = f (w). ∂x ∂y Z c(βx – λy) dw = + Φ(u), where u = aβeλy – bλeβx . General solution: f (w) u 14.

aeλy

∂w

∂w + g(y) = h(w). ∂x ∂y Z Z Z Z dx dx dy dw = + Φ(u), where u = – . General solution: h(w) f (x) f (x) g(y) 15.

f (x)

∂w

∂w + g(x) = h(w). ∂x ∂y Z Z The transformation ξ = g(x) dx, η = f (y) dy leads to an equation of the form 1.2.1.8: 16.

f (y)

∂w

∂w ∂w + = F (ξ)G(η)h(w), ∂ξ ∂η

17.

f1 (x)

∂w

∂x General solution: w

where

1–k

+ f2 (y)

∂w ∂y

F (ξ) =

1 , g(x)

G(η) =

  = aw + g1 (x) + g2 (y) w k.

= E1 (x)Φ(u) + (1 – k)E1 (x)

 Z E1 (x) = exp a(1–k)

where

Z

g1 (x) dx + (1 – k)E2 (y) f1 (x)E1 (x)

Z

1 . f (y)

g2 (y) dy , f2 (y)E2 (y)

   Z Z Z dx dy dx dy , E2 (y) = exp a(1–k) , u= – . f1 (x) f2 (y) f1 (x) f2 (y)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 21

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

22

Page 22

FIRST-ORDER QUASILINEAR EQUATIONS

  ∂w ∂w + f2 (y) = a + g1 (x) + g2 (y) eλw . ∂x ∂y General solution: Z Z g1 (x) dx g2 (y) dy –λw e = E1 (x)Φ(u) – λE1 (x) – λE2 (y) , f1 (x)E1 (x) f2 (y)E2 (y) 18.

f1 (x)

where

 Z E1 (x) = exp –aλ

   Z Z Z dx dy dx dy , E2 (y) = exp –aλ , u= – . f1 (x) f2 (y) f1 (x) f2 (y)

+ G(x, y, w) ∂w = H(x, y, w). ◮ Equations of the form F (x, y, w) ∂w ∂x ∂y ∂w ∂w + aw = f (x). ∂x ∂y General solution: Z  
y = ax w – F (x) + a F (x) dx + Φ w – F (x) ,

19.

∂w ∂w + aw = f (y). ∂x ∂y General solution: Z y dz x=± + Φ(u), √ 2aF (z) – 2au y0 Z where F (y) = f (y) dy.

where

F (x) =

Z

f (x) dx.

20.

∂w

1 2 aw , 2

∂w

= f (y – bx). ∂x ∂y This is a model equation describing nonlinear waves issuing from a moving source (the variables x and y play the role of time and the spatial coordinate, respectively, and b is the source velocity). 21.

+ aw

u = F (y) –

1◦ . General solution: Z y–bx x=± p z0

where F (z) =

Z

dz b2

+ Φ(u),

+ 2aF (z) – 2au

u = F (y – bx) –

1 2 aw + bw, 2

f (z) dz; z0 may be taken arbitrary.

2◦ . Solution with steady profile:  Z 2 w = b – (b – w0 ) – 2

∞ ξ

f (z) dz

1/2

,

ξ = y – bx,

where w0 is the constant of integration. References for equation 1.2.1.21: A. L. Hoffman (1967) and G. B. Whitham (1974).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 22

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.2. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY FUNCTIONS

22.

Page 23

23

 ∂w ∂w  + aw + f (x) = g(x). ∂x ∂y

General solution:

  y = ax w – G(x) + a

where F (x) =

23.

Z

Z

f (x) dx and G(x) =

Z


G(x) dx + F (x) + Φ w – G(x) , g(x) dx.

  ∂w + aw + yf (x) = 0. ∂x ∂y

∂w

General solution:

yF (x) – aw

Z

F (x) dx = Φ(w),

where

 Z  F (x) = exp – f (x) dx .

  ∂w + aw + f (y) = 0. ∂x ∂y Z y dt General solution: x = + Φ(w). y0 f (t) + aw

24.

∂w

  ∂w + ayw + f (x) = 0. ∂x ∂y Z x f (t) exp(–atw) dt = Φ(w). General solution: y exp(–axw) – 25.

∂w

x0

26.

∂w ∂x

– f (y)w

∂w ∂y

= g(x)w + h(x).

Complete integral:* w = ϕ(x)

Z

dy + ψ(x), f (y)

where  –1 Z ϕ(x) = G(x) C1 – G(x) dx ,   Z h(x) dx , ψ(x) = S(x) C2 + S(x)

G(x) = exp

Z



g(x) dx , Z  S(x) = G(x) exp ϕ(x) dx .

* The general integral for equations 1.2.1.26–1.2.1.28 can be constructed according to formulas presented in Subsection 24.1.1.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 23

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

24

Page 24

FIRST-ORDER QUASILINEAR EQUATIONS

∂w ∂w + f (y)w = aw 2 + g(x)w + h(x). ∂x ∂y Complete integral:  Z w(x, y) = ϕ(x) + C1 ψ(x) exp a 27.

 dy , f (y)

where C1 is an arbitrary constant, and the functions ϕ(x) and ψ(x) are determined by the following system of first order ordinary differential equations: ϕ′x = aϕ2 + g(x)ϕ + h(x),   ψx′ = aϕ + g(x) ψ.

(1) (2)

The Riccati equation (1) is integrable in quadrature for a lot of specific functions g(x) and h(x) [e.g., for h(x) ≡ 0 and any g(x)]. For details, see the books by Kamke (1976) and Polyanin and Zaitsev (2003). If the solution of equation (1) is known, equation (2) is easy to integrate—it is linear in the unknown function ψ. 28.

∂w

– f (y)w

∂x Complete integral:

∂w ∂y

= g(y)w 2 + h(x)w.

   –1 Z Z 1 Ψ(y) C2 – dy , w = H(x) C1 + H(x) dx Ψ(y) f (y) where H(x) = exp

Z



h(x) dx ,

Ψ(y) = exp

Z

  ∂w – aw + yf (x) + g(x) – h(x)w = 0. ∂x ∂y The transformation Z w(x, y) = H(x)u(t, z), z = yF (x) + g(x)F (x) dx, 29.

∂w

where F (x) = exp

Z

 f (x) dx ,

H(x) = exp

leads to a simpler equation of the form 1.1.1.18: ∂w

+ f (x)w k

Z

t=

Z

F (x)H(x) dx, 

h(x) dx ,

∂u ∂u – au = 0. ∂t ∂z

∂w

= g(x)w. ∂x ∂y The transformation Z 1 ξ = f (x)Gk (x) dx, u(ξ, y) = w(x, y), G(x) 30.

 g(y) dy . f (y)

where

leads to a special case of equation 1.1.1.47 with b = c = 0:

G(x) = exp

Z

 g(x) dx ,

∂u ∂u + uk = 0. ∂ξ ∂y

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 24

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.2. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY FUNCTIONS

31.

1 w(x, y), u(x, η) = Ψ(y)

where

leads to a special case of equation 1.1.1.47 with b = c = 0: ∂w ∂x

+ f (x)eλw

∂w ∂y

The transformation Z ξ = f (x)eλG(x) dx,

∂w

+ f (y)eλw

u(ξ, y) = w(x, y) – G(x),

leads to a special case of equation 1.1.2.18 with b = 0: 

f (x) + aw

General solution:

 g(y) dy , f (y)

∂u ∂u + uk = 0. ∂x ∂η

where

G(x) =

Z

g(x) dx,

∂u ∂u + eλu = 0. ∂ξ ∂y

∂w

= g(y)eλw. ∂x ∂y The transformation Z dy , u(x, η) = w(x, y) – Ψ(y), η= f (y)eλΨ(y)

34.

Ψ(y) = exp

Z

= g(x).

leads to a special case of equation 1.1.2.18 with b = 0:

33.

25

∂w ∂w + f (y)w k = g(y)w k+1. ∂x ∂y

The transformation Z dy η= , f (y)Ψk (y)

32.

Page 25

where

Ψ(y) =

Z

g(y) dy, f (y)

∂u ∂u + eλu = 0. ∂x ∂η

  ∂w + g(y) + bw = 0. ∂x ∂y

 ∂w

Z

dx – f (x) + aw

Z

dy = Φ(w). g(y) + bw

In the integration, w is considered a parameter.   ∂w + g(x) + bw = 0. ∂x ∂y Z Z General solution: (bx – ay)w + g(x) dx – f (y) dy = Φ(w).

35.

36.



f (y) + aw

∂w ∂x

 ∂w

+ f (w)

∂w ∂y

= 0.

Model equation of gas dynamics. This equation is also encountered in hydrodynamics, multiphase flows, wave theory, acoustics, chemical engineering, and other applications.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 25

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

26

Page 26

FIRST-ORDER QUASILINEAR EQUATIONS

1◦ . General solution: Φ(y – xf (w), w) = 0,

or

y = xf (w) + Φ(w),

where Φ and Φ are arbitrary functions. 2◦ . The solution of the Cauchy problem with the initial condition w = ϕ(y)

at x = 0

can be represented in the parametric form
where F(ξ) = f ϕ(ξ) .

y = ξ + F(ξ)x,

w = ϕ(ξ),

3◦ . Consider the Cauchy problem with the discontinuous initial condition n w1 for y < 0, w(0, y) = w2 for y > 0.

It is assumed that x ≥ 0, f > 0, and f ′ > 0 for w > 0, w1 > 0, and w2 > 0. Generalized solution for w1 < w2 : ( for y/x < V1 , w1 where V1 = f (w1 ), V2 = f (w2 ). w(x, y) = f –1 (y/x) for V1 ≤ y/x ≤ V2 , w2 for y/x > V2 ,
Here f –1 is the inverse of the function f , i.e., f –1 f (w) ≡ w. This solution is continuous in the half-plane x > 0 and describes a “rarefaction wave.” Generalized solution for w1 > w2 :  Z w2 1 w1 for y/x < V , where V = f (w) dw. w(x, y) = w2 for y/x > V , w2 – w1 w1 This solution undergoes a discontinuity along the line y = V x and describes a “shock wave.”

4◦ . In Section 23.3, qualitative features of solutions to model equations of gas dynamics are considered, including the wave breaking effect and shock waves. This subsection also presents general formulas that permit one to construct generalized (discontinuous) solutions for arbitrary initial conditions. The references listed below contain a lot of solutions to Cauchy problems which describe the merging and decay of discontinuities, periodic waves, and other nonlinear physical effects. References for equation 1.2.1.36: E. Hopf (1950), P. D. Lax (1954), O. A. Oleinik (1954, 1957, 1959), R. Courant (1962), G. B. Whitham (1974), J. Lighthill (1978), B. L. Rozhdestvenskii and N. N. Yanenko (1978), R. J. LeVeque (1992), A. G. Kulikovskii and E. I. Sveshnikova (1995).

∂w ∂w + f (w) = g(x). ∂x ∂y General solution: Z x

y= f G(t) – G(x) + w dt + Φ w – G(x) , 37.

x0

where

G(x) =

Z

g(x) dx.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 26

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.2. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY FUNCTIONS

Page 27

27

∂w ∂w + f (w) = g(y). ∂x ∂y General solution: Z y

x= ψ G(t) – G(y) + F (w) dt + Φ F (w) – G(y) ,

38.

where G(y) =

Z

y0

g(y) dy and F (w) =

parametrically by ψ =

1 , z = F (w). f (w)

Z

f (w) dw. The function ψ = ψ(z) is defined

∂w ∂w + f (w) = g(y – ax). ∂x ∂y This is a model equation describing nonlinear waves issuing from a moving source (the variables x and y play the role of time and the spatial coordinate, respectively, and a is the source velocity).

39.

1◦ . The substitution ξ = y – ax leads to an equation of the form 1.2.1.38:  ∂w ∂w  + f (w) – a = g(ξ). ∂x ∂ξ

2◦ . Solution with steady profile: Z Z f (w) dw – aw = g(ξ) dξ + C,

ξ = y – ax,

where C is the constant of integration.

∂w ∂w + f (w) = g(w). ∂x ∂y   Z Z f (w) dw General solution: y = dw + Φ x – . g(w) g(w) 40.

41.

  ∂w + f (w) + ax = 0. ∂x ∂y

∂w

General solution: y = xf (w) + 12 ax2 + Φ(w).   ∂w + f (w) + ay = 0. ∂x ∂y 1 General solution: x = ln ay + f (w) + Φ(w). a

42.

∂w

∂w ∂w + f (w + ax + by) = 0. ∂x ∂y The substitution u = w + ax + by leads to an equation of the form 1.2.1.38: 43.

∂u ∂u + f (u) = a + bf (u). ∂x ∂y

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 27

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

28

Page 28

FIRST-ORDER QUASILINEAR EQUATIONS

 ∂w ∂w  + f (w) + g(x) = 0. ∂x ∂y Z General solution: y = xf (w) + g(x) dx + Φ(w).

44.

 ∂w ∂w  + f (w) + g(x) = h(x). ∂x ∂y General solution: Z x

y= f H(t) – H(x) + w dt + G(x) + Φ w – H(x) ,

45.

x0

where G(x) =

Z

g(x) dx and H(x) =

Z

h(x) dx.

  ∂w + f (w) + g(x) = h(w). ∂x ∂y General solution:
Z w Z
g H(t) – H(w) + x f (w) dw + dt + Φ x – H(w) , y= h(w) h(t) w0 Z dw where H(x) = . h(w)

46.

∂w

  ∂w + f (w) + g(y) = 0. ∂x ∂y Z y dt General solution: x = + Φ(w). y0 g(t) + f (w) 47.

∂w

  ∂w + f (w) + yg(x) = 0. ∂x ∂y General solution: Z yG(x) – f (w) G(x) dx = Φ(w), 48.

∂w

where

 Z  G(x) = exp – g(x) dx .

  ∂w + f (w) + yg(x) = h(x). ∂x ∂y General solution: Z

yG(x) – G(x)f H(t) – H(x) + w dx = Φ w – H(x) ,

49.

∂w

 Z  Z where G(x) = exp – g(x) dx and H(x) = h(x) dx.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 28

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.2. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY FUNCTIONS

Page 29

29

 ∂w ∂w  + yf (w) + g(x) = 0. ∂x ∂y Z x     General solution: y exp –xf (w) – g(t) exp –tf (w) dt = Φ(w), where x0 may be

50.

x0

taken arbitrary. 51.

 ∂w ∂w  + xf (w) + yg(w) = 0. ∂x ∂y

General solution: y + x

52.

 ∂w ∂w  + xf (w) + yg(w) + h(w) = 0. ∂x ∂y

General solution: y +

53.

  f (w) f (w) + 2 = exp g(w)x Φ(w). g(w) g (w)

  f (w) xf (w) + h(w) + 2 = exp g(w)x Φ(w). g(w) g (w)

  ∂w + x2n+1 y mf (w) + xnyg(w) = 0. ∂x ∂y

∂w

The transformation ξ = xn+1 , η = y 1–m leads to an equation of the form 1.2.1.51:  ∂w ∂w  + ξF (w) + ηG(w) = 0, ∂ξ ∂η 54.

where

F (w) =

1–m f (w), 1+n

G(w) =

1–m g(w). 1+n

  ∂w + eλxf (w) + eβy g(w) = 0. ∂x ∂y

∂w

General solution for λβ ≠ 0: –βy

e

55.

F (x, w) + βg(w) ∂w ∂x

Z

x

F (t, w) dt = Φ(w),

+ f (x)g(y)h(w)

∂w ∂y

∂w ∂x

+ f (x)g(y)h(w)

∂w ∂y

 β λx F (x, w) = exp e f (w) . λ

= 0.

General solution: Φ(u, w) = 0, where u =

56.

where

x0



Z

dy – h(w) g(y)

Z

f (x) dx.

= p(x).

General solution: Z Z x

dy = f (t)h P (t) – P (x) + w dt + Φ w – P (x) , g(y) x0

where

P (x) =

Z

p(x) dx.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 29

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

30

Page 30

FIRST-ORDER QUASILINEAR EQUATIONS

∂w ∂w + f (x)g(y)h(w) = p(w). ∂x ∂y General solution: Z w Z

dy h(t) = f P (t) – P (w) + x dt + Φ x – P (w) , g(y) w0 p(t) 57.

where

P (w) =

Z

dw . p(w)

 ∂w   ∂w f (x) + g1 (w) + h(y) + g2 (w) = 0. ∂x ∂y Z Z dy dx – = Φ(w). In the integration, w is General solution: f (x) + g1 (w) h(y) + g2 (w) considered a parameter. 58.



  ∂w + h(x) + g2 (w) = 0. ∂x ∂y Z Z General solution: g2 (w)x – g1 (w)y + h(x) dx – f (y) dy = Φ(w).

59.



f (y) + g1 (w)

 ∂w

1.2.2. Equations Contain Arbitrary Functions of Two Variables ◮ Equations of the form F (x, y) ∂w + G(x, y) ∂w = H(x, y, w). ∂x ∂y ∂w ∂w + f (y) = g(x, w). ∂x ∂y General solution: Z x–

1.

dy = I(x, w), f (y) where I(x, u) = C is a general integral of the ordinary differential equation u′x = g(x, u). ∂w

∂w

= h2 (x, y)w 2 + h1 (x, y)w + h0 (x, y). ∂x ∂y Suppose a particular solution w0 = w0 (x, y) of the equation is known. Then the substitution ζ = w – w0 leads to an equation of the form 1.2.2.3 with k = 2:
∂ζ ∂ζ +g = h2 ζ 2 + 2h2 w0 + h1 ζ. f ∂x ∂y For hn = const (n = 0, 1, 2), a particular solution w0 of the original equation can be determined by solving the quadratic equation h2 w02 + h1 w0 + h0 = 0. If the functions f and hn (n = 0, 1, 2) are independent of y, then a particular solution should be sought in the form w0 = w0 (x). 2.

f (x, y)

3.

f (x, y)

+ g(x, y)

∂w ∂w + g(x, y) = h1 (x, y)w + h2 (x, y)w k. ∂x ∂y

The substitution ξ = w1–k leads to the nonhomogeneous linear equation ∂ξ ∂ξ + g(x, y) = (1 – k)h1 (x, y)ξ + (1 – k)h2 (x, y). ∂x ∂y Equations of this type are considered in Polyanin, Zaitsev, and Moussiaux (2002, Chapter 5). f (x, y)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 30

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.2. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY FUNCTIONS

4.

f (x, y)

Page 31

31

∂w ∂w + g(x, y) = h1 (x, y) + h2 (x, y)eλw . ∂x ∂y

The substitution ξ = e–λw leads to the nonhomogeneous linear equation f (x, y)

∂ξ ∂ξ + g(x, y) = –λh1 (x, y)ξ – λh2 (x, y). ∂x ∂y

Equations of this type are considered in Polyanin, Zaitsev, and Moussiaux (2002, Chapter 5). 5.

f (x, y)

∂w ∂w + g(x, y) = h2 (x, y)eλw + h1 (x, y) + h0 (x, y)e–λw . ∂x ∂y

The substitution ξ = eλw leads to an equation of the form 1.2.2.2: f (x, y)

6.

f (x, y)

∂w ∂x

∂ξ ∂ξ + g(x, y) = λh2 (x, y)ξ 2 + λh1 (x, y)ξ + λh0 (x, y). ∂x ∂y

+ g(x, y)

∂w ∂y

= h1 (x, y) cosh(λw) + h2 (x, y).

Applying the formula cosh(λw) = 1.2.2.5. 7.

f (x, y)

1 λw 2 (e

+ e–λw ), we arrive at an equation of the form

∂w ∂w + g(x, y) = h1 (x, y) sinh(λw) + h2 (x, y). ∂x ∂y

Applying the formula sinh(λw) = 12 (eλw –e–λw ), we arrive at an equation of the form 1.2.2.5. 8.

f (x, y)

∂w ∂x

+ g(x, y)

∂w ∂y

= h1 (x, y)w ln w + h2 (x, y)w.

The substitution w = eξ leads to the nonhomogeneous linear equation f (x, y)

∂ξ ∂ξ + g(x, y) = h1 (x, y)ξ + h2 (x, y). ∂x ∂y

Equations of this type are considered in Polyanin, Zaitsev, and Moussiaux (2002, Chapter 5). 9.

f (x, y)

∂w ∂w + g(x, y) = h1 (x, y)w ln w + h2 (x, y)w lnk w. ∂x ∂y

The substitution w = eξ leads to an equation of the form 1.2.2.3: f (x, y)

∂ξ ∂ξ + g(x, y) = h1 (x, y)ξ + h2 (x, y)ξ k . ∂x ∂y

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 31

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

32

Page 32

FIRST-ORDER QUASILINEAR EQUATIONS

∂w ∂w + g(x, y) = h(x, y)ϕ(w). ∂x ∂y Z dw The substitution ξ = leads to the nonhomogeneous linear equation ϕ(w)

10.

f (x, y)

f (x, y)

∂ξ ∂ξ + g(x, y) = h(x, y). ∂x ∂y

Equations of this type are considered in Polyanin, Zaitsev, and Moussiaux (2002, Chapter 5); the equation can be solved using the method of characteristics described in changed to Subsection 24.3.5. ◮ Equations of the form F (x, y, w) ∂w + G(x, y, w) ∂w = H(x, y, w). ∂x ∂y ∂w

+ f (x, w)

∂w

= g(x). ∂x ∂y General solution: Z x

y= f t, G(t) – G(x) + w dt + Φ w – G(x) ,

11.

where

G(x) =

x0

∂w

13.

∂w ∂x

+ f (x, w)

+ f (x, w)g(y, w) Z

g(x) dx.

∂w

= g(w). ∂x ∂y General solution:
Z w
f G(t) – G(w) + x, t y= dt + Φ x – G(w) , g(t) w0

12.

Z

∂w ∂y

where

G(w) =

Z

dw . g(w)

= 0.

x

Z

y

dt + Φ(w), where Φ(w) is an arbitrary g(t, w) x0 y0 function, and x0 and y0 are arbitrary numbers.

General solution:

f (ξ, w) dξ =

  ∂w + f (x, w) + yg(x, w) = 0. ∂x ∂y General solution: Z x yG(x, w) – f (t, w)G(t, w) dt = Φ(w), where

14.

∂w

x0

15.

 Z G(x, w) = exp –

x



g(t, w) dt . x0

  ∂w + y kf (x, w) + yg(x, w) = 0. ∂x ∂y

∂w

The substitution z = y 1–k leads to an equation of the form 1.2.2.14:   ∂w ∂w + (1 – k) f (x, w) + zg(x, w) = 0. ∂x ∂z

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 32

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.2. EQUATIONS WITH TWO INDEPENDENT VARIABLES CONTAINING ARBITRARY FUNCTIONS

16.

Page 33

33

 ∂w ∂w  λy + e f (x, w) + g(x, w) = 0. ∂x ∂y

The substitution z = e–λy leads to an equation of the form 1.2.2.14:

17.

∂w

  ∂w ∂w – λ f (x, w) + zg(x, w) = 0. ∂x ∂z + f (ax + by, w)

∂w

∂x General solution for b ≠ 0: Φ(u, w) = 0,

= 0.

∂y

Z

u=

where

z

c

18.

∂w

+f



y

∂x x General solution:

,w



∂w

z = ax + by,

c is any.

= 0.

∂y

Φ(u, w) = 0,

dt – x, a + bf (t, w)

u=

where

Z

c

y/x

dt – ln |x|, f (t, w) – t

y ∂w ∂w + f (xny m, w) = 0. ∂x x ∂y General solution: Z z dt   – ln |x|, Φ(u, w) = 0, where u = c t mf (t, w) + n

c is any.

19.

20.

∂w

+ e–axf (eaxy, w)

∂w

∂x General solution:

Φ(u, w) = 0,

∂y

where

z = xn y m ,

c is any.

= 0.

u=

Z

z c

dt – x, f (t, w) + at

z = eax y,

∂w ∂w + eay f (xeay , w) = 0. ∂x ∂y General solution: Z z dt   – ln |x|, Φ(u, w) = 0, where u = c t atf (t, w) + 1

c is any.

21.

22.

∂w

+ yf (eaxy m, w)

∂x General solution: Φ(u, w) = 0,

where

∂w ∂y

u=

z = xeay ,

c is any.

= 0. Z

z c

dt  – x, t a + mf (t, w) 

z = eax y m ,

c is any.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 33

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

34

Page 34

FIRST-ORDER QUASILINEAR EQUATIONS

∂w ∂w + xn–1 eay f (xneay , w) = 0. ∂x ∂y General solution: Z z dt   – ln |x|, Φ(u, w) = 0, where u = c t atf (t, w) + n 23.

z = xn eay ,

c is any.

 ∂w ∂w  k + y f (x, w)g(eaxy, w) – ay = 0. ∂x ∂y General solution: Z Z dv Φ(u, w) = 0, where u = – f (x, w) exp[a(1 – k)x] dx, v k g(v, w) 24.

v = eax y.

In the integration, w is considered a parameter. ∂w ∂w + eαx–βy f (aeαx + beβy , w) = 0. ∂x ∂y General solution: Z z 1 dt – eαx , Φ(u, w) = 0, where u = α c aα + bβf (t, w)

25.

z = aeαx + beβy ,

 ∂w ∂w  + f (x, w)g(xney , w) – n = 0. ∂x ∂y General solution: Z Z dv f (x, w) Φ(u, w) = 0, where u = – dx, vg(v, w) x 26.

c is any.

x

v = xn ey .

In the integration, w is considered a parameter. 27.



f (y, w) + amxny m–1

General solution: Φ(u, w) = 0,

 ∂w

 ∂w  – g(x, w) + anxn–1 y m = 0. ∂x ∂y

where

u=

Z

f (y, w) dy +

Z

g(x, w) dx + axn y m .

In the integration, w is considered a parameter. 28.



eαxf (y, w) + cβ

General solution: Φ(u, w) = 0,

 ∂w

  ∂w – eβy g(x, w) + cα = 0. ∂x ∂y

where

u=

Z

e–βy f (y, w) dy +

Z

e–αx g(x, w) dx – ce–αx–βy .

In the integration, w is considered a parameter.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 34

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

1.3. OTHER QUASILINEAR EQUATIONS

29.

Page 35

35

∂w ∂w + f (x, y, w) = 0. ∂x ∂y


General solution: Φ u(x, y, w), w = 0, where u(x, y, a) = C is the general integral of the ordinary differential equation yx′ = f (x, y, a) with parameter a. Solutions for a large number of ordinary differential equations with various right-hand sides dependent on free parameters can be found in the handbook Polyanin and Zaitsev (2005); see also Kamke (1977) and Murphy (1960). ∂w – fx(x, y, w) = 0. ∂x ∂y
General solution: Φ w, f (x, y, w) = 0.

30.

fy (x, y, w)

f (x, w)

∂w

∂w

+ g(x, w)

∂w

= h(x, w). ∂x ∂y Solutions are sought in the implicit form (hodograph transformations) y = y(x, w), which means that x and w are taken to be the independent variables and y the dependent variable. Taking into account the fact that the old derivatives are expressed in terms of the new ones as wy = 1/yw and wx = –yx /yw , one arrives at the linear equation ∂y ∂y + h(x, w) = g(x, w). f (x, w) ∂x ∂w Equations of this type are considered in Polyanin, Zaitsev, and Moussiaux (2002, Chapter 5); this equation can be solved using the method of characteristics described in Subsection 24.3.5. 31.

1.3. Other Quasilinear Equations 1.3.1. Equations with Three Independent Variables ∂w

+ f (x, w)

∂w

+ g(x, w)

∂w

= h(x). ∂x ∂y ∂z General solution: Φ(u1 , u2 , u3 ) = 0, where 1.

u1 = w – H(x), u2 = y –

Z

x a


f t, H(t) – H(x) + w dt,

H(x) =

Z

h(x) dx, Z x
u3 = z – g t, H(t) – H(x) + w dt. a

In the integration, w is treated as a parameter; a is an arbitrary number. ∂w ∂w ∂w + f (x, w) + g(x, w) = h(w). ∂x ∂y ∂z General solution: Φ(u1 , u2 , u3 ) = 0, where 2.

Z

dw u1 = x – H(w), H(w) = , h(w)

Z w Z w f H(t) – H(w) + x, t g H(t) – H(w) + x, t u2 = y – dt, u3 = z – dt. h(t) h(t) a a In the integration, w is treated as a parameter; a is an arbitrary number.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 35

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

36

Page 36

FIRST-ORDER QUASILINEAR EQUATIONS

∂w ∂w ∂w + f (x, w) + g(x, w) = h1 (x)h2 (w). ∂x ∂y ∂z Z The substitution x = h1 (x) dx leads to an equation of the form 1.3.1.2. The substitution Z dw w= leads to an equation of the form 1.3.1.1 for w. h2 (w) 3.

∂w ∂w ∂w + f1 (x, w)g1 (y, w) + f2 (x, w)g2 (z, w) = 0. ∂x ∂y ∂z General solution: Φ(u1 , u2 , w) = 0, where Z Z Z Z dy dz u1 = f1 (x, w) dx – , u2 = f2 (x, w) dx – . g1 (y, w) g2 (z, w) In the integration, w is treated as a parameter. 4.

  ∂w + f2 (x, w) + zg2 (x, w) = 0. ∂x ∂y ∂z General solution: Φ(u1 , u2 , w) = 0, where Z Z dy , u1 = f1 (x, w) dx – g1 (y, w)  Z  Z u2 = zG2 (x, w) – f2 (x, w)G2 (x, w) dx, G2 (x, w) = exp – g2 (x, w) dx . 5.

∂w

+ f1 (x, w)g1 (y, w)

∂w

In the integration, w is treated as a parameter. 6.

 ∂w ∂w ∂w  n + f1 (x, w)g1 (y, w) + z f2 (x, w) + zg2 (x, w) = 0. ∂x ∂y ∂z

The substitution η = z 1–n leads to an equation of the form 1.3.1.5:   ∂w ∂w ∂w + f1 (x, w)g1 (y, w) + (1 – n) f2 (x, w) + ηg2 (x, w) = 0. ∂x ∂y ∂η 7.

∂w ∂x

+ f1 (x, w)g1 (y, w)

  ∂w + eβz f2 (x, w) + g2 (x, w) = 0. ∂y ∂z

∂w

The substitution η = e–βz leads to an equation of the form 1.3.1.5:   ∂w ∂w ∂w + f1 (x, w)g1 (y, w) – β f2 (x, w) + ηg2 (x, w) = 0. ∂x ∂y ∂η

 ∂w   ∂w ∂w  + f1 (x, w) + yg1 (x, w) + f2 (x, w) + zg2 (x, w) = 0. ∂x ∂y ∂z General solution: Φ(u1 , u2 , w) = 0, where  Z  Z u1 = yG1 (x, w) – f1 (x, w)G1 (x, w) dx, G1 (x, w) = exp – g1 (x, w) dx ,  Z  Z u2 = zG2 (x, w) – f2 (x, w)G2 (x, w) dx, G2 (x, w) = exp – g2 (x, w) dx .

8.

In the integration, w is treated as a parameter.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 36

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

37

1.3. OTHER QUASILINEAR EQUATIONS

9.

Page 37

 ∂w  n  ∂w ∂w  k + y f1 (x, w) + yg1 (x, w) + z f2 (x, w) + zg2 (x, w) = 0. ∂x ∂y ∂z

The transformation ξ = y 1–k , η = z 1–n leads to an equation of the form 1.3.1.8:

   ∂w  ∂w ∂w + (1 – k) f1 (x, w) + ξg1 (x, w) + (1 – n) f2 (x, w) + ηg2 (x, w) = 0. ∂x ∂ξ ∂η 10.

 ∂w  βz  ∂w ∂w  αy + e f1 (x, w) + g1 (x, w) + e f2 (x, w) + g2 (x, w) = 0. ∂x ∂y ∂z

The transformation ξ = e–αy , η = e–βz leads to an equation of the form 1.3.1.8:

  ∂w   ∂w ∂w – α f1 (x, w) + ξg1 (x, w) – β f2 (x, w) + ηg2 (x, w) = 0. ∂x ∂ξ ∂η 11.

 ∂w  βz  ∂w ∂w  k + y f1 (x, w) + yg1 (x, w) + e f2 (x, w) + g2 (x, w) = 0. ∂x ∂y ∂z

The transformation ξ = y 1–k , η = e–βz leads to an equation of the form 1.3.1.8:

  ∂w   ∂w ∂w + (1 – k) f1 (x, w) + ξg1 (x, w) – β f2 (x, w) + ηg2 (x, w) = 0. ∂x ∂ξ ∂η 12.

∂w ∂x

+ f (a1 x + b1 y, w)

∂w ∂y

+ g(a2 x + b2 z, w)

∂w ∂z

= 0.

General solution: Φ(u1 , u2 , w) = 0, where u1 = x – u2 = x –

Z

ξ1

c1 Z ξ2 c2

dt , a1 + b1 f (t, w)

ξ1 = a1 x + b1 y

(b1 ≠ 0),

dt , a2 + b2 g(t, w)

ξ2 = a2 x + b2 z

(b2 ≠ 0),

and c1 and c2 are arbitrary numbers. For b1 = 0 or b2 = 0, one should use the formulas 1 u1 = y – a1 13.

Z

a1 x

f (t, w) dt,

c1

1 u2 = z – a2

Z

a2 x

g(t, w) dt. c2

  ∂w ∂w z ∂w + f (y + ax, w) +g ,w = 0. ∂x ∂y x ∂z

General solution: Φ(u1 , u2 , w) = 0, where u1 = x –

Z

y+ax

c1

dt , f (t, w) + a

u2 =

Z

z/x

c2

dt – ln |x|, g(t, w) – t

and c1 and c2 are arbitrary numbers.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 37

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

38 14.

Page 38

FIRST-ORDER QUASILINEAR EQUATIONS     y ∂w z ∂w ∂w +f ,w +g ,w = 0. ∂x x ∂y x ∂z

General solution: Φ(u1 , u2 , w) = 0, where u1 =

Z

y/x a

dt – ln |x|, f (t, w) – t

u2 =

Z

z/x

dt – ln |x|, g(t, w) – t

b

and a and b are arbitrary numbers. 15.

∂w ∂x

+

y x

f (xny m, w)

∂w

+

∂y

z x

g(xkz p, w)

∂w

= 0.

∂z

General solution: Φ(u1 , u2 , w) = 0, where u1 =

Z

xn y m a

dt – ln |x|, t[mf (t, w) + n]

u2 =

Z

xk z p

dt – ln |x|, t[pg(t, w) + k]

b

and a and b are arbitrary numbers. 16.

∂w ∂x

+ eαy f (xeαy , w)

∂w ∂y

∂w

+ eβz g(xeβz , w)

= 0.

∂z

General solution: Φ(u1 , u2 , w) = 0, where u1 =

Z

xeαy

c1

dt – ln |x|, t[αtf (t, w) + 1]

u2 =

Z

xeβz c2

dt – ln |x|, t[βtg(t, w) + 1]

and c1 and c2 are arbitrary numbers. 17.

∂w ∂x

+ eαy f (xeαy , w)

∂w ∂y

+ e–βx g(zeβx, w)

∂w

= 0.

∂z

General solution: Φ(u1 , u2 , w) = 0, where u1 =

Z

xeαy

c1

dt – ln |x|, t[αtf (t, w) + 1]

u2 =

Z

zeβx c2

dt – x, g(t, w) + βt

and c1 and c2 are arbitrary numbers. 18.

∂w ∂x

+ yf (eαxy k, w)

∂w ∂y

+ zg(eβx z n, w)

∂w ∂z

= 0.

General solution: Φ(u1 , u2 , w) = 0, where u1 =

Z

eαx y k

c1

dt – x, t[α + kf (t, w)]

u2 =

Z

eβx z n

c2

dt – x, t[β + ng(t, w)]

and c1 and c2 are arbitrary numbers.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 38

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 39

39

1.3. OTHER QUASILINEAR EQUATIONS

∂w ∂w ∂w + e–αxf (yeαx, w) + e–βxg(zeβx, w) = 0. ∂x ∂y ∂z General solution: Φ(u1 , u2 , w) = 0, where

19.

u1 =

Z

yeαx c1

dt – x, f (t, w) + αt

u2 =

Z

zeβx

dt – x, g(t, w) + βt

c2

and c1 and c2 are arbitrary numbers. 20.

∂w

1

+

f (xkeαy , w)

∂w

+

1

g(xneβz , w)

∂w

∂x x ∂y x General solution: Φ(u1 , u2 , w) = 0, where u1 =

Z

xk eαy

c1

dt – ln |x|, t[k + αf (t, w)]

u2 =

Z

= 0.

∂z xn eβz

c2

dt – ln |x|, t[n + βg(t, w)]

and c1 and c2 are arbitrary numbers. 21.

f (x, w)

∂w

+ g(x, w)

∂w

+ h(x, w)

∂x ∂y General solution: Φ(u1 , u2 , w) = 0, where Z g(x, w) u1 = y – dx, f (x, w)

∂w ∂z

= 0.

u2 = z –

Z

h(x, w) dx. f (x, w)

In the integration, w is treated as a parameter. 22.

f (x, w)

∂w

+ g(y, w)

∂w

+ h(z, w)

∂x ∂y General solution: Φ(u1 , u2 , w) = 0, where Z Z dy dx u1 = – , g(y, w) f (x, w)

∂w ∂z

u2 =

= 0.

Z

dz – h(z, w)

Z

dx . f (x, w)

In the integration, w is treated as a parameter.

1.3.2. Equations with Arbitrary Number of Independent Variables 1.

∂w

+ f2 (x1 , w)

∂w

+ · · · + fn(x1 , w)

∂w

∂x1 ∂x2 ∂xn General solution: Φ(u1 , u2 , . . . , un ) = 0, where

= g(x1 ). Z

u1 = w – G(x1 ), G(x) = g(x) dx, Z x1
uk = xk – fk t, G(t) – G(x1 ) + w dt, k = 2, . . . , n. a

In the integration, w is treated as a parameter; a is an arbitrary number.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 39

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

40 2.

Page 40

FIRST-ORDER QUASILINEAR EQUATIONS ∂w ∂w ∂w + f2 (x1 , w) + · · · + fn(x1 , w) = g(w). ∂x1 ∂x2 ∂xn

General solution: Φ(u1 , u2 , . . . , un ) = 0, where Z

dw , G(w) = g(w)
fk G(t) – G(w) + x1 , t dt, k = 2, . . . , n. g(t)

u1 = x1 – G(w), uk = xk –

Z

w

a

In the integration, w is treated as a parameter; a is an arbitrary number. 3.

∂w ∂x1

+ f2 (x1 , w) Z

∂w ∂x2

+ · · · + fn(x1 , w)

∂w ∂xn

= g(x1 )h(w).

The substitution x = g(x1 ) dx1 leads to an equation of the form 1.3.2.2. The substitution Z dw leads to an equation of the form 1.3.2.1 for w. w= h(w) 4.

∂w ∂x1

+ f2 (x1 , w)g2 (x2 , w)

∂w ∂x2

+ · · · + fn(x1 , w)gn(xn, w)

General solution: Φ(w, u2 , . . . , un ) = 0, where Z Z dxk , uk = fk (x1 , w) dx1 – gk (xk , w)

∂w ∂xn

= 0.

k = 2, . . . , n.

In the integration, w is treated as a parameter. 5.

 ∂w   ∂w ∂w  + f2 (x1 , w)+x2 g2 (x1 , w) +· · ·+ fn(x1 , w)+xngn(x1 , w) = 0. ∂x1 ∂x2 ∂xn

General solution: Φ(w, u2 , . . . , un–1 ) = 0, where uk = xk Gk (x1 , w) –

Z

fk (x1 , w)Gk (x1 , w) dx1 ,

 Z  Gk (x1 , w) = exp – gk (x1 , w) dx1 .

In the integration, w is treated as a parameter; k = 2, . . . , n. 6.

 2  ∂w + xk 2 f2 (x1 , w) + x2 g2 (x1 , w) ∂x1 ∂x2  kn  ∂w + · · · + xn fn(x1 , w) + xngn(x1 , w) = 0. ∂xn ∂w

2 n The transformation z1 = x1 , z2 = x1–k , . . . , zn = x1–k leads to an equation of the form n 2 1.3.2.5:

  ∂w   ∂w ∂w + (1 – k2 ) f2 (z1 , w) + z2 g2 (z1 , w) + · · · + (1 – kn ) fn (z1 , w) + zn gn (z1 , w) = 0. ∂z1 ∂z2 ∂zn

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 40

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 41

41

1.3. OTHER QUASILINEAR EQUATIONS

 ∂w ∂w  a2 x2 + e f2 (x1 , w) + g2 (x1 , w) ∂x1 ∂x2   ∂w + · · · + ean xn fn(x1 , w) + gn(x1 , w) = 0. ∂xn The transformation z1 = x1 , z2 = e–a2 x2 , . . . , zn = e–an xn leads to an equation of the form 1.3.2.5:

7.

   ∂w  ∂w ∂w – a2 f2 (z1 , w) + z2 g2 (z1 , w) – · · · – an fn (z1 , w) + zn gn (z1 , w) = 0. ∂z1 ∂z2 ∂zn 8.

∂w

+ f2 (a2 x1 + b2 x2 , w)

∂w

+ · · · + fn(anx1 + bnxn, w)

∂x1 ∂x2 General solution: Φ(w, u2 , . . . , un ) = 0, where Z zk dt uk = x1 – , zk = ak x1 + bk xk ck ak + bk fk (t, w) Z zk 1 uk = xk – fk (t, w) dt, zk = ak x1 ak ck

∂w

∂xn

= 0.

if bk ≠ 0, if bk = 0,

and the ck are arbitrary numbers. In the integration, w is treated as a parameter. 9.

    ∂w x2 ∂w xn ∂w + f2 ,w + · · · + fn ,w = 0. ∂x1 x1 ∂x2 x1 ∂xn

General solution: Φ(w, u2 , . . . , un ) = 0, where Z zk dt xk – ln |x1 |, uk = zk = , x1 ck fk (t, w) – t

ck are any,

k = 2, . . . , n.

In the integration, w is treated as a parameter. x2 ∂w xn ∂w ∂w 2 m2 n mn + f2 (xk + ··· + fn(xk = 0. 1 x2 , w) 1 xn , w) ∂x1 x1 ∂x2 x1 ∂xn General solution: Φ(w, u2 , . . . , un ) = 0, where Z zi dt i ui = – ln |x1 |, zi = xk1 i xm ci are any, i = 2, . . . , n. i , ci t[mi fi (t, w) + ki ] 10.

In the integration, w is treated as a parameter. ∂w ∂w ∂w + ea2 x2 f2 (x1 ea2 x2 , w) + · · · + ean xn fn(x1 ean xn , w) = 0. ∂x1 ∂x2 ∂xn General solution: Φ(w, u2 , . . . , un ) = 0, where Z zk dt uk = – ln |x1 |, zk = x1 eak xk , ck are any, k = 2, . . . , n. ck t[ak tfk (t, w) + 1]

11.

In the integration, w is treated as a parameter.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 41

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

42

Page 42

FIRST-ORDER QUASILINEAR EQUATIONS

∂w ∂w ∂w 2 n + x2 f2 (ea2 x1 xk + · · · + xnfn(ean x1 xk = 0. n , w) 2 , w) ∂x1 ∂x2 ∂xn General solution: Φ(w, u2 , . . . , un ) = 0, where Z zi dt – x1 , ui = zi = eai x1 xki i , ci are any, i = 2, . . . , n. t[a + k f (t, w)] i i i ci 12.

In the integration, w is treated as a parameter. 13.

∂w

+ e–a2 x1 f2 (x2 ea2 x1 , w)

∂w

+ · · · + e–an x1 fn(xnean x1 , w)

∂x1 ∂x2 General solution: Φ(w, u2 , . . . , un ) = 0, where Z zk dt uk = – x1 , zk = xk eak x1 , ck fk (t, w) + ak t

ck are any,

∂w ∂xn

= 0.

k = 2, . . . , n.

In the integration, w is treated as a parameter. 14.

∂w

+

1

2 a2 x2 f2 (xk , w) 1 e

∂w

+ ··· +

1

n an xn fn(xk , w) 1 e

∂w

= 0. ∂x1 x1 ∂x2 x1 ∂xn General solution: Φ(w, u2 , . . . , un ) = 0, where Z zm dt um = – ln |x1 |, zm = xk1 m eam xm , cm are any, m = 2, . . . , n. t[k + a f (t, w)] m m m cm In the integration, w is treated as a parameter. 15.

f1 (x1 , w)

∂w

∂x1 General solution:

+ f2 (x1 , w)

∂w ∂x2

Φ(u2 , u3 , . . . , un , w) = 0,

+ · · · + fn(x1 , w)

where

uk = xk –

∂w ∂xn

Z

= 0.

fk (x1 , w) dx1 . f1 (x1 , w)

In the integration, w is treated as a parameter; k = 2, . . . , n. ∂w ∂w ∂w + f2 (x2 , w) + · · · + fn(xn, w) = 0. ∂x1 ∂x2 ∂xn General solution: Z Z dxk dxn Φ(u1 , u2 , . . . , un–1 , w) = 0, where uk = – . fk (xk , w) fn (xn , w) 16.

f1 (x1 , w)

In the integration, w is treated as a parameter; k = 1, . . . , n – 1.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 42

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 43

Chapter 2

First-Order Equations with Two Independent Variables Quadratic in Derivatives Preliminary remarks. For brevity, we only give complete integrals for most equations. Subsection 24.1.1 presents formulas that allow one to derive the general integral using a complete integral.

2.1. Equations Containing Arbitrary Parameters 2.1.1. Equations of the Form ∂w ∂w = f (x, y, w)

∂x ∂y

1.

∂w ∂w ∂x ∂y

= a.

Complete integral: w = aC1 x +

y + C2 . C1

Reference: E. Kamke (1965).

2.

∂w ∂w ∂x ∂y

= axy + b.

Complete integral: w = y

p

ax2

+ C1 + b

Z

Reference: E. Kamke (1965).

3.

∂w ∂w ∂x ∂y

p

dx ax2 + C1

+ C2 .

= axy + bx + cy + s.

Z p (bx + s) dx 2 Complete integral: w = y ax + 2cx + C1 + + C2 . p ax2 + 2cx + C1 4.

∂w ∂w

= axky + bxn.

∂x ∂y Complete integral:

w = ϕ(x)y + b

Z



xn dx + C1 , ϕ(x)

2a k+1 ϕ(x) = ± x + C2 k+1

1/2

.

43

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 43

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

44 5.

Page 44

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES ∂w ∂w = axny k + bxmy 2k+1 . ∂x ∂y

This is a special case of equation 2.2.1.3 with f (x) = axn and g(x) = bxm . 6.

∂w ∂w ∂x ∂y

= axneλy + bxme2λy .

This is a special case of equation 2.2.1.4 with f (x) = axn and g(x) = bxm . 7.

∂w ∂w = aw + bxn. ∂x ∂y

Complete integral: w = (ax + C1 )(y + C2 ) + b(ax + C1 )

8.

∂w ∂w ∂x ∂y

Z

xn dx . (ax + C1 )2

= bw a.

  2  2–a
 2 – a 2   bC1 x + y + C2 2C1 Complete integral: w =     y   C2 exp bC1 x + C1

if a ≠ 2, if a = 2.

Reference: E. Kamke (1965).

∂w ∂w = Axay bw c. ∂x ∂y   2 
2–c 2–c  2   AC1 u + v + C2 if c ≠ 2, 2C1 Complete integral: w =     v   C2 exp AC1 u + if c = 2,   C1  xa+1  y b+1 if a ≠ –1, where u = v = b + 1 if b ≠ –1,  a+1  ln |y| if b = –1. ln |x| if a = –1, 9.

Reference: E. Kamke (1965).

10.

∂w ∂w bw 2 = . ∂x ∂y w+a

√
Complete integral in implicit form: x + bC12 y + C2 = C1 2 w + a + u , where  √ √ √ w+a– a    a ln √ √ w+a+ a u= q    2√–a arctan w + a –a

if a > 0, if a < 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 44

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 45

45

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS

∂w ∂w = (xy + a)w k. ∂x ∂y 2–k ∂z ∂z 2 w 2 leads to an equation of the form 2.1.1.2: = xy + a. The substitution z = 2–k ∂x ∂y

11.

12.

∂w ∂w = aw k+1 + bxnw 2k, ∂x ∂y

k ≠ 1.

The substitution u = w1–k leads to an equation of the form 2.1.1.7: ∂u ∂u = a(1 – k)2 u + b(1 – k)2 xn . ∂x ∂y 13.

∂w ∂w ∂x ∂y

= aeλw + bxne2λw.

The substitution u = e–λw leads to an equation of the form 2.1.1.7:

∂u ∂u = aλ2 u + bλ2 xn . ∂x ∂y

2.1.2. Equations of the Form

f (x, y, w) ∂w ∂w + g(x, y, w) ∂w = h(x, y, w) ∂x ∂y ∂x 1.

∂w ∂w ∂x ∂y

+a

∂w ∂x

= bw.

  a Complete integral: w = (by + C1 ) x + C2 – ln |by + C1 | . b ∂w ∂w

+ (x + a)

∂w

+ bx + c = 0. ∂x ∂y ∂x Complete integral: w = C1 y – bx + (ab + bC1 – c) ln |x + a + C1 | + C2 . 2.

∂w ∂w ∂w + a(x + b) + cy + s = 0. ∂x ∂y ∂x Complete integral for a ≠ 0:  1 w = –(x + b)(ay + C1 ) + 2 acy + (as – cC1 ) ln |ay + C1 | + C2 . a Complete integrals for a = 0 and c ≠ 0: q y(cy + 2s) cy + s w = –C1 (x + b) + + C2 , w=± C12 – 2c(x + b) + C2 . 2C1 c 3.

∂w ∂w ∂w + axy + b = 0, ∂x ∂y ∂x Complete integral:

4.

1 1 b w = – axy 2 – aC1 x + ϕ(y) + C2 , 2 2 a

a ≠ 0.  2 y    √ arctan √ C1 C1 √ ϕ(y) = y – –C1 1   ln √ √ –C1 y + –C1

if C1 > 0, if C1 < 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 45

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

46 5.

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES ∂w ∂w ∂w + axy + by k = 0. ∂x ∂y ∂x

1 1 Complete integral: w = – axy 2 – C1 x + 2b 2 2 6.

∂w ∂w ∂x ∂y

+ axk

∂w ∂x

∂w ∂w ∂x ∂y

+ ay k

∂w ∂x

Z

y k dy + C2 . ay 2 + C1

+ bxn = 0.

Complete integral: w = C1 y – b

7.

Page 46

Z

xn dx + C2 . axk + C1

+ bx + c = 0.

 c p a k+1 Complete integral: w = ± x + C1 – 2by – y + C2 . b k+1 8.

∂w ∂w ∂x ∂y

+ axy k

∂w ∂x

+ by n = 0.

Complete integral: w = –xϕ(y) + b

9.

a k+1 y + C1 . k+1

b ≠ 0.

∂w ∂w ∂w + a sin x + b = 0, ∂x ∂y ∂x 2b

b ≠ 0.

arctan

C12 – a2

a + C1 tan(x/2) + C1 y + C2 . q C12 – a2

∂w ∂w ∂w + a sin(µy) = b sin(λx). ∂x ∂y ∂x

Complete integral: w =

12.

ϕ(y) =


b ln |C1 + aex | – x + C1 y + C2 . C1

Complete integral: w = – q 11.

y n dy + C2 , ϕ(y)

∂w ∂w ∂w + aex + b = 0, ∂x ∂y ∂x

Complete integral: w =

10.

Z

a

∂w ∂w ∂x ∂y

bC1 a y cos(λx) + cos(µy) – + C2 . λ µ C1

+ (bx + cy + sw)

∂w ∂x

= 1,

s ≠ 0.

The substitution z = bx + cy + sw – ac/s leads to an equation of the form 2.1.3.14: (zx – b)(azy + sz) = s2 . Reference: E. Kamke (1965).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 46

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS

13.

ay

Page 47

47

∂w ∂w ∂w +w + b = 0. ∂x ∂y ∂x


y –1/a 2C12 x – by 2/a + C2 . 2C1 Another complete integral for a = –2: w2 = 2(x – C1 )(C2 y – b). Complete integral: w =

14.

ay

∂w ∂w

+ (bx + cy + sw)

∂x ∂y

∂w ∂x

= 1,

a ≠ 0.

Complete integral:  c – y + C1 y –s/a + ϕ(x) a + s w=  c y(ln y – 1) + C y + ϕ(x) 1 s

if a + s ≠ 0; if a + s = 0.

1p 2sx + C2 for b = 0. If b ≠ 0, the function ϕ is defined implicitly by s x = C2 ebϕ – sb–2 (bϕ + 1).

Here ϕ(x) =

15.

w

∂w ∂w ∂x ∂y

+a

∂w ∂x

+ bw = 0.

Complete integral in implicit form: a + pa2 – 4bC w2 p 1 ±a ln a2 – 4bC1 w2 – a ln w = 2b(x + C1 y + C2 ). w ±

2.1.3. Equations of the Form

f (x,y,w) ∂w ∂w + g(x,y,w) ∂w + h(x,y,w) ∂w = s(x,y,w) ∂x ∂y ∂x ∂y 1.

∂w ∂w ∂x ∂y

=a

∂w ∂x

+b

∂w ∂y

.

Complete integrals:

2.

∂w ∂w ∂x ∂y

w = C1 + C2 x + C3 y, where C2 C3 = aC2 + bC3 , p w = bx + ay ± C1 x + 4abxy + C2 , p w = bx + ay ± (2ay + C1 )(2bx + C1 ) + C2 . +a

∂w ∂x

+ bx

Complete integral: w = –

∂w ∂y

= 0,

ab ≠ 0.

bC1 x2 + C1 y + C2 . 2(a + C1 )

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 47

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

48

Page 48

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES

∂w ∂w ∂w ∂w = ax + ay . ∂x ∂y ∂x ∂y Four different complete integrals: p p

a w= (x + C1 y)2 + C2 , w = ax y + y 2 + C1 + C2 , w = ay x + x2 + C1 + C2 , 2C1   Z p Z p 1 1 2 2 u + C1 du + v + C1 dv + C2 , u = x + y, v = x – y. w = a xy + 2 2 3.

4.

∂w ∂w

+ ay

∂w

∂x ∂y ∂x 1◦ . Complete integral:

+ bx

∂w ∂y

= 0,

ab ≠ 0.

  a 2 C1 – 1 2 y + C2 . w= bx – 2 C1

2◦ . If ab > 0, then numbers α and β can be defined such that a = ±α2 and b = ±β 2 , with both upper signs or both lower signs being taken at a time. If we set u = βx + αy and v = βx – αy, then we arrive at an equation of the form 2.2.6.10: wu2 ± uwu = wv2 ± vwv . Thus, we obtain another complete integral. 5.

∂w ∂w

=x

∂w

∂x ∂y ∂x Complete integral:

∂w ∂w

±

w = xy

+y

∂w ∂y

+ ax + by + c.

p p (x + b) y 2 + 2ay + C1 ± (ab – c) ln y + y 2 + 2ay + C1 + a + C2 .

=x

∂w

+y

∂w

+ axy + b. ∂x ∂y ∂x ∂y Complete integral for a ≠ –1: p p √ b w = xy ± x (a + 1)y 2 + C1 ± √ ln a + 1 y + (a + 1)y 2 + C1 + C2 . a+1 6.

Complete integral for a = –1:

w = xy – C1 x + ∂w ∂w

∂w

+y

∂w

+ axy + bx. ∂x ∂y ∂x ∂y Complete integrals for a ≠ –1: q   w = x y ± (a + 1)y 2 + 2by + C1 + C2 , p b p w = xy ± y (a + 1)x2 + C1 ± (a + 1)x2 + C1 + C2 . a+1 Complete integrals for a = –1: p x y w= (y – bx) – + C2 , w = xy ± x 2by + C1 + C2 . C1 2 7.

=x

b y + C2 . C1

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 48

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

49

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS

∂w ∂x

+ ax



∂w ∂y

+ by



= c.



= b.

q    Complete integral: w = – 12 x ax + C1 ± C12 – 4c – 12 y by + C1 9.



∂w ∂x

+ ay



∂w ∂y

+ ax

Complete integral: w = –axy + C1 x +

10.

∂w ∂w ∂x ∂y

+ ay

∂w ∂x

+ bx

∂w ∂y

±

8.



Page 49

q

 C12 – 4c + C2 .

b y + C2 . C1

+ cx2 + sy 2 = 0,

4cs = a2 b2 .

The transformation bx2 ay 2 z=w+ + , 2 2

λx – λ–1 y u= , √ 2

λx + λ–1 y v= , √ 2

λ=

r

2c , ab

leads to an equation of the form 2.1.6.4: zu2 – zv2 = 2abu2 . 11.

a

∂w ∂w ∂w ∂w +x +y = w. ∂x ∂y ∂x ∂y

Complete integral: w = C1 x + C2 y + aC1 C2 . 12.

∂w ∂w ∂x ∂y

+ b(ax + by)

∂w ∂x

+ a(ax + by)

∂w ∂y

+ abc = 0.

Complete integral: q (ax + by)2 1 w=– – aC1 x + bC1 y ± (ax + by) (ax + by)2 – c + C12 2 2 q 1 ± (C12 – c) ln ax + by + (ax + by)2 – c + C12 + C2 . 2 ∂w ∂w ∂x ∂y

+ b(ax + by)

∂w ∂x

– a(ax + by)

Complete integral: w = C1 (by – ax)

14.



∂w ∂x

+a



∂w ∂y

+ bw



±

13.

∂w ∂y

+ abc = 0.

1 (2aC1 x + 2bC1 y – c + C12 )3/2 + C2 . 3C1

= c.

The transformation x = u, y = ln v, w = cz + v – au leads to an equation of the form 2.1.2.14: cv

∂z ∂z ∂z + [–abu + (b + 1)v + bcz] = 1. ∂u ∂v ∂u

Reference: E. Kamke (1965).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 49

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

50 15.

Page 50

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES 2x

∂w ∂w ∂w ∂w –w +a = 0. ∂x ∂y ∂y ∂x

Complete integral in implicit form: (aC1 w + C12 x2 )3/2 – 32 aC12 (xw + ay) = C13 x3 + C2 . y

∂w ∂w

–w

∂w

+a

∂x ∂y ∂x Complete integrals: p w = C1 y ± 2aC1 x + C2 ,

∂w ∂y

w=

= 0,

a ≠ 0.

 p x – C2  y ± y 2 + aC1 , C1

∂w ∂w ∂w ∂w +b +a = 0. ∂x ∂y ∂x ∂y Complete integral: q w = C1 (ax – by) – ln |ax + by| ± 1 + C12 (ax + by)2

17.

18.

w=

C2 y ax + C1 – . ax + C1 2y

(ax + by)

(x2 + a)

∂w

∂x Complete integral:



 ∂w – 1 + xy 2 = 0. ∂y ∂y

∂w

±

16.

arctanh q

 y  C arctan 2  1 C C1 w = ± 1 ln(x2 + a) + C2 + y  2  C1 arctanh C1

1 1 + C12 (ax + by)2

+ C2 .

for the upper sign, for the lower sign.

∂w ∂w ∂w ∂w + b(ax – cw) + a(cw – by) = 0. ∂x ∂y ∂x ∂y General integral in parametric form: i i z 1h z 1h Φ(z) + (2u – v + 1)Φ′z (z) , y = Φ(z) + (2v – u + 1)Φ′z (z) , x= a u b v  1 1 w= xu + yv – (u + v – 1)Φ(z) , z= (u + v – 1)3 . c uv   ∂w ∂w ∂w ∂w 20. w =a x +y . ∂x ∂y ∂x ∂y a Complete integral: w2 = (C1 x + C2 y)2 + C3 . C1 C2 19.

c(ax – by)

21.

aw

∂w ∂w ∂x ∂y

+ x2 y

∂w ∂x

+ xy 2

∂w ∂y

= xyw.

Complete integral: w2 = C1 x2 + C2 y 2 + aC1 C2 . ∂w ∂w ∂w ∂w + by + ax = 0. ∂x ∂y ∂x ∂y aC2 + bC1 Complete integral: w2 = C3 – (C1 x2 + C2 y 2 ). (a + b)C1 C2 22.

(a + b)w

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 50

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

51

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS



2.1.4. Equations of the Form ∂w + f (x, y, w) ∂w

∂x

∂y

Page 51

2

= g(x, y, w)

⊲ Equations of this type are encountered in mechanics, where x plays the role of time and y plays the role of the spatial coordinate. 1.

  ∂w ∂w 2 = by. +a ∂x ∂y

This equation governs the free vertical drop of a point body near the Earth’s surface (y is the 1 vertical coordinate measured downward, x time, m = 2a the mass of the body, and g = 2ab the gravitational acceleration).   2a by + C1 3/2 Complete integral: w = –C1 x ± + C2 . 3b a Reference: A. P. Markeev (1990).

2.

∂w ∂x

+a



∂w ∂y

2

+ by 2 = 0.

This equation governs free oscillations of a point body of mass m = 1/(2a) in an elastic field with elastic coefficient k = 2b (x is time and r y is the displacement from the equilibrium). Z C1 – by 2 Complete integral: w = –C1 x + C2 ± dx + C2 . a Reference: F. R. Gantmakher (1966).

3.

  ∂w ∂w 2 +a = bxk + cy n. ∂x ∂y

This is a special case of equation 2.2.2.1 with f (x) = bxk and g(y) = cy n . Complete integral: b w = –C1 x + xk+1 + k+1 4.

Z r

cy n + C1 dy + C2 . a

  ∂w ∂w 2 +a = bxky + cxn. ∂x ∂y

This is a special case of equation 2.2.2.2 with f (x) = bxk and g(x) = cxn . Complete integral: 

bxk+1 + C1 w=y k+1 5.

∂w ∂x

+a



∂w ∂y

2



+

cxn+1 ab2 x2k+3 2abC1 xk+2 – – aC12 x + C2 . – n + 1 (k + 1)2 (2k + 3) (k + 1)(k + 2)

= bxky 2 + cxny + sxm.

This is a special case of equation 2.2.2.3 with f (x) = bxk , g(x) = cxn , and h(x) = sxm .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 51

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

52 6.

Page 52

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES ∂w

+a



∂w

2

= k(bx + cy)n.

∂x ∂y This is a special case of equation 2.2.2.4 with f (ξ) = kξ n . Complete integral: Z q b 1 w = C1 x + C2 – (bx + cy) ± 4ac2 kξ n + b2 – 4ac2 C1 dξ, ξ = bx + cy. 2ac2 2ac2 7.

  ∂w ∂w 2 = bxk + ceβy . +a ∂x ∂y

This is a special case of equation 2.2.2.1 with f (x) = bxk and g(y) = ceβy . Complete integral: Z r βy b ce + C1 xk+1 + dy + C2 . w = –C1 x + k+1 a 8.

  ∂w ∂w 2 +a = beλx + ceβy . ∂x ∂y

This is a special case of equation 2.2.2.1 with f (x) = beλx and g(y) = ceβy . Complete integral: Z r βy ce + C1 b λx w = –C1 x + e + dy + C2 . λ a 9.

  ∂w 2 ∂w +a = beλx + cy k. ∂x ∂y

This is a special case of equation 2.2.2.1 with f (x) = beλx and g(y) = cy k . Complete integral: Z r k b λx cy + C1 w = –C1 x + e + dy + C2 . λ a 10.

  ∂w ∂w 2 +a = bxky + ceβx. ∂x ∂y

This is a special case of equation 2.2.2.2 with f (x) = bxk and g(x) = ceβx . 11.

  ∂w ∂w 2 +a = beλx y + ceβx. ∂x ∂y

This is a special case of equation 2.2.2.2 with f (x) = beλx and g(x) = ceβx . Complete integral:   b λx c ab2 2abC1 λx w=y e + C1 + eβx – 3 e2λx – e – aC12 x + C2 . λ β 2λ λ2 12.

∂w ∂x

+a



∂w ∂y

2

= beλx y + cxk.

This is a special case of equation 2.2.2.2 with f (x) = beλx and g(x) = cxk .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 52

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS

13.

∂w ∂x

+a



∂w ∂y

2

Page 53

53

= kebx+cy .

This is a special case of equation 2.2.2.4 with f (ξ) = keξ . Complete integral: Z p b 1 (bx + cy) ± 4ac2 keξ + b2 – 4ac2 C1 dξ, ξ = bx + cy. w = C1 x + C2 – 2ac2 2ac2 14.

  ∂w ∂w 2 +a = bxkw + cxn. ∂x ∂y

This is a special case of equation 2.2.2.7 with f (x) = bxk and g(x) = cxn . Complete integral:   Z  n  dx b 2 2 k+1 cx – aC2 F (x) , F (x) = exp x . w = F (x)(C1 + C2 y) + F (x) F (x) k+1 15.

∂w ∂x

+a



∂w ∂y

2

= bxkw + ceβx.

2

= beλx w + cxk.

2

= beλx w + ceβx.

This is a special case of equation 2.2.2.7 with f (x) = bxk and g(x) = ceβx . Complete integral:   Z  βx  dx b 2 2 k+1 ce – aC2 F (x) w = F (x)(C1 + C2 y) + F (x) , F (x) = exp x . F (x) k+1 16.

∂w ∂x

+a



∂w ∂y

This is a special case of equation 2.2.2.7 with f (x) = beλx and g(x) = cxk . Complete integral:   Z  k  dx b λx 2 2 , F (x) = exp e w = F (x)(C1 + C2 y) + F (x) cx – aC2 F (x) . F (x) λ 17.

∂w ∂x

+a



∂w ∂y

This is a special case of equation 2.2.2.7 with f (x) = beλx and g(x) = ceβx . Complete integral:   Z  βx  dx b λx 2 2 w = F (x)(C1 + C2 y) + F (x) ce – aC2 F (x) , F (x) = exp e . F (x) λ 18.

  ∂w ∂w 2 +a = bw 2 + cxkw + sxn. ∂x ∂y

This is a special case of equation 2.2.2.8 with f (x) = cxk and g(x) = sxn . 19.

  ∂w ∂w 2 +a = bw 2 + cxkw + seβx . ∂x ∂y

This is a special case of equation 2.2.2.8 with f (x) = cxk and g(x) = eβx .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 53

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

54 20.

Page 54

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES ∂w ∂x

+a



∂w ∂y

2

= bw 2 + ceλxw + seβx.

This is a special case of equation 2.2.2.8 with f (x) = ceλx and g(x) = eβx . 21.

∂w ∂x

+ ax

k



∂w ∂y

2

= bxnw + c1 xmy + c2 xl.

This is a special case of equation 2.2.2.9 with f (x) = axk , g(x) = bxn , h(x) = c1 xm , and s(x) = c2 xl . 22.

 2 ∂w k ∂w + ay = bw 2 + c1 xnw + c2 xm. ∂x ∂y

This is a special case of equation 2.2.2.10 with f (y) = ay k , g(x) = c1 xn , and h(x) = c2 xm . 23.

2  ∂w λx ∂w + ae = beβx w + c1 eγxy + c2 eµx. ∂x ∂y

This is a special case of equation 2.2.2.9 with f (x) = aeλx , g(x) = beβx , h(x) = c1 eγx , and s(x) = c2 eµx . 24.

∂w ∂x

λy

+ ae



∂w ∂y

2

= bw 2 + c1 eβxw + c2 eγx.

This is a special case of equation 2.2.2.10 with f (y) = aeλy , g(x) = c1 eβx , and h(x) = c2 eγx . 25.

∂w ∂x

k

+ ax y + bx

n

 
∂w 2 ∂y

= 0.

This is a special case of equation 2.2.2.11 with f (x) = axk and g(x) = bxn . Complete integral: w = ϕ(x)y – b

26.

Z

n 2

x ϕ (x) dx + C1 ,

ϕ(x) =



a xk+1 + C2 k+1

–1

.

 
∂w 2 ∂w k n + ax y + bx = cxmw + s0 xl. ∂x ∂y

This is a special case of equation 2.2.2.12 with f (x) = axk , g(x) = bxn , h(x) = cxm , r(x) = 0, and s(x) = s0 xl . 27.

∂w ∂x

k

λx

+ ax y + be

 
∂w 2 ∂y

= 0.

This is a special case of equation 2.2.2.11 with f (x) = axk and g(x) = beλx . Complete integral: w = ϕ(x)y – b

Z

λx 2

e ϕ (x) dx + C1 ,

ϕ(x) =



a xk+1 + C2 k+1

–1

.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 54

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS

∂w

n k

m 2k–2

∂w

n k

m 2k–2

Page 55

55

 
∂w 2

+ ax y + bx y = 0. ∂x ∂y This is a special case of equation 2.2.2.13 with f (x) = axn and g(x) = bxm .

28.

 
∂w 2

+ ax y + bx y = cxq w + s0 xl. ∂x ∂y This is a special case of equation 2.2.2.14 with f (x) = axn , g(x) = bxm , h(x) = cxq , and s(x) = s0 xl . 29.

30.

∂w ∂x

+ A(ax + by)

k



∂w ∂y

2

= B(ax + by)n.

2

= Beax+by .

This is a special case of equation 2.2.2.17 with f (ξ) = Aξ k and g(ξ) = Bξ n . Complete integral: Z q a aξ 1–k ± 2 ξ –k 4b2 ABξ k+n – 4b2 AC1 ξ k + a2 dξ, ξ = ax+by. w = C1 x+C2 – 2 2b A(1 – k) 2b A 31.

∂w ∂x

+ A(ax + by)

k



∂w ∂y

This is a special case of equation 2.2.2.17 with f (ξ) = Aξ k and g(ξ) = Beξ . 32.

∂w ∂x

λx

+ ae

βx

y + be

 
∂w 2 ∂y

= 0.

This is a special case of equation 2.2.2.11 with f (x) = aeλx and g(x) = beβx . Complete integral:  –1 Z a λx w = ϕ(x)y – b eβx ϕ2 (x) dx + C1 , ϕ(x) = e + C2 . λ 33.

 
∂w 2 ∂w λx k + ae y + bx = 0. ∂x ∂y

This is a special case of equation 2.2.2.11 with f (x) = aeλx and g(x) = bxk . Complete integral:  –1 Z a λx w = ϕ(x)y – b xk ϕ2 (x) dx + C1 , ϕ(x) = e + C2 . λ 34.

 
∂w 2 ∂w + aeλxy + beβx = ceγxw + s0 eµx. ∂x ∂y

This is a special case of equation 2.2.2.12 with f (x) = aeλx , g(x) = beβx , h(x) = ceγx , r(x) = 0, and s(x) = s0 eµx . 35.

∂w ∂x

βx

+ ae

λy+µx

+ be

 
∂w 2 ∂y

= 0.

This is a special case of equation 2.2.2.15 with f (x) = aeβx and g(x) = beµx ,

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 55

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

56 36.

Page 56

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES ∂w ∂x

λx k

+ ae

βx 2k–2

y + be

y

 
∂w 2 ∂y

= 0.

This is a special case of equation 2.2.2.13 with f (x) = aeλx and g(x) = beβx .  
∂w 2 ∂w λx k βx 2k–2 37. + ae y + be y = ceγxw + s0 eµx. ∂x ∂y

This is a special case of equation 2.2.2.14 with f (x) = aeλx , g(x) = beβx , h(x) = ceγx , and s(x) = s0 eµx .  
∂w 2 ∂w k n λy + ax + bx e = 0. 38. ∂x ∂y This is a special case of equation 2.2.2.15 with f (x) = axk and g(x) = bxn . 39.

∂w ∂x

βx

+ ae

λy+µx

+ be

 
∂w 2 ∂y

= cw + s0 eγx.

This is a special case of equation 2.2.2.16 with f (x) = aeβx , g(x) = beµx , h(x) = c, and s(x) = s0 eγx . 2  ∂w ax+by ∂w + Ae 40. = Beλ(ax+by) . ∂x ∂y

This is a special case of equation 2.2.2.17 with f (ξ) = Aeξ and g(ξ) = Beλξ . Complete integral: Z p a –ξ 1 e–ξ 4b2 ABe(λ+1)ξ – 4b2 AC1 eξ + a2 dξ, ξ = ax+by. w = C1 x+C2 + 2 e ± 2 2b A 2b A 2  ∂w ax+by ∂w 41. + Ae = B(ax + by)k. ∂x ∂y This is a special case of equation 2.2.2.17 with f (ξ) = Aeξ and g(ξ) = Bξ k . 42.

 
∂w 2 ∂w k2 k1 k0 + a2 x w + a1 x y + a0 x = b2 xn2 w + b1 xn1 y + b0 xn0 . ∂x ∂y

This is a special case of equation 2.2.2.19 with fm (x) = am xkm and gm (x) = bm xnm (m = 0, 1, 2).  
∂w 2 ∂w λ2 x λ1 x λ0 x 43. + a2 e w + a1 e y + a0 e = b2 eβ2 xw + b1 eβ1 xy + b0 eβ0 x. ∂x ∂y This is a special case of equation 2.2.2.19 with fm (x) = am eλm x and gm (x) = bm eβm x (m = 0, 1, 2).  2 ∂w k ∂w 44. – aw = 0. ∂x ∂y   1
k+1 k+1 2 C1 x + C1 y + C2 . Complete integral: w = a

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 56

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS

45.

∂w ∂x

– aw

k



∂w ∂y

2

Page 57

57


∂w – bxny + cxm = 0. ∂y

This is a special case of equation 2.2.2.24 with f (w) = awk , g(x) = bxn , and h(x) = cxm . 46.

 2 ∂w n k ∂w = bxmw. + ax w ∂x ∂y

This is a special case of equation 2.2.2.20 with f (x) = axn and g(x) = bxm . 47.

∂w ∂x

k

+ ay w



n

∂w ∂y

2

= bw m.

This is a special case of equation 2.2.2.25 with f (y) = ay k , g(w) = wn , and h(w) = bwm . 48.

  ∂w 2 ∂w + by nw k = aw k+2 + cxmw. ∂x ∂y

This is a special case of equation 2.2.2.21 with f (y) = by n and g(x) = cxm . 49.

∂w ∂x

+ A(w + ax + by)

k



∂w ∂y

2

= B(w + ax + by)n.

This is a special case of equation 2.2.2.27 with f (u) = Auk and g(u) = Bun . 50.

∂w ∂x

λx

+ ae

w

k



∂w ∂y

2

= beβx w.

This is a special case of equation 2.2.2.20 with f (x) = aeλx and g(x) = beβx . 51.

 2 ∂w λy k ∂w + ae w = beβw . ∂x ∂y

This is a special case of equation 2.2.2.25 with f (y) = aeλy , g(w) = wk , and h(w) = beβw . 52.

∂w ∂x

λy

+ be

w

k



∂w ∂y

2

= aw k+2 + ceβxw.

This is a special case of equation 2.2.2.21 with f (y) = beλy and g(x) = ceβx .  2 ∂w βw ∂w 53. – ae = 0. ∂x ∂y  1  Complete integral: w = ln β(aC12 x + C1 y + C2 ) . β 54.

∂w ∂x

k βw

– ax e



∂w ∂y

2

– bxn = 0.

This is a special case of equation 2.2.2.22 with f (x) = axk and g(x) = bxn .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 57

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

58 55.

Page 58

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES ∂w ∂x

λx+βw

– ae



∂w ∂y

2

– beγx = 0.

This is a special case of equation 2.2.2.22 with f (x) = aeλx and g(x) = beγx . 56.

 2
∂w ∂w βw ∂w – beλx y + ceµx – ae = 0. ∂x ∂y ∂y

This is a special case of equation 2.2.2.24 with f (w) = aeβw , g(x) = beλx , and h(x) = ceµx . 57.

∂w ∂x

λy+βw

+ ae



∂w ∂y

2

= beγw.

This is a special case of equation 2.2.2.25 with f (y) = aeλy , g(w) = eβw , and h(w) = beγw . 58.

∂w ∂x

k n

– ax y w



m

∂w ∂y

2

= 0.

   1 m+1 aC12 k+1 2C1 2–n Complete integral: w = (m + 1) x + y 2 + C2 . k+1 2–n   ∂w 2 – axkeλy w m = 0. ∂x ∂y  1   m+1 aC12 k+1 2C1 – 1 λy x – e 2 + C2 Complete integral: w = (m + 1) . k+1 λ 59.

∂w

  ∂w 2 = 0. – aeλxy nw m ∂x ∂y   2  1 m+1 aC1 λx 2C1 2–n Complete integral: w = (m + 1) e + y 2 + C2 . λ 2–n 60.

61.

∂w

∂w

k n λw

– ax y e



∂w

2

= 0. ∂y  2  aC1 λ k+1 2C1 λ 2–n 1 Complete integral: w = ln x + y 2 + C2 . λ k+1 2–n ∂x

 2 ∂w λx+βy+γw ∂w 62. – ae = 0. ∂x ∂y  2  aC1 γ λx 2C1 γ – 1 βy 1 2 Complete integral: w = ln e – e + C2 . γ λ β 63.

∂w ∂x

w+ax+by

+ Ae



∂w ∂y

2

= Beλ(w+ax+by) .

This is a special case of equation 2.2.2.27 with f (u) = Aeu and g(u) = Beλu .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 58

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 59

59

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS

2.1.5. Equations of the Form

  ∂w + f (x, y, w) ∂w 2 + g(x, y, w) ∂w = h(x, y, w) ∂x ∂x ∂y

⊲ Equations of this type are encountered in mechanics, where x plays the role of time and y plays the role of the spatial coordinate. 1.

  ∂w 2 ∂w ∂w +a +b = cxk + sy n. ∂x ∂y ∂y

c b 1 xk+1 – y± Complete integral: w = –C1 x + C2 + k+1 2a 2a 2.

∂w ∂x

+a



∂w ∂y

2

+b

∂w ∂y

3.

∂x

+a



∂w ∂y

2

+b

∂w ∂y

  ∂w ∂w 2 ∂w +a = ceλx + seβy . +b ∂x ∂y ∂y

1 c b y± Complete integral: w = –C1 x + C2 + eλx – λ 2a 2a 5.

∂w ∂x

+a



∂w ∂y

2

+b

∂w ∂y

Z p

4aseβy + b2 + 4aC1 dy.

= ceλx + sy k.

c b 1 Complete integral: w = –C1 x + C2 + eλx – y± λ 2a 2a 4.

4asy n + b2 + 4aC1 dy.

= cxk + seβy .

c b 1 xk+1 – y± Complete integral: w = –C1 x + C2 + k+1 2a 2a ∂w

Z q

Z q

4asy k + b2 + 4aC1 dy.

Z p

4aseβy + b2 + 4aC1 dy.

= cxky + sxn.

Complete integral: 

 cxk+1 sxn+1 ac2 x2k+3 c(2aC1 + b)xk+2 w=y + C1 + – – – (aC12 + bC1 )x + C2 . k+1 n + 1 (k + 1)2 (2k + 3) (k + 1)(k + 2) 6.

  ∂w ∂w 2 ∂w +a +b = cxky + seβx . ∂x ∂y ∂y

This is a special case of equation 2.2.3.2 with f (x) = cxk and g(x) = seβx . 7.

  ∂w ∂w 2 ∂w +a +b = ceλxy + sxk. ∂x ∂y ∂y

This is a special case of equation 2.2.3.2 with f (x) = ceλx and g(x) = sxk .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 59

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

60 8.

Page 60

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES ∂w

+a



+a



∂w

2

+b

2

+b

∂w

= ceλxy + seβx .

∂x ∂y ∂y Complete integral:   c λx s ac2 c(2aC1 + b) λx w=y e + C1 + eβx – 3 e2λx – e – (aC12 + bC1 )x + C2 . λ β 2λ λ2

9.

∂w

∂w

∂x ∂y Complete integral:

w = (C1 y+C2 )F (x)+F (x)

∂w

= cxkw + sxn.

∂y

Z



n

sx

   dx c k+1 , F (x) = exp x . F (x) k+1

–aC12 F 2 (x)–bC1 F (x)

  ∂w ∂w ∂w 2 +b +a = cxkw + seβx . ∂x ∂y ∂y Complete integral:   Z  βx  dx c 2 2 k+1 w = (C1 y+C2 )F (x)+F (x) se –aC1 F (x)–bC1 F (x) , F (x) = exp x . F (x) k+1 10.

11.

∂w

+a



∂w

∂x ∂y Complete integral:

2

+b

w = (C1 y + C2 )F (x) + F (x)

∂w ∂y Z

= ceλxw + sxk.

 k  dx sx – aC12 F 2 (x) – bC1 F (x) , F (x)

  ∂w ∂w ∂w 2 12. +b +a = ceλxw + seβx. ∂x ∂y ∂y Complete integral: Z  βx  dx w = (C1 y + C2 )F (x) + F (x) se – aC12 F 2 (x) – bC1 F (x) , F (x) 13.

∂w ∂x

+ ax

k



∂w ∂y

2

+ bxn

∂w ∂y



 c λx F (x) = exp e . λ

F (x) = exp



 c λx e . λ

= cxm.

aC12 k+1 bC1 n+1 c Complete integral: w = C1 y – x – x + xm+1 + C2 . k+1 n+1 m+1  2 ∂w ∂w k ∂w 14. + ax + bxn = ceβx. ∂x ∂y ∂y Complete integral: w = C1 y –

aC12 k+1 bC1 n+1 c βx x – x + e + C2 . k+1 n+1 β

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 60

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS

15.

∂w ∂x

+ ax

k



∂w ∂y

2

+ bxn

∂w ∂y

Page 61

61

= cxmw + βxν y + γxµ.

This is a special case of equation 2.2.3.5 with f (x) = axk , g(x) = bxn , h(x) = cxm , p(x) = βxν , and s(x) = γxµ . 16.

∂w +a2 xk2 ∂x



∂w ∂y

2

∂w = bxnw+c2 xm2 y 2 +c1 xm1 y+c0 xm0 . ∂y

+(a1 xk1 y+a0 xk0 )

This is a special case of equation 2.2.3.6. 17.

∂w ∂x

λx

+ ae



∂w ∂y

2

+ bxk

Complete integral: w = C1 y –

18.

= ceβx.

∂y

aC12 λx bC1 k+1 c βx e – x + e + C2 . λ k+1 β

  ∂w 2 ∂w ∂w + aeλx = ceγx. + beβx ∂x ∂y ∂y

Complete integral: w = C1 y –

19.

∂w

∂w ∂x

λx

+ ae



∂w ∂y

2

aC12 λx bC1 βx c γx e – e + e + C2 . λ β γ

+ beβx

∂w ∂y

= ceγxw + keνxy + meµx.

This is a special case of equation 2.2.3.5 with f (x) = aeλx , g(x) = beβx , h(x) = ceγx , p(x) = keνx , and s(x) = meµx . 20.

 2 ∂w ∂w k ∂w + by n + ay = cy m. ∂x ∂y ∂y

Complete integral: by n–k+1 1 w = –C1 x + C2 – ± 2a(n – k + 1) 2a 21.

∂w ∂x

+ ay

k



∂w ∂y

2

+ by n

∂w ∂y

Z

y

–k

q

b2 y 2n + 4acy k+m + 4aC1 y k dy.

= ceβy .

This is a special case of equation 2.2.3.7 with f (y) = ay k , g(y) = by n , and h(y) = ceβy . 22.

∂w ∂x

+ ay

k



∂w ∂y

2

+ by n

∂w ∂y

= cxm + sy l.

Complete integral: cxm+1 by n–k+1 1 w = –C1 x + C2 + – ± m + 1 2a(n – k + 1) 2a

Z

y –k

q

b2 y 2n + 4asy k+l + 4aC1 y k dy.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 61

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

62 23.

Page 62

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES ∂w ∂x

λy

+ ae



∂w ∂y

2

+ beβy

∂w ∂y

= ceγy .

Complete integral: be(β–λ)y 1 w = –C1 x + C2 – ± 2a(β – λ) 2a 24.

∂w + aeλy ∂x



∂w ∂y

2

+ beβy

Z

e–λy

p

b2 e2βy + 4ace(λ+γ)y + 4C1 aeλy dy.

∂w = ceγx + seµy . ∂y

Complete integral: w = –C1 x + C2 +

25.

∂w ∂x

c γx be(β–λ)y 1 e – ± γ 2a(β – λ) 2a

+ A(ax + by)k



∂w ∂y

2

Z

e–λy

p

b2 e2βy + 4ase(λ+µ)y + 4C1 aeλy dy.

+ B(ax + by)n

∂w ∂y

= C(ax + by)m.

This is a special case of equation 2.2.3.9 with f (z) = Az k , g(z) = Bz n , and h(z) = Cz m . 26.

∂w ∂x

ax+by

+ Ae



∂w ∂y

2

+ Beλ(ax+by)

∂w

= Ceβ(ax+by) .

∂y

This is a special case of equation 2.2.3.9 with f (z) = Aez , g(z) = Beλz , and h(z) = Ceβz . 27.

∂w ∂x

+ aw

k



∂w ∂y

2

+ bw n

∂w ∂y

= 0.

Complete integral in implicit form: C1 x + C2 y +

28.

 2 ∂w ∂w k ∂w + aw = 0. + beλw ∂x ∂y ∂y

Complete integral in implicit form: C1 x + C2 y +

29.

∂w ∂x

+ aw

k



∂w ∂y

2

+ bw n

∂w ∂y

Z

Z

aC22 wk dw = C3 . C1 + bC2 wn

aC22 wk dw C1 + bC2 eλw

= C3 .

= cw m.

This is a special case of equation 2.2.3.12 with f (w) = awk , g(w) = bwn , and h(w) = cwm . 30.

∂w ∂x

– aw

k



∂w ∂y

2

– (bxny + cxm)

∂w ∂y

= 0.

This is a special case of equation 2.2.3.13 with f (w) = awk , g(x) = bxn , and h(x) = cxm .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 62

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

63

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS

∂w

31.

∂x

λw

+ ae



∂w ∂y

2

+ bw n

∂w ∂y

= 0.

Complete integral in implicit form: C1 x + C2 y + ∂w

32.

∂x

λw

+ ae



∂w ∂y

2

+ beβw

∂w ∂y

Page 63

Z

aC22 eλw dw = C3 . C1 + bC2 wn

Z

aC22 eλw dw

= 0.

Complete integral in implicit form: C1 x + C2 y +

C1 + bC2 eβw

= C3 .

 2 ∂w ∂w λw ∂w + beβw + ae = ceγw. ∂x ∂y ∂y

33.

This is a special case of equation 2.2.3.12 with f (w) = aeλw , g(w) = beβw , and h(w) = ceγw . ∂w

34.

∂x

λw

– ae



∂w ∂y

2

– (beβxy + ceγx)

∂w ∂y

= 0.

This is a special case of equation 2.2.3.13 with f (w) = aeλw , g(x) = beβx , and h(x) = ceγx .   ∂w ∂w 2 ∂w + A(w + ax + by)k + B(w + ax + by)n = C(w + ax + by)m. ∂x ∂y ∂y

35.

This is a special case of equation 2.2.3.14 with f (z) = Az k , g(z) = Bz n , and h(z) = Cz m . ∂w

36.

∂x

w+ax+by

+ Ae



∂w ∂y

2

+ Beλ(w+ax+by)

∂w ∂y

= Ceβ(w+ax+by) .

This is a special case of equation 2.2.3.14 with f (z) = Aez , g(z) = Beλz , and h(z) = Ceβz .

2.1.6. Equations of the Form

 2  2 ∂w ∂w f (x, y, w) + g(x, y, w) = h(x, y, w) ∂x ∂y

⊲ Equations of this type are encountered in mechanics,

geometric optics, and differen∂w 2 ∂w 2 tial geometry. In particular, the equation ∂x + ∂y = f (x, y) describes the twodimensional wave front in the process of propagation of light in an inhomogeneous medium with variable refractive index f (x, y). 1.

a



∂w ∂x

2

  ∂w 2 +b = c. ∂y

For a = b, this is a differential equation of light rays. Complete integral: w = C1 x + C2 y + C3 , where aC12 + bC22 = c. w2 (x – C1 )2 (y – C2 )2 An alternative form of the complete integral: = + . c a b

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 63

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

64

Page 64

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES 

∂w

2



∂w

2

+ = a – 2by. ∂x ∂y This equation governs parabolic motion of a point mass in vacuum (the coordinate x is measured along the Earth’s surface, the coordinate y is measured vertically upward from the Earth’s surface, and a is the gravitational acceleration). 1 Complete integral: w = C1 x ± 3b (a – C12 – 2by)3/2 + C2 . 2.

Reference: P. Appell (1953).

3.



∂w ∂x

2

+a



∂w ∂y

2

= bx + cy.

Complete integral for b2 + ac2 ≠ 0: w=

bC1 2 [(b2 + ac2 )(bx + cy) – ab2 C12 ]3/2 (by – acx) – + C2 . 3 b2 + ac2 (b2 + ac2 )2

Complete integral for b2 + ac2 = 0: w=

C1 C1 b c (bx + cy)2 – y+ x + C2 . 2 2 2c 4C1 b

Another complete integral for a = 1, b = k2 , and c = ±k2 : ±

w = 23 k(x + C1 )3/2 ± 23 k(y

C1 )3/2 + C2 .

   ∂w 2 ∂w 2 +a = b1 x2 + b2 y 2 + c1 x + c2 y + s. 4. ∂x ∂y A separable equation. Complete integral: Z p Z r b2 y 2 + c2 y + C1 2 w=± b1 x + c1 x + s – C1 dx ± dy + C2 . a 



   ∂w 2 ∂w 2 5. + = ax2 + bxy + cy 2 + s. ∂x ∂y Let the numbers A, B, k1 , and k2 satisfy the following algebraic system of equations (k1 or k2 can be set equal to unity): Ak12 + Bk22 = a,

2(A – B)k1 k2 = b,

Ak22 + Bk12 = c.

Then the transformation ξ = k1 x + k2 y, η = k2 x – k1 y leads to an equation of the form 2.1.6.4:     ∂w 2 ∂w 2 A B s + = 2 ξ2 + 2 η2 + 2 . 2 2 ∂ξ ∂η k1 + k2 k1 + k2 k1 + k22 

∂w

2



∂w

2

= ax2 + bxy + cy 2 + βx + γy + s. ∂x ∂y The transformation used for solving equation 2.1.6.5 brings the above equation to a separable equation of the form 2.1.6.4. 6.

+

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 64

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS

7.



∂w ∂x

2

+a



∂w ∂y

2

Page 65

65

= bxk.

Complete integral: w = C1 y + C2 ± 

Z q

bxk – aC12 dx.

   ∂w 2 ∂w 2 8. + = axk + by n + c. ∂x ∂y A separable equation. Complete integral: Z p Z p k ax + C1 dx ± by n + c – C1 dy + C2 . w=± Reference: E. Kamke (1965).



∂w ∂x

2



∂w ∂y

2

a

+ b. x2 + y 2 This equation arises from the solution of the two-body problem in celestial mechanics. By using the polar coordinates x = r cos θ and y = r sin θ, one can obtain a separable equation. This equation has the complete integral s Z a C2 w=± b + – 21 dr + C1 θ + C2 . r r 9.

+

= p

References: P. Appell (1953), R. Courant (1962), and E. Kamke (1965).

10.



∂w ∂x

2

+



∂w ∂y

2

= a(x2 + y 2 )k.

2

= a(x2 + y 2 )(xy)k.

2

= a(x2 + y 2 )(x2 – y 2 )k.

2

= A(ax + by)k + B(bx – ay)n + s.

This is a special case of equation 2.2.4.4 with f (z) = az k . 11.



∂w ∂x

2

+



∂w ∂y

This is a special case of equation 2.2.4.5 with f (z) = az k . 12.



∂w ∂x

2

+

2

+



∂w ∂y

This is a special case of equation 2.2.4.6 with f (z) = az k . 13.



∂w ∂x



∂w ∂y

This is a special case of equation 2.2.4.3 with f (z) = Az k and g(u) = Bun + s. 

  ∂w 2 14. +b = cw. ∂y c Complete integral: w = (C1 x + C2 y + C3 )2 . 2 4(aC1 + bC22 ) ∂w a ∂x

2

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 65

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

66 15.

Page 66

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES 

∂w ∂x

2

+



∂w ∂y

2

= aw 2 + b.

Complete integral in implicit form: x cos C1 + y sin C1 + C2 = ±

Z

√

dw

. aw2 + b
e 2 exp x cos C1 + y sin C1 . Whence, in particular, for a = 1 and b = 0 we have w = C 16.



∂w ∂x

2

+



∂w ∂y

2

+1=

a2

w2

.

This equation describes a family of spherical surfaces of radius a with centers at points of the xy-plane. Complete integral in implicit form: (x – C1 )2 + (y – C2 )2 + w2 = a2 . (y – C1 x – C2 )2 Another complete integral: + w2 = a2 . 1 + C12 17.



∂w ∂x

2

+



∂w ∂y

2

= aw k.

This is a special case of equation 2.2.4.7 with f (w) = awk . 18.



∂w ∂x

2

+



∂w ∂y

The substitution u =

2

2–k 2 w 2 leads to an equation of the form 2.1.6.5: 2–k

 19.



∂w ∂x

2

+ axy



20.

∂w ∂x

2

2

+

2

= b.

∂u ∂x

∂w ∂y

Complete integral: w = – 


= ax2 + bxy + cy 2 + s w k.



∂u ∂y

2


3/2 2 b – aC12 x 3aC12

+ A(ax + by)

k



∂w ∂y

2

= ax2 + bxy + cy 2 + s.

√ + 2C1 y + C2 .

= B(ax + by)n.

This is a special case of equation 2.2.4.15 with f (z) = 1, g(z) = Az k , and h(z) = Bz n . 21.

a



∂w ∂x

2

+ bw

Complete integral:



∂w ∂y

bC22 w

=

2

= c2 .

–aC12



2/3 3bcC22 + (C1 x + C2 y + C3 ) . 2

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 66

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS

x

22.



∂w ∂x

2

+ ay



∂w ∂y

2

Page 67

67

= bx + cy.

Complete integral: w=

23.

p

p C1 x(bx + C1 ) + √ ln 2bx + 2 bx(bx + C1 ) + C1 2 b r p y(cy – C1 ) C1 – √ ln 2cy + 2 cy(cy – C1 ) – C1 + C2 . + a 2 ac

ax

2



∂w ∂x

2

+ by

2



∂w ∂y

2

= w c.

The transformation u = ln x,

v = ln y,

 2   2 – c  2–c z w=  z 2 e

if c ≠ 2, if c = 2

leads to an equation of the form 2.1.6.1: azu2 + bzv2 = 1. 24.

2

2

(x – a )



∂w ∂x

2

2

2

– (y – a )



∂w ∂y

2

= b(x2 – y 2 ) + kx.

This equation arises from the solution of the plane problem of attraction between two identical fixed point bodies (the variables x and y play the role of elliptic coordinates and a is the distance between the bodies). s Z Z s 2 bx2 + kx + C1 by + C1 Complete integral: w = ± dx ± dy + C2 . x2 – a2 y 2 – a2 Reference: V. I. Arnold (1980).

25.

(x + a1 )(x + a2 )



∂w ∂x

2

– (y + a1 )(y + a2 )



∂w

2

∂y √ √ = k x + a1 + b y + a2 + c(x – y).

Complete integral:

w=

Z

s

√ √ Z s C1 + cy – b y + a1 C1 + cx + k x + a1 dx + dy + C2 . (x + a1 )(x + a2 ) (y + a1 )(y + a2 )

For 2k = m1 + m2 , 2b = m1 – m2 , x = λ1 , and y = λ2 , this equation governs the motion of a point body of unit mass in the xy-plane under the action of the gravitational forces produced by two point masses m1 and m2 located at the points (x = ±1, y = 0) in elliptic coordinates (λ1 , λ2 ). Reference: E. Kamke (1965).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 67

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

68

Page 68

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES ∂w

2



∂w

2

– 4x(a – y)(b – y)(c – y) = xy(x – y). ∂x ∂y This equation is encountered in searching for geodesic lines on an ellipsoid with semiaxes a, b, and c. Complete integral: Z s Z s 1 x(x + C1 ) 1 y(y + C1 ) dx + dy + C2 , w= F (t) = (a – t)(b – t)(c – t). 2 F (x) 2 F (y) 26.

4y(a – x)(b – x)(c – x)



Reference: E. Kamke (1965).



∂w

2

ax

n



∂w

2

+ by = cxm + sy l + β. ∂x ∂y A separable equation. Complete integral: Z r m Z s l cx + C1 sy + β – C1 w=± dx ± dy + C2 . by n axk

27.

k

The signs before each integral can chosen independently of each other.     ∂w 2 ∂w 2 –a +b = 0. 28. (sin x) ∂x ∂y This equation arises in introducing orthogonal geodesic parametric lines on a unit sphere. Complete integral: s Z bC12 w = C1 y + C2 ± a– dx. (sin x)2 2



Reference: E. Kamke (1965).

29.

A(ax + by)

k



∂w ∂x

2

+ B(ax + by)

n



∂w ∂y

2

= C(ax + by)m.

This is a special case of equation 2.2.4.15 with f (z) = Az k , g(z) = Bz n , and h(z) = Cz m . 30.

A(w + ax + by)

k



∂w ∂x

2

+ B(w + ax + by)

n



∂w ∂y

2

= C(w + ax + by)m.

This is a special case of equation 2.2.4.16 with f (z) = Az k , g(z) = Bz n , and h(z) = Cz m . 31.

k

ax w

n1



∂w ∂x

2

m

+ by w

n2



∂w ∂y

2

= cw n3 + s.

This is a special case of equation 2.2.4.17 with f1 (x) = axk , g1 (w) = wn1 , f2 (y) = by m , g2 (w) = wn2 , and h(w) = cwn3 + s. 32.

ax+by

Ae



∂w ∂x

2

λ(ax+by)

+ Be



∂w ∂y

2

= Ce(ax+by) .

This is a special case of equation 2.2.4.15 with f (z) = Aez , g(z) = Beλz , and h(z) = Ceβz .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 68

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS

33.

w+ax+by

Ae



∂w ∂x

2

λ(w+ax+by)

+ Be



∂w ∂y

2

Page 69

69

= Ce(w+ax+by) .

This is a special case of equation 2.2.4.16 with f (z) = Aez , g(z) = Beλz , and h(z) = Ceβz . 34.

aeλ1 x+β1 w



∂w ∂x

2

+ beλ2 y+β2 w



∂w ∂y

2

= ceγw + s.

This is a special case of equation 2.2.4.17 with f1 (x) = aeλ1 x , g1 (w) = eβ1 w , f2 (y) = beλ2 y , g2 (w) = eβ2 w , and h(w) = ceγw + s.

2.1.7. Equations of the Form



 2 ∂w f (x, y) + g(x, y) ∂w ∂w = h(x, y, w) ∂x ∂x ∂y

∂w

2

+a

∂w ∂w

+ bx + cy = 0. ∂x ∂x ∂y 1◦ . Complete integral for b ≠ 0:

1.

w=

3ab2 x + b(ac – 2b)y + aC1 (b + ac) p a(aC1 – 2by) + C2 . 3a2 b2

2◦ . Complete integral for b ≠ –ac:

–3/2  (aC1 )2 – 4b(b + ac)(bx + cy) abC1 x – (ac + 2b)C1 y w= – + C2 . 2b(b + ac) 12b2 (b + ac)2 3◦ . Complete integral for b = 0: w = C1 x –

y (cy + 2C12 ) + C2 . 2aC1

4◦ . Complete integral for b = –ac: w=

2.



3.



∂w

2

+a

2

+a

c2 (2y – ax)x (c3 y + 2C12 )y – + C2 . 2C1 2acC1

∂w ∂w

+ bxy = 0. ∂x ∂y r q √  y  b C1 Complete integral: w = x – C1 – y 2 – √ ln –ab y + a(aC1 – by 2 ) + C2 . 2a a 2 –ab ∂x

∂w

∂w ∂w

∂x ∂x ∂y Complete integrals: w=

+ bw = 0.

(by + C1 )(y – ax + C2 ) , a2

w=–

b(C1 x + y + C2 )2 . 4C1 (a + C1 )

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 69

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

70 4.

Page 70

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES 

∂w ∂x

2

+a

2

+a

∂w ∂w ∂x ∂y

+ bw + cy k = 0.

  Z by + C1 y k dy 2 y – ax + a c Complete integral: w = + C2 . a2 (by + C1 )2 5.



∂w ∂x

∂w ∂w ∂x ∂y

+ by kw = 0.

  1 b k+1 y Complete integral: w = 2 (y – ax + C1 ) + C2 . k+1 a 6.

a



∂w ∂x

2

+b

∂w ∂w ∂x ∂y

= cw 2 .


Complete integral: w = C3 exp C1 x + C2 y , where aC12 + bC1 C2 = c. 

∂w

2

8.



∂w ∂x

2

+ ay

∂w ∂w

+ (bx + c)y = 0. ∂x ∂x ∂y Complete integral: s  √ p  abx + ac – 2b 2 2by aC  1  –a(2by + C1 ) + arctan –1 – + C2  2 2 C1 a b a w=  cy + C12 ln |y|    C1 x – + C2 aC1 7.

+ ay

∂w ∂w ∂x ∂y

9.

∂w ∂x

2

+ ay

if b = 0.

+ bxy + c = 0.

ax – 2 Complete integral: w = a 

if b ≠ 0,

r

2by 2(c + C1 ) C1 – – arctan a a(–C1 )1/2

s

2by – 1 + C2 . aC1

∂w ∂w + bw = 0. ∂x ∂y

Complete integral: w =

(ln |y| – ax + C1 )(b ln |y| + C2 ) . a2



 ∂w 2 ∂w ∂w 10. + (ay + b) + cyw = 0. ∂x ∂x ∂y Complete integral:

 ln |ay + b| – ax + C1 acy – bc ln |ay + b| + C2    a3 w= 2    (y – bx + C1 )(cy + C2 ) 2b2

if a ≠ 0, if a = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 70

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

71

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS



∂w

2

+ axy

2

+

Page 71

∂w ∂w

+ b = 0. ∂x ∂x ∂y Complete integral: q q  1  b w = – x aC1 x – a2 C12 x2 – 4b + C1 ln |y| – ln aC1 x + a2 C12 x2 – 4b + C2 . 4 aC1

11.



∂w

ax ∂w ∂w

+ b = 0. ∂x y ∂x ∂y Complete integral: q q  1 1  b w = – x aC1 x – a2 C12 x2 – 4b + C1 y 2 – ln aC1 x + a2 C12 x2 – 4b + C2 . 4 2 aC1

12.

 a ∂w ∂w ∂w 2 + b = 0. 13. + ∂x xy ∂x ∂y Complete integral: q aC1 aC1 C1 2 1 aC1 w= a2 C1 2 – 4bx2 – – ln |x| + arctanh p y + C2 . 2 2 2 2 2 a2 C1 – 4bx2 



 ∂w 2 ∂w ∂w 14. + xb = 0. + ya ∂x ∂x ∂y Complete integral:  Z q 1 C1 C1 y 1–a   + C2 C12 – 4xb dx – – x + 2 2 a–1 w= Z q  1 C  – 1 x + C12 – 4xb dx + C1 ln |y| + C2 2 2 15.



∂w ∂x

2

+ A(ax + by)k

2

+ aw

∂w ∂w ∂x ∂y

if a ≠ 1, if a = 1.

+ B(ax + by)n = 0.

This is a special case of equation 2.2.5.14 with f (z) = az k , g(z) = bz n , and h(w) = 1. 16.



∂w ∂x

∂w ∂w ∂x ∂y

C1 Complete integral: w = + a

+ b = 0. 

9b 4aC1

1/3

(y – C1 x + C2 )2/3 .



 ∂w 2 ∂w ∂w 17. + aw 2 = b2 w 2 . ∂x ∂x ∂y Complete integral in implicit form: ±b(C1 x + C2 y + C3 ) = R + C1 ln

R – C1 , w

where

R2 = aC1 C2 w2 + C12 .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 71

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

72 18.

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES x



2

∂w ∂x

+a

∂w ∂w ∂x ∂y

+ by 2 = 0.

Complete integral: w = –C1 xe–y/a + x



2

∂w ∂x

+ ay

b 2 (y – 2ay + 2a2 )ey/a + C2 . C1

∂w ∂w

= b. ∂x ∂y b 1/a Complete integral: w = y + C1 xy –1/a + C2 . C1 Other complete integrals for a = 1: 19.

2 p bC1 (C1 x + y) + C2 , w= C1 20.

Page 72

ax



∂w ∂x

2

– (ay + b)

w=

∂w ∂w ∂x ∂y

Complete integral: w = C1 x(ay + b) + 

∂w 21. (x + y) ∂x Complete integral:



2

+ ay

2 4bx + C1 + C1 ln

+ cy(ay + b)2 = 0.

q

4bx + C12 – C1 + C2 . y

cy 2 + C2 . 2C1

∂w ∂w + b = 0. ∂x ∂y

 y b 1/a y –1/a – y + C2 a–1 C1 w=  C x by   1 – – C1 ln |y| + C2 y C1      C1 x –

if a ≠ 1, if a = 1.

∂w ∂w + b = 0. ∂x ∂y   a 9b 1/3 Complete integral: w = – (C1 x – y + C2 )2/3 . C1 4C12

22.

∂w w ∂x

2

q



2

∂w ∂w + b = 0. ∂x ∂y 1/3  9b Complete integral: w = (y – C1 x + C2 )2/3 . 4C1 (a – C1 )

23.

w

∂w ∂x

+a



∂w

2

∂w ∂w

= cx2 y. ∂x ∂x ∂y Complete integral in implicit form: Z p √ C1 (C1 w2 + bC2 w + aC1 ) dw = 12 c(C1 x2 + C2 y 2 ) + C3 . 24.

2

+ aw

y(w + a)

+ bxw

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 72

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS

Page 73

73

2.1.8. Other Equations 

∂w

2

+a

∂w ∂w

∂x

x 1 w = – aC1 x + 2 4 3.



4.



∂w ∂x

2

+a

2

+a

q

a2 C12

+ by

∂w

+ cx + d = 0. ∂x ∂y ∂x
3/2 b 2 1 1 – Complete integral: w = – aC1 x – a2 C12 – 4d – 4cx y + C1 y + C2 . 2 12c 2a   ∂w 2 ∂w ∂w ∂w 2. +a + by 2 – cx2 = 0. ∂x ∂x ∂y ∂x Complete integral: 1.

+ 4cx2

q a2 C12 √ b 3 2 2 2 – y + C1 y + √ ln 2 c x + a C1 + 4cx + C2 . 3a 8 c

∂w ∂w ∂w + becw = 0. ∂x ∂y ∂y 1 C1 (a – C1 ) Complete integral: w = ln . c bc(C1 x – y + C2 ) ∂w ∂x

∂w ∂w ∂x ∂y

+b

Complete integral: w = C1 x –

5.



∂w ∂x

2

6.

∂w ∂x

2

∂w ∂y

+ d = 0.

C12 + bC1 + d y + C2 . aC1 + c

  ∂w ∂w +a +b x +y = 0. ∂x ∂y ∂x ∂y

+a

2

+a

2

+a

b(x + C1 y)2 + C2 . 2(1 + aC1 )

∂w ∂w ∂x ∂y

Complete integral: w = 

∂x

+c

∂w ∂w

Complete integral: w = – 

∂w

+ (bw + c)

∂w ∂x

+ (pw + q)

∂w ∂y

= 0.

    1 p – bC1 exp (y – C1 x + C2 ) + cC1 – q . p – bC1 C1 (a – C1 )

∂w ∂w ∂w ∂w + bw 2 + cw = 0. ∂x ∂y ∂x ∂y   –1 c Complete integral: w = c bC1 – exp (C1 x – y + C2 ) . C1 (a – C1 )

7.

  ∂w ∂w + bw x +y = 0. ∂x ∂x ∂y ∂x ∂y   b(C1 x – y)2 + C2 Complete integral: w = exp . 2C1 (a – C1 ) 8.



∂w ∂x

∂w

∂w ∂w

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 73

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

74 9.

Page 74

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES 

∂w

2

y ∂w ∂w

10.



∂w ∂x

2

+ aw

+

a ∂w

= 0. x ∂x ∂y x ∂y a Complete integral: w = – ln |C1 x + y| + C2 . C1 ∂x

+

∂w ∂w ∂x ∂y

+ bw 2

= 0.

∂y

aw ln |w| + C1 y = – x + C2 . bw C1

Complete integral in implicit form:

  ∂w ∂w +b x +y = 0. 11. + aw ∂x ∂x ∂y ∂x ∂y p C12 ± abC1 (y – C1 x)2 + C2 Complete integral: w = . aC1 

12.



∂w

∂w ∂x

2

∂w

2

∂w ∂w



∂w +a ∂y

2

= bw

∂w + cw 2 + d. ∂x

Complete integral in implicit form: x + C1 y + C2 =

13.



∂w ∂x

2



∂w +a ∂y

2

2(aC12

+b

+ 1)

Z

bw ±

q

dw

.

(acC12 + b2 )w2 + adC12

∂w ∂w +c + d = 0. ∂x ∂y

Complete integral: w = Ax + C1 y + C2 ,

14.

a



∂w ∂x

2

A=

– 12 b

+

1 2

q

b2 – 4d – 4cC1 – 4aC12 .

  ∂w 2 ∂w ∂w +b + cx + sy = k. ∂y ∂x ∂y

This is a special case of equation 2.2.6.10. 15.



∂w a + cy ∂x

2

 2 ∂w +b + cx = k. ∂y

Complete integral: w = –cxy + C1 x + C2 y + C3 , where aC12 + bC22 = k. 16.



∂w a ∂x

2

    ∂w 2 ∂w ∂w +b =c x +y –w . ∂y ∂x ∂y

aC12 + bC22 Complete integral: w = C1 x + C2 y – . c

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 74

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

75

2.1. EQUATIONS CONTAINING ARBITRARY PARAMETERS

17.



∂w ∂x

2

+



∂w ∂y

2

+ (x + a)

2

=a

∂w

+ (y + b)

∂x

∂w ∂y

Page 75

= w + c.

Complete integral: w = C1 (x + a) + C2 (y + b) + C12 + C22 – c. 18.



∂w ∂x

2

+



∂w ∂y

∂w ∂w . ∂x ∂y

Complete integral: w = C1 x + C2 y + C3 , 19.

x



∂w ∂x

2

+ ay



∂w ∂y

2

+b

C12 + C22 = aC1 C2 .

where

∂w ∂w ∂x ∂y

= 0.

Complete integral:



x

∂w ∂x

∂w

+y

∂y

2

–a

2



∂w ∂x

2

+



∂w ∂y

±

20.

q bC1 y w = C1 (y – ax) + ln ± C1 b2 – 4axy 2 x

2

|b|C1 arctanh

p

b2 – 4axy + C2 . |b|

 + 1 = 0.

Complete integral: x cos C1 + y sin C1 + C2 . a By passing to the polar coordinates x = ρ cos θ and y = ρ sin θ, one arrives at an alternative form of the complete integral: r ρ2 + C12 σ+1 aσ w = a ln – C1 arctan + C1 θ + C2 , where σ 2 = 2 . σ–1 C1 ρ – a2 w = a arccosh



  ∂w 2 21. ay – bx =a +b + ab. ∂x ∂y ∂x ∂y √ √ By passing to the polar coordinates x b = ρ cos θ and y a = ρ sin θ, one arrives at a separable equation. A complete integral of this equation is ∂w

∂w

2

w = σ – C1 arctan

22.



y

∂w ∂x

–x

∂w ∂y

2



∂w

2

σ + C1 θ + C2 , C1 2

2

= a(x + y )



∂w ∂x

where 2

+



σ 2 = ρ2 (C12 – 1) – C12 . ∂w ∂y

2

 +1 .

By passing to the polar coordinates x = ρ cos θ and y = ρ sin θ, one arrives at a separable equation. A complete integral of this equation (for 0 ≤ a < 1) is r a C1 σ – C1 w = C1 θ+σ+ ln + C2 , where σ 2 = C12 – ρ2 1–a 2 σ + C1 (hence, one must require that ρ2 < C12 ).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 75

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

76

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES w



∂w

∂w

2



∂w

∂w

2

– +a + = b. ∂x ∂y ∂x ∂y Complete integral:     C1 + C2 2 1/3 3(C1 x + C2 y + C3 ) 2/3 w = –a +b C1 – C2 2(C1 – C2 ) 23.

Page 76

if

C1 ≠ C2 .



   2  2  ∂w ∂w 2 2 2 ∂w 2 ∂w 24. x +y = (1 – w ) a +b . ∂x ∂y ∂x ∂y The general integral can be found by solving the equation s ay p b2 x2 + a2 y 2 arctan = 1 – w2 Φ(w) – 1 – arctan bx a2 b2 (1 – w2 )

for w.     ∂w 2 ∂w ∂w ∂w –x +a x +y = b. 25. y ∂x ∂y ∂x ∂y

b – C12 y Complete integral: w = ln(x2 + y 2 ) – C1 arctan + C2 . 2a x  2 ∂w ∂w ∂w ∂w 26. x +y –w =a . ∂x ∂y ∂x ∂y p Complete integral: w = C1 x + C2 y ± aC1 C2 .    ∂w 2 ∂w 2 27. =a +b + c. ∂x ∂y q Complete integral: w = C1 x + C2 y ± aC12 + bC22 + c.

28.



∂w ∂w x +y –w ∂x ∂y



x

∂w ∂x

+y

Complete integral: 

∂w ∂y √

–w

2



2

= ax



∂w

2

+ by



∂w

2

. ∂x ∂y p √ C3 w = C1 x + C2 y – A, where aC1 + bC2 = A2 .

2      ∂w ∂w ∂w 2 ∂w 2 2 2 2 2 2 29. x +y – w = a (x + y + w ) + +1 . ∂x ∂y ∂x ∂y Complete integral in implicit form: (x – C1 )2 + (y – C2 )2 + (w – C3 )2 = 14 a–2 ,

where

C12 + C22 + C32 = 14 a–2 .

2 ∂w ∂w ∂w 30. x x +y – w = ay 2 . ∂x ∂y ∂y This equation describes a family of cones whose vertices lie on the w-axis. Complete C 2 x2 integral: w = – 1 + C1 + C2 x. ay 2



C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 76

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

77

2.2. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

31.



w

∂w ∂x

+x

2

+



w

∂w ∂y

+y

2

2

–a w

2



Complete integral: (x – C1 )2 + (y – C2 )2 + w2 =

∂w ∂x

C12

+

2



+

C22

∂w ∂y

Page 77

2

 + 1 = 0.

(w ≠ 0).

a2

2.2. Equations Containing Arbitrary Functions 2.2.1. Equations of the Form ∂w ∂w = f (x, y, w)

∂x ∂y

1.

∂w ∂w ∂x ∂y

= ay + f (x).

Z p f (x) dx Complete integral: w = ±y 2ax + C1 ± + C2 . √ 2ax + C1 ∂w ∂w

= f (x)y + g(x). ∂x ∂y Complete integral: Z g(x) w = ϕ(x)y + dx + C1 , ϕ(x)

2.

∂w ∂w

= f (x)y k + g(x)y 2k+1 . ∂x ∂y Complete integral: Z 1 f (x) k+1 w = ϕ(x)y + dx + C1 , k+1 ϕ(x)

3.

 Z 1/2 ϕ(x) = ± 2 f (x) dx + C2 .

∂w ∂w = f (x)eλy + g(x)e2λy . ∂x ∂y Complete integral: Z 1 f (x) λy dx + C1 , w = ϕ(x)e + λ ϕ(x)



2 ϕ(x) = ± k+1

Z

g(x) dx + C2

1/2

.

4.



2 ϕ(x) = ± λ

Z

g(x) dx + C2

1/2

.

∂w ∂w

= f (x)g(y). Z Z 1 Complete integral: w = C1 f (x) dx + g(y) dy + C2 . C1

5.

∂x ∂y

∂w ∂w = f (ax + by). ∂x ∂y Complete integral:

6.

1/2 Z  1 2 w = C1 (ax – by) ± C1 + f (z) dz + C2 , ab

z = ax + by.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 77

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

78

Page 78

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES

∂w ∂w = f (xy). ∂x ∂y Complete integral: 7.

x w = C1 ln ± y

Z

∂w ∂w f (xay b) = . ∂x ∂y xy Complete integral:

1/2 1 2 dz + C2 , C1 + zf (z) z

z = xy.

8.

a –b

w = C1 ln x y

9.




±

Z

 1/2 1 2 1 C + f (z) dz + C2 , z 1 ab

∂w ∂w = aw + f (x). ∂x ∂y

Complete integral: w = (ax + C1 )(y + C2 ) + (ax + C1 )

Z

z = xa y b .

f (x) dx . (ax + C1 )2

∂w ∂w

= f (x)w + g(x)y + h(x). ∂x ∂y Complete integral: w = ϕ(x)y + ψ(x), 10.

where the functions ϕ = ϕ(x) and ψ = ψ(x) are determined by the ordinary differential equations ϕϕ′x = f (x)ϕ + g(x), ϕψy′ = f (x)ψ + h(x).

(1) (2)

The exact solutions of Abel’s equation (1) for various f (x) and g(x) can be found in the book by Zaitsev and Polyanin (1993, 1994) and Polyanin and Zaitsev (2003). Equation (2) is linear in ψ and, hence, integrable for any f (x) and h(x), provided that the function ϕ is known. For the simplest cases where f and/or g are identically zero, equation (1) has the solutions Z ϕ(x) = f (x) dx + C1 for g(x) ≡ 0 and any f (x),  Z 1/2 ϕ(x) = ± 2 g(x) dx + C1

11.

∂w ∂w ∂x ∂y

for f (x) ≡ 0 and any g(x).

= aw k+1 + f (x)w 2k.

The substitution u = w1–k leads to an equation of the form 2.2.1.9: ∂u ∂u = a(1 – k)2 u + (1 – k)2 f (x). ∂x ∂y

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 78

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

79

2.2. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

12.

∂w ∂w = aeλw + f (x)e2λw. ∂x ∂y

The substitution u = e–λw leads to an equation of the form 2.2.1.9:

13.

Page 79

∂u ∂u = aλ2 u+λ2 f (x). ∂x ∂y

∂w ∂w = f (ax + by)g(w). ∂x ∂y

Complete integral in implicit form: Z 14.

dw = C1 (ax – by) ± √ g(w)

∂w ∂w ∂x ∂y

1/2 Z  1 C12 + dz + C2 , f (z) ab

= f (xy)g(w).

Complete integral in implicit form: Z Z 1/2 dw x 1 2 = C1 ln ± C1 + zf (z) dz + C2 , √ y z g(w) 15.

z = ax + by.

z = xy.

∂w ∂w 1 = f (xay b)g(w). ∂x ∂y xy

Complete integral in implicit form: Z 16.


dw = C1 ln xa y –b ± √ g(w)

∂w ∂w ∂x ∂y

Z

 1/2 1 2 1 C + f (z) dz + C2 , z 1 ab

Z

dw = C1 √ h(w)

z = xa y b .

= f (x)g(y)h(w).

Complete integral in implicit form:

Z

1 f (x) dx + C1

Z

g(y) dy + C2 .

2.2.2. Equations of the Form

 2 ∂w ∂w f (x, y) + g(x, y, w) = h(x, y, w) ∂x ∂y

⊲ Equations of this type are encountered in mechanics, where x plays the role of time and y plays the role of the spatial coordinate.   ∂w ∂w 2 1. +a = f (x) + g(y). ∂x ∂y Z Z r g(y) + C1 dy + C2 . Complete integral: w = –C1 x + f (x) dx + a

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 79

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

80 2.

Page 80

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES ∂w ∂x

+a



∂w ∂y

2

= f (x)y + g(x).

Complete integral: w = ϕ(x)y +

3.

Z



 g(x) – aϕ2 (x) dx + C1 ,

ϕ(x) =

Z

f (x) dx + C2 .

  ∂w ∂w 2 +a = f (x)y 2 + g(x)y + h(x). ∂x ∂y

Complete integral: w = ϕ(x)y 2 + ψ(x)y + χ(x), where the functions ϕ(x), ψ(x), and χ(x) are determined by solving the following system of ordinary differential equations: ϕ′x = –4aϕ2 + f (x), ψx′ = –4aϕψ + g(x),

(1) (2)

χ′x = –aψ 2 + h(x).

(3)

Riccati’s equation (1) can be integrated in quadrature for numerous f (x). For details, see Kamke (1976) and Polyanin p and Zaitsev (2003). For example, if f (x) = const, then the 1 particular solution ϕ = 2 f can be used. Given a solution of equation (1), equations (2) and (3) are easy to integrate, since they are linear in the unknowns ψ and χ. 4.

∂w ∂x

+a



∂w ∂y

2

= f (bx + cy).

Complete integral: w = C1 x + C2 –

5.

b 1 (bx + cy) ± 2 2ac 2ac2

    y ∂w ∂w 2 =f +a . ∂x ∂y x

1◦ . Complete integral:

Z q

4ac2 f (ξ) + b2 – 4ac2 C1 dξ,

ξ = bx + cy.

  y w = xϕ , C1 + C2 , x

where the function ϕ = ϕ(z, C1 ) is the general solution of the first-order ordinary differential equation ϕ – zϕ′z + a(ϕ′z )2 = f (z). 2◦ . Apply the Lagrange–Charpit method (see Subsection 24.1.3). The characteristic system has a first integral of the form x

∂w ∂w +y – w = C1 . ∂x ∂y

(1)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 80

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 81

81

2.2. EQUATIONS CONTAINING ARBITRARY FUNCTIONS ∂w ∂x

from the original equation, thus obtaining the equation     ∂w 2 ∂w y ax –y + w + C1 – xf = 0, ∂y ∂y x

We use (1) to eliminate

which can be treated as an ordinary differential equation for w with parameter x. Making the change of variables 1 1 2 xu2 (ξ, x) + xξ + xf (ξ) – C1 , 4a 4a we arrive at an Abel equation of the second kind for u: w(y, x) = –

ξ=

y , x

(2)

2a ′ f (ξ), (3) x ξ where x is treated as a parameter. The constant of integration in the general solution of equation (3) will depend on x, i.e., C2 = C2 (x). This dependence can be found by substituting the general solution of equation (3) into (1) and by taking into account relation (2). The books by Polyanin and Zaitsev (2003) and Zaitsev and Polyanin (1994) present cases where equation (3) is integrable in quadrature.     ∂w ∂w 2 y 6. +a = f (x) + g . ∂x ∂y x Z The substitution w(x, y) = u(x, y) + f (x) dx leads to an equation of the form 2.2.2.5: uu′ξ ± u =

 2   ∂u ∂u y +a =g . ∂x ∂y x

  ∂w 2 ∂w +a = f (x)w + g(x). ∂x ∂y Complete integral: Z   dx w = F (x)(C1 + C2 y) + F (x) g(x) – aC22 F 2 (x) , F (x)

7.

∂w



∂w

2

F (x) = exp

Z

 f (x) dx .

= bw 2 + f (x)w + g(x). ∂x ∂y For b = 0, see equation 2.2.2.7. Complete integral for b ≠ 0: p
w = ϕ(x) + C1 ψ(x) exp ±y b/a . 8.

+a

The function ϕ = ϕ(x) is determined by the first order ordinary differential equation ϕ′ = bϕ2 + f (x)ϕ + g(x),

and the function ψ(x) is related to ϕ(x) by Z    ψ(x) = exp 2bϕ(x) + f (x) dx .

(1)

(2)

Riccati’s equation (1) can be integrated in quadrature for various f and g, in particular, for g(x) ≡ 0 and arbitrary f (x) and for f (x) = const and g(x) = const. For details, see the books by Kamke (1976) and Polyanin and Zaitsev (2003).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 81

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

82

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES 

∂w

∂w

+ f (x) ∂x ∂y Complete integral:

9.

Page 82

2

= g(x)w + h(x)y + s(x).

  w = G(x) yϕ(x) + ψ(x) ,

where

ϕ(x) = C1 + ∂w

Z



h(x) dx, G(x)

∂w

ψ(x) = C2 +

G(x) = exp

Z

2



Z

 g(x) dx ,

 dx s(x) – f (x)G2 (x)ϕ2 (x) . G(x)

= bw 2 + g(x)w + h(x). ∂y Z dy leads to an equation of the form 2.2.2.8. The substitution ξ = p |f1 (y)| 10.

∂x

∂w

+ f (y)



+ f (x)y + g(x) ∂x Complete integral:

11.

w = ϕ(x)y – ∂w



Z

∂y

= 0.

2

g(x)ϕ (x) dx + C1 ,

+ f (x)y + g(x) ∂x Complete integral:

12.

   ∂w 2

   ∂w 2 ∂y

 –1 Z ϕ(x) = C2 + f (x) dx .

= h(x)w + r(x)y + s(x).

w = ϕ(x)y + ψ(x). The functions ϕ(x) and ψ(x) are determined by solving the ordinary differential equations ϕ′ + f (x)ϕ2 = h(x)ϕ + r(x), ψ ′ + g(x)ϕ2 = h(x)ψ + s(x),

(1) (2)

where the prime denotes the derivative with respect to x. Riccati’s equation (1) can be integrated in quadrature for various f (x), h(x), and g(x). For details, see the books by Kamke (1976) and Polyanin and Zaitsev (2003). In particular, for r(x) ≡ 0 the general solution of equation (1) has the form –1  Z ϕ(x) = H(x) C1 + f (x)H(x) dx ,

H(x) = exp

Z



h(x) dx .

Equation (2) is linear in ψ and is easy to integrate (for known ϕ). We have Z   dx ψ(x) = C2 H(x) + H(x) s(x) – g(x)ϕ2 (x) . H(x)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 82

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

83

2.2. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

13.

∂w ∂x



k

+ f (x)y + g(x)y

2k–2

   ∂w 2 ∂y

Page 83

= 0.

The substitution ξ = y 2–k leads to an equation of the form 2.2.2.11:     ∂w 2 ∂w 2 = 0. + (2 – k) f (x)ξ + g(x) ∂x ∂ξ 14.

   ∂w 2 ∂w  k 2k–2 + f (x)y + g(x)y = h(x)w + s(x). ∂x ∂y

1◦ . If k ≠ 2, then the substitution ξ = y 2–k leads to an equation of the form 2.2.2.12 with r(x) = 0:     ∂w 2 ∂w 2 + (2 – k) f (x)ξ + g(x) = h(x)w + s(x). ∂x ∂ξ 2◦ . If k = 2, then by introducing the new variable ξ = ln |y|, we arrive at an equation of the form 2.2.2.12.    ∂w 2 ∂w  λy = 0. 15. + f (x) + g(x)e ∂x ∂y The substitution ξ = e–λy leads to an equation of the form 2.2.2.11:     ∂w 2 ∂w – λ f (x)ξ + g(x) = 0. ∂x ∂ξ 16.

∂w ∂x



λy

+ f (x) + g(x)e

   ∂w 2 ∂y

= h(x)w + s(x).

The substitution ξ = e–λy leads to an equation of the form 2.2.2.12 with r(x) = 0:     ∂w 2 ∂w – λ f (x)ξ + g(x) = h(x)w + s(x). ∂x ∂ξ   ∂w ∂w 2 17. + f (ax + by) = g(ax + by). ∂x ∂y Complete integral: p Z –a ± 4b2 f (ξ)g(ξ) – 4b2 C1 f (ξ) + a2 w = C1 x + C2 + dξ, 2b2 f (ξ)

ξ = ax + by.

     ∂w y ∂w 2 y 18. +f =g . ∂x x ∂y x Complete integral:   y w = xϕ , C1 + C2 , x where the function ϕ = ϕ(z, C1 ) is the general solution of the first-order ordinary differential equation ϕ – zϕ′z + f (z)(ϕ′z )2 = g(z).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 83

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

84

Page 84

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES ∂w

   ∂w 2



+ f2 (x)w + f1 (x)y + f0 (x) = g2 (x)w + g1 (x)y + g0 (x). ∂x ∂y Complete integral: w = ϕ(x)y + ψ(x), where the functions ϕ(x) and ψ(x) are determined by the following system of ordinary differential equations:

19.

ϕ′x + f2 (x)ϕ3 + f1 (x)ϕ2 – g2 (x)ϕ – g1 (x) = 0,   ψx′ + f2 (x)ϕ2 – g2 (x) ψ + f0 (x)ϕ2 – g0 (x) = 0.

(1) (2)

Equation (1) can be integrated in quadrature for a lot of functions fn (x) and gn (x), in particular, for f2 (x) = g1 (x) = 0 and f1 (x) = g1 (x) = 0. For details, see Kamke (1976) and Polyanin and Zaitsev (2003). Given a solution of equation (1), equation (2) is easy to integrate, since it is linear in the unknown ψ.   ∂w ∂w 2 20. + f (x)w k = g(x)w. ∂x ∂y 1◦ . The substitution u =

1 k+1 k+1 w

leads to an equation of the form 2.2.2.9:  2 ∂u ∂u + f (x) = (k + 1)g(x)u. ∂x ∂y

2◦ . The transformation w(x, y) = G(x)v(z, y),

z=

Z

k+1

f (x)G

(x) dx,

G(x) = exp

leads to a simpler equation of the form 2.1.4.44 with a = –1:  2 ∂v k ∂v +v = 0. ∂z ∂y 21.



g(x) dx ,

 2 ∂w k ∂w + f (y)w = aw k+2 + g(x)w. ∂x ∂y

The substitution u =

∂w

1 k+1 k+1 w

leads to an equation of the form 2.2.2.10 with h(x) ≡ 0:  2 ∂u ∂u + f (y) = a(k + 1)2 u2 + (k + 1)g(x)u. ∂x ∂y

βw

– f (x)e ∂x The transformation 22.

Z



∂w ∂y

2

w(x, y) = u(z, y) + G(x),

– g(x) = 0.

z=

Z

  f (x) exp βG(x) dx,

G(x) =

Z

g(x) dx,

 2 ∂u βu ∂u leads to a simpler equation, –e = 0, which has the complete integral ∂z ∂y

βu = 2 ln C1 – 12 βy – ln C2 – βz .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 84

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

85

2.2. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

23.

∂w ∂x

– f (w)



∂w ∂y

2

= 0. Z

Complete integral in implicit form:

24.

∂w

– f (w)



∂w

2

Page 85

f (w) dw = C12 x + C1 y + C2 .

  ∂w – yg(x) + h(x) = 0. ∂y

∂x ∂y The transformation Z Z 2 t = ϕ (x) dx, z = ϕ(x)y + h(x)ϕ(x) dx,

ϕ(x) = exp

Z



g(x) dx ,

leads to a simpler equation of the form 2.2.2.23:   ∂w ∂w 2 – f (w) = 0. ∂t ∂z 25.

∂w ∂x

+ f (y)g(w)



∂w

Complete integral: x + C1

∂y Z

2

= h(w).

dy + p f (y)

Z

2C 2 g(w) dw = C2 . q 1 1 + 1 + 4C12 g(w)h(w)

  ∂w ∂w 2 = 0. 26. – f (x)g(y)h(w) ∂x ∂y Z Z Z dy Complete integral in implicit form: h(w) dw = C12 f (x) dx + C1 + C2 . √ g(y) ∂w

+ f (w + ax + by) ∂x Complete integral: 27.



∂w ∂y

2

= g(w + ax + by).

w = –ax – by + ϕ(ξ) + C3 ,

ξ = C1 x + C2 y,

where ϕ(ξ) =

Z 2bC2 f (ξ) – C1 ±

q

4C22 f (ξ)g(ξ) + 4C2 (aC2 – bC1 )f (ξ) + C12 2C22 f (ξ)

dξ.

One of the constants C1 or C2 can be set equal to unity. 28.

f1 (x)

∂w ∂x

+ f2 (y)

Complete integral: w =



Z

∂w ∂y

2

= g1 (x) + g2 (y).

g1 (x) – C1 dx + f1 (x)

Z s

g2 (y) + C1 dy + C2 . f2 (y)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 85

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

86

Page 86

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES f (x)

∂w

+ g(y) ∂x Complete integral: 29.



∂w ∂y

2

= aw + h1 (x) + h2 (y). w = ϕ(x) + ψ(y).

Here

  Z h1 (x) dx ϕ(x) = E(x) C1 + , f (x)E(x)

 Z E(x) = exp a

 dx . f (x)

The function ψ(y)Zis determined by the ordinary differential equation g(y)(ψy′ )2 = aψ+h2 (y). dy By the change z = p , this equation can be reduced to an equation which is discussed |g(y)| in detail in Zaitsev and Polyanin (1994) and Polyanin and Zaitsev (2003).

2.2.3. Equations of the Form

  ∂w + f (x, y, w) ∂w 2 + g(x, y, w) ∂w = h(x, y, w) ∂x ∂x ∂y

⊲ Equations of this type are encountered in mechanics, where x plays the role of time and y plays the role of the spatial coordinate. 1.

∂w

+a



∂w

∂x ∂y Complete integral:

2

+b

∂w ∂y

w = –C1 x + C2 + ∂w

∂w

2

3.

∂w

+a



∂w

∂x ∂y Complete integral:

2

+b

f (x) dx –

b 1 y± 2a 2a

Z q

+b

w = (C1 y+C2 )F (x)+F (x) ∂w

∂w

∂w

Z

2

∂y 

ϕ(x) =

Z

f (x) dx + C2 .

= f (x)w + g(x).

 dx , F (x) = exp F (x)

g(x)–aC12 F 2 (x)–bC1 F (x)

∂x

∂y

+ g(x)

Z

 f (x) dx .

∂w

= h(x). ∂y Z   Complete integral: w = C1 y + C2 + h(x) – C12 f (x) – C1 g(x) dx.

4.

+ f (x)



4ag(y) + b2 + 4aC1 dy.

∂w

= f (x)y + g(x). ∂x ∂y ∂y Complete integral: Z   w = ϕ(x)y + g(x) – aϕ2 (x) – bϕ(x) dx + C1 ,

2.

+a



Z

= f (x) + g(y).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 86

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

2.2. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

5.

∂w

+ f (x)



∂w

∂x ∂y Complete integral:

2

+ g(x)

∂w ∂y

Page 87

87

= h(x)w + p(x)y + s(x).

w = yϕ(x) + ψ(x), where

Z  p(x) dx, H(x) = exp h(x) dx , ϕ(x) = C1 H(x) + H(x) H(x) Z   dx ψ(x) = C2 H(x) + H(x) . s(x) – f (x)ϕ2 (x) – g(x)ϕ(x) H(x)    ∂w ∂w ∂w 2  6. +f (x) + g1 (x)y +g0(x) = s(x)w +h2 (x)y 2 +h1 (x)y +h0 (x). ∂x ∂y ∂y Complete integral: w = ϕ(x)y 2 + ψ(x)y + χ(x), where the functions ϕ(x), ψ(x), and χ(x) are determined by solving the following system of ordinary differential equations:   ϕ′x + 4f (x)ϕ2 + 2g1 (x) – s(x) ϕ – h2 (x) = 0, (1)   ′ ψx + 4f (x)ϕ + g1 (x) – s(x) ψ + 2g0 (x)ϕ – h1 (x) = 0, (2) Z

χ′x – s(x)χ + f (x)ψ 2 + g0 (x)ψ – h0 (x) = 0.

(3)

Riccati’s equation (1) can be integrated in quadrature for a lot of functions f (x), g1 (x), s(x), and h2 (x), in particular, for h2 (x) ≡ 0 (the other functions can be arbitrary). For details, see Kamke (1976) and Polyanin and Zaitsev (2003). Given a solution of equation (1), equations (2) and (3) are easy to integrate, since they are linear in the unknowns ψ and χ.   ∂w 2 ∂w ∂w 7. + f (y) + g(y) = h(y). ∂x ∂y ∂y p Z –g(y) ± g2 (y) + 4f (y)h(y) + 4C1 f (y) Complete integral: w = –C1 x + C2 + dy. 2f (y)   ∂w ∂w 2 ∂w 8. + f (y) + g(y) = h(x) + r(y). ∂x ∂y ∂y Complete integral: p Z Z –g(y) ± g2 (y) + 4f (y)r(y) + 4C1 f (y) w = –C1 x + C2 + h(x) dx + dy. 2f (y)   ∂w ∂w 2 ∂w + f (ax + by) + g(ax + by) = h(ax + by). 9. ∂x ∂y ∂y For b = 0, see equation 2.2.3.4. For b ≠ 0, by passing from x and y to the new variables x and ξ = ax + by, we arrive at an equation of the form 2.2.3.7:  2   ∂w ∂w ∂w 2 + b f (ξ) + a + bg(ξ) = h(ξ). ∂x ∂ξ ∂ξ

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 87

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

88 10.

Page 88

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES ∂w ∂x



+ f2 (x)w + f1 (x)y + f0 (x) 

   ∂w 2 ∂y

+ g2 (x)w + g1 (x)y + g0 (x)

Complete integral:

 ∂w ∂y

= h2 (x)w + h1 (x)y + h0 (x).

w = ϕ(x)y + ψ(x), where the functions ϕ(x) and ψ(x) are determined by solving the following system of ordinary differential equations:     ϕ′x + f2 (x)ϕ3 + f1 (x) + g2 (x) ϕ2 + g1 (x) – h2 (x) ϕ – h1 (x) = 0, (1)   ′ 2 2 (2) ψx + f2 (x)ϕ + g2 (x)ϕ – h2 (x) ψ + f0 (x)ϕ + g0 (x)ϕ – h0 (x) = 0.

Equation (1) can be integrated in quadrature for a lot of functions fn (x), gn (x), and hn (x), in particular, for f2 (x) = 0, h1 (x) = 0 and f1 (x) = –g2 (x), h1 (x) = 0. For details, see Kamke (1976) and Polyanin and Zaitsev (2003). Given a solution of equation (1), equation (2) is easy to integrate, since it is linear in the unknown ψ. 11.

∂w ∂x

+ f (w)



∂w ∂y

2

+ g(w)

∂w ∂y

= 0. Z

C22 f (w) dw = C3 . Complete integral in implicit form: C1 x + C2 y + C1 + C2 g(w) One of the constants C1 , C2 , or C3 can be set equal to ±1. 12.

  ∂w ∂w ∂w 2 + f (w) + g(w) = h(w). ∂x ∂y ∂y

Complete integral in implicit form: C1 x + C2 y +

Z

2C22 f (w) dw = C3 . q C1 + C2 g(w) ± [C1 + C2 g(w)]2 + 4C22 f (w)h(w)

One of the constants C1 , C2 , or C3 can be set equal to ±1. 13.

∂w

– f (w)



∂w

2

  ∂w – yg(x) + h(x) = 0. ∂y

∂x ∂y The transformation Z Z 2 t = ϕ (x) dx, z = ϕ(x)y + h(x)ϕ(x) dx,

ϕ(x) = exp

Z



g(x) dx ,

leads to a simpler equation of the form 2.2.2.23:  2 ∂w ∂w – f (w) = 0. ∂t ∂z

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 88

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

2.2. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

14.

∂w ∂x

+ f (w + ax + by)



∂w ∂y

2

+ g(w + ax + by)

∂w ∂y

Page 89

89

= h(w + ax + by).

The substitution u = w + ax + by leads to an equation of the form 2.2.3.12:  2   ∂u ∂u ∂u + g(u) – 2bf (u) + f (u) = h(u) – b2 f (u) + bg(u) + a. ∂x ∂y ∂y

2.2.4. Equations of the Form

 2  2 ∂w ∂w f (x, y, w) + g(x, y, w) = h(x, y, w) ∂x ∂y

⊲ Equations of this type are encountered in mechanics, geometric optics, and differential geometry. 1.



∂w ∂x

2

+



∂w ∂y

2

= f (x).

Complete integral: w = C1 y + C2 ± Reference: E. Kamke (1965).

2.



∂w ∂x

2

+



∂w ∂y

2

Complete integral: w = ±

Z q

f (x) – C12 dx.

= f (x) + g(y). Z p

f (x) + C1 dx ±

Z p

g2 (y) – C1 dy + C2 . The signs before

each of the integrals can be chosen independently of each other. 3.



∂w ∂x

2

+



∂w ∂y

2

= f (ax + by) + g(bx – ay).

The transformation ξ = ax + by, η = bx – ay leads to an equation of the form 2.2.4.2:  4.



∂w ∂x

2

+



∂w ∂y

∂w ∂ξ

2

2

= f (x2 + y 2 ).

+



∂w ∂η

2

=

a2

1 1 f (ξ) + 2 2 g(η). 2 +b a +b

Hamilton’s equation for the plane motion of a point mass under the action of a central force. Complete integral: x 1 w = C1 arctan + C2 ± y 2

Z q

zf (z) – C12

dz , z

z = x2 + y 2 .

Reference: E. Kamke (1965).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 89

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

90 5.

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES 

∂w

2

+



∂w

∂x ∂y Complete integral:

2

= (x2 + y 2 )f (xy).

2

2

w = C1 (x – y ) + C2 ±

6.



∂w

2



∂w

+ ∂x ∂y Complete integral:

2



∂w ∂x

2

+



∂w ∂y

2

1 2

∂w

2



∂w

f (z) – 4C12 dz,

z = xy.

Z q

f (z) – C12 dz,

z = x2 – y 2 .

= f (w).

Complete integral in implicit form: 

Z q

= (x2 + y 2 )f (x2 – y 2 ).

w = C1 xy + C2 ±

7.

Page 90

Z

2

dw =± p f (w)

q

(x + C1 )2 + (y + C2 )2 .

= f (x, y)g(w). ∂y  2  2 Z dw ∂u ∂u The substitution u = leads to a simpler equation: + = f (x, y). √ ∂x ∂y g(w) For solutions of this equation for some types of the right-hand side, see 2.2.4.1–2.2.4.6. 8.



∂x

∂w

+

2

1



∂w

2

+ 2 = f (x). ∂x x ∂y This equation governs the plane motion of a point mass in a central force field, with x and y being the polar coordinates. s Z C2 Complete integral: w = C1 y ± f (x) – 21 dx + C2 . x

9.

Reference: P. Appell (1953).

10.



∂w ∂x

2

+ f (x)



∂w ∂y

2

= g(x).

2

= g(y).

Z q Complete integral: w = C1 y + C2 + g(x) – C12 f (x) dx. 11.



∂w ∂x

2



∂w + f (y) ∂y

Complete integral: w = C1 x + C2 +

Z

s

g(y) – C12 dy. f (y)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 90

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

91

2.2. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

12.



∂w ∂x

2

+ f (w)



2

∂w ∂y

= g(w). s

Z

Complete integral in implicit form:

C12 + C22 f (w)

g(w) One of the constants C1 or C2 can be set equal to ±1.

13.



∂w ∂x

2



Page 91

∂w – f (x, y, w) ∂y

2

dw = C1 x + C2 y + C3 .

= 0.

The equation can be factorized. Equating each of the factors to zero, we obtain two simpler ∂w ∂w p equations: ± f (x, y, w) = 0. ∂x ∂y 14.

f1 (x)



2

∂w ∂x

+ f2 (y)



∂w ∂y

2

= g1 (x) + g2 (y).

A separable equation. This equation is encountered in differential geometry in studying geodesic lines of Liouville surfaces. Complete integral: w=±

Z s

g1 (x) + C1 dx ± f1 (x)

Z s

g2 (y) – C1 dy + C2 . f2 (y)

The signs before each of the integrals can be chosen independently of each other. References: P. Appell (1953), E. Kamke (1965).

15.

f (ax + by)



∂w ∂x

2

+ g(ax + by)



∂w ∂y

2

= h(ax + by).

Complete integral:

w = C1 x+C2 +

16.

Z –aC1 f (z) ±

f (w + ax + by)



∂w ∂x

q

a2 f (z)h(z) + b2 g(z)h(z) – b2 C12 f (z)g(z)

2



a2 f (z) + b2 g(z) + g(w + ax + by)

∂w ∂y

2

dz, z = ax+by.

= h(w + ax + by).

Complete integral: w = –ax – by + ϕ(ξ),

ξ = C1 x + C2 y + C3 ,

where the function ϕ = ϕ(ξ) is defined implicitly by  2  Z C1 f (ϕ) + C22 g(ϕ) dϕ ξ= . q 2 2 2 aC1 f (ϕ) + bC2 g(ϕ) ± C1 f (ϕ)h(ϕ) + C2 g(ϕ)h(ϕ) – (aC2 – bC1 ) f (ϕ)g(ϕ) One of the constants C1 or C2 can be set equal to ±1.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 91

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

92

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES

17.



f1 (x)g1 (w)

∂w ∂x

2

+ f2 (y)g2 (w)

Complete integral: s Z C12 g1 (w) + C22 g2 (w) h(w)



dw = C1

∂w ∂y

Z

2

p

Page 92

= h(w).

dx f1 (x)

+ C2

Z

One of the constants C1 or C2 can be set equal to ±1.

p

dy f2 (y)

+ C3 .

2.2.5. Equations of the Form

1.





∂w ∂x

∂w ∂x

2

2

+a

+ f (x, y, w) ∂w ∂w = g(x, y, w) ∂x ∂y ∂w ∂w ∂x ∂y

= f (x).

Complete integral: w = C1 (ax – 2y) ±

2.



∂w ∂x

2

+a

∂w ∂w ∂x ∂y



∂w ∂x

2

+a

2

+a

∂w ∂w ∂x ∂y

a2 C12 + f (x) dx + C2 .

= f (y).

Complete integral: w = C1 (y – ax) –

3.

Z q

1 a2 C1

Z

f (y) dy + C2 .

+ bw + f (y) = 0.

  Z by + C1 f (y) dy 2 Complete integral: w = + C2 . y – ax + a a2 (by + C1 )2 4.



∂w ∂x

∂w ∂w ∂x ∂y

+ f (y)w = 0.

y – ax + C1 Complete integral: w = a2 

∂w ∂x

2

+a



∂w ∂x

2

+a

Z

 f (y) dy + C2 .

∂w ∂w = f (w). ∂x ∂y Z 1 – C12 dw Complete integral in implicit form: = C1 x + y + C2 . p aC1 f (w)

5.

∂w ∂w = f (x)g(w). ∂x ∂y Z dw The substitution z = leads to an equation of the form 2.2.5.1. p |g(w)|

6.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 92

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

2.2. EQUATIONS CONTAINING ARBITRARY FUNCTIONS



∂w

2

+a

2

+ f (x)

2

+ f (y)

Page 93

93

∂w ∂w

= f (y)g(w). ∂x ∂y Z dw The substitution z = leads to an equation of the form 2.2.5.2. p |g(w)|

7.



∂x

∂w ∂w + g(x) = 0. ∂x ∂y Z h q i Complete integral: w = 2C1 y + C2 + –C1 f (x) ± C12 f 2 (x) – g(x) dx. 8.



∂w ∂x

∂w

∂w ∂w

+ g(x) = 0. ∂x ∂y Z q Z C1 1 dy 2 Complete integral: w = – x ± C1 – 4g(x) + C1 + C2 . 2 2 f (y) 9.

∂x



2

∂w ∂w + g(y) = 0. ∂x ∂y Z g(y) + C12 Complete integral: w = C1 x + C2 – dy. f (y)

10.

∂w ∂x

+ f (y)



 ∂w ∂w ∂w 2 + f (y) = g(y)w + xh(y) + r(y). 11. ∂x ∂x ∂y Complete integral: w = xϕ(y) + ψ(y), where the function ϕ(y) and ψ(y) are determined by the following system of ordinary differential equations: f (y)ϕϕ′y = g(y)ϕ + h(y), f (y)ϕψy′

2

= g(y)ψ – ϕ + r(y).

(1) (2)

Abel’s equation (1) can be considered independently. Given its solution, the solution of equation (2), which is linear in ψ, is easy to find. For the simplest cases where g and/or h are identically zero, equation (1) has the solutions Z g(y) dy + C1 for h(y) ≡ 0 and any g(y), ϕ(y) = f (y)  Z 1/2 h(y) ϕ(y) = ± 2 dy + C1 for g(y) ≡ 0 and any h(y). f (y) For other solutions of Abel’s equation (1) see Polyanin and Zaitsev (2003). 

∂w ∂x

2

∂w ∂w + f (y) + g(x)h(w) = 0. ∂x ∂y Z dw The substitution z = leads to an equation of the form 2.2.5.9. p |g(w)| 12.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 93

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

94 13.

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES 

∂w ∂x

2

+ f (y)

The substitution z =

14.

Page 94



∂w ∂x

2

Z

∂w ∂w ∂x ∂y p

dw |g(w)|

+ f (ax + by)

+ g(y)h(w) = 0. leads to an equation of the form 2.2.5.10.

∂w ∂w ∂x ∂y

+ g(ax + by)h(w) = 0.

Complete integral in implicit form: Z

dw = C1 y + √ h(w)

Z –aC1 f (z) ±

where z = ax + by.

15.



∂w ∂x

2

+ f (w + ax + by)

q   a2 C12 f 2 (z) – 4ag(z) a + bf (z)   dz + C2 , 2a a + bf (z)

∂w ∂w ∂x ∂y

+ g(w + ax + by) = 0.

Complete integral: w = –ax – by + ϕ(z),

z = C1 x + C2 y + C3 ,

where the function ϕ = ϕ(z) is defined implicitly by z=

Z

p

2C1 [C1 + C2 f (ϕ)] dϕ ∆2 (ϕ)

∆(ϕ) ± – 4C1 [C1 + C2 f (ϕ)][a2 + abf (ϕ) + g(ϕ)] ∆(ϕ) = 2aC1 + (aC2 + bC1 )f (ϕ).

,

One of the constants C1 or C2 can be set equal to ±1. 16.



∂w ∂x

2

+ f (x)g(y)

∂w ∂w ∂x ∂y

+ h(x)ϕ(w) = 0.

Complete integral in implicit form: Z 17.



∂w ∂x

dw = √ ϕ(w) 2

Z h

C1 f (x) ±

+ f (y)g(w)

q

∂w ∂w ∂x ∂y

Complete integral in implicit form:

Z i C12 f 2 (x) – h(x) dx – 2C1

dy + C2 . g(y)

– h(w) = 0.

Z s

1 + C1 g(w) dw = x + C1 h(w)

Z

dy + C2 . f (y)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 94

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 95

95

2.2. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

2.2.6. Other Equations 1.

∂w ∂w ∂x ∂y

+ f (x)

∂w ∂x

+ g(x) Z

∂w ∂y

+ h(x) = 0.

C1 g(x) + h(x) dx + C1 y + C2 . C1 + f (x)    ∂w ∂w 2. + ay + ax = f (x)g(y). ∂x ∂y Z Z 1 Complete integral: w = –axy + C1 f (x) dx + g(y) dy + C2 . C1    ∂w ∂w + f1 (x) + f2 (y) = g1 (x)g2 (y). 3. ∂x ∂y  Z  Z   1 g2 (y) – f2 (y) dy + C2 . Complete integral: w = C1 g1 (x) – f1 (x) dx + C1 Complete integral: w = –

4.



∂w ∂x

2

+ f (y)

∂w ∂w ∂x ∂y

+ g(x)

C1 1 Complete integral: w = – x– 2 2 5.



∂w ∂x

2

+ f (y)

∂w ∂w ∂x ∂y

Complete integral: w = C1 x ± 

∂w

2

+ f (x)

∂w ∂w

Z n

∂x

g(x)–

+ g(y) Z q

∂w

∂w ∂x

= h(x). q



g(x) + C1

2

o

+ 4h(x) dx+C1

Z

dy +C2 . f (y)

= h(x).

C12 + h(x) dx –

2C1 + g(y) dy + C2 . f (y)

∂w

= h(x). ∂x Z Z h q i Complete integral: w = 2C1 y – g(y) dy + –C1 f (x) ± C12 f 2 (x) + h(x) dx + C2 .

6.

∂x

∂x ∂y

+ f (x)g(y)

Z



 ∂w ∂w ∂w ∂w ∂w 2 7. + f (x) + g(x) + h(x) + s(x) = 0. ∂x ∂x ∂y ∂x ∂y Complete integral: Z q o  2 1 n w= –C1 f (x) – g(x) + C1 f (x) + g(x) – 4C1 h(x) – 4s(x) dx + C1 y + C2 . 2 

2

∂w ∂w ∂w ∂w + g(y) + h(y) + s(y) = 0. ∂x ∂y ∂x ∂y Z C12 + C1 g(y) + s(y) Complete integral: w = C1 x – dy + C2 . C1 f (y) + h(y) 8.

∂w ∂x

+ f (y)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 95

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

96 9.

Page 96

FIRST-ORDER EQUATIONS QUADRATIC IN DERIVATIVES 

∂w

2

+



∂w

∂x ∂y Complete integral: "

2

2

2



= f (x + y ) x

x 1 w = C1 exp C2 arcsin √ ± z 2

Z

∂w ∂x

s

+y

∂w ∂y

2

+ g(x2 + y 2 )w 2 .

# zg(z) – C22 dz , 1 – zf (z) z

z = x2 + y 2 .



   ∂w 2 ∂w 2 ∂w ∂w 10. f1 (x) + f2 (y) + g1 (x) + g2 (y) = h1 (x) + h2 (y). ∂x ∂y ∂x ∂y A separable equation. Complete integral: q Z –g1 (x) ± g2 (x) + 4f1 (x)h1 (x) + 4C1 f1 (x) 1 w= dx 2f1 (x) q Z –g2 (y) ± g2 (y) + 4f2 (y)h2 (y) – 4C1 f2 (y) 2 + dy + C2 . 2f2 (y) The signs before the roots can be chosen independently of each other.     ∂w 2 ∂w ∂w ∂w 2 11. f1 (x) + f2 (x) + f3 (x) = g(w). ∂x ∂x ∂y ∂y Complete integral in implicit form: q Z Z C1 f2 (x) ± C 2 f 2 (x) + 4f1 (x) – 4C 2 f1 (x)f3 (x) dw 1 2 1 = C1 y + C2 – dx. √ 2f1 (x) g(w)     ∂w 2 ∂w ∂w ∂w 2 12. f1 (x) + f2 (x) + f3 (x) ∂x ∂x ∂y ∂y ∂w ∂w + g1 (x) + g2 (x) + h(x) = 0. ∂x ∂y Complete integral: p Z g1 (x) + C1 f2 (x) ± [g1 (x) + C1 f2 (x)]2 – 4f1 (x)∆(x) w = C1 y + C2 – dx, 2f1 (x) ∆(x) = C12 f3 (x) + C1 g2 (x) + h(x).     ∂w 2 ∂w ∂w ∂w 2 13. f1 (ax + by) + f2 (ax + by) + f3 (ax + by) ∂x ∂x ∂y ∂y ∂w ∂w + g2 (ax + by) + h(ax + by) = 0. + g1 (ax + by) ∂x ∂y For b = 0, see equation 2.2.6.12. For b ≠ 0, the transformation w(x, y) = u(ξ, x), ξ = ax + by leads to an equation of the form 2.2.6.12:    2  2  ∂u 2   ∂u ∂u ∂u 2 a f1 (ξ) + abf2 (ξ) + b f3 (ξ) + 2af1 (ξ) + bf2 (ξ) + f1 (ξ) ∂ξ ∂ξ ∂x ∂x   ∂u ∂u + ag1 (ξ) + bg2 (ξ) + g1 (ξ) + h(ξ) = 0. ∂ξ ∂x

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 96

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

97

2.2. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

14.

2

2

f (x + y )



∂w ∂x

2

+



∂w ∂y

2 

=



x

∂w ∂x

+y

∂w ∂y

–w

Page 97

2

.

Complete integral: 

w = C1 exp C2 arcsin p

x x2

+ y2



Ψ(x2 + y 2 ),

where Ψ = Ψ(z) is a solution of the nonlinear ordinary differential equation z(2zΨ′z – Ψ)2 = f (z)[4z 2 (Ψ′z )2 + C22 Ψ2 ],

z = x2 + y 2 .

2

In particular, if f (z) = z, then Ψ(z) = z (1–C2 )/4 . 15.



∂w f1 (w) ∂x

2

  ∂w ∂w ∂w 2 + f2 (w) + f3 (w) ∂x ∂y ∂y ∂w ∂w + g1 (w) + g2 (w) + h(w) = 0. ∂x ∂y

Complete integral in implicit form: Z 2F (w) dw C1 x + C2 y + = C3 , p G(w) ± G2 (w) – 4F (w)h(w)

where F (w) = C12 f1 (w) + C1 C2 f2 (w) + C22 f3 (w) and G(w) = C1 g1 (w) + C2 g2 (w). 16.

f1 (w + ax + by)



∂w ∂x

2

+ f2 (w + ax + by)

∂w ∂w ∂x ∂y

+ f3 (w + ax + by)



∂w

2

∂y ∂w ∂w + g2 (w + ax + by) + h(w + ax + by) = 0. + g1 (w + ax + by) ∂x ∂y

The substitution u(x, y) = w(x, y) + ax + by leads to an equation of the form 2.2.6.15:  2  2   ∂u ∂u ∂u ∂u ∂u f1 (u) + f2 (u) + f3 (u) + g1 (u) – 2af1 (u) – bf2 (u) ∂x ∂x ∂y ∂y ∂x  ∂u  + g2 (u)–2bf3 (u)–af2 (u) +h(u)+a2 f1 (u)+abf2 (u)+b2 f3 (u)–ag1 (u)–bg2 (u) = 0. ∂y

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 97

This page intentionally left blank

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 99

Chapter 3

First-Order Nonlinear Equations with Two Independent Variables of General Form 3.1. Nonlinear Equations Containing Arbitrary Parameters 3.1.1. Equations Contain the Fourth Powers of Derivatives 

 ∂w 4 ∂w + bx + c. 1. =a ∂x ∂y Complete integrals: 4 (bx + aC1 + c)5/4 + C1 y + C2 , 5b   ∂w 4 ∂w =a + by + c. ∂x ∂y

w=

2.

Complete integral: w = C1 x – 3.



∂w ∂x

4

+a

∂w ∂y



∂w ∂x

4

+a

aC1 x – bxy – cy (aC1 – by)5 – + C2 . a 5a5 b

y(by + 2c – 2C14 ) + C2 . 2a

= bxk.

Complete integral: w =

4.

w=

Z

bxk + aC1


1/4

dy – C1 y + C2 .

∂w = by k. ∂y

Complete integral: w = aC1 x – a3 C14 y +

b y k+1 + C2 . a(k + 1)



 ∂w 4 ∂w 5. +a = bxy k + cy n. ∂x ∂y Complete integral: Z c 1 n+1 w = xϕ(y) + y – ϕ4 (y) dy + C1 , a(n + 1) a

ϕ(y) =

b y k+1 + C2 . a(k + 1)

99

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 99

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

100 6.

Page 100

FIRST-ORDER NONLINEAR EQUATIONS WITH TWO INDEPENDENT VARIABLES 

∂w ∂x

4

+a

∂w ∂y

= bxk + cy n.

Complete integral: w =

7.



8.



∂w

Z

4

– aw k

4

– axny mw k

bxk + aC1


1/4

dx – C1 y +

c y n+1 + C2 . a(n + 1)

∂w

= 0. ∂y  3 
3–k 3–k 3 4 aC1 x + a C1 y + C2 . Complete integral: w = 3 ∂x

∂w

∂w

= 0. ∂y   3  3–k 3 – k 4aC1 n+4 a3 C14 1–m x 4 + y Complete integral: w = + C2 . 3 n+4 1–m ∂x

  ∂w 2 ∂w + d = 0. +b +c 9. ∂x ∂y r q x Complete integral: w = ± –2b + 2 b2 – 4d – 4aC13 – 4cC1 + C1 y + C2 . 2   2 2   ∂w 3 2 ∂w 10. a – 2w = b . ∂x ∂y  2 4 1 x Complete integral: w = + C1 + (y + C2 )3 . 2 a 27b 

∂w ∂x

k

11.

ay

12.

∂w ∂x

4





∂w +a ∂y

∂w



3

4

=

3

= ay + bxk.

∂w

2

∂w

+b

. ∂y Z q  1  Complete integral: w = C1 x – b ± b2 + 4aC14 y k dy + C2 . 2 

∂x

∂w ∂y

∂y

1/4

Complete integral: w = (4ax + C1 )

13.

∂w



∂w

3

∂x ∂y Complete integral:

y+b

Z

xk (4ax + C1 )–3/4 dx + C2 .

= axky + bxn.

w = ϕ(x)y + b

Z

xn dx + C2 , [ϕ(x)]3

ϕ(x) =



4a k+1 x + C1 k+1

1/4

.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 100

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

3.1. NONLINEAR EQUATIONS CONTAINING ARBITRARY PARAMETERS

∂w

14.

∂x



∂w ∂y

3

15.

∂x



∂w ∂y

3

n+3 aC13 k+1 3 x + y 3 + C2 . k+1 C1 (n + 3)

= aw + bxk. 1/3

Complete integral: w = (3ax + C1 ) ∂w

16.

∂x



∂w ∂y

3

101

= axky n.

Complete integral: w = ∂w

Page 101

1/3

(y + C2 ) + b(3ax + C1 )

Z

xk (3ax + C1 )–4/3 dx.

= axnw + bxmy + cxl.

This is a special case of equation 3.2.1.22 with k = 3, f (x) = axn , g(x) = bxm , and h(x) = cxl . 17.

w

2



∂w ∂x

2 

∂w ∂y 2

2

Complete integral: w =

2 2

=a y

p



∂w ∂x

2

2 2

+b x



∂w ∂y

2

.

C12 + 1 (bx2 + aC1 y 2 ) + C2 . C1

3.1.2. Equations Contain Derivatives in Radicands 1.

∂w ∂x

+

s

  ∂w 2 a+b = 0. ∂y

This equation is encountered in optimal control and differential games; see Subbotin (1991, 1995). q

Complete integral: w = C1 y – a + bC12 x + C2 . See also Example 24.7 in Subsection 24.3.6, where the Cauchy problem for this equation with a = b = 1 is considered.

2.

∂w ∂x

+

s

  ∂w 2 a+b –x ∂y

s

c+k

∂w ∂y

2

= 0.

q Complete integral: w = C1 y + c+ – x a + bC12 + C2 . The Cauchy problem for this equation with b = k = 1, a > 0, and c > 0 was considered in the book by Melikyan (1998). 1 2 2x

3.

x

∂w ∂x

+y

∂w ∂y

–w+

s

a



q



∂w ∂x

Complete integral: w = C1 x + C2 y +

kC12

2

q

  ∂w 2 +b + c = 0. ∂y aC12 + bC22 + c.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 101

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

102

Page 102

FIRST-ORDER NONLINEAR EQUATIONS WITH TWO INDEPENDENT VARIABLES

3.1.3. Equations Contain Arbitrary Powers of Derivatives 1.



∂w ∂x

k

+a

∂w ∂y

= bxn.

Complete integral: w =

2.



∂w ∂x

k

+a

∂w ∂y

Z

(bxn + aC1 )1/k dy – C1 y + C2 .

= by n.

Complete integral: w = aC1 x – ak–1 C1k y + 3.



∂w ∂x

k

+a

∂w ∂y

= bxn + cy m.

Complete integral: w = 

b y n+1 + C2 . a(n + 1)

Z

(bxn + aC1 )1/k dx – C1 y +

c y m+1 + C2 . a(m + 1)

 ∂w k ∂w 4. +a = bxy n + cy m. ∂x ∂y Complete integral: Z c 1 w = xϕ(y) + y m+1 – ϕk (y) dy + C1 , a(m + 1) a 

∂w ∂x

k

– aw n

6.



∂w ∂x

k

– axny mw l

7.



k

+ ay n

ϕ(y) =

b y n+1 + C2 . a(n + 1)

∂w = 0. ∂y  1–k 
n–k+1 n–k+1 k–1 k Complete integral: w = aC1 x + a C1 y + C2 . 1–k 5.

∂w = 0. ∂y    1–k l–k+1 l – k + 1 akC1 n+k ak–1 C1k 1–m Complete integral: w = x k + y + C2 . 1–k n+k 1–m ∂w ∂x

∂w ∂w ∂x ∂y

= by mw + cxy l + dy s.

This is a special case of equation 3.2.1.17 with f (y) = ay n , g(y) = by m , h(y) = cy l , and r(y) = dy s . 

∂w ∂x

k



y

∂w =a ∂y

2

∂w . ∂y Z q b 1 y± Complete integral: w = C1 x + C2 – b2 + 4aC1k y n dy. 2a 2a

8.

n

+b

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 102

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

103

3.1. NONLINEAR EQUATIONS CONTAINING ARBITRARY PARAMETERS

9.

n

m

(axy + by )

Complete integral:



w = xϕ(y) + b

10.

m

ay w



n

∂w

∂x The transformation

∂w ∂x Z

k

k

–

∂w ∂y

= 0.

w(x, y) = G(y)u(x, z),

∂w ∂y

11.

∂x



∂w ∂y

k

z=a

Z

m

n+k–1

y G



∂w



1 1–k

.

(y) dy,



∂u ∂x

+b

Z

k



 b s+1 y G(y) = exp , s+1 – u–n

∂u = 0. ∂z

= ay + bxn. 

Complete integral: w = a(k + 1)x + C1 ∂w

a(1 – k) n+1 y ϕ(y) = + C2 n+1

+ by sw = 0.

leads to a simpler equation of the form 3.1.3.5: ∂w



y m ϕk (y) dy + C1 ,

–

Page 103

k



1 k+1 y

 – k xn a(k + 1)x + C1 k+1 dx + C2 .

= axny + bxm. ∂x ∂y This is a special case of equation 3.2.1.19 with f (x) = axn and g(x) = bxm . 12.

13.

∂w ∂x



∂w ∂y

k

= axny m.

Complete integral: w =

14.

∂w ∂x



∂w ∂y

k

m+k aC1k n+1 k x + y k + C2 . n+1 C1 (m + k)

= aw + bxn.

Complete integral: w = (akx +

15.

∂w ∂x



∂w ∂y



∂w ∂y

1 C1 ) k

(y + C2 ) + b(akx

k

= axnw + bxmy + cxl.

k

+a

1 + C1 ) k

Z

xn (akx + C1 )–

k+1 k dx.

This is a special case of equation 3.2.1.22 with f (x) = axn , g(x) = bxm , and h(x) = cxl . ∂w + bx + c = 0. ∂y bx + 2aC1 + 2c Complete integral: w = – x + C1 y + C2 . 2C1k 16.

∂w ∂x

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 103

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

104

Page 104

FIRST-ORDER NONLINEAR EQUATIONS WITH TWO INDEPENDENT VARIABLES ∂w



∂w

k

+a

k

=x

∂w

+ bx + c = 0. ∂x bx + 2c Complete integral: w = – x + C1 y + C2 . 2(a + C1k ) 17.

18.

∂x



∂y

∂w ∂w

∂x ∂y Complete integral:

x

∂w

∂w

–y ∂y ∂x Complete integral: 19.

∂x

–y

∂w ∂y

.

1 x 1 R – 1 = ln ± R ± ln + C2 , 2 y 2 R + 1

C1–k w 

∂w

k

=



∂w ∂x

y w = C1 arctan + C2 + x

2

Z q

+



∂w ∂y

C1k R2

–

where

2

C12

21.





4xyC11–2k + 1.

.

dR , R

R=

where

     ∂w 2 ∂w 2 ∂w ∂w k = – . +x ∂x ∂y ∂x ∂y Complete integral: 1 R – 1 1 x + y C1 w = ln ± R ± ln + C2 , where 2 x–y 2 R +1

20.

q

R=

q

x2 + y 2 .

y

∂w ∂x

k

+a



∂w ∂y

n

R=

q

(x + y)(x – y)C12–k + 1.

= b.

1/n

Complete integral: w = (b – aC1 )1/k x + C1 y + C2 . 22.



∂w ∂x

k

+a



∂w ∂y

k

Complete integral: w = 1

23.

ax



m

∂w ∂x

k

+ by

Complete integral: w =

24.

axy



m

∂w ∂x

k

l


1 + aC1k 1–k



Z 

+ by

= w.

l

∂w ∂y

n

∂w ∂x

 k k–1 k–1 (x + C1 y + C2 ) . k

= cxp + sy q .

cxp – C1 axm





n

1/k –

dx +

Z 

sy q + C1 by l

1/n

dy + C2 .

∂w = 0. ∂y

This is a special case of equation 3.2.2.5 with f (y) = ay m and g(y) = by l .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 104

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

3.1. NONLINEAR EQUATIONS CONTAINING ARBITRARY PARAMETERS

25.

ax

e



∂w ∂x

k

by

+e



∂w ∂y

k

Page 105

105

= c.

Complete integral:

26.

a



∂w ∂x

 k(c – C1k )1/k –by/k  kC   – 1 e–ax/k – e + C2 a b w=  k(c – C1k )1/k –by/k   C1 x – + C2 e b + cy

k

if a ≠ 0, if a = 0.

n  ∂w + cx +b = cxm + sy l. ∂y

This is a special case of equation 3.2.2.10 with f (x) = cxm and g(y) = sy l . 27.



∂w ∂x

k

∂w ∂y

n

Complete integral: w =

28.



∂w ∂x

k

∂w ∂y

= axmy l. m+k nC1k n+l ka1/k k + x y n + C2 . (m + k)C1n n+l

n

= axmy lw s.

This is a special case of equation 3.2.2.8 with f (x) = axm , g(y) = y l , and h(w) = ws . 29.



∂w ∂x

k

∂w ∂y

n

∂w

= axm

∂y Z

–n/k

Complete integral: w = C1 y + C1

30.



∂w + ay ∂x

k

∂w + ax ∂y

n

+ bxl. (aC1 xm + bxl )1/k dx + C2 .

= bxmy l.

This is a special case of equation 3.2.2.11 with f (x) = bxm and g(y) = y l .

3.1.4. More Complicated Equations 1.

∂w ∂x



– a exp λ

∂w ∂y



= 0.

Complete integral: w = aeλC1 x + C1 y + C2 . 2.

  ∂w – a exp λ = bxk. ∂x ∂y

∂w

Complete integral: w = C1 y + aeλC1 x +

b xk+1 + C2 . k+1

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 105

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

106 3.

Page 106

FIRST-ORDER NONLINEAR EQUATIONS WITH TWO INDEPENDENT VARIABLES   ∂w ∂w k – (ay + bx ) exp λ = 0. ∂x ∂y

This is a special case of equation 3.3.1.6 with f (u) = exp(λu) and g(x) = bxk . 4.

  ∂w ay – exp + bxy = 0. ∂x ∂y ∂w


2C1 x – bx2 + y ln C1 y – y + C2 . 2a   ∂w ∂w – a sinh λ = 0. ∂x ∂y

Complete integral: w = 5.

Complete integral: w = a sinh(λC1 )x + C1 y + C2 . 6.

  ∂w – ax sinh λ = bxn. ∂x ∂y

∂w

k

Complete integral: w = C1 y + 7.

a sinh(λC1 ) k+1 b x + xn+1 + C2 . k+1 n+1

  ∂w – aw sinh λ = 0. ∂x ∂y

∂w

This is a special case of equation 3.3.1.10 with f (u) = a sinh(λu) and β = 0. 8.

∂w ∂x



– a ln λ

∂w ∂y



= 0.

Complete integral: w = a ln(λC1 )x + C1 y + C2 . 9.

  ∂w – a ln λ = bxk. ∂x ∂y

∂w

Complete integral: w = C1 y + a ln(λC1 )x + 10.

∂w ∂x

k



– ax ln λ

∂w ∂y



Complete integral: w = C1 y +

b xk+1 + C2 . k+1

= bxn. a ln(λC1 ) k+1 b x + xn+1 + C2 . k+1 n+1

  ∂w λ ∂w 11. – aw ln = 0. ∂x w ∂y   Complete integral: w = C1 exp C2 y + a ln(λC2 )x . 12.

  ∂w ∂w – a cos λ = 0. ∂x ∂y

Complete integral: w = a cos(λC1 )x + C1 y + C2 .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 106

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

3.2. EQUATIONS CONTAINING ARBITRARY FUNCTIONS OF INDEPENDENT VARIABLES

13.

∂w ∂x



+ a cos λ

∂w ∂y



Page 107

107

= bxky + cxn.

This is a special case of equation 3.3.1.3 with f (u) = a cos(λu), g(x) = bxk , h(x) = cxn .   ∂w ∂w ∂w k 14. – ax sin λ – bxn = 0. ∂x ∂y ∂y a sin(λC1 ) k+1 bC1 n+1 Complete integral: w = x + x + C1 y + C2 . k+1 n+1   ∂w ∂w k n 15. – (ax y + bx ) sin λ = 0. ∂x ∂y

This is a special case of equation 3.3.1.7 with f (u) = sin(λu), g(x) = axk , and h(x) = bxn .   ∂w ∂w 16. – a sin λx = b. ∂x ∂y a cos(λC1 x) + bx – C1 y + C2 . Complete integral: w = λC1   ∂w λ ∂w 17. – aw sin = 0. ∂x w ∂y   Complete integral: w = C1 exp C2 y + a sin(λC2 )x . ⊲ For solutions of other nonlinear equations, see also Sections 3.2 and 3.3, where equations of a general form with arbitrary functions are considered.

3.2. Equations Containing Arbitrary Functions of Independent Variables 3.2.1. Equations Contain One Arbitrary Power of Derivative 1.

∂w ∂x

+a



∂w ∂y

k

= f (x).

Complete integral: w =

2.

∂w ∂x

+a



∂w ∂y

k

–aC1k x

+

Z

f (x) dx + C1 y + C2 .

= f (y).

Complete integral: w = –C1 x +

Z 

f (y) + C1 a

1/k

dy + C2 .

  ∂w ∂w k 3. +a = yf (x) + g(x). ∂x ∂y Complete integral: Z   w = yϕ(x) + g(x) – aϕk (x) dx + C1 ,

ϕ(x) =

Z

f (x) dx + C2 .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 107

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

108 4.

FIRST-ORDER NONLINEAR EQUATIONS WITH TWO INDEPENDENT VARIABLES ∂w ∂x

+a



∂w ∂y

k

= f (x) + g(y).

Complete integral: w = –C1 x + ∂w



∂w

+a ∂x ∂y Complete integral: 5.

k

∂w



∂w

+ f (x) ∂x ∂y Complete integral:

where

Z

f (x) dx +

Z 

g(y) + C1 a

1/k

dy + C2 .

= f (x)w + g(x).

w = C1 F (x) + C2 yF (x) + F (x)

6.

Page 108

k

Z



 dx , F (x)

g(x) – aC2k F k (x)

F (x) = exp

Z

 f (x) dx .

= g(x)w + h(x)y + s(x).

  w = G(x) yϕ(x) + ψ(x) ,

G(x) = exp

Z



g(x) dx ,

Z

Z   dx h(x) ϕ(x) = C1 + dx, ψ(x) = C2 + s(x) – f (x)Gk (x)ϕk (x) . G(x) G(x)   ∂w ∂w k ∂w 7. + f (x) + g(x) = 0. ∂x ∂y ∂y Z  k  Complete integral: w = C1 y + C2 – C1 f (x) + C1 g(x) dx.    ∂w k ∂w  8. – yf (x) + g(x) = 0. ∂x ∂y Complete integral: w = yϕ(x) +

Z

k

ϕ (x)g(x) dx + C1 ,

  1 Z 1–k ϕ(x) = (1 – k) f (x) dx + C2 .

k  ∂w n ∂w – f (x)w – g(x)w = 0. 9. ∂x ∂y The transformation Z w(x, y) = G(x)u(z, y), z = f (x)Gn+k–1 (x) dx,

G(x) = exp

Z



g(x) dx ,

 k ∂u n ∂u leads to a simpler equation, –u = 0. For n + k ≠ 1, the last equation has the ∂z ∂y complete integral
λ k–1 u = C1 y + C1k λk–1 z + C2 , λ= . n+k–1

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 108

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

3.2. EQUATIONS CONTAINING ARBITRARY FUNCTIONS OF INDEPENDENT VARIABLES

∂w

βw

– f (x)e ∂x The transformation 10.



∂w ∂y

k

w(x, y) = u(z, y) + G(x),

Page 109

109

– g(x) = 0.

z=

Z



 f (x) exp βG(x) dx,

G(x) =

Z

g(x) dx,

 k ∂u ∂u – eβu = 0, which has the complete integral ∂z ∂y   k β 1 u = ln C1 – y – ln(C2 – βz). β k β

leads to a simpler equation,

11.

  ∂w k ∂w – f (w) = 0. ∂x ∂y

Complete integral in implicit form:

Z



 1 f (w) k–1 dw = C1k x + C1 y + C2 .

   ∂w ∂w ∂w k  – f (w) – yg(x) + h(x) = 0. ∂x ∂y ∂y The transformation Z  Z Z k ϕ(x) = exp g(x) dx , t = ϕ (x) dx, z = ϕ(x)y + h(x)ϕ(x) dx, 12.

  ∂w ∂w k leads to a simpler equation of the form 3.2.1.11: – f (w) = 0. ∂t ∂z

  ∂w ∂w ∂w k 13. – f (w) + g(w) = 0. ∂x ∂y ∂y  1 Z  k–1 f (w) Complete integral in implicit form: dw = C1 x + y + C2 . C1 + g(w)   ∂w ∂w k – f (x)g(y)h(w) = 0. ∂x ∂y Complete integral in implicit form: Z Z Z   1 – 1 k k–1 dw = C1 f (x) dx + C1 [g(y) k dy + C2 . h(w) 14.

  ∂w ∂w k 15. f (x) + g(y) + h(x) = 0. ∂x ∂y Z Z C1k + h(x) dy Complete integral: w = C1 dx + C2 .  1/k – f (x) g(y)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 109

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

110 16.

FIRST-ORDER NONLINEAR EQUATIONS WITH TWO INDEPENDENT VARIABLES 

∂w ∂x

2

∂w

+ f (x)

∂x

= g(x)

1 Complete integral: w = C1 y – 2 17.

Page 110



∂w ∂x

k

+ f (y)

∂w ∂w ∂x ∂y



∂w ∂y

k

+ h(x).

 Z  q k 2 f (x) ± f (x) + 4C1 g(x) + 4h(x) dx + C2 . = g(y)w + xh(y) + r(y).

Complete integral: w = xϕ(y) + ψ(y), where the function ϕ(y) and ψ(y) are determined by solving the following system of ordinary differential equations: f (y)ϕϕ′y = g(y)ϕ + h(y), f (y)ϕψy′

k

= g(y)ψ – ϕ + r(y).

(1) (2)

Abel’s equation (1) can be considered independently. Given its solution, the solution of equation (2), which is linear in ψ, is easy to find. For the simplest cases where g and/or h are identically zero, equation (1) has the solutions Z g(y) ϕ(y) = dy + C1 for h(y) ≡ 0 and any g(y), f (y)  Z 1/2 h(y) for g(y) ≡ 0 and any h(y). ϕ(y) = ± 2 dy + C1 f (y) 18.

∂w ∂x



∂w ∂y

k

= ay + f (x). 

Complete integral: w = a(k + 1)x + C1

19.

∂w ∂x



∂w ∂y

k



1 k+1 y

+

Z

– k  f (x) a(k + 1)x + C1 k+1 dx + C2 .

= f (x)y + g(x).

Complete integral:

w = ϕ(x)y +

20.

∂w ∂x



∂w ∂y

k

Z

g(x) dx + C1 , [ϕ(x)]k

  1 Z k+1 ϕ(x) = (k + 1) f (x) dx + C2 .

= f (x)g(y).

Complete integral: w =

C1k

Z

1 f (x) dx + C1

Z



1/k g(y) dy + C2 .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 110

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

3.2. EQUATIONS CONTAINING ARBITRARY FUNCTIONS OF INDEPENDENT VARIABLES

21.

∂w ∂x



∂w ∂y

k

Page 111

111

= aw + f (x).

Complete integral: w = (akx +

1 C1 ) k

(y + C2 ) + (akx

1 + C1 ) k

Z

f (x)(akx + C1 )–

k+1 k dx.

  ∂w ∂w k 22. = f (x)w + g(x)y + h(x). ∂x ∂y Complete integral: w = yϕ(x) + ψ(x), where the function ϕ(x) and ψ(x) are determined by solving the following system of ordinary differential equations: ϕk ϕ′x = f (x)ϕ + g(x), k

ϕ

ψx′

(1)

= f (x)ψ + h(x).

(2)

Equation (1) can be considered independently. Given its solution, the solution of equation (2), which is linear in ψ, is easy to find.

3.2.2. Equations Contain Two or Three Arbitrary Powers of Derivatives 1.

f (x)



∂w ∂x

k

+ g(y)

Complete integral: w =

2.

f (x)



∂w

k

∂x Complete integral:



Z 

+ g(y)

∂w ∂y

n

= h1 (x) + h2 (y).

h1 (x) + C1 f (x)



∂w ∂y

n

1/k

dx +

Z 

h2 (y) – C1 g(y)

1/n

dy + C2 .

= aw + h1 (x) + h2 (y).

w = ϕ(x) + ψ(y), where the functions ϕ(x) and ψ(y) are determined from the ordinary differential equations f (x)(ϕ′x )k – aϕ – h1 (x) = C1 , g(y)(ψy′ )n – aψ – h2 (y) = –C1 . Z Z –1/k –1/n By the substitutions z = f (x) dx and t = g(y) dy, respectively, these equations can be reduced to equations some solvable cases of which can be found in Zaitsev and Polyanin (1994) and Polyanin and Zaitsev (1995).

3.



∂w ∂x

k

– f (w)



∂w ∂y

n

n ≠ k.

= 0,

Complete integral in implicit form:

Z



 1 f (w) n–k dw = C1n x + C1k y + C2 .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 111

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

112 4.

Page 112

FIRST-ORDER NONLINEAR EQUATIONS WITH TWO INDEPENDENT VARIABLES 

∂w ∂x

k

– f (x)g(y)h(w)



∂w ∂y

n

n ≠ k.

= 0,

Complete integral in implicit form: Z 5.

xf (y)





 1 h(w) n–k dw = C1n

∂w ∂x

k

+ g(y)



∂w ∂x

Z



n

1 f (x) k dx + C1k ∂w

–

Complete integral: w = xϕ(y) +

∂y Z

Z



= 0.

g(y)ϕn (y) dy + C1 .

  1 Z 1–k ϕ(y) = (1 – k) f (y) dy + C2 Z  f (y) dy ϕ(y) = C2 exp 6.

f (x)w



∂w ∂x

n

k

+ g(y)w

k



∂w ∂y

– 1 g(y) n dy + C2 .

n

if

k ≠ 1,

if

k = 1.

= w n+k ln w.

Complete integral:

w = exp

7.

∂w ∂x





∂w ∂y

k–1 k k

Z



– 1 f (x) k dx + C1



+ f1 (x)y +f2(x)



k k–1

n–1 + n

   ∂w n ∂y



+f3 (x)

Z



– 1 g(y) n dy + C2

∂w ∂x



n  n–1

.

= g1 (x)w +g2 (x)y +g3 (x).

A complete integral of this equation can be sought in the form w = yϕ(x) + ψ(x). 8.



∂w ∂x

k

∂w ∂y

n

= f (x)g(y)h(w).

Complete integral in implicit form: Z 

k

– 1 h(w) k+n dw = C1n

∂w ∂y

n

Z



1 1 f (x) k dx + k C1

Z



1 g(y) n dy + C2 .

∂w + g(x). ∂y Z  1/k –n/k Complete integral: w = C1 y + C1 C1 f (x) + g(x) dx + C2 . 9.

∂w ∂x



= f (x)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 112

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

3.3. EQUATIONS CONTAINING ARBITRARY FUNCTIONS OF DERIVATIVES



Page 113

113

 n ∂w 10. a + cy + b + cx = f (x) + g(y). ∂x ∂y   Z  Z  f (x) + C1 1/k g(y) – C1 1/n Complete integral: w = –cxy + dx + dy + C2 . a b 11.



k

∂w

∂w ∂x

+ ay

k

∂w ∂y

+ ax

n

Complete integral: w = –axy + C1n 

= f (x)g(y).

Z

1/k 1 dx + k f (x) C1

Z



1/n dy + C2 . g(y)

n ∂w 12. + f2 (y) = g1 (x)g2 (y). ∂y  Z  Z   n 1/k 1 1/n g (y) – f2 (y) dy + C2 . Complete integral: w = C1 g1 (x) – f1 (x) dx + C1k 2 13.

∂w + f1 (x) ∂x

∂w ∂x

 k



s

+ f1 (x) g1 (x) +



∂w ∂y

k

s

+ f2 (x) g2 (x) +



∂w ∂y

n

= 0.

Such equations are encountered in optimal control and differential games (k = n = 2); see Melikyan (1998). Z h q i q Complete integral: w = C1 y + C2 – f1 (x) g1 (x) + C1k + f2 (x) g2 (x) + C1n dx. 14.



∂w ∂x

k

+ f (y)



∂w ∂x

k–n

Complete integral in implicit form:

15.



∂w ∂x

k

∂w ∂y

n

= f (x)



Z

∂y

n

= g(w).

  Z 1 – C1k 1/n dw dy = C1 x+ +C2 . k–n 1/k [g(w)] [f (y)]1/n C1

∂w ∂y Z

–n/k

Complete integral: w = C1 y + C1

∂w

m 

+ g(x).

1/k C1m f (x) + g(x) dx + C2 .

3.3. Equations Containing Arbitrary Functions of Derivatives 3.3.1. Equations Contain Arbitrary Functions of One Variable 1.

∂w ∂x

+f



∂w ∂y



= 0.

This equation is encountered in optimal control and differential games; see Subbotin (1991, 1995).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 113

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

114

Page 114

FIRST-ORDER NONLINEAR EQUATIONS WITH TWO INDEPENDENT VARIABLES

1◦ . Complete integral: w = C1 y – f (C1 )x + C2 . 2◦ . On differentiating the equation with respect to y, we arrive at the quasilinear equation ∂u ∂u + f ′ (u) = 0, ∂x ∂y

u=

∂w , ∂y

which is discussed in detail in Section 23.3 (see equation (23.3.1.1)). 3◦ . The solution of the Cauchy problem with the initial condition w(0, y) = ϕ(y) can be written in parametric form as   y = f ′ (ζ)x + ξ, w = ζf ′ (ζ) – f (ζ) x + ϕ(ξ), where ζ = ϕ′ (ξ). See also Examples 24.7 and 24.8 in Subsection 24.3.6. 2.

∂w ∂x

+f



∂w ∂y



= g(x).

Complete integral: w = C1 y – f (C1 )x +

3.

∂w +f ∂x



∂w ∂y



Z

g(x) dx + C2 .

= g(x)y + h(x).

Complete integral: w = ϕ(x)y +

4.

∂w ∂x

+f



∂w ∂y



Z




 h(x) – f ϕ(x) dx + C1 ,

Z

ϕ(x) =

g(x) dx + C2 .

= g(x)w + h(x).

Complete integral: w = (C1 y + C2 )ϕ(x) + ϕ(x)

5.

∂w ∂x

+f



∂w ∂y



Z



Z  ϕ(x) = exp g(x) dx .

 dx h(x) – f (C1 ϕ(x)) , ϕ(x)

= g(x)w + h(x)y + s(x).

This is a special case of equation 3.3.2.4 with F (x, u) = f (u) – s(x). 6.

∂w ∂x



 – ay + g(x) f



∂w ∂y



= 0.

Complete integral:

w = ϕ(x)y +

Z


f ϕ(x) g(x) dx + C1 ,

where the function ϕ(x) is defined implicitly by the relation

Z

dϕ = ax + C2 . f (ϕ)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 114

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

3.3. EQUATIONS CONTAINING ARBITRARY FUNCTIONS OF DERIVATIVES

7.

∂w



 – g(x)y + h(x) f

∂x Complete integral:



∂w ∂y



Page 115

115

= 0.

Z


f ϕ(x) h(x) dx + C1 , Z Z dϕ = g(x) dx + C2 . where the function ϕ(x) is defined implicitly by the relation f (ϕ)    ∂w  ∂w 8. + g1 (x)y + g0 (x) f + h2 (x)w + h1 (x)y + h0 (x) = 0. ∂x ∂y 1◦ . Complete integral: w = ϕ(x)y + ψ(x), w = ϕ(x)y +

where the functions ϕ = ϕ(x) and ψ = ψ(x) are determined by solving the ordinary differential equations ϕ′x + g1 (x)f (ϕ) + h2 (x)ϕ + h1 (x) = 0, ψx′ + g0 (x)f (ϕ) + h2 (x)ψ + h0 (x) = 0.

(1) (2)

2◦ . For g1 (x) ≡ 0, the general solutions of equations (1) and (2) are  Z  Z h1 (x) ϕ(x) = C1 H(x) – H(x) dx, H(x) = exp – h2 (x) dx , H(x) Z h0 (x) + g0 (x)f (ϕ(x)) ψ(x) = C2 H(x) – H(x) dx. H(x)   ∂w 1 ∂w – wf = 0. 9. ∂x w ∂y   Complete integral: w = C1 exp C2 y + f (C2 )x .   β ∂w 10. – wf w = 0. ∂x ∂y For β = –1, see equation 3.3.1.9. Complete integral for β ≠ –1: ∂w

  1 w = (1 + β)y + C1 1+β ϕ(x + C2 ),

where the function ϕ(x) is defined implicitly by the relation x = ∂w



βw ∂w

–f e ∂x Complete integral:

11.

∂y



Z

dϕ
. ϕf ϕβ+1

= 0.

w=

1 ln(βy + C1 ) + ϕ(x + C2 ), β

where the function ϕ(x) is defined implicitly by the relation x =

Z

dϕ
. f eβϕ

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 115

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 116

116

FIRST-ORDER NONLINEAR EQUATIONS WITH TWO INDEPENDENT VARIABLES   ∂w 1 ∂w β 12. – g(x)w f = h(x)w. ∂x w ∂y The transformation Z  Z β–1 w(x, y) = H(x)u(z, y), z = g(x)H (x) dx, H(x) = exp h(x) dx ,

  1 ∂u ∂u ∂u β –u f , may be leads to a simpler equation, = 0, which, by solving for ∂z u ∂y ∂y rewritten in the form of equation 3.3.1.10.   ∂w ∂w βw 13. – g(x)e f = h(x). ∂x ∂y The transformation Z Z   w(x, y) = u(z, y) + H(x), z = g(x) exp βH(x) dx, H(x) = h(x) dx,

  ∂u ∂u ∂u βu leads to a simpler equation, –e f = 0, which, by solving for , may be ∂z ∂y ∂y rewritten in the form of equation 3.3.1.11. k    ∂w ∂w 14. x +y = f . ∂x ∂y ∂x ∂y ∂x k  1 k – 1 Complete integral: w = f (C1 ) 1–k (x + C1 y) k–1 + C2 . k  2  2   ∂w ∂w ∂w ∂w 15. + =f x –y . ∂x ∂y ∂y ∂x Z q p y dR f (C1 )R2 – C12 Complete integral: w = C1 arctan + C2 + , where R = x2 + y 2 . x R     ∂w ∂w 16. Φ + f1 (x) Ψ + f2 (y) = g1 (x)g2 (y). ∂x ∂y Complete integral: Z Z 
 
 w= ϕ C1 g1 (x) – f1 (x) dx + ψ C1–1 g2 (y) – f2 (y) dy + C2 , ∂w

∂w



∂w

where ϕ is the inverse of Φ and ψ is the inverse of Ψ.

3.3.2. Equations Contain Arbitrary Functions of Two Variables   ∂w ∂w – F x, = 0. ∂x ∂y Z Complete integral: w = F (x, C1 ) dx + C1 y + C2 .

1.

Reference: E. Kamke (1965).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 116

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

117

3.3. EQUATIONS CONTAINING ARBITRARY FUNCTIONS OF DERIVATIVES

2.

∂w ∂x



+ F x,

∂w ∂y



= aw.

Complete integral: w = eax (C1 y + C2 ) – eax

3.

∂w ∂x



+ F x,

∂w ∂y

Page 117



Z

e–ax F (x, C1 eax ) dx.

= g(x)w.

Complete integral: w = ϕ(x)(C1 y + C2 ) – ϕ(x)

4.

Z


dx , F x, C1 ϕ(x) ϕ(x)

ϕ(x) = exp

Z



g(x) dx .

  ∂w ∂w + F x, = g(x)w + h(x)y. ∂x ∂y

Complete integral: w = yϕ(x) + ψ(x), where ϕ(x) = C1 G(x) + G(x) ψ(x) = C2 G(x) – G(x)

5.

∂w –F ∂x



y ∂w , x ∂y



Z

Z

h(x) dx, G(x)

Z  G(x) = exp g(x) dx ,


dx . F x, ϕ(x) G(x)

= 0.

This is a special case of equation 3.3.4.6. 6.

∂w ∂x



– wF x,

1 ∂w w ∂y



= 0.

  Z Complete integral: w = C1 exp C2 y + F (x, C2 ) dx . 7.

∂w ∂x



– g(x)F w,

∂w ∂y



– h(x)

∂w ∂y

= 0.

Z Z By passing to the new variables t = g(x) dx, z = y + h(x) dx, we obtain a simpler   ∂w ∂w equation, – F w, = 0, which is a special case of equation 3.3.3.4. ∂t ∂z 8.



∂w

2

+ g(x)

∂w



= F x,

∂w



. ∂y Z q i 1 h g(x) ± g2 (x) + 4F (x, C1 ) dx + C1 y + C2 . Complete integral: w = – 2 ∂x

∂x

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 117

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 118

118

FIRST-ORDER NONLINEAR EQUATIONS WITH TWO INDEPENDENT VARIABLES   ∂w ∂w 9. F , = 0. ∂x ∂y Complete integral: w = C1 x + C2 y + C3 , where C1 and C3 are arbitrary constants and the constant C2 is related to C1 by F (C1 , C2 ) = 0. Reference: E. Kamke (1965).

 ∂w ∂w 10. w = x +y +F , . ∂x ∂y ∂x ∂y Clairaut’s equation. Complete integral: w = C1 x + C2 y + F (C1 , C2 ). ∂w



∂w

References: R. Courant (1962), E. Kamke (1965).







∂w



= F2 y, . ∂x ∂y A separable equation. Complete integral: 11.

F1 x,

∂w

w = ϕ(x) + ψ(y) + C1 , where the functions ϕ = ϕ(x) and ψ = ψ(y) are determined from the ordinary differential equations

F1 x, ϕ′x = C2 , F2 y, ψy′ = C2 . Reference: E. Kamke (1965).

    ∂w ∂w F1 x, + F2 y, + aw = 0. ∂x ∂y A separable equation. Complete integral: 12.

w = ϕ(x) + ψ(y), where the functions ϕ = ϕ(x) and ψ = ψ(y) are determined from the ordinary differential equations
F1 x, ϕ′x + aϕ = C1 ,
F2 y, ψy′ + aψ = –C1 , where C1 is an arbitrary constant. If a ≠ 0, one can set C1 = 0 in these equations. Reference: E. Kamke (1965).



1 ∂w 13. F1 x, w ∂x Complete integral:



k

+ w F2



1 ∂w y, w ∂y



= 0.

w(x, y) = ϕ(x)ψ(y). The functions ϕ = ϕ(x) and ψ = ψ(y) are determined by solving the ordinary differential equations     ψy′ ϕ′x –k k ϕ F1 x, = C, ψ F2 y, = –C, ϕ ψ where C is an arbitrary constant.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 118

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

3.3. EQUATIONS CONTAINING ARBITRARY FUNCTIONS OF DERIVATIVES

14.



F1 x,

∂w ∂x



λw

+e



F2 y,

∂w ∂y



Page 119

119

= 0.

Complete integral: w(x, y) = ϕ(x) + ψ(y). The functions ϕ = ϕ(x) and ψ = ψ(y) are determined by solving the ordinary differential equations

e–λϕ F1 x, ϕ′x = C, eλψ F2 y, ψy′ = –C, where C is an arbitrary constant. 15.

    1 ∂w 1 ∂w F1 x, + F2 y, = k ln w. w ∂x w ∂y

Complete integral: w(x, y) = ϕ(x)ψ(y). The functions ϕ = ϕ(x) and ψ = ψ(y) are determined by solving the ordinary differential equations     ψy′ ϕ′x – k ln ϕ = C, F2 y, – k ln ψ = –C, F1 x, ϕ ψ where C is an arbitrary constant. 16.

    ∂w ∂w ∂w + yF1 x, + F2 x, = 0. ∂x ∂y ∂y

Complete integral: w = ϕ(x)y –

Z


F2 x, ϕ(x) dx + C1 ,

where the function ϕ(x) is determined by solving the ordinary differential equation ϕ′x + F1 (x, ϕ) = 0. 17.

      ∂w ∂w ∂w ∂w + wF1 x, + yF2 x, + F3 x, = 0. ∂x ∂y ∂y ∂y

Complete integral: w = ϕ(x)y + ψ(x), where the functions ϕ = ϕ(x) and ψ = ψ(x) are determined by solving the following system of ordinary differential equations: ϕ′x + F1 (x, ϕ)ϕ + F2 (x, ϕ) = 0, ψx′ + F1 (x, ϕ)ψ + F3 (x, ϕ) = 0.

(1) (2)

Equation (1) can be considered independently. Given its solution, the integration of equation (2), which is linear in ψ, is easy.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 119

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

120 18.

Page 120

FIRST-ORDER NONLINEAR EQUATIONS WITH TWO INDEPENDENT VARIABLES         ∂w ∂w ∂w ∂w ∂w F1 y, + wF2 y, + xF3 y, + F4 y, = 0. ∂x ∂y ∂x ∂x ∂x

Complete integral: w = xϕ(y) + ψ(y), where the functions ϕ(y) and ψ(y) are determined by solving the following system of ordinary differential equations: F1 (y, ϕ)ϕ′y + F2 (y, ϕ)ϕ + F3 (y, ϕ) = 0,

(1)

F1 (y, ϕ)ψy′

(2)

+ F2 (y, ϕ)ψ + F4 (y, ϕ) = 0.

Equation (1) can be considered independently. Given its solution, the solution of equation (2), which is linear in ψ, is easy to find. 19.

F



∂w

+ ay,

∂x

∂w

+ ax

∂y



= 0.

Complete integral: w = –axy + C1 x + C2 y + C3 , where F (C1 , C2 ) = 0. 20.

F



∂w ∂w + akxk–1 y n, + anxky n–1 ∂x ∂y



= 0.

Complete integral: w = –axk y n + C1 x + C2 y + C3 , where F (C1 , C2 ) = 0. 21.



∂w

2



∂w

+ ∂x ∂y Complete integral:

  ∂w ∂w 2 2 =F x +y , y –x . ∂x ∂y

2

w = –C1 arctan

y 1 + x 2

Z q

ξF (ξ, C1 ) – C12

dξ + C2 , ξ

ξ = x2 + y 2 .

Reference: E. Kamke (1965).

3.3.3. Equations Contain Arbitrary Functions of Three Variables ∂w



∂w

+ F x, y, ∂x ∂y The substitution 1.



= g(x)w.

w(x, y) = G(x)u(x, y),

G(x) = exp

Z

 g(x) dx ,

leads to a similar equation which does not depend on u explicitly:  

∂u ∂u 1 + F x, y, = 0, where F x, y, p = F x, y, G(x)p . ∂x ∂y G(x)

It follows that the original equation has a complete integral of the following structure: w = C1 G(x) + ϕ(x, y, C2 ).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 120

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

3.3. EQUATIONS CONTAINING ARBITRARY FUNCTIONS OF DERIVATIVES

2.



∂w ∂w F x, , ∂x ∂y



Page 121

121

= 0.

Complete integral: w = C1 y +ϕ(x, C1 )+C2 , where the
function ϕ = ϕ(x, C1 ) is determined from the ordinary differential equation F x, ϕ′x , C1 = 0. 3.



∂w ∂w F ax + by, , ∂x ∂y



= 0.

For b = 0, see equation 3.3.3.2. Complete integral for b ≠ 0: w = C1 x + ϕ(z, C1 ) + C2 ,

z = ax + by,

where the function ϕ
= ϕ(z) is determined from the nonlinear ordinary differential equation F z, aϕ′z + C1 , bϕ′z = 0. 4.



∂w ∂w , F w, ∂x ∂y



= 0.

We seek a particular solution of the equation in the form w = w(z),

z = C1 x + C2 y,

where C1 and C2 are arbitrary constants. We arrive at the following ordinary differential equation for w = w(z):
F w, C1 wz′ , C2 wz′ = 0.

Solving this equation for the derivative yields wz′ = f (w). By integrating the last equation, we obtain a complete integral of the original partial differential equation in implicit form: Z dw = C1 x + C2 y + C3 . f (w)

This equation defines cylindrical surfaces whose elements are parallel to the xy-plane. Reference: E. Kamke (1965).

5.



F ax + by + cw,

∂w ∂x

,

∂w ∂y



= 0.

For c = 0, see equation 3.3.3.3. If c ≠ 0, then the substitution cu = ax + by + cw leads to an  ∂u a ∂u b  equation of the form 3.3.3.4: F cu, – , – = 0. ∂x c ∂y c 6.



F x,

∂w ∂x

+ ay,

∂w ∂y

+ ax



= 0.

Complete integral: w = –axy + C1 y + ϕ(x) + C2 , where the function
ϕ(x) is determined by solving the first-order ordinary differential equation F x, ϕ′x , C1 = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 121

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 122

122

FIRST-ORDER NONLINEAR EQUATIONS WITH TWO INDEPENDENT VARIABLES   ∂w ∂w 7. F w, f (x) , g(y) = 0. ∂x ∂y Z Z dx dy The transformation ξ = ,η= leads to the equation of the form 3.3.3.4: f (x) g(y)
F w, wξ , wη = 0. Reference: E. Kamke (1965).



 ∂w ∂w ∂w ∂w 8. F , ,x +y = 0. ∂x ∂y ∂x ∂y This is a special case of equation 3.3.4.3.   ∂w ∂w ∂w ∂w 9. F , , w–x –y = 0. ∂x ∂y ∂x ∂y Clairaut’s equation (in implicit form). Complete integral: w = C1 x + C2 y + C3 , where the constants C1 , C2 , and C3 are related by F (C1 , C2 , C3 ) = 0. References: R. Courant (1962), E. Kamke (1965).



∂w



, + f (y)w + xg(y) + h(y) = 0. ∂x ∂y Complete integral:  Z  Z g(y) dy, F (y) = exp – f (y) dy . w = xϕ(y) + ψ(y), where ϕ(y) = C1 F (y) – F (y) F (y) 10.

F y,

∂w

The function ψ(y) is determined
by solving the nonlinear ordinary differential equation F y, ϕ(y), ψy′ + f (y)ψ + h(y) = 0.       ∂w ∂w ∂w ∂w ∂w ∂w ∂w 11. xF y, ,x –w + G y, ,x –w = H y, ,x –w . ∂x ∂x ∂y ∂x ∂x ∂x ∂x The Euler transformation x = WX , y = Y , w = XWX – W , where W = W (X, Y ), brings the nonlinear equation in question to the quasilinear equation ∂W ∂W F (Y , X, W ) – G(Y , X, W ) = H(Y , X, W ). ∂X ∂Y Equations of this type are considered in Chapter 23. Reference: E. Kamke (1965).

      ∂w ∂w ∂w ∂w ∂w ∂w 12. xF u, , + yG u, , = H u, , , ∂x ∂y ∂x ∂y ∂x ∂y ∂w ∂w u=x +y – w. ∂x ∂y The Legendre transformation x = WX , y = WY , w = XWX + Y WY – W , where W = W (X, Y ), brings the nonlinear equation in question to the quasilinear equation ∂W ∂W + G(W , X, Y ) = H(W , X, Y ). ∂X ∂Y Equations of this type are considered in Chapter 23. F (W , X, Y )

Reference: E. Kamke (1965).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 122

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

3.3. EQUATIONS CONTAINING ARBITRARY FUNCTIONS OF DERIVATIVES

Page 123

123

3.3.4. Equations Contain Arbitrary Functions of Four Variables 1.



∂w ∂w ∂w F x, , ,w – y ∂x ∂y ∂y



= 0.

Complete integral: w = C1 y + ϕ(x), where the function ϕ(x) is determined from the nonlinear ordinary differential equation
F x, ϕ′x , C1 , ϕ = 0. Reference: E. Kamke (1965).

2.



∂w ∂w ∂w F y, , ,w – x ∂x ∂y ∂x



= 0.

Complete integral: w = C1 x + ϕ(y), where the function ϕ(y) is determined from the nonlinear ordinary differential equation
F y, C1 , ϕ′y , ϕ = 0. Reference: E. Kamke (1965).

3.

  ∂w ∂w ∂w ∂w F w, , ,x +y = 0. ∂x ∂y ∂x ∂y

Complete integral: w = ϕ(ξ),

ξ = C1 x + C2 y,

where the function ϕ(ξ) is
determined by solving the nonlinear ordinary differential equation F ϕ, C1 ϕ′ξ , C2 ϕ′ξ , ξϕ′ξ = 0. 4.



∂w ∂w ∂w ∂w , , w–x –y F ax + by, ∂x ∂y ∂x ∂y



= 0.

Complete integral: w = C1 x + C2 y + ϕ(ξ),

ξ = ax + by,

where the function ϕ(ξ) is determined
by solving the nonlinear ordinary differential equation F ξ, aϕ′ξ + C1 , bϕ′ξ + C2 , ϕ – ξϕ′ξ = 0. 5.



  ∂w F x, , G y, = 0. ∂x ∂y ∂w

Complete integral: w = ϕ(x, C1 ) + ψ(y, C1 ) + C2 , where the functions ϕ and ψ are determined by the ordinary differential equations F (x, ϕ′x , C1 ) = 0,

G(y, ψy′ ) = C1 .

On solving these equations for the derivatives, we obtain linear separable equations, which are easy to integrate.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 123

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

124 6.

Page 124

FIRST-ORDER NONLINEAR EQUATIONS WITH TWO INDEPENDENT VARIABLES   y ∂w ∂w ∂w ∂w F , , ,x +y – w = 0. x ∂x ∂y ∂x ∂y

1◦ . Complete integral:

  y w = xϕ , C1 , C2 + C2 , x

where the function ϕ = ϕ(z, C1 , C2 ) is the general solution of the first-order ordinary differential equation
F z, ϕ – zϕ′z , ϕ′z , C2 = 0. 2◦ . A complete integral, w = w(x, y, C1 , C2 ), may be obtained by solving the equations   ∂w ∂w y ∂w ∂w +y – w = C1 , , , , C1 = 0, x F ∂x ∂y x ∂x ∂y

∂w for the derivatives ∂w ∂x and ∂y followed by determining w with the aid of the Lagrange– Charpit method (see Subsection 24.1.3).

Reference: E. Kamke (1965).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 124

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 125

Chapter 4

First-Order Nonlinear Equations with Three or More Independent Variables 4.1. Nonlinear Equations with Three Variables Quadratic in Derivatives 4.1.1. Equations Contain Squares of One or Two Derivatives 1.

∂w ∂x

+a

∂w ∂y

+b

∂w ∂z

+c



∂w ∂x

2

= k.

Complete integral: w = C1 x + C2 y + C3 z + C4 , where C1 + aC2 + bC3 + cC12 = k. 2.

  ∂w ∂w ∂w ∂w 2 + ax + bx +c = k. ∂x ∂y ∂z ∂y

Complete integral: w = C1 y + C2 z – 12 (aC1 + bC2 )x2 + (k – cC12 )x + C3 . 3.

∂w ∂x

+ ay

∂w ∂y

+ by

∂w ∂z

+c



∂w ∂y

2

= k.

Complete integral: w = aC1 z + (C2 e–ax – bC1 )y + (k – b2 cC12 )x –

4.

cC22 –2ax 2bc C1 C2 e–ax + e + C3 . a 2a

  ∂w ∂w ∂w ∂w 2 + ay + bz +c = k. ∂x ∂y ∂z ∂x

Complete integral: w = C1 x + C2 ln |y| + C3 |z| + C4 , where C1 + aC2 + bC3 + cC12 = k. 5.

∂w ∂x

+ az

∂w ∂y

+ by

∂w ∂z

+c



∂w ∂y

2

= kx + s.

Complete integral:

λ w = C1 eλx + C2 e–λx y – C1 eλx – C2 e–λx z b cC12 2λx cC22 –2λx 2 1 + 2 kx + (s – 2cC1 C2 )x – e + e + C3 , 2λ 2λ

λ=

√

ab.

125

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 125

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

126

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES ∂w

∂w

∂w

+ axz + bxy +c ∂x ∂y ∂z Complete integral: 6.

Page 126



∂w ∂y

2

= kx + s.

  2λ   w = y C1 exp(λx2 ) + C2 exp(–λx2 ) – z C1 exp(λx2 ) – C2 exp(–λx2 ) b Z  2 + 12 kx2 + sx – c C1 exp(λx2 ) + C2 exp(–λx2 ) dx + C3 , ∂w

1



∂w

2

1



∂w

λ=

1 2

√

ab.

2

+ + az = 0. + ∂x 2 ∂y 2 ∂z A problem of the motion of a material point under gravity is reduced to this equation; x is time, y and z are the longitudinal and transverse spatial coordinates, and a is the acceleration due to gravity. 1 Complete integral: w = –C1 x + C2 y + C3 ± (2C1 – C22 – 2az)3/2 . 3a 7.

Reference: G. K. Suslov (1946).

8.

∂w ∂x

+

1 2



∂w ∂y

2

+

1 2



∂w ∂z

2

a = p . y2 + z2

A problem on the motion of two point bodies attracting to each other according to Newton’s law of universal gravitation is reduced to this equation. By passing to the polar coordinates y = r cos θ and z = r sin θ, we arrive at the separable equation     1 ∂w 2 a ∂w 1 ∂w 2 + 2 = . + ∂x 2 ∂r ∂θ r 2r Complete integral: ±

w = –C1 x – C2 θ

Z 

2a C22 – 2 2C1 + r r

1/2

dr + C3 .

Reference: R. Courant (1962).

  ∂w 2 9. +a +b = cw + kxn. ∂x ∂y ∂z This is a special case of equation 4.3.1.2 with f (x) = c and g(x) = kxn . Complete integral: Z 1 cx 2 2 2cx cx w = (C1 + C2 y + C3 z)e – (aC2 + bC3 )e + ke xn e–cx dx. c ∂w

10.

∂w ∂x



+a

∂w



2

∂w ∂y

2

  ∂w 2 +b = cw + keβx. ∂z

This is a special case of equation 4.3.1.2 with f (x) = c and g(x) = keβx . Complete integral: w = (C1 + C2 y + C3 z)ecx –

1 k βx (aC22 + bC32 )e2cx + e . c β–c

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 126

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.1. NONLINEAR EQUATIONS WITH THREE VARIABLES QUADRATIC IN DERIVATIVES

11.

∂w

+ a1



∂w

2

+ a2



∂w

2

Page 127

127

= bw 2 + cxnw + kxm.

∂x ∂y ∂z This is a special case of equation 4.3.1.3 with f (x) = cxn and g(x) = kxm .     ∂w ∂w 2 ∂w 2 ∂w ∂w 12. + a1 + a2 + b1 + b2 = 0. ∂x ∂y ∂z ∂y ∂z

Complete integral: w = –(a1 C12 + a2 C22 + b1 C1 + b2 C2 )x + C1 y + C2 z + C3 . 13.

∂w

+ a1



∂w

∂x ∂y Complete integral:

2

+ a2



∂w ∂z

2

+ b1 z

w = yϕ1 (x) + zϕ2 (x) – where ϕ1 (x) = C1 eλx + C2 e–λx ,

Z



∂w ∂y

+ b2 y

∂w ∂z

= 0.

 a1 ϕ21 (x) + a2 ϕ22 (x) dx + C3 ,

ϕ2 (x) = –


λ C1 eλx – C2 e–λx , b2

λ=

p

b1 b2 .

   
∂w ∂w ∂w 2 ∂w 2 14. + a2 + b11 y + b12 z + a1 ∂x ∂y ∂z ∂y
∂w + b21 y + b22 z = sw + c1 y + c2 z + c0 . ∂z This is a special case of equation 4.3.1.17.     ∂w ∂w ∂w 2 ∂w 2 ∂w + a2 + b1 xkz 15. + a1 + b2 xky = cxn. ∂x ∂y ∂z ∂y ∂z Complete integral: Z   c w = yϕ1 (x) + zϕ2 (x) + xn+1 – a1 ϕ21 (x) + a2 ϕ22 (x) dx + C3 , n+1 where √ b1 b2 , ϕ1 (x) = C1 exp(λxk+1 ) + C2 exp(–λxk+1 ), λ = k+1  λ(k + 1)  ϕ2 (x) = – C1 exp(λxk+1 ) – C2 exp(–λxk+1 ) . b2     ∂w ∂w 2 ∂w 2 ∂w ∂w + a1 + a2 + b1 eβxz + b2 eβx y = 0. 16. ∂x ∂y ∂z ∂y ∂z Complete integral: Z   w = yϕ1 (x) + zϕ2 (x) – a1 ϕ21 (x) + a2 ϕ22 (x) dx + C3 , where



 ϕ1 (x) = b2 C1 exp λeβx + C2 exp –λeβx , λ = 

 ϕ2 (x) = βλ –C1 exp λeβx + C2 exp –λeβx .

√

b1 b2 , β

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 127

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

128 17.

Page 128

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES ∂w ∂x

+ ax

k



∂w ∂y

2

+ bx

n



∂w ∂z

2

= cxmw.

Complete integral: w = ϕ(x)(C1 +C2 y+C3 z)–ϕ(x)

18.

Z

aC22 xk +bC32 xn

ϕ(x)




dx,

  w = ebx yϕ1 (x) + zϕ2 (x) + ψ(x) ,

where

Z

m –bx

ϕ1 (x) = C1 + c1 x e dx, ϕ2 (x) = C2 + c2 Z   ψ(x) = C3 – ebx a1 xk ϕ21 (x) + a2 xn ϕ22 (x) dx. ∂w ∂x

+ ax

k



∂w ∂y

+ cz

2

+ bx

n



∂w ∂z

Complete integral: w = –cyz + C1 y + C2 z –

20.

 c m+1 x ϕ(x) = exp . m+1

 2  2 ∂w k ∂w n ∂w + a2 x = bw + c1 xmy + c2 xsz. + a1 x ∂x ∂y ∂z

Complete integral:

19.



∂w ∂x

+ a1 y

k



∂w ∂y

2

+ a2 z

n



∂w ∂z

2

+ cy

2

Z

xs e–bx dx,

= sxm.

aC12 k+1 bC22 n+1 s x – x + xm+1 + C3 . k+1 n+1 m+1 = bxm + c1 y p + c2 z q .

Complete integral: b w = C1 – (C2 + C3 )x + xm+1 + m+1 21.

∂w ∂x

+ a1 y k



∂w ∂y

2

+ a2 z n



∂w ∂z

Z r

2

C2 + c1 y p dy + a1 y k

Z r

C3 + c2 z q dz. a2 z n

= bw 2 + cxmw + sxq .

This is a special case of equation 4.3.1.10 with f1 (y) = a1 y k , f2 (z) = a2 z n , g(x) = cxm , and h(x) = sxq . 22.

   
∂w 2
∂w 2 ∂w k1 n1 k2 n2 + a1 x y + b1 x + a2 x z + b2 x = c1 xmw + c2 xs. ∂x ∂y ∂z

This is a special case of equation 4.3.1.12. 23.

 2  2 ∂w k ∂w n ∂w +A(ax+by+cz) +B(ax+by+cz) = D(ax+by+cz)m. ∂x ∂y ∂z

This is a special case of equation 4.3.1.14 with f (u) = Auk , g(u) = Bun , and h(u) = Dum .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 128

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

129

4.1. NONLINEAR EQUATIONS WITH THREE VARIABLES QUADRATIC IN DERIVATIVES

24.

∂w ∂x

λx

+ ae



∂w ∂y

2

βx

+ be



∂w ∂z

2

= cw.

Complete integral: w = C1 ecx + ecx (C2 y + C3 z) –

25.

Page 129

aC22 (2c+λ)x bC32 (2c+β)x e e – . c+λ c+β

  2 2 ∂w λx ∂w βx ∂w + be = ceγxw. + ae ∂x ∂y ∂z

Complete integral: w = ϕ(x)(C1 + C2 y + C3 z) – ϕ(x)

26.

∂w ∂x

λx

+ a1 e



∂w ∂y

2

Complete integral:

where

Z

ϕ(x)

βx

+ a2 e



aC22 eλx + bC32 eβx ∂w ∂z

2




dx,



 c γx e ϕ(x) = exp . γ

= bw + c1 eγxy + c2 eµxz.

  w = ebx yϕ1 (x) + zϕ2 (x) + ψ(x) ,

c1 (γ–b)x c2 (µ–b)x e , ϕ2 (x) = C2 + e , γ –b µ–b Z   ψ(x) = C3 – e2bx a1 eλx ϕ21 (x) + a2 eβx ϕ22 (x) dx.

ϕ1 (x) = C1 +

27.

 2  2 ∂w ∂w ∂w + aeλx + cz + beβx + cy = keγx + s. ∂x ∂y ∂z

Complete integral: w = –cyz + C1 y + C2 z –

28.

a ∂w + ∂x sin2 z



∂w ∂y

2

+a



∂w ∂z

2

a 2 λx b 2 βx k γx C e – C2 e + e + sx + C3 . λ 1 β γ

+ b cos z = 0.

A problem of the motion of a rod leaning on a horizontal plane and a vertical axis is reduced to this equation (x is time, and y and z are angular coordinates). 1/2 Z  C22 b Complete integral: w = –aC1 x + C2 y ± C1 – – cos z dz + C3 . sin2 z a Reference: A. P. Markeev (1990).

29.

 2  2 ∂w k m ∂w n s ∂w + ay w + bz w = 0. ∂x ∂y ∂z

aC12 m+1 bC22 s+1 2C1 2–k 2C2 2–n Complete integral: x + w + w = y 2 + z 2 + C3 . m+1 s+1 2–k 2–n

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 129

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

130

Page 130

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES ∂w

30.

k

+ ay w



m

∂x Complete integral: Z x+

∂w ∂y

2

n

+ bz w

s



∂w ∂z

2

= cw q.

2ϕ(w) dw 2C2 2–n 2C1 2–k = y 2 + z 2 + C3 , p 2–n 1 + 1 + 4cwq ϕ(w) 2 – k

where ϕ(w) = aC12 wm + bC22 ws . 31.

x

k ∂w

+a

∂x



∂w ∂y

2

  ∂w 2 +b = cxn. ∂z

aC12 + bC22 1–k c x + xn–k+1 + C3 . 1–k n–k+1     ∂w ∂w 2 ∂w 2 w +a +b = cw 2 . ∂x ∂y ∂z

Complete integral: w = C1 y + C2 z –

32.

Complete integral: ln |w| = (c – aC12 – bC22 )x + C1 y + C2 z + C3 .

4.1.2. Equations Contain Squares of Three Derivatives 1.

a



∂w ∂x

2

    ∂w 2 ∂w 2 +b +c = 1. ∂y ∂z

Complete integral: w = C1 x + C2 y + C3 z + C4 , where aC12 + bC22 + cC32 = 1. Reference: E. Kamke (1965).



∂w

2



∂w

2



∂w

2

+ + = axk + by n + cz m. ∂x ∂y ∂z This is a special case of equation 4.3.1.23. Complete integral: Z p Z p Z p w=± axk + C1 dx + ± by n + C2 dy + ± cz m – C1 – C2 dz + C3 . 2.

    ∂w 2 ∂w 2 +b +c = w k. ∂x ∂y ∂z Complete integral:

3.

a



∂w

2

2–k 2 w 2 = C1 x + C2 y + C3 z + C4 , 2–k

4.

a1 x

k1



∂w ∂x

2

+ a2 y

k2



∂w ∂y

2

+ a3 z

k3

where 

∂w ∂z

2

aC12 + bC22 + cC32 = 1.

= b1 xn1 + b2 y n2 + b3 z n3 .

This is a special case of equation 4.3.1.23 with f1 (x) = a1 xk1 , f2 (y) = a2 y k2 , f3 (z) = a3 z k3 , g1 (x) = b1 xn1 , g2 (y) = b2 y n2 , and g3 (z) = b3 z n3 .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 130

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.1. NONLINEAR EQUATIONS WITH THREE VARIABLES QUADRATIC IN DERIVATIVES

5.

a1 x

k1



∂w ∂x

2

+ a2 y

k2



∂w ∂y

2

+ b1 x

n1

+ b2 y

n2

 
∂w 2 ∂z

Page 131

131

= c1 xm1 + c2 y m2 .

xk1 ,

f2 (y) = a2 y k2 , g1 (x) = b1 xn1 , This is a special case of equation 4.3.1.24 with f1 (x) = a1 n m m 2 1 2 g2 (y) = b2 y , h1 (x) = c1 x , and h2 (y) = c2 y .    2 2 2
k1 ∂w k2 ∂w k3 ∂w 6. a1 x + a2 y + a3 z = b1 xn1 + b2 y n2 + b3 z n3 w m. ∂x ∂y ∂z This is a special case of equation 4.3.1.26 with f1 (x) = a1 xk1 , f2 (y) = a2 y k2 , f3 (z) = a3 z k3 , g1 (x) = b1 xn1 , g2 (y) = b2 y n2 , g3 (z) = b3 z n3 , and h(w) = wm .  2  2  2 k ∂w n ∂w m ∂w +B(ax+by+c) +C(ax+by+c) +s = 0. 7. A(ax+by+c) ∂x ∂y ∂y This is a special case of equation 4.3.1.27.  2  2  2 k1 n1 ∂w k2 n2 ∂w k3 n3 ∂w 8. a1 x w + a2 y w + a3 z w = bw m. ∂x ∂y ∂z This is a special case of equation 4.3.1.28.

4.1.3. Equations Contain Products of Derivatives with Respect to Different Variables 1.

∂w ∂x

+ axk



∂w ∂y

+ bz



∂w ∂z

+ by



Complete integral: w = –byz + C1 y + C2 z +

= cxn. c aC1 C2 k+1 xn+1 – x + C3 . n+1 k+1

   ∂w ∂w λx ∂w 2. + ae + bz + by = ceβx + k. ∂x ∂y ∂z c a Complete integral: w = –byz + C1 y + C2 z + eβx + kx – C1 C2 eλx + C3 . β λ 3.

∂w ∂w

+a

∂w ∂w

∂x ∂y ∂x ∂z Complete integral:

+ by

∂w ∂y

+ cz

∂w ∂z

= kxn + s.

Z

kxn + s dx + C3 , ϕ(x) + aψ(x) where the functions ϕ = ϕ(x) and ψ = ψ(x) are determined by the following autonomous system of ordinary differential equations: w = yϕ(x) + zψ(x) +

(ϕ + aψ)ϕ′ + bϕ = 0,

(ϕ + aψ)ψ ′ + cψ = 0.

If bc ≠ 0, the general solution of this system is given by ϕ + abC1 ϕc/b + bx = C2 ,

ψ = cC1 ϕc/b .

In two degenerate cases, we have ϕ = C1 , ψ = C1 ,

C1 ln |ψ| + aψ + cx = C2 aC1 ln |ϕ| + ϕ + bx = C2

if b = 0; if c = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 131

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

132

Page 132

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES

∂w ∂w ∂w ∂w ∂w ∂w +a + bz + cy = kxn + s. ∂x ∂y ∂x ∂z ∂y ∂z Complete integral: Z kxn + s dx + C3 , w = yϕ(x) + zψ(x) + ϕ(x) + aψ(x) where the functions ϕ = ϕ(x) and ψ = ψ(x) are determined by the following autonomous system of ordinary differential equations: 4.

(ϕ + aψ)ϕ′ + cψ = 0,

(ϕ + aψ)ψ ′ + bϕ = 0.

If bc ≠ 0, the general solution of this system is given by bcϕ2 + cC1 = (C2 + bcx + aϕ)2 ,

cψ 2 = bϕ2 + C1 .

In the degenerate cases, we have ψ = C1 , ϕ2 + 2aC1 ϕ + 2cC1 x = C2 ϕ = C1 , ∂w ∂w

2

aψ + 2C1 ψ + 2bC1 x = C2

if b = 0; if c = 0.

∂w ∂w

+ x + cy 2 – s = 0. ∂x ∂y ∂x ∂z The Legendre transformation (both direct and inverse transformations are given) ∂W ∂W ∂W ∂W ∂W ∂W x= , y= , z= , w=X +Y +Z – W; ∂X ∂Y ∂Z ∂X ∂Y ∂Z ∂w ∂w ∂w ∂w ∂w ∂w X= , Y = , Z= , W =x +y +z – w, ∂x ∂y ∂z ∂x ∂y ∂z brings this equation to an equation of the form 4.1.1.6 (with k = 0):   ∂W ∂W ∂W ∂W 2 + bXZ + aXY +c = s. ∂X ∂Y ∂Z ∂Y 5.

az

∂w ∂w

+ by

∂w ∂w

z

+ ay

= bxnw k

∂w

+ cxmw k+1 . ∂x ∂y ∂x ∂z ∂x There is a solution in the form of the product of a function of one variable and a function of two variables: w = f (x)g(y, ξ), ξ = ay 2 – z 2 . The dependence f = f (x) can be found by solving the first-order ordinary differential equation (bxn f k–1 – C1 )fx′ + cxm f k = 0, which is integrable by quadrature, e.g., for n = 0 or k = 1. The function g = g(y, ξ) is determined by    1 Z y  1–k  dt   if k ≠ 1, p  Φ(ξ) ± C1 (1 – k) y0 at2 – ξ g(y, ξ) =   Z y   dt  Φ(ξ) exp ±C  if k = 1, p  1 y0 at2 – ξ where Φ = Φ(ξ) is an arbitrary function. To obtain a complete integral, one should set Φ = C3 (the constant C2 occurs in the solution of the equation for f ). 6.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 132

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.1. NONLINEAR EQUATIONS WITH THREE VARIABLES QUADRATIC IN DERIVATIVES

Page 133

133

∂w ∂w ∂w ∂w ∂w ∂w +b +c = k. ∂x ∂y ∂x ∂z ∂y ∂z Complete integral: w = C1 x + C2 y + C3 z + C4 , where aC1 C2 + bC1 C3 + cC2 C3 = k.

7.

a

Reference: E. Kamke (1965).

8.

∂w ∂w ∂x ∂y

+a

∂w ∂w ∂x ∂z

+b

∂w ∂w ∂y ∂z

= cw.

 2 y z cC1 C2 Complete integral: w = x+ + + C3 . 4(C1 + aC2 + b) C2 C1 9.

az k

∂w ∂w

+ by n

∂w ∂w

+ cxm

∂w ∂w

= 0. ∂x ∂z ∂y ∂z cC3 m+1 aC1 k+1 bC2 n+1 w= z + y + x + C4 , k+1 n+1 m+1 where C1 C2 + C1 C3 + C2 C3 = 0.

10.

∂x ∂y

aw k

∂w ∂w

+ bw n

∂w ∂w

+ cw m

∂w ∂w

= 1. ∂x ∂y ∂x ∂z ∂y ∂z Complete integral: Z p aC1 C2 wk + bC1 C3 wn + cC2 C3 wm dw = C1 x + C2 y + C3 z + C4 .

One of the constants C1 , C2 , or C3 can be set equal to unity.

4.1.4. Equations Contain Squares and Products of Derivatives  2  2 ∂w k1 ∂w k2 ∂w ∂w k3 ∂w 1. + a1 w + a2 w + a3 w = bw n + c. ∂x ∂y ∂y ∂z ∂z Complete integral: Z f (w) dw C1 x + C2 y + C3 z + 2 = C4 , q C1 + C12 + 4(bwn + c)f (w)

where f (w) = a1 C22 wk1 + a2 C2 C3 wk2 + a3 C32 wk3 . One of the constants C1 , C2 , or C3 can be set equal to ±1. 

  2 ∂w 2 ∂w ∂w k ∂w ∂w n ∂w 2. + ax + bxz + cxz m = sxp. ∂x ∂x ∂y ∂y ∂y ∂z Complete integral: Z q aC1 k+1 1 2 2 2k p w = C1 y – x ± 4 a C1 x + sx – C2 x dx 2(k + 1) C2 bC1 + z 1–m – z n–m+1 . cC1 (1 – m) c(n – m + 1)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 133

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

134 3.

Page 134

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES 

∂w ∂x

2

+a1

∂w ∂w ∂x ∂y

+a2

∂w ∂w ∂x ∂z

+a3

∂w ∂w

=w

∂y ∂z

k



b1

∂w ∂x

+b2

∂w ∂y

+b3

∂w ∂z



.

Complete integral:
b1 C1 + b2 C2 + b3 C3 C x + C y + C z = 1 2 3 C12 + a1 C1 C2 + a2 C1 C3 + a3 C2 C3 4.



a1

∂w ∂x

2



+a1

∂w ∂y

2



+a2

∂w ∂z

2

+b1



1 1–k 1–k w

| ln w|

if k ≠ 1, if k = 1.

∂w ∂w ∂w ∂w ∂w ∂w +b2 +b3 = cw+s. ∂x ∂y ∂x ∂z ∂y ∂z

Complete integral: w=



1 4 (C1 x

+ C2 y + C3 z + C4 )2 – s/c C1 x + C2 y + C3 z + C4

if c ≠ 0, if c = 0,

where the constants C1 , C2 , and C3 are related by the one constraint a1 C12 + a2 C22 + a3 C32 + b1 C1 C2 + b2 C1 C3 + b3 C2 C3 = c. 5.



2       c ∂w 2 ∂w 2  ∂w ∂w 2 ∂w 2 2 2 + + = a(x + y + z ) + b +y +z . x ∂x ∂y ∂z ∂x ∂y ∂z

∂w

This is a special case of equation 4.3.2.3; see item 2◦ for k = 2 and f (r) = (ar + b)c [for c = –1, see also E. Kamke (1965)].

4.2. Other Nonlinear Equations with Three Variables Containing Parameters 4.2.1. Equations Cubic in Derivatives 1.



∂w ∂x

3

Z

Complete integral:

2.



∂w ∂x

3

– aw

Complete integral:

3.

∂w

– aw k

∂w ∂w ∂w ∂x ∂y ∂z

k

∂x



Z

p

– bw n

∂w

– cw m

∂y

∂w ∂z

dw awk + bC1 wn + cC2 wm

∂w ∂x

2

– bw



n

dw

awk + bC12 wn

∂w ∂y

2

+ cC22 wm

= 0. = x + C1 y + C2 z + C3 .

– cw



m

∂w ∂z

2

= 0.

= x + C1 y + C2 z + C3 .

= axky nz m.

This is a special case of equation 4.3.2.1 with f (x) = axk , g(y) = y n , and h(z) = z m . aC1 k+1 C2 n+1 1 Complete integral: w = x + y + z m+1 + C3 . k+1 n+1 C1 C2 (m + 1)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 134

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.2. OTHER NONLINEAR EQUATIONS WITH THREE VARIABLES CONTAINING PARAMETERS

4.

Page 135

135

∂w ∂w ∂w = axky nz mw s. ∂x ∂y ∂z

This is a special case of equation 4.3.2.2 with f (x) = axk , g(y) = y n , h(z) = z m , and ϕ(w) = ws .

5.

∂w ∂w ∂w ∂x ∂y ∂z

= ax



∂w ∂x

2

+ by



∂w ∂y

2

+ cz



∂w ∂z

2

.

The Legendre transformation (both direct and inverse transformations are specified) ∂W ∂W ∂W , y= , z= , ∂X ∂Y ∂Z ∂w ∂w ∂w X= , Y = , Z= , ∂x ∂y ∂z

∂W ∂W ∂W +Y +Z – W; ∂X ∂Y ∂Z ∂w ∂w ∂w W =x +y +z – w, ∂x ∂y ∂z

x=

w=X

leads to the linear equation aX 2

∂W ∂W ∂W + bY 2 + cZ 2 = XY Z. ∂X ∂Y ∂Z

The general solution of this equation is (Polyanin, Zaitsev, and Moussiaux (2002, p. 178)) W = W0 (X, Y , Z) + Φ



 1 1 1 1 – , – , aX bY aX cZ

where W0 = W0 (X, Y , Z) is a particular solution, W0 = XY Z

6.

∂w ∂x





 aX ln(aX) bY ln(bY ) cZ ln(cZ) + + . (aX – bY )(aX – cZ) (bY – aX)(bY – cZ) (cZ – aX)(cZ – bY )

∂w + az ∂y



∂w + ay ∂z



= b.

Complete integral: w = –ayz + C1 x + C2 y +

7.



∂w ∂x

+ ayz



∂w ∂y

+ axz



∂w ∂z

b z + C3 . C1 C2

+ axy

Complete integral: w = –axyz + C1 x + C2 y +

8.



∂w – aw ∂x



∂w – bw ∂y



∂w – cw ∂z



= b.

b z + C3 . C1 C2



=k

∂w ∂w ∂w . ∂x ∂y ∂z

Complete integral: ln |w| = C1 x+C2 y +C3 z +C4 , where (C1 –a)(C2 –b)(C3 –c) = kC1 C2 C3 .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 135

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

136

Page 136

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES

4.2.2. Equations Contain Roots and Moduli of Derivatives 1.

∂w + ∂x

s

    ∂w 2 ∂w 2 a+b +c = 0. ∂y ∂z

Equations of this type are encountered in differential games [e.g., q see Subbotin (1991, 1995)]. Complete integral: w = –Ax + C1 y + C2 z + C3 , where A = a + bC12 + cC22 . 2.

∂w + a1 ∂x

s

1 + b1



∂w ∂y

2

+ a2

s

1 + b2



∂w ∂z

2

= 0.

Complete integral: w = –Ax + C1 y + C2 z + C3 , where A = a1

3.

∂w ∂x

+

s

2 2   ∂w ∂w a+b + kz + c + ky = 0. ∂y ∂z

q

1 + b1 C12 + a2

Complete integral: w = –kyz – Ax + C1 y + C2 z + C3 , where A =

q

q 1 + b2 C22 .

a + bC12 + cC22 .

∂w ∂w ∂w ∂w +z + – = 0. 4. ∂x ∂y ∂y ∂z This equation is encountered in differential games. An exact closed-form solution of this equation subject to the initial condition w = max{|y|, |z|}

at

x=2

was obtained in Taras’ev (1985). In the domain 1 ≤ x ≤ 2, this solution is given by ( max{ϕ1 , ϕ3 , ϕ6 } if ϕ1 + ϕ3 ≥ 0, w(x, y, z) = max{ϕ2 , ϕ4 , ϕ5 } if ϕ2 + ϕ4 ≥ 0, max{ϕ5 , ϕ6 } if ϕ1 + ϕ3 ≤ 0 and ϕ2 + ϕ4 ≤ 0, where

ϕ1 = ϕ1 (x, y, z) = x + y + 2z – 12 x2 – xz, ϕ3 = ϕ3 (x, y, z) = 2 – x + z, ϕ5 = ϕ5 (x, y, z) = 2 – ϕ3 – 2(1 – ϕ1 – ϕ3 )1/2 , ϕk = ϕk (x, y, z) = ϕk–1 (x, –y, –z), k = 2, 4, 6.

At the points (x, y, z) of matching various ϕn and ϕm , the function w is continuous but can be nondifferentiable. In the domain 0 ≤ x ≤ 1, the solution has a more sophisticated structure and is defined by 12 different functions, 6 of which are defined implicitly. References: A. M. Taras’ev (1985) and A. I. Subbotin (1991).

4.2.3. Equations Contain Arbitrary Powers of Derivatives   ∂w n +b = cw + sxm. ∂x ∂y ∂z This is a special case of equation 4.3.2.4 with f (x) = c and g(x) = sxm . 1.

∂w

+a



∂w

k

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 136

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.2. OTHER NONLINEAR EQUATIONS WITH THREE VARIABLES CONTAINING PARAMETERS

2.

∂w ∂x

+ a1



∂w ∂y

k

+ a2



∂w ∂z

n

+ b1 xmz

∂w ∂y

+ b2 xmy

∂w ∂z

Page 137

137

= 0.

This is a special case of equation 4.3.2.5 with f (x) = xm . 3.

    ∂w ∂w ∂w ∂w k ∂w n + a2 + b1 eλxz + a1 + b2 eλxy = 0. ∂x ∂y ∂z ∂y ∂z

This is a special case of equation 4.3.2.5 with f (x) = eλx . 4.

∂w ∂x

+ a1 xn1



∂w ∂y

k1

+ a2 xn2



∂w ∂z

k2

= bxsw + c1 xm1 y + c2 xm2 z.

This is a special case of equation 4.3.2.6 with f1 (x) = a1 xn1 , f2 (x) = a2 xn2 , g(x) = bxs , h1 (x) = c1 xm1 , and h2 (x) = c2 xm2 . 5.

  k1 k2 ∂w λ1 x ∂w λ2 x ∂w + a1 e + a2 e = beβxw + c1 eγ1 xy + c2 eγ2 xz. ∂x ∂y ∂z

This is a special case of equation 4.3.2.6 with f1 (x) = a1 eλ1 x , f2 (x) = a2 eλ2 x , g(x) = beβx , h1 (x) = c1 eγ1 x , and h2 (x) = c2 eγ2 x .  n1  n2 ∂w ∂w ∂w 6. + a1 xk1 + bz + a2 xk2 + by = cxm. ∂x ∂y ∂z a1 C1n1 k +1 a2 C2n2 k +1 c xm+1 – x 1 – x 2 + C3 . m+1 k1 + 1 k2 + 1  n1  n2 ∂w λ1 x ∂w λ2 x ∂w + a2 e = ceβx. 7. + a1 e + bz + by ∂x ∂y ∂z c a1 n1 λ1 x a2 n2 λ2 x Complete integral: w = –byz + C1 y + C2 z + eβx – C e – C e + C3 . β λ1 1 λ2 2

Complete integral: w = –byz + C1 y + C2 z +

8.

∂w ∂x

+ ax

k



∂w ∂y

+ bz

n

∂w ∂z

+ by

m

= cxs.

aC1n C2m k+1 c xs+1 – x + C3 . s+1 k+1  n m ∂w ∂w λx ∂w 9. + ae + bz + by = ceβx. ∂x ∂y ∂z c a Complete integral: w = –byz + C1 y + C2 z + eβx – C1n C2m eλx + C3 . β λ Complete integral: w = –byz + C1 y + C2 z +



      ∂w k ∂w k ∂w ∂w ∂w n 10. +b +c = x +y +z . ∂y ∂z ∂x ∂y ∂z
1
k k–n Complete integral: w = aC1k + bC2k + cC3k n–k C1 x + C2 y + C3 z k–n + C4 . k One of the constants C1 , C2 , or C3 can be set equal to ±1. ∂w a ∂x

k

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 137

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

138 11.

Page 138

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES  n  n  n k1 ∂w k2 ∂w k3 ∂w a1 w + a2 w + a3 w ∂x ∂y ∂z  m     ∂w ∂w m ∂w m – b1 w s1 – b2 w s2 – b3 w s3 = 0. ∂x ∂y ∂z

Complete integral: Z 

a1 C1n wk1 + a2 C2n wk2 + a3 C3n wk3 b1 C1m ws1 + b2 C2m ws2 + b3 C3m ws3



1 n–m

dw = C1 x + C2 y + C3 z + C4 .

One of the constants C1 , C2 , or C3 can be set equal to unity. 12.

a



∂w ∂x

k

    ∂w n ∂w m +b +c = 1. ∂y ∂z

Complete integral: w = C1 x + C2 y + C3 z + C4 , where aC1k + bC2n + cC3m = 1. 13.

a



∂w ∂x

k

    ∂w n ∂w m ∂w ∂w ∂w +b +c +x +y +z = w. ∂y ∂z ∂x ∂y ∂z

Complete integral: w = C1 x + C2 y + C3 z + aC1k + bC2n + cC3m . 14.

a1



∂w ∂x

+ byz

k1

+ a2



∂w

+ bxz

∂y

k2

+ a3



∂w

+ bxy

∂z

k3

= c.

Complete integral: w = –bxyz + C1 x + C2 y + C3 z + C4 , where a1 C1k1 + a2 C2k2 + a3 C3k3 = c. 15.

a1 x

k1



n1

∂w ∂x

+ a2 y

k2



∂w ∂y

n2

+ a3 z

k3



∂w ∂z

n3

= b1 xm1 + b2 y m2 + b3 z m3 .

This is a special case of equation 4.3.2.9. 16.



∂w ∂x

k1 

∂w ∂y

k2 

∂w ∂z

k3

= axn1 y n2 z n3 w m.

This is a special case of equation 4.3.2.10. 17.



∂w a ∂x

k

∂w ∂y

n

∂w ∂z

m

=x

∂w ∂w ∂w +y +z . ∂x ∂y ∂z

Complete integral: w=

18.

a




k+n+m k+n+m–1 C1 x + C2 y + βz k+n+m–1 + C3 , k+n+m

∂w ∂x

k

∂w ∂y

n

∂w ∂z

m

+x

∂w ∂x

+y

∂w ∂y

+z

β = aC1k C2n ∂w ∂z


–

1 m.

= w.

Complete integral: w = C1 x + C2 y + C3 z + aC1k C2n C3m .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 138

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.3. NONLINEAR EQUATIONS WITH THREE VARIABLES CONTAINING ARBITRARY FUNCTIONS

19.



∂w ∂x

+ ayz

k

∂w ∂y

+ axz

n

∂w ∂z

+ axy

m

Page 139

139

= b.

Complete integral: w = –axyz + C1 x + C2 y + C3 z + C4 , where C1k C2n C3m = b. 20.

∂w + a1 xn1 ∂x



∂w ∂y

k1 

∂w ∂z

m1

+ a2 x

n2



∂w ∂y

k2 

∂w ∂z

m2

= bxsw.

This is a special case of equation 4.3.4.3 with F (x, u, v) = a1 xn1 uk1 v m1 + a2 xn2 uk2 v m2 , and g(x) = bxs .

21.

∂w ∂x

+ a1 x

n1



∂w ∂y

k1 

∂w ∂z

m1

+ a2 x

n2



∂w ∂y

k2 

∂w

m2

∂z = bxsw + c1 xq1 y + c2 xq2 z.

This is a special case of equation 4.3.4.4.

22.

∂w ∂x

λ1 x

+ a1 e



∂w ∂y

k1 

∂w ∂z

m1

λ2 x

+ a2 e



∂w ∂y

k2 

∂w ∂z

m2

= beβxw.

This is a special case of equation 4.3.4.3.

4.3. Nonlinear Equations with Three Variables Containing Arbitrary Functions 4.3.1. Equations Quadratic in Derivatives 1.



∂w ∂x

2

+ f (x)

∂w ∂w ∂w + g1 (y) + g2 (z) + h(x) = 0. ∂x ∂y ∂z

Complete integral: 1 w= 2

2.

Z h

∂w ∂x

–f (x) +

+a



∂w ∂y

q

f 2 (x) –

2

  ∂w 2 +b = f (x)w + g(x). ∂z

i

4h(x) – 4C1 – 4C2 dx + C1

Z

dy + C2 g1 (y)

Z

dz + C3 . g2 (z)

Complete integral: w = C1 F (x) + (C2 y + C3 z)F (x) + F (x) where F (x) = exp

Z

 f (x) dx ,

Z

g(x) – λF 2 (x) dx, F (x)

λ = aC22 + bC32 .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 139

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

140 3.

Page 140

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES ∂w ∂x

+ a1



∂w ∂y

2

+ a2



∂w ∂z

2

= bw 2 + f (x)w + g(x).

For b = 0, see equation 4.3.1.2. Complete integral for b ≠ 0: w = ϕ(x) + C3 ψ(x) exp(C1 y + C2 z). The constants C1 and C2 are related by the constraint a1 C12 + a2 C22 = b. The function ϕ = ϕ(x) is determined by the ordinary differential equation ϕ′x = bϕ2 + f (x)ϕ + g(x),

(1)

and the function ψ = ψ(x) is expressed via ϕ(x) as follows: ψ(x) = exp

Z

   2bϕ(x) + f (x) dx .

(2)

The Riccati equation (1) can be integrated by quadrature for many f and g, in particular, for g(x) ≡ 0 and any f (x), as well as for f (x) = const and g(x) = const. For details, see the books Kamke (1976) and Polyanin and Zaitsev (2003). 4.

∂w ∂x

+ a1



∂w ∂y

2

+ a2



∂w ∂z

2

+ b1 f (x)z

∂w ∂y

+ b2 f (x)y

∂w ∂z

= 0.

This is a special case of equation 4.3.1.17, item 2◦ . Complete integral: w = yϕ1 (x) + zϕ2 (x) –

Z



 a1 ϕ21 (x) + a2 ϕ22 (x) dx + C3 ,

where the functions ϕ1 (x) and ϕ2 (x) are given by

 Z   Z  ϕ1 (x) = C1 b2 exp λ f (x) dx + C2 b2 exp –λ f (x) dx ,  Z   Z  ϕ2 (x) = –C1 λ exp λ f (x) dx + C2 λ exp –λ f (x) dx . 5.

∂w ∂x

+ a1



∂w ∂y

2

+ a2



∂w ∂z

2

+ f (x)z

∂w ∂y

+ g(x)y

∂w ∂z

λ=

p

b1 b2 ,

= 0.

Complete integral: w = yϕ1 (x) + zϕ2 (x) –

Z



 a1 ϕ21 (x) + a2 ϕ22 (x) dx + C3 ,

ϕ2 (x) = –

1 dϕ1 , g(x) dx

where the function ϕ1 = ϕ1 (x) is determined by the second-order linear ordinary differential equation   d 1 dϕ1 – f (x)ϕ1 = 0. dx g(x) dx

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 140

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.3. NONLINEAR EQUATIONS WITH THREE VARIABLES CONTAINING ARBITRARY FUNCTIONS

6.

∂w ∂x

+ f (x)



∂w ∂y

2

+ g(x)



∂w ∂z

2



∂w

+ f (x) ∂x ∂y Complete integral:

7.

2

+ g(x)



∂w ∂z

2

Z

Z

  eax C22 f (x) + C32 g(x) dx.

= h(x)w.

w = ϕ(x)(C1 + C2 y + C3 z) – ϕ(x) where ϕ(x) = exp

141

= aw.

Complete integral: w = C1 eax + eax (C2 y + C3 z) – eax ∂w

Page 141

 h(x) dx .

Z

  ϕ(x) C22 f (x) + C32 g(x) dx,

    ∂w 2 ∂w 2 ∂w + f2 (x) = g(x)w + h1 (x)y + h2 (x)z. 8. + f1 (x) ∂x ∂y ∂z Complete integral:   w = G(x) yϕ1 (x) + zϕ2 (x) + ψ(x) , where

Z

Z h1 (x) h2 (x) ϕ1 (x) = C1 + dx, ϕ2 (x) = C2 + dx, G(x) G(x) Z  Z   2 2 ψ(x) = C3 – G(x) f1 (x)ϕ1 (x) + f2 (x)ϕ2 (x) dx, G(x) = exp g(x) dx . ∂w

+ f1 (y)



∂w

2

∂w



∂w

+ f2 (z)



∂w

2

= g(x) + h1 (y) + h2 (z). ∂x ∂y ∂z Equations of this type are often encountered in mechanics; the variable x plays the role of time, and y and z are generalized coordinates. Complete integral: Z Z r Z r C2 + h1 (y) C2 + h2 (z) w = C1 – (C2 + C3 )x + g(x) dx + dy + dz. f1 (y) f2 (z) 9.

2



∂w

2

+ f2 (z) = bw 2 + g(x)w + h(x). ∂y ∂z Z Z dy dz The transformation ξ = , η= leads to an equation of the form p p |f1 (y)| |f2 (z)| 4.3.1.3.  2  2 ∂w ∂w ∂w 11. + f (x) + az + g(x) + ay = h(x). ∂x ∂y ∂z Z   Complete integral: w = –ayz + C1 y + C2 z + h(x) – C12 f (x) – C22 g(x) dx + C3 . 10.

∂x

+ f1 (y)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 141

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

142 12.

Page 142

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES ∂w ∂x



+ f1 (x)y + g1 (x)

   ∂w 2 ∂y



+ f2 (x)z + g2 (x)

   ∂w 2

∂z = a(x)w + b(x) + h1 (x)y + h2 (x)z.

Complete integral: w = ϕ1 (x)y + ϕ2 (x)z + ψ(x), where the functions ϕ1 (x), ϕ2 (x), and ψ(x) are determined by solving the ordinary differential equations ϕ′k + fk (x)ϕ2k = a(x)ϕk + hk (x), ψ

′

+ g1 (x)ϕ21

+

g2 (x)ϕ22

k = 1, 2;

(1)

= a(x)ψ + b(x).

(2)

The prime denotes the derivative with respect to x. The Riccati equations (1) can be integrated by quadrature for many fk (x), hk (x), and a(x); for details, see the books Kamke (1976) and Polyanin and Zaitsev (1995). In particular, for hk ≡ 0 the solutions of equations (1) are given by  –1 Z  Z ϕk (x) = A(x) Ck + A(x)fk (x) dx , A(x) = exp a(x) dx . Equation (2) is linear with respect to ψ and can be readily integrated (for known ϕk ): Z   dx ψ(x) = C3 A(x) + A(x) b(x) – g1 (x)ϕ21 (x) – g2 (x)ϕ22 (x) . A(x) 13.

      ∂w 2   ∂w 2 + f1 (x)y k + g1 (x)y 2k–2 + f2 (x)z n + g2 (x)z 2n–2 ∂x ∂y ∂z = a(x)w + b(x).

∂w

1◦ . If k ≠ 2 and n ≠ 2, then the transformation ξ = y 2–k , η = z 2–n leads to an equation of the form 4.3.1.12:       ∂u 2   ∂u 2 ∂u 2 2 + (2 – k) f1 (x)ξ + g1 (x) + (2 – n) f2 (x)η + g2 (x) = a(x)u + b(x). ∂x ∂ξ ∂η 2◦ . If k = 2 and n ≠ 2, then we introduce the new variables ξ = ln |y| and η = z 2–n to arrive at an equation of the form 4.3.1.12. The case k ≠ 2 and n = 2 is treated analogously. ∂w

+ f (ax + by + cz) ∂x Complete integral: 14.



∂w ∂y

2

+ g(ax + by + cz)

w = C1 x + C2 y + C3 z + ϕ(ξ) + C4 ,



∂w ∂z

2

= h(ax + by + cz).

ξ = ax + by + cz,

where Z

p

G2 (ξ) – 4F (ξ)H(ξ) dξ, F (ξ) = b2 f (ξ) + c2 g(ξ), 2F (ξ) G(ξ) = a + 2bC2 f (ξ) + 2cC3 g(ξ), H(ξ) = C1 + C22 f (ξ) + C32 g(ξ) – h(ξ). ϕ(ξ) =

–G(ξ) ±

One of the constants C1 , C2 , or C3 can be set equal to unity.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 142

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.3. NONLINEAR EQUATIONS WITH THREE VARIABLES CONTAINING ARBITRARY FUNCTIONS

15.

∂w ∂x

+ f (x)



∂w ∂y

+ az



∂w ∂z

+ ay

Complete integral: w = –ayz + C1 y + C2 z +

16.

∂w ∂x

+ f11 (x)



∂w

2

 Z

Page 143

143

= g(x). 

∂w ∂w

 g(x) – C1 C2 f (x) dx + C3 . 

∂w

2

+ f12 (x) + f22 (x) ∂y ∂y ∂z ∂z 2 2 = g(x)w + h11 (x)y + h12 (x)yz + h22 (x)z + h1 (x)y + h2 (x)z + h0 (x).

This is a special case of equation 4.6.1.10. A complete integral can be sought in the form w = ϕ11 (x)y 2 + ϕ12 (x)yz + ϕ22 (x)z 2 + ψ1 (x)y + ψ2 (x)z + ψ0 (x). Substituting this expression into the original partial differential equation yields a system of ordinary differential equations for ϕkm (x) and ψk (x). If all hk (x) ≡ 0, then we may set ψk (x) ≡ 0.      ∂w ∂w 2 ∂w ∂w ∂w 2  ∂w +f12 (x) + g11 (x)y+g12 (x)z 17. +f11 (x) +f22 (x) ∂x ∂y ∂y ∂z ∂z ∂y   ∂w = s(x)w + h1 (x)y + h2 (x)z + h0 (x). + g21 (x)y + g22 (x)z ∂z This is a special case of equation 4.3.4.5. 1◦ . Complete integral: w = ϕ1 (x)y + ϕ2 (x)z + ψ(x). The functions ϕ1 (x), ϕ2 (x), and ψ(x) are determined by solving the system of ordinary differential equations ϕ′1 + (g11 – s)ϕ1 + g21 ϕ2 = h1 , ϕ′2 + g12 ϕ1 + (g22 – s)ϕ2 = h2 ,

(1) (2)

ψ ′ – sψ = h0 – f11 ϕ21 – f12 ϕ1 ϕ2 – f22 ϕ22 ,

(3)

where fij = fij (x), gij = gij (x), hk = hk (x), and s = s(x); the prime denotes the derivative with respect to x. Having found ϕ1 = ϕ1 (x) and ϕ2 = ϕ2 (x) from the linear system of equations (1) and (2), one can determine the solution of equation (3) according to the formula Z  Z 2 2 dx ψ(x) = C3 S + S (h0 – f11 ϕ1 – f12 ϕ1 ϕ2 – f22 ϕ2 ) , S = exp s dx . S 2◦ . Consider the special case h1 ≡ 0 and h2 ≡ 0. In addition, assume that g11 = a1 g(x) + s(x),

g12 = a2 g(x),

g21 = a3 g(x),

g22 = a4 g(x) + s(x),

(4)

where g = g(x) and s = s(x) are arbitrary functions and a1 , a2 , a3 , and a4 are arbitrary numbers. With these assumptions, the general solution of the linear homogeneous system (1), (2) has the form  Z  Z   ϕ1 = C1 exp λ1 g dx + C2 exp λ2 g dx ,  Z   Z  (5) a1 + λ1 a1 + λ2 C1 exp λ1 g dx – C2 exp λ2 g dx , ϕ2 = – a3 a3 where λ1 and λ2 are the roots of the quadratic equation (λ + a1 )(λ + a4 ) – a2 a3 = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 143

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

144

Page 144

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES

3◦ . Let h1 = h1 (x) and h2 = h2 (x) be arbitrary functions and let the gij = gij (x) be defined by (4). In this case, the general solution of the linear nonhomogeneous system of equations (1) and (2) can be obtained by variation of constants or with the aid of the Wronskian determinant [see, for example, Murphy (1960), Kamke (1977), and Zwillinger (1998)].     ∂w 2 ∂w 2 ∂w k k + w g(x) = h(x)w. 18. + w f (x) ∂x ∂y ∂z 1 The substitution u = wk+1 leads to an equation of the form 4.3.1.7: k+1   2  ∂u ∂u 2 ∂u + f (x) + g(x) = (k + 1)h(x)u. ∂x ∂y ∂z 19.

∂w ∂x

k

+ w f1 (y)



∂w ∂y

2

k

+ w f2 (z)



∂w ∂z

2

= aw k+2 + g(x)w.

1 wk+1 leads to an equation of the form 4.3.1.10 with h(x) ≡ 0: k+1  2  2 ∂u ∂u ∂u + f1 (y) + f2 (z) = a(k + 1)2 u2 + (k + 1)g(x)u. ∂x ∂y ∂z

The substitution u =

20.

∂w

+ f1 (y)g1 (w)



∂w

2

+ f2 (z)g2 (w)



∂w

2

= 0. ∂x ∂y ∂z Complete integral: Z Z Z  2  dy dz 2 + C2 + C3 . x+ C1 g1 (w) + C2 g2 (w) dw = C1 p p f1 (y) f2 (z)

It has been assumed in the products f1 g1 and f2 g2 that both f1 and f2 are nonnegative, and g1 and g2 can have any sign. 

∂w

∂w

2



∂w

2

+ f1 (y)g1 (w) + f2 (z)g2 (w) = h(w). ∂x ∂y ∂z Complete integral: Z Z Z dy dz 2ϕ(w) dw x + C1 + C2 + = C3 , p p p f1 (y) f2 (z) 1 + 1 + 4h(w)ϕ(w) 21.

where ϕ(w) = C12 g1 (w) + C22 g2 (w).

      ∂w 2 ∂w 2 22. + g1 (y) + g2 (z) – h1 (y) – h2 (z) = 0. ∂x f1 (y) + f2 (z) ∂y ∂z Liouville’s equation. It is encountered in problems of mechanics (x is time, and y and z are generalized coordinates). Complete integral: Z r Z r C1 f1 (y) + h1 (y) + C2 C1 f2 (z) + h2 (z) – C2 w = –C1 x + dy + dz + C3 . g1 (y) g2 (z) ∂w

1

Reference: E. N. Berezkin (1968).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 144

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

145

4.3. NONLINEAR EQUATIONS WITH THREE VARIABLES CONTAINING ARBITRARY FUNCTIONS

23.

f1 (x)



∂w ∂x

2

+ f2 (y)



∂w ∂y

2

+ f3 (z)



2

∂w ∂z

Page 145

= g1 (x) + g2 (y) + g3 (z).

Complete integral: w=±

24.

Z 

1/2

g1 (x) + C1 f1 (x)

f1 (x)



∂w ∂x

2

Z 

dx ±

+ f2 (y)



g2 (y) + C2 f2 (y)

∂w ∂y

2

1/2

dy ±

Z 



+ g1 (x) + g2 (y)

g3 (z) – C1 – C2 f3 (z)

   ∂w 2 ∂z

1/2

dz + C3 .

= h1 (x) + h2 (y).

A problem of the motion of a point mass being attracted by two fixed centers according to Newton’s law of universal gravitation can be reduced to a special case of this equation [see Appell (1953)]. Complete integral: w = C1 z + C2 ±

Z r

h1 (x) – C12 g1 (x) + C3 dx ± f1 (x)

Z r

h2 (y) – C12 g2 (y) – C3 dy. f2 (y)

The signs before integrals may be chosen independently. 25.

f1 (x)



∂w ∂x

2

+ f2 (y)



∂w ∂y

2

+ f3 (z)



∂w ∂z

2

= aw + g1 (x) + g2 (y) + g3 (z).

This is a special case of equation 4.6.1.17 with n = 3. 

    ∂w 2 ∂w 2  26. + f2 (y) + f3 (z) = g1 (x) + g2 (y) + g3(z) h(w). ∂y ∂z Z dw The substitution u = leads to an equation of the form 4.3.1.23: √ h(w) ∂w f1 (x) ∂x



2

 2  2  2 ∂u ∂u ∂u f1 (x) + f2 (y) + f3 (z) = g1 (x) + g2 (y) + g3 (z). ∂x ∂y ∂z 27.

f (ax+by +cz)



∂w ∂x

2

+g(ax+by +cz)



∂w ∂y

2

+h(ax+by +cz)



∂w ∂z

2

= k.

Complete integral: w = C1 x + C2 y + C3 z + ϕ(ξ) + C4 ,

ξ = ax + by + cz,

where Z

p

G2 (ξ) – 4F (ξ)H(ξ) dξ, F (ξ) = a2 f (ξ) + b2 g(ξ) + c2 h(ξ), 2F (ξ) G(ξ) = 2aC1 f (ξ) + 2bC2 g(ξ) + 2cC3 h(ξ), H(ξ) = C12 f (ξ) + C22 g(ξ) + C32 h(ξ) – k.

ϕ(ξ) =

–G(ξ) ±

One of the constants C1 , C2 , or C3 can be set equal to unity.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 145

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

146 28.

Page 146

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES f1 (x)g1 (w)



∂w ∂x

2

+ f2 (y)g2 (w)

Complete integral: s Z C12g1(w)+C22g2(w)+C32g3(w) h(w)

dw = C1



Z

∂w ∂y

p

2

dx

+ f3 (z)g3 (w)

+C2

f1(x) One of the constants C1 , C2 , or C3 can be set equal to ±1.

Z

p



dy f2(y)

∂w ∂z

2

Z

+C3

= h(w).

p

dz f3(z)

+C4.

∂w ∂w ∂w ∂w ∂w ∂w + f (x) + g(x) = h(x). ∂x ∂y ∂x ∂z ∂y ∂z Z h(x) – C1 C2 g(x) Complete integral: w = C1 y + C2 z + dx + C3 . C1 + C2 f (x)

29.

∂w ∂w ∂w ∂w + g(z) = h(x). ∂x ∂y ∂x ∂z Z Z Z 1 dy dz Complete integral: w = h(x) dx + C1 + C2 + C3 . C1 + C2 f (y) g(z) 30.

f (y)

∂w ∂w ∂w ∂w ∂w ∂w + g(y) + h(x) = 0. ∂x ∂y ∂x ∂z ∂y ∂z Z Z Z C1 C1 Complete integral: w = C1 h(x) dx – g(y) dy + f (z) dz + C3 . C2 + 1 C2

31.

f (z)

4.3.2. Equations with Power Nonlinearity in Derivatives ∂w ∂w ∂w = f (x)g(y)h(z). ∂x ∂y ∂z Z Z Complete integral: w = C1 f (x) dx + C2 g(y) dy +

1.

1 C1 C2

Z

h(z) dz + C3 .

∂w ∂w ∂w

= f (x)g(y)h(z)ϕ(w). ∂x ∂y ∂z Complete integral: Z Z Z Z dw 1 = C1 f (x) dx + C2 g(y) dy + h(z) dz + C3 . C1 C2 ϕ1/3 (w)        
∂w 2 ∂w 2 ∂w 2 ∂w ∂w ∂w k 2 2 2 3. + + =f x +y +z x +y +z . ∂x ∂y ∂z ∂x ∂y ∂z 1◦ . By passing from x, y, z to the spherical coordinates r, θ, ϕ, x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ, we arrive at the equation  2       ∂w k ∂w 2 1 ∂w 2 2 ∂w 2 2 r – r f (r ) r + + = 0. ∂r ∂r ∂θ sin2 θ ∂ϕ A complete integral of this equation can be found by the method of separation of variables as the sum of functions depending on different variables (linear in ϕ).

2.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 146

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.3. NONLINEAR EQUATIONS WITH THREE VARIABLES CONTAINING ARBITRARY FUNCTIONS

Page 147

147

2◦ . Complete integral for k = 2: w = C1

4.

Z

r

p

dr r 2 f (r 2 ) – 1

+

s

Z

C12 –

C22 sin2 θ

dθ + C2 ϕ + C3 .

    ∂w k ∂w n ∂w +b = f (x)w + g(x). +a ∂x ∂y ∂z

Complete integral: Z

w = C1 F (x) + (C2 y + C3 z)F (x) + F (x) where F (x) = exp

5.

Z

 f (x) dx .



 dx g(x) – aC2k F k (x) – bC3n F n (x) , F (x)

    ∂w k ∂w n ∂w ∂w ∂w + a2 + b1 f (x)z + a1 + b2 f (x)y = 0. ∂x ∂y ∂z ∂y ∂z

Complete integral: w = yϕ1 (x) + zϕ2 (x) –

Z

where



 a1 ϕk1 (x) + a2 ϕn2 (x) dx + C3 ,

 Z   Z  ϕ1 (x) = C1 b2 exp λ f (x) dx + C2 b2 exp –λ f (x) dx ,  Z   Z  ϕ2 (x) = –C1 λ exp λ f (x) dx + C2 λ exp –λ f (x) dx . 6.

∂w ∂x

+ f1 (x)



∂w ∂y

k

Complete integral:

G(x) = exp

ψ(x) = C3 –

Z



g(x) dx , 

∂w ∂z

n

p

b1 b2 ,

= g(x)w + h1 (x)y + h2 (x)z.

  w = G(x) yϕ1 (x) + zϕ2 (x) + ψ(x) ,

where Z

+ f2 (x)



λ=

ϕ1 (x) = C1 +

Z

h1 (x) dx, G(x)

ϕ2 (x) = C2 +

 f1 (x)Gk–1 (x)ϕk1 (x) + f2 (x)Gn–1 (x)ϕn2 (x) dx.

Z

h2 (x) dx, G(x)

 k  n ∂w ∂w ∂w 7. + f (x) + az + g(x) + ay = h(x). ∂x ∂y ∂z Z   Complete integral: w = –ayz + C1 y + C2 z + h(x) – C1k f (x) – C2n g(x) dx + C3 .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 147

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

148 8.

Page 148

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES ∂w ∂x

+ f (x)



∂w ∂y

+ az

k

∂w ∂z

+ ay

Complete integral: w = –ayz + C1 y + C2 z + 

∂w

k



∂w

k

+ f2 (y)



∂w

n

Z

∂w

n

∂w

+ f3 (z)

= g(x).



 g(x) – C1k C2n f (x) dx + C3 .



∂w

m

= g1 (x) + g2 (y) + g3 (z). ∂x ∂y ∂z Complete integral:    Z  Z  Z  g1 (x) + C1 1/k g2 (y) + C2 1/n g3 (z) – C1 – C2 1/m w= dx + dy + dz + C3 . f1 (x) f2 (y) f3 (z)

9.

f1 (x)

n

m

= f (x)g(y)h(z)ϕ(w). ∂x ∂y ∂z This is a special case of equation 4.6.2.4. 10.

4.3.3. Equations with Arbitrary Dependence on Derivatives 

      ∂w k ∂w n  ∂w 1. f1 (x) + f2 (y) + g1 (x) + g2 (y) F = h1 (x) + h2 (y). ∂x ∂y ∂z Complete integral:   Z  Z  h1 (x) – F (C1 )g1 (x) + C3 1/k h2 (y) – F (C1 )g2 (y) – C3 1/n w = C1 z+C2 + dx+ dy. f1 (x) f2 (y)     ∂w ∂w ∂w 2. + f (x)F ay + + g(x)G bz + = h(x). ∂x ∂y ∂z Complete integral: Z   w=– F (C1 )f (x) + G(C2 )g(x) – h(x) dx – 12 ay 2 – 12 bz 2 + C1 y + C2 z + C3 .   a ∂w 3. + f (x)F y + g(x)G z = h(x). ∂x ∂y ∂z 1◦ . Complete integral for a ≠ 1 and b ≠ 1: Z C1 1–a C2 1–b w = – [F (C2 )f (x) + G(C1 )g(x) – h(x)] dx + z + y + C3 . 1–a 1–b ∂w



b ∂w



2◦ . Complete integral for a = 1 and b ≠ 1: Z C2 1–b w = – [F (C2 )f (x) + G(C1 )g(x) – h(x)] dx + C1 ln |z| + y + C3 . 1–b

3◦ . Complete integral for a = 1 and b = 1: Z w = – [F (C2 )f (x) + G(C1 )g(x) – h(x)] dx + C1 ln |z| + C2 ln |y| + C3 .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 148

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.3. NONLINEAR EQUATIONS WITH THREE VARIABLES CONTAINING ARBITRARY FUNCTIONS



∂w

∂w



Page 149

149

  a ∂w + g(x)G z = h(x). ∂z

+ f (x)F by + ∂x ∂y ◦ 1 . Complete integral for a ≠ 1: Z C1 1–a b 2 w = – [F (C2 )f (x) + G(C1 )g(x) – h(x)] dx + z – y + C2 y + C3 . 1–a 2

4.

2◦ . Complete integral for a = 1: Z b w = – [F (C2 )f (x) + G(C1 )g(x) – h(x)] dx + C1 ln |z| – y 2 + C2 y + C3 . 2 

∂w

2

∂w



∂w



  ∂w + g(x)G bz + = h(x). ∂z

+ f (x)F ay + + p(x) ∂x ∂x ∂y Complete integral: Z q i 1 h w= –p(x) + p2 (x) – 4F (C2 )f (x) – 4G(C1 )g(x) + 4h(x) dx 2 a b – y 2 – z 2 + C1 z + C2 y + C3 . 2 2       ∂w 2 ∂w a ∂w b ∂w 6. + p(x) + f (x)F y + g(x)G z = h(x). ∂x ∂x ∂y ∂z 1◦ . Complete integral for a ≠ 1 and b ≠ 1: Z q i 1 h C1 z 1–b C2 y 1–a w= –p(x)+ p2 (x) – 4F (C2 )f (x) – 4G(C1 )g(x) + 4h(x) dx+ + +C3 . 2 1–b 1–a 5.

2◦ . Complete integral for a ≠ 1 and b = 1: Z q i 1 h C2 1–a w= –p(x)+ p2(x)–4F (C2)f (x)–4G(C1)g(x)+4h(x) dx+C1 ln |z|+ y +C3. 2 1–a

3◦ . Complete integral for a = 1 and b = 1: Z q i 1 h w= –p(x)+ p2 (x) – 4F (C2 )f (x) – 4G(C1 )g(x) + 4h(x) dx+C1 ln |z|+C2 ln |y|+C3 . 2 

     ∂w 2 ∂w ∂w b ∂w 7. + p(x) + f (x)F ay + + g(x)G z = h(x). ∂x ∂x ∂y ∂z 1◦ . Complete integral for b ≠ 1: Z q i 1 h C1z 1–b a 2 w= –p(x)+ p2(x)–4F (C2)f (x)–4G(C1)g(x)+4h(x) dx+ – y +C2y+C3. 2 1–b 2

2◦ . Complete integral for b = 1: Z q i 1 h a w= –p(x)+ p2(x)–4F (C2)f (x)–4G(C1)g(x)+4h(x) dx+C1 ln |z|– y 2 +C2y+C3. 2 2

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 149

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

150

Page 150

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES

4.3.4. Nonlinear Equations of General Form 1.

  ∂w ∂w + F x, , = 0. ∂x ∂y ∂z

∂w

Complete integral: w = C1 y + C2 z + C3 –

2.

  ∂w ∂w ∂w + F x, , = aw. ∂x ∂y ∂z

Z

F (x, C1 , C2 ) dx.

ax

ax

Complete integral: w = e (C1 y + C2 z + C3 ) – e ∂w



∂w ∂w 3. + F x, , ∂x ∂y ∂z Complete integral:



where ϕ(x) = exp ∂w



e–ax F (x, C1 eax , C2 eax ) dx.

= g(x)w.

w = ϕ(x)(C1 y + C2 z + C3 ) – ϕ(x) Z

Z



Z


dx F x, C1 ϕ(x), C2 ϕ(x) , ϕ(x)

g(x) dx .

∂w ∂w 4. + F x, , ∂x ∂y ∂z Complete integral:



= g(x)w + h1 (x)y + h2 (x)z. w = yϕ(x) + zψ(x) + χ(x),

where Z h2 (x) ψ(x) = C2 G(x) + G(x) dx, G(x) Z  Z dx χ(x) = C3 G(x) – G(x) F (x, ϕ, ψ) , G(x) = exp g(x) dx . G(x)

ϕ(x) = C1 G(x) + G(x)

Z

h1 (x) dx, G(x)

  ∂w   ∂w + f12 (x)y + f11 (x)z + f10 (x) + f22 (x)y + f21 (x)z + f20 (x) ∂x ∂y ∂z   ∂w ∂w + F x, , = g(x)w + h1 (x)y + h2 (x)z. ∂y ∂z Complete integral: w = yϕ1 (x) + zϕ2 (x) + ψ(x). 5.

∂w

The functions ϕ1 (x), ϕ2 (x), and ψ(x) are determined by solving the system of ordinary differential equations ϕ′1 + (f12 – g)ϕ1 + f22 ϕ2 = h1 , ϕ′2 + f11 ϕ1 + (f21 – g)ϕ2 = h2 , ψ ′ – gψ = –f10 ϕ1 – f20 ϕ2 – F (x, ϕ1 , ϕ2 ),

(1) (2) (3)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 150

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.3. NONLINEAR EQUATIONS WITH THREE VARIABLES CONTAINING ARBITRARY FUNCTIONS

Page 151

151

where fij = fij (x), g = g(x), and hk = hk (x); the prime denotes the derivative with respect to x. Having found ϕ1 = ϕ1 (x) and ϕ2 = ϕ2 (x) from the linear system of equation (1) and (2), one can determine the solution of equation (3) according to the formula Z  Z   dx , G = exp g dx . f10 ϕ1 + f20 ϕ2 + F (x, ϕ1 , ϕ2 ) ψ(x) = C3 G – G G 2◦ . Consider the special case h1 ≡ 0 and h2 ≡ 0. In addition, assume that f11 = a1 f (x),

f12 = a2 f (x) + g(x),

f21 = a3 f (x) + g(x),

f22 = a4 f (x),

(4)

where f = f (x) and g = g(x) are arbitrary functions and a1 , a2 , a3 , and a4 are arbitrary numbers. With these assumptions, the general solution of the linear homogeneous system (1), (2) has the form  Z   Z  ϕ1 = C1 exp λ1 f dx + C2 exp λ2 f dx ,    Z  Z (5) a2 + λ2 a2 + λ1 ϕ2 = – C1 exp λ1 f dx – C2 exp λ2 f dx , a4 a4

where λ1 and λ2 are the roots of the quadratic equation (λ + a2 )(λ + a3 ) – a1 a4 = 0.

3◦ . Let h1 = h1 (x) and h2 = h2 (x) be arbitrary functions and let the fij = fij (x) be defined by relations (4). In this case, the general solution of the linear nonhomogeneous system of equations (1), (2) can be obtained by variation of constants or with the aid of the Wronskian determinant [see, for example, Murphy (1960), Kamke (1977), and Zwillinger (1998)].     ∂w ∂w ∂w ∂w ∂w 6. + wF x, , + yG x, , ∂x ∂y ∂z ∂y ∂z     ∂w ∂w ∂w ∂w , + R x, , = 0. + zH x, ∂y ∂z ∂y ∂z A complete integral can be sought in the form w = ϕ(x)y + ψ(x)z + χ(x).   ∂w ∂w ∂w 7. + F x, + az, + ay = 0. ∂x ∂y ∂z Z Complete integral: w = –ayz + C1 y + C2 z – F x, C1 , C2 ) dx + C3 .   ∂w ∂w ∂w 8. + F x, g1 (y) + h1 (y), g2 (z) + h2 (z) = 0. ∂x ∂y ∂z Complete integral: Z Z Z C2 – h2 (z) C1 – h1 (y) dy + dz – F (x, C1 , C2 ) dx + C3 . w= g1 (y) g2 (z)       ∂w 2 ∂w ∂w ∂w ∂w ∂w F x, , + G x, , = 0. 9. + ∂x ∂x ∂y ∂z ∂y ∂z Complete integral: Z i p 1 h w = C1 y + C2 z + C3 + –F (x, C1 , C2 ) + F 2 (x, C1 , C2 ) – 4G(x, C1 , C2 ) dx. 2

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 151

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 152

152

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES   ∂w ∂w ∂w 10. F , , = 0. ∂x ∂y ∂z Complete integral: w = C1 x + C2 y + C3 z + C4 , where the first three constants are related by the one constraint F (C1 , C2 , C3 ) = 0. Reference: E. Kamke (1965).



 ∂w ∂w ∂w ∂w ∂w ∂w 11. F , , +x +y +z = 0. ∂x ∂y ∂z ∂x ∂y ∂z Complete integral: w = ϕ(ξ) + C4 , ξ = C1 x + C2 y + C3 z, where the function ϕ = ϕ(ξ) is determined by the ordinary differential equation
F C1 ϕ′ξ , C2 ϕ′ξ , C3 ϕ′ξ + ξϕ′ξ = 0.

One of the constants C1 , C2 , or C3 can be set equal to unity.   ∂w ∂w ∂w ∂w ∂w ∂w 12. F , , +x +y +z = w. ∂x ∂y ∂z ∂x ∂y ∂z Clairaut’s equation. Complete integral: w = C1 x + C2 y + C3 z + F (C1 , C2 , C3 ). Reference: E. Kamke (1965).



,

∂w

,

∂w



+y

∂w

+z

∂w

= w. ∂x ∂y ∂z ∂y ∂z Complete integral: w = C2 y + C3 z + ϕ(x), where the function ϕ(x)
= ϕ(x, C1 , C2 , C3 ) is determined by the ordinary differential equation F x, ϕ′x , C2 , C3 = ϕ.   ∂w ∂w ∂w 14. F ax + by + cz, , , = 0. ∂x ∂y ∂z Complete integral: 13.

F x,

∂w

w = C1 x + C2 y + C3 + ϕ(ξ, C1 , C2 ),

ξ = ax + by + cz,

where the function ϕ = ϕ(ξ, C1 , C2 ) is determined by the ordinary differential equation
F ξ, aϕ′ξ + C1 , bϕ′ξ + C2 , cϕ′ξ = 0. ∂w

∂w

∂w



+ ayz, + axz, + axy = 0. ∂x ∂y ∂z Complete integral: w = –axyz + C1 y + C2 z + C3 z + C4 , where F (C1 , C2 , C3 ) = 0.   ∂w ∂w ∂w ∂w ∂w ∂w 16. F ax + by + cz, , , +x +y +z = w. ∂x ∂y ∂z ∂x ∂y ∂z Complete integral: 15.

F



w = C1 x + C2 y + C3 z + ϕ(ξ),

ξ = ax + by + cz,

where the function ϕ = ϕ(ξ) is determined by the ordinary differential equation
F ξ, aϕ′ξ + C1 , bϕ′ξ + C2 , cϕ′ξ + C3 + ξϕ′ξ = ϕ.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 152

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.3. NONLINEAR EQUATIONS WITH THREE VARIABLES CONTAINING ARBITRARY FUNCTIONS



∂w

∂w

153



, , = 0. ∂x ∂y ∂z Complete integral: w = ϕ(ξ), ξ = C1 x + C2 y + C3 z, where the function ϕ(ξ) is determined by solving the first-order autonomous ordinary
differential equation F ϕ, C1 ϕ′ξ , C2 ϕ′ξ , C3 ϕ′ξ = 0. 17.

F w,

∂w

Page 153

Reference: E. Kamke (1965).



∂w

∂w

∂w



, , = 0. ∂x ∂y ∂z For k = 0, see equation 4.3.4.14. For k ≠ 0, the substitution ku = ax + by + cz + kw leads  ∂u a ∂u b ∂u c  to an equation of the form 4.3.4.17: F ku, – , – , – = 0. ∂x k ∂y k ∂z k   ∂w ∂w ∂w 19. F x, , + az, + ay = 0. ∂x ∂y ∂z Complete integral: w = –ayz +C1 y +C2 z +ϕ(x)+C3 , where the function ϕ(ξ) is determined
by the first-order ordinary differential equation F x, ϕ′x , C1 , C2 = 0.   ∂w ∂w ∂w ∂w ∂w ∂w 20. F w, , , ,x +y +z = 0. ∂x ∂y ∂z ∂x ∂y ∂z Complete integral: w = ϕ(ξ), ξ = C1 x + C2 y + C3 z, where the function ϕ(ξ) is determined by the first-order ordinary differential equation
F ϕ, C1 ϕ′ξ , C2 ϕ′ξ , C3 ϕ′ξ , ξϕ′ξ = 0.      ∂w ∂w ∂w 21. F x, , G y, , H z, = 0. ∂x ∂y ∂z Complete integral: w = ϕ(x, C1 , C2 ) + ψ(y, C1 ) + χ(z, C2 ) + C3 . The functions ϕ, ψ, and χ are determined by the ordinary differential equations F (x, ϕ′x , C1 , C2 ) = 0, G(y, ψy′ ) = C1 , H(z, χ′z ) = C2 . Solving these equations for the derivatives, we obtain linear separable equations, which are easy to integrate.       ∂w ∂w ∂w 22. F G H x, , y, , z, = 0. ∂x ∂y ∂z Complete integral: w = ϕ(x, C1 ) + ψ(y, C2 , C1 ) + χ(z, C2 ) + C3 . The functions ϕ, ψ, and χ are determined by the ordinary differential equations


H x, ϕ′x = C1 , G C1 , y, ψy′ = C2 , F C2 , z, χ′z = 0. Solving these equations for the derivatives, we obtain linear separable equations, which are easy to integrate. 18.

F ax + by + cz + kw,

Reference: A. P. Markeev (1990).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 153

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

154

Page 154

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES

4.4. Nonlinear Equations with Four Independent Variables ⊲ This section deals with specific nonlinear equations with four independent variables which contain arbitrary parameters. For nonlinear equations containing arbitrary functions, see Section 4.5 with n = 4.

4.4.1. Equations Quadratic in Derivatives 1.

∂w

+a



∂w

∂x1 ∂x2 Complete integral:

2

    ∂w 2 ∂w 2 m +b +c = kxn 1 w + sx1 . ∂x3 ∂x4

w = C1 F (x1 ) + (C2 x2 + C3 x3 + C4 x4 )F (x1 ) + F (x1 ) 

 k n+1 where F (x) = exp x , λ = aC22 + bC32 + cC42 . n+1 1

Z

2 sxm 1 – λF (x1 ) dx1 , F (x1 )

      ∂w 2 ∂w 2 ∂w 2 ∂w +a +b +c = kw 2 + sxm 2. 1 w. ∂x1 ∂x2 ∂x3 ∂x4 This is a special case of equation 4.5.1.2. ∂w

1



∂w

2

1



∂w

2



1

∂w

∂w

2

–cos x3 +k cos x3 = 0. ∂x2 2a sin2 x3 ∂x4 This equation governs the motion of a rigid body with one fixed point in the Lagrange– Poisson case (x1 is time; x2 , x3 , and x4 are Eulerian angles; w is the Hamiltonian function; a = b ≠ c are mass moments of inertia; and k is the body weight times the distance from the center of mass to the fixed point). Complete integral: Z s a (C3 – C2 cos x3 )2 w = –C1 x1 + C2 x2 + C3 x4 + C4 ± 2aC1 – 2ak cos x3 – C22 – dx3 . c sin2 x3 3.

+ ∂x1 2c

∂x2

+

2a

∂x3

+

Reference: E. N. Berezkin (1962).

4.

∂w 1 + ∂x1 2



∂w ∂x2

2

+

1 2x22



∂w ∂x3

2

+

1 2x22 sin2 x3



∂w ∂x4

2

–

a = 0. x2

This equation governs the motion of a planet in a central Newtonian gravitational field (x1 is time, x2 is the radial coordinate, x3 and x4 are the angular coordinates, and w is the Hamiltonian function). Complete integral: v s Z u Z 2 u C C22 2a w = –C1 x1 + C2 x4 + C3 ± t2C1 + – 24 dx2 ± C42 – dx3 . x2 x2 sin2 x3 Reference: P. Appell (1953) and E. N. Berezkin (1962).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 154

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

155

4.4. NONLINEAR EQUATIONS WITH FOUR INDEPENDENT VARIABLES

5.

∂w ∂x1

+

axk 1



∂w ∂x2

2

+

bxn 1



∂w ∂x3

2

+



cxm 1

2

∂w ∂x4

Page 155

= sxq1 w.

Complete integral: w = ϕ(x1 )(C1 + C2 x2 + C3 x3 + C4 x4 ) – ϕ(x1 )   s q+1 x1 . where ϕ(x1 ) = exp q+1 6.

∂w ∂x1

2 + a 2 xk 1



∂w ∂x2

2

3 + a 3 xk 1



∂w

Z


ϕ(x1 ) aC22 xk1 + bC32 xn1 + cC42 xm 1 dx1 ,

2

4 + a 4 xk 1



2

4 a 4 xk 4



∂w

2

∂x3 ∂x4 m3 m4 n 2 = bx1 w + c2 xm 1 x2 + c 3 x1 x3 + c 4 x1 x4 .

This is a special case of equation 4.6.1.5. 7.

∂w ∂x1

+

2 a 2 xk 2



∂w ∂x2

2

+

3 a 3 xk 3



∂w ∂x3

+

∂w

2

∂x4 n3 n2 n4 1 = c 1 xn 1 + c 2 x2 + c 3 x3 + c 4 x4 .

This is a special case of equation 4.2.2.6. 8.

∂w + a1 eλ1 x1 ∂x1



∂w ∂x2

2

λ2 x1

+ a2 e



∂w ∂x3

2

λ3 x1

+ a3 e



∂w ∂x4

2

= beβx1 w.

Complete integral: Z
w = ϕ(x1 )(C1 +C2 x2 +C3 x3 +C4 x4 )–ϕ(x1 ) ϕ(x1 ) a1 C22 eλ1 x1 +a2 C32 eλ2 x1 +a3 C42 eλ3 x1 dx1 ,

  b βx1 e . where ϕ(x1 ) = exp β 9.

∂w ∂x1

n2 2 + a 2 xk 2 w



∂w ∂x2

2

n3 3 + a 3 xk 3 w



∂w ∂x3

2

n4 4 + a 4 xk 4 w



∂w ∂x4

2

= 0.

Complete integral: C1 – x1 +

10.

∂w ∂x1

2 3 4 2C2 2–k 2C3 2–k 2C4 2–k x2 2 + x3 2 + x4 2 2 – k2 2 – k3 2 – k4 a2 C22 n +1 a3 C32 n +1 a4 C42 n +1 = w 2 + w 3 + w 4 . n2 + 1 n3 + 1 n4 + 1

+

n2 2 a 2 xk 2 w



∂w ∂x2

2

+

n3 3 a 3 xk 3 w



∂w ∂x3

2

+

n4 4 a 4 xk 4 w



∂w ∂x4

2

= bw m.

This is a special case of equation 4.6.1.15.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 155

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

156

Page 156

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES

4.4.2. Equations Contain Power-Law Functions of Derivatives 1.

∂w ∂x1

+

s

k+a



∂w ∂x2

2

    ∂w 2 ∂w 2 +b +c = 0. ∂x3 ∂x4

Complete integral: w = –Ax1 + C1 x2 + C2 x3 + C3 x4 + C4 ,

2.

∂w ∂x1

+ a1

s

1 + b1



∂w ∂x2

2

+ a2

s

1 + b2

A=



∂w ∂x3

2

q

k + aC12 + bC22 + cC32 .

+ a3

s

1 + b3



∂w ∂x4

2

= 0.

Complete integral: where A = a1 3.

p

w = –Ax1 + C1 x2 + C2 x3 + C3 x4 + C4 , p p 1 + b1 C12 + a2 1 + b2 C22 + a3 1 + b3 C32 .

∂w ∂w ∂w ∂w

∂x1 ∂x2 ∂x3 ∂x4

Complete integral: w =

4.

n m s = axk 1 x2 x3 x4 .

C1 k+1 C2 n+1 C3 m+1 a x1 + x2 + x3 + xs+1 . k+1 n+1 m+1 C1 C2 C3 (s + 1) 4

 k1  k2  k3 ∂w ∂w ∂w ∂w + a1 + bx3 x4 + a2 + bx2 x4 + a3 + bx2 x3 = cxn 1. ∂x1 ∂x2 ∂x3 ∂x4

Complete integral:
w = –bx2 x3 x4 + C1 x2 + C2 x3 + C3 x4 – a1 C1k1 + a2 C2k2 + a3 C3k3 x1 + 5.

1 a 1 xk 1



∂w ∂x1

k1

+

2 a 2 xk 2



∂w ∂x2

k2

3 a 3 xk 3



3 + a 3 xk 3



+

∂w

c xn+1 + C4 . n+1 1

k3

4 a 4 xk 4



k3

4 + a 4 xk 4



∂w

k4

+ ∂x3 ∂x4 n2 n4 1 3 = b1 xn + b x + b3 xn 2 2 1 3 + b4 x4 .

This is a special case of equation 4.6.2.6. 6.

1 a 1 xk 1



∂w ∂x1

k1

2 + a 2 xk 2



∂w ∂x2

k2

∂w ∂x3

∂w ∂x4

k4

= bw n.

This is a special case of equation 4.6.2.7. 7.

a1



∂w ∂x1

+bx2

k1

+a2



∂w ∂x2

+bx1

k2

+a3



∂w ∂x3

+cx4

k3

+a4



∂w ∂x4

+cx3

k4

= s.

Complete integral: w = –bx1 x2 – cx3 x4 + C1 x1 + C2 x2 + C3 x3 + C4 x4 + C5 , where a1 C1k1 + a2 C2k2 + a3 C3k3 + a4 C4k4 = s.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 156

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

157

4.4. NONLINEAR EQUATIONS WITH FOUR INDEPENDENT VARIABLES

8.

a



∂w

∂x1

k

∂w

∂x2

n

Page 157

    ∂w m ∂w s ∂w ∂w ∂w ∂w +b +x1 +x2 +x3 +x4 = w. ∂x3 ∂x4 ∂x1 ∂x2 ∂x3 ∂x4

Complete integral: w = C1 x1 + C2 x2 + C3 x3 + C4 x4 + aC1k C2n + bC3m C4s . 

∂w 9. a1 + bx2 ∂x1 Complete integral:

k1 

∂w + bx1 ∂x2

k2

+ a2



∂w + cx4 ∂x3

n1 

∂w + cx3 ∂x4

n2

= s.

w = –bx1 x2 – cx3 x4 + C1 x1 + C2 x2 + C3 x3 + C4 x4 + C5 , where a1 C1k1 C2k2 + a2 C3n1 C4n2 = s. 

∂w

+ bx2 ∂x1 Complete integral: 10.

a1

k1 

∂w ∂x3

+ cx4

k2

+ a2



∂w ∂x2

+ bx1

n1 

∂w ∂x4

+ cx3

n2

= s.

w = –bx1 x2 – cx3 x4 + C1 x1 + C2 x2 + C3 x3 + C4 x4 + C5 , where a1 C1k1 C3k2 + a2 C2n1 C4n2 = s. 11.



∂w

k1 

∂w

k2 

∂w

k3 

∂w

∂x1 ∂x2 ∂x3 ∂x4 This is a special case of equation 4.6.2.4.

k4

m 1 n2 n3 n4 = axn 1 x2 x3 x4 w .



       ∂w k1 ∂w k2 ∂w k3 ∂w k4 ∂w ∂w ∂w ∂w 12. a = x1 +x2 +x3 +x4 . ∂x1 ∂x2 ∂x3 ∂x4 ∂x1 ∂x2 ∂x3 ∂x4 Complete integral: w=


k1 +k2 +k3 +k4 k1 + k2 + k3 + k4 – 1 C1 x1 + C2 x2 + C3 x3 + Ax4 k1 +k2 +k3 +k4 –1 + C4 , k1 + k2 + k3 + k4

where A = aC1k1 C2k2 C3k3 13.

a



∂w

∂x1

k1 

∂w

∂x2


–1/k4

k2 

.

∂w

∂x3

k3 

∂w

∂x4

k4

+x1

∂w ∂x1

+x2

∂w ∂x2

+x3

∂w ∂x3

+x4

∂w ∂x4

= w.

Complete integral: w = C1 x1 + C2 x2 + C3 x3 + C4 x4 + aC1k1 C2k2 C3k3 C4k4 . 

∂w 14. + ax2 ∂x1 Complete integral:

k1 

∂w + ax1 ∂x2

k2 

∂w + bx4 ∂x3

k3 

∂w + bx3 ∂x4

k4

= c.

w = –ax1 x2 – bx3 x4 + C1 x1 + C2 x2 + C3 x3 + C4 x4 + C5 , where C1k1 C2k2 C3k3 C4k4 = c.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 157

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

158

Page 158

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES

4.5. Nonlinear Equations with Arbitrary Number of Variables Containing Arbitrary Parameters 4.5.1. Equations Quadratic in Derivatives

1.

    ∂w ∂w 2 ∂w 2 + a2 + · · · + an = bw + cxk 1. ∂x1 ∂x2 ∂xn

This is a special case of equation 4.6.1.1 with f (x1 ) = b and g(x1 ) = cxk1 . Complete integral: bx1

w = (C1 + C2 x2 + · · · + Cn xn )e 2.

Z 1 2 2 2bx1 bx1 e–bx1 xk1 dx1 . – (a2 C2 + · · · + an Cn )e + ce b

    ∂w ∂w 2 ∂w 2 + a2 + · · · + an = bw 2 + cxk 1 w. ∂x1 ∂x2 ∂xn

This is a special case of equation 4.6.1.2 with f (x1 ) = cxk1 and g(x1 ) = 0. Complete integral: w = ϕ(x1 ) + C1 ψ(x1 ) exp(C2 x2 + · · · + Cn xn ). The arbitrary constants C2 , . . . , Cn are related by the one constraint a2 C22 + · · · + an Cn2 = b. The function ϕ(x) and ψ(x) are given by  –1 Z ϕ(x) = F (x) Cn+1 – b F (x) dx ,  Z   k ψ(x) = exp 2bϕ(x) + cx dx . 3.

∂w ∂x1

+

2 a 2 xk 1



∂w ∂x2

2

+ ··· +

n a n xk 1



F (x) = exp

∂w ∂xn

2



 c xk+1 , k+1

= bw.

Complete integral: Z
w = C1 ebx1 + ebx1 (C2 x2 + · · · + Cn xn ) – ebx1 ebx1 a2 C22 xk1 2 + · · · + an Cn2 xk1 n dx1 . 4.

∂w ∂x1

+

2 a 2 xk 1



∂w ∂x2

2

+ ··· +

n a n xk 1



∂w ∂xn

2

= bxm 1 w.

This is a special case of equation 4.6.1.4 with fi (x1 ) = ai xk1 i and g(x1 ) = bxm 1 . Complete integral: w = ϕ(x1 )(C1 + C2 x2 + · · · + Cn xn ) – ϕ(x1 )  where ϕ(x1 ) = exp

 b m+1 x . m+1 1

Z


ϕ(x1 ) a2 C22 xk1 2 + · · · + an Cn2 xk1 n dx1 ,

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 158

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.5. NONLINEAR EQUATIONS WITH ARBITRARY NUMBER OF VARIABLES CONTAINING PARAMETERS

5.

∂w ∂x1

2 + a 2 xk 1



∂w ∂x2

2

n + · · · + a n xk 1



2

∂w ∂xn

Page 159

159

mn 2 = βw + b2 xm 1 x2 + · · · + bnx1 xn.

i This is a special case of equation 4.6.1.5 with fi (x1 ) = ai xk1 i , g(x1 ) = β, and hi (x1 ) = bi xm . 1 Complete integral: w = x2 ϕ2 (x1 ) + · · · + xn ϕn (x1 ) + ψ(x1 ),

where the functions ϕk (x1 ) and ψ(x1 ) are given by Z k ϕk (x1 ) = Ck eβx1 + bk eβx1 e–βx1 xm dx1 , k = 2, . . . , n; 1 Z   ψ(x1 ) = C1 eβx1 – eβx1 e–βx1 a2 xk1 2 ϕ22 (x1 ) + · · · + an xk1 n ϕ2n (x1 ) dx1 .

∂w 2 6. + a 2 xk 2 ∂x1 Complete integral:



∂w ∂x2

2

+ ··· +

n a n xk n



∂w ∂xn

2

β

mn 2 = cx1 + b2 xm 2 + · · · + bnxn .

v Z u Z s m2 n u C + b x c Cn + bn xm 2 2 n 2 t xβ+1 + dx +· · ·+ dxn . w = C1 –(C2 +· · ·+Cn )x1 + 2 β+1 1 an xknn a2 xk2 2 7.

∂w ∂x1

λ2 x1

+ a2 e



∂w ∂x2

2

λn x1

+ · · · + a ne



∂w ∂xn

2

= beβx1 w.

This is a special case of equation 4.6.1.4 with fi (x1 ) = ai eλi x1 and g(x1 ) = beβx1 . Complete integral: Z
w = ϕ(x1 )(C1 + C2 x2 + · · · + Cn xn ) – ϕ(x1 ) ϕ(x1 ) a2 C22 eλ2 x1 + · · · + an Cn2 eλn x1 dx1 ,   b βx1 where ϕ(x1 ) = exp e . β 8.

∂w ∂x1

λ2 x1

+ a2 e



∂w ∂x2

2

λn x1

+ · · · + a ne



∂w

2

∂xn = cw + b2 eβ2 x1 x2 + · · · + bneβn x1 xn.

This is a special case of equation 4.6.1.5 with fi (x1 ) = ai eλi x1 , g(x1 ) = c, and hi (x1 ) = bi eβi x1 . 9.

∂w ∂x1

+

n X k=2

k mk a k xβ 1 xk

+


k –2 bkxγ1 k x2m k



∂w ∂xk

2

µ0 1 = c 1 xµ 1 w + c 0 x1 .

This is a special case of equation 4.6.1.9 with fk (x1 ) = ak xβ1 k , gk (x1 ) = bk xγ1 k , a(x1 ) = c1 xµ1 1 , and b(x1 ) = c0 xµ1 0 . 10.

n n X X ∂w ∂w ∂w β γ + akmx1 km = bw + ckmx1 km xkxm. ∂x1 k,m=2 ∂xk ∂xm k,m=2

This is a special case of equation 4.6.1.10.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 159

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

160

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES ∂w

11.

Page 160

∂x1

+

β 2 a 2 xk 1 w



2

∂w ∂x2

+ ··· +

β n a n xk 1 w



∂w ∂xn

2

= bxm 1 w.

1 wβ+1 leads to an equation of the form 4.5.1.4: β+1     ∂u ∂u 2 ∂u 2 k2 kn + a2 x1 + · · · + an x1 = b(β + 1)xm 1 u. ∂x1 ∂x2 ∂xn

The substitution u =

∂w

12.

∂x1

+

k 2 a 2 xm 1 w



∂w ∂x2

2

+ ··· +

k n a n xm 1 w



∂w ∂xn

2

= bw k+2 + cxs1 w.

i and h(x1 ) = cxs1 . This is a special case of equation 4.6.1.13 with fi (x1 ) = ai xm 1

∂w m2 2 + a 2 xk 2 w ∂x1

13.



∂w ∂x2

2

+ ··· +

mn n a n xk n w



∂w ∂xn

2

= 0.

This is a special case of equation 4.6.1.14 with fi (xi ) = ai xki i , gi (w) = wmi . Complete integral for ki ≠ 2 and mi ≠ –1: x1 +

14.

2–k2 2–kn a2 (2 – k2 )2 C22 m +1 an (2 – kn )2 Cn2 mn +1 w 2 +· · ·+ w = C1 + 2C2 x2 2 + · · · + 2Cn xn 2 . m2 + 1 mn + 1

a1



∂w ∂x1

2

+ a2



Complete integral: w =

2

+ · · · + an

4ak


2 xk + Ck .

∂w

∂x2 n X 1 k=1



∂w ∂xn

2

= w.

Reference: A. M. Vinogradov and I. S. Krasilshchik (1997).

15.

1 a 1 xk 1



∂w ∂x1

2

+ ··· +

n a n xk n



∂w ∂xn

2

mn 1 = b1 xm 1 + · · · + bnxn .

i This is a special case of equation 4.6.1.16 with fi (xi ) = ai xki i and gi (xi ) = bi xm i .

16.

1 a 1 xk 1



∂w ∂x1

2

+ ··· +

n a n xk n



∂w ∂xn

2


β mn 1 = b1 xm w . 1 + · · · + bnxn

β i This is a special case of equation 4.6.1.18 with fi (xi ) = ai xki i , gi (xi ) = bi xm i , and h(w) = w .

17.

λ1 x1

a1 e



∂w ∂x1

2

λn xn

+ · · · + a ne



∂w ∂xn

2

= b1 eβ1 x1 + · · · + bneβn xn .

This is a special case of equation 4.6.1.16 with fk (xk ) = ak eλk xk and gk (xk ) = bk eβk xk . 18.

λ1 x1

a1 e



∂w ∂x1

2

λn xn

+ · · · + a ne



∂w ∂xn

2


= b1 eβ1 x1 + · · · + bneβn xn eγw.

This is a special case of equation 4.6.1.18 with fk (xk ) = ak eλk xk , gk (xk ) = bk eβk xk , and h(w) = eγw .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 160

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.5. NONLINEAR EQUATIONS WITH ARBITRARY NUMBER OF VARIABLES CONTAINING PARAMETERS

Page 161

161

4.5.2. Equations with Power-Law Nonlinearity in Derivatives 1.

∂w ∂w

∂w

...

∂x1 ∂x2 ∂xn Complete integrals:

= x1 x2 . . . x n .

1 2

w=

(a)

n–1 X

Ck x2k + 12 Ax2n + Cn ,

A=

k=1

1 , C1 C2 . . . Cn–1

 n n–1 Y n (w – Cn ) = (x2n – A) (x2k – Ck ). 2 n

(b)

k=1

2.

∂w ∂w

∂w

...

∂x1 ∂x2 ∂xn Complete integral:

kn m 1 k2 = axk 1 x2 . . . x n w .

n–m n C1 k1 +1 Cn–1 kn–1 +1 a w n = x + ··· + x + xkn +1 + Cn . n–m k1 + 1 kn–1 + 1 (kn + 1)C1 C2 . . . Cn–1

3.

∂w ∂w

∂w

...

∂x1 ∂x2 ∂xn Complete integral: w=

= x1

∂w ∂x1

+ x2

∂w ∂x2

+ · · · + xn

∂w ∂xn

.

1 n n–1 (C1 . . . Cn ) 1–n (C1 x1 + · · · + Cn xn ) n–1 + Cn+1 . n

One of the constants C1 , . . . , Cn can be set equal to unity. 4.

∂w ∂w ∂x1 ∂x2

...

∂w ∂xn

=

Complete integral: ln |w| =



a1

n P

∂w ∂x1

–w



a2

∂w ∂x2





– w . . . an

∂w ∂xn

 –w .

Ck xk + Cn+1 , where the arbitrary constants C1 , . . . , Cn are

k=1

related by the one constraint C1 C2 . . . Cn = (a1 C1 – 1)(a2 C2 – 1) . . . (an Cn – 1). 5.

m1 1 a 1 xk 1 w



∂w ∂x1

β

+ ··· +

mn n anxk n w



β

= bw γ .

mn

= w.

∂w ∂xn

This is a special case of equation 4.6.2.7. 6.

a1



∂w ∂x1

m1

+ a2



∂w ∂x2

m2



+ · · · + an

∂w ∂xn

Complete integral: w=

n X k=1

bk xk + Ck




mk mk –1 ,

where

bk =

1 ak1–mk



mk mk – 1



mk 1–mk

.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 161

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

162 7.

Page 162

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES       ∂w ∂w m1 ∂w m2 ∂w mn w= xk +a ... . ∂xk ∂x1 ∂x2 ∂xn k=1 n X

Complete integral: w=

n X

Ck xk + aC1m1 C2m2 . . . Cnmn .

k=1

⊲ Complete integrals of some other equations containing arbitrary parameters can be obtained by utilizing the results of Section 4.6, where equations containing arbitrary functions are discussed.

4.6. Nonlinear Equations with Arbitrary Number of Variables Containing Arbitrary Functions 4.6.1. Equations Quadratic in Derivatives ∂w



∂w

+ a2 ∂x1 ∂x2 Complete integral:

1.

2

+ · · · + an



∂w ∂xn

2

= f (x1 )w + g(x1 ).

w = C1 F (x1 ) + (C2 x2 + · · · + Cn xn )F (x1 ) + F (x1 ) Z  where F (x) = exp f (x) dx , b = a2 C22 + · · · + an Cn2 .

Z

g(x1 ) – bF 2 (x1 ) dx1 , F (x1 )

    ∂w ∂w 2 ∂w 2 2. + a2 + · · · + an = bw 2 + f (x1 )w + g(x1 ). ∂x1 ∂x2 ∂xn 1◦ . For b = 0, see equation 4.6.1.1. Complete integral for b ≠ 0: w = ϕ(x1 ) + C1 ψ(x1 ) exp(C2 x2 + · · · + Cn xn ).

The arbitrary constants C2 , . . . , Cn are related by the one constraint a2 C22 + · · · + an Cn2 = b. The function ϕ = ϕ(x) is determined by the ordinary differential equation ϕ′x = bϕ2 + f (x)ϕ + g(x), and the function ψ = ψ(x) is expressed via ϕ(x) as Z    ψ(x) = exp 2bϕ(x) + f (x) dx .

(1)

(2)

Riccati’s equation (1) can be integrated by quadrature for many f and g; for details, see the books Kamke (1976) and Polyanin and Zaitsev (2003). Consider two special cases. 2◦ . For g(x) ≡ 0 and arbitrary f (x), equation (1) has the general solution  –1 Z  Z ϕ(x) = F (x) Cn+1 – b F (x) dx , F (x) = exp f (x) dx .

3◦ . For f (x) = f = const and g(x) = g = const, equation (1) is separable. Integrating yields Z dϕ = x + Cn+1 . 2 bϕ + f ϕ + g

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 162

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.6. EQUATIONS WITH ARBITRARY NUMBER OF VARIABLES CONTAINING ARBITRARY FUNCTIONS

3.

∂w ∂x1

+ f2 (x1 )



∂w ∂x2

2

+ · · · + fn(x1 )



∂w ∂xn

2

Page 163

163

= aw.

Complete integral: w = C1 eax1 + eax1 (C2 x2 + · · · + Cn xn ) – eax1 + · · · + fn(x1 )



w = ϕ(x1 )(C1 + C2 x2 + · · · + Cn xn ) – ϕ(x1 )

Z

4.

∂w ∂x1

+ f2 (x1 )



Z

∂w ∂x2

2

  eax1 C22 f2 (x1 ) + · · · + Cn2 fn (x1 ) dx1 .

∂w ∂xn

2

= g(x1 )w.

Complete integral:

Z  g(x1 ) dx1 . where ϕ(x1 ) = exp 5.

∂w ∂x1

+ f2 (x1 )



∂w ∂x2

2

+ · · · + fn(x1 )



  ϕ(x1 ) C22 f2 (x1 ) + · · · + Cn2 fn (x1 ) dx1 ,

∂w

2

∂xn = g(x1 )w + h2 (x1 )x2 + · · · + hn(x1 )xn.

Complete integral: w = x2 ϕ2 (x1 ) + · · · + xn ϕn (x1 ) + ψ(x1 ), where the functions ϕk (x1 ) and ψ(x1 ) are given by Z ϕk (x1 ) = Ck G(x1 ) + G(x1 ) Z ψ(x1 ) = C1 G(x1 ) – G(x1 ) 6.

∂w ∂x1

+ f2 (x2 )



∂w ∂x2

2

Z  hk (x1 ) dx1 , G(x1 ) = exp g(x1 ) dx1 , k = 2, . . . , n; G(x1 )   dx1 f2 (x1 )ϕ22 (x1 ) + · · · + fn (x1 )ϕ2n (x1 ) . G(x1 )

+ · · · + fn(xn)



∂w ∂xn

2

= g(x1 ) + h2 (x2 ) + · · · + hn(xn).

Equations of this form are often encountered in mechanics; the variable x1 plays the role of time, and x2 , . . . , xn are generalized coordinates. Complete integral: w = C1 –(C2 +· · ·+Cn)x1 + ∂w



∂w

Z

g(x1) dx1 +

2

Z s

C2 +h2(x2) dx2 +· · ·+ f2(x2) 

∂w

Z s

Cn +hn(xn) dxn. fn(xn)

2

+ · · · + fn(xn) = bw 2 + g(x1 )w + h(x1 ). ∂x2 ∂xn Z dxk The transformation ξk = (k = 2, . . . , n) leads to an equation of the form p |fk (xk )| 4.6.1.2. 7.

∂x1

+ f2 (x2 )

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 163

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

164 8.

Page 164

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES   n n X  ∂w 2 ∂w X + fk(x1 )xk + gk(x1 ) = a(x1 )w + b(x1 ) + hk(x1 )xk. ∂x1 k=2 ∂xk k=2

Complete integral: w=

n X

ϕk (x1 )xk + ψ(x1 ),

k=2

where the functions ϕk (x1 ) and ψ(x1 ) are determined by the ordinary differential equations ϕ′k + fk (x1 )ϕ2k = a(x1 )ϕk + hk (x1 ), n P gk (x1 )ϕ2k = a(x1 )ψ + b(x1 ), ψ′ +

k = 2, . . . , n;

(1) (2)

k=2

where the prime denotes the derivative with respect to x1 . Riccati’s equation (1) can be integrated by quadrature for many fk (x1 ), hk (x1 ), and a(x1 ), in particular, for hk (x1 ) = 0; see the books Kamke (1976), Polyanin and Zaitsev (2003). Equation (2) is linear with respect to ψ and can be readily integrated (for known ϕk ). 9.

  n ∂w X ∂w 2 mk 2mk –2  + fk(x1 )xk + gk(x1 )xk = a(x1 )w + b(x1 ). ∂x1 k=2 ∂xk

k 1◦ . If all mk ≠ 2, then the transformation ξk = x2–m (k = 2, . . . , n) leads to an equation of k the form 4.6.1.8:

  n   ∂w 2 ∂w X 2 + (2 – mk ) fk (x1 )ξk + gk (x1 ) = a(x1 )w + b(x1 ). ∂x1 ∂ξk k=2

2◦ . If there is an ml = 2, then we introduce the new variable ξl = ln |xl | instead of xl and transform the other variables just as in item 1◦ . As a result, we obtain an equation of the form 4.6.1.8. 10.

∂w ∂x1

+

n X

fkm(x1 )

k,m=2

∂w ∂w ∂xk ∂xm

= ψ(x1 )w +

+

n X

n  X n X k=2

gkm(x1 )xm + gk(x1 )

m=2

hkm(x1 )xkxm +

k,m=2

n X



∂w ∂xk

hk(x1 )xk + h0 (x1 ).

k=2

A complete integral can be sought in the form w=

n X

k,m=2

ϕkm (x1 )xk xm +

n X

ϕk (x1 )xk + ϕ0 (x1 ).

k=2

Substituting this expression into the original partial differential equation yields a system of ordinary differential equations for ϕkm (x1 ), ϕk (x1 ), and ϕ0 (x1 ). If all gk (x1 ) ≡ 0, hk (x1 ) ≡ 0, and h0 (x1 ) ≡ 0, then one may set ϕk (x1 ) ≡ 0 and ϕ0 (x1 ) ≡ 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 164

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.6. EQUATIONS WITH ARBITRARY NUMBER OF VARIABLES CONTAINING ARBITRARY FUNCTIONS

11.

Page 165

165

   n  X ∂w 1 ∂w 2 + n gk(xk) – hk(xk) = 0. P ∂x1 ∂xk fk(xk) k=2 k=2

Liouville’s equation. Complete integral: w = –C1 x1 +

n Z X k=2

s

C1 fk (xk ) + hk (xk ) + Ck dxk + Cn+1 , gk (xk )

where the arbitrary constants C2 , . . . , Cn are related by the one constraint C2 + · · · + Cn = 0. Note that the equation in question governs the motion of a mechanical system with ideal holonomic constraints in the case where the kinetic energy T and the force function U have the form n  n  n n P 1 P P 1 P ′ 2 T = (xk ) , U = fk (xk ) hk (xk ) fk (xk ). 4 k=2 k=2 gk (xk ) k=2 k=2 Here x1 is time and x2 , . . . , xn are generalized coordinates. References: E. N. Berezkin (1968), V. V. Kozlov (1995).

12.

∂w ∂x1

k

+ w f2 (x1 )



∂w ∂x2

2

+ · · · + w fn(x1 )

2

+ · · · + w kfn(xn)

k



∂w ∂xn

2

= g(x1 )w.

2

= aw k+2 + g(x1 )w.

1 wk+1 leads to an equation of the form 4.6.1.4: k+1     ∂u ∂u 2 ∂u 2 + f2 (x1 ) + · · · + fn (x1 ) = (k + 1)g(x1 )u. ∂x1 ∂x2 ∂xn

The substitution u =

13.

∂w ∂x1

+ w kf2 (x2 )



∂w ∂x2



∂w ∂xn

1 wk+1 leads to an equation of the form 4.6.1.7 with h(x1 ) ≡ 0: k+1     ∂u 2 ∂u 2 ∂u + f2 (x2 ) + · · · + fn (xn ) = a(k + 1)2 u2 + (k + 1)g(x1 )u. ∂x1 ∂x2 ∂xn

The substitution u =

14.

    ∂w ∂w 2 ∂w 2 + f2 (x2 )g2 (w) + · · · + fn(xn)gn(w) = 0. ∂x1 ∂x2 ∂xn

Complete integral: Z Z Z  2  dx2 dxn 2 x1 + C2 g2 (w) + · · · + Cn gn (w) dw = C1 + C2 + · · · + Cn . p p f2 (x2 ) fn (xn )

It has been assumed in the products fk gk that the fk are all nonnegative and the gk can have any sign.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 165

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

166 15.

Page 166

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES ∂w ∂x1

+ f2 (x2 )g2 (w)



∂w ∂x2

2

+ · · · + fn(xn)gn(w)



∂w ∂xn

2

= h(w).

Complete integral: Z Z Z dx2 dxn 2ϕ(w) dw x1 + C2 + · · · + C + = C1 , p p p n f2 (x2 ) fn (xn ) 1 + 1 + 4h(w)ϕ(w)

where ϕ(w) = C22 g2 (w) + · · · + Cn2 gn (w). 16.



∂w f1 (x1 ) ∂x1

2



∂w + · · · + fn(xn) ∂xn

2

= g1 (x1 ) + · · · + gn(xn).

Complete integral: w=±

Z 

g1 (x1 ) + C1 f1 (x1 )

1/2

dx1 ± · · · ±

Z 

gn (xn ) + Cn fn (xn )

1/2

dxn + Cn+1 ,

where the arbitrary constants C1 , . . . , Cn are related by the one constraint C1 + · · · + Cn = 0. 17.



∂w f1 (x1 ) ∂x1

2



∂w + · · · + fn(xn) ∂xn

2

= aw + g1 (x1 ) + · · · + gn(xn).

Complete integral: w = ϕ1 (x1 , C1 ) + · · · + ϕn (xn , Cn ). The functions ϕk = ϕk (xk , Ck ) are determined by the √ ordinary differential equations fk (xk )(ϕ′k )2 – aϕk – gk (xk ) = 0. By the substitutions ψk = aϕk – gk (xk ), these equations can be reduced to Abel’s equations of the second kind for ψk [integrable cases of these equations can be found in Polyanin and Zaitsev (2003) and Zaitsev and Polyanin (1994)]. 18.

f1 (x1 )



∂w ∂x1

2

The substitution u =

Z

+ · · · + fn(xn)



∂w ∂xn

2

  = g1 (x1 ) + · · · + gn(xn) h(w).

dw leads to an equation of the form 4.6.1.16: √ h(w)

    ∂u 2 ∂u 2 f1 (x1 ) + · · · + fn (xn ) = g1 (x1 ) + · · · + gn (xn ). ∂x1 ∂xn 19.

f1 (x1 )g1 (w)



∂w ∂x1

2

+ · · · + fn(xn)gn(w)



∂w ∂xn

2

= h(w).

Complete integral: Z 

C12g1(w)+· · ·+Cn2 gn(w) h(w)

1/2

dw = C1

Z

dx1 +· · ·+Cn [f1(x1)]1/2

Z

dxn +Cn+1. [fn(xn)]1/2

One of the constants C1 , . . . , Cn can be set equal to unity.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 166

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.6. EQUATIONS WITH ARBITRARY NUMBER OF VARIABLES CONTAINING ARBITRARY FUNCTIONS

Page 167

167

4.6.2. Equations with Power-Law Nonlinearity in Derivatives ∂w ∂w ∂w ... = f1 (x1 )f2 (x2 ) . . . fn(xn). ∂x1 ∂x2 ∂xn Complete integral: Z Z Z 1 w = C1 f1 (x1 ) dx1 + · · · + Cn–1 fn–1 (xn–1 ) dxn–1 + fn (xn ) dxn + Cn . C1 C2 . . . Cn–1

1.

∂w ∂w

...

∂w

= f1 (x1 )f2 (x2 ) . . . fn(xn)ϕ(w). ∂x1 ∂x2 ∂xn Complete integral: Z Z Z dw = C1 f1 (x1 ) dx1 + · · · + Cn–1 fn–1 (xn–1 ) dxn–1 ϕ1/n (w) Z 1 fn (xn ) dxn + Cn . + C1 . . . Cn–1     ∂w ∂w ∂w ∂w ∂w ∂w 3. fn(xn) ... + · · · + fk(xk) ... ... ∂x1 ∂xn–1 ∂x1 ∂xk–1 ∂xk+1 ∂xn   ∂w ∂w + · · · + f1 (x1 ) ... = 0. ∂x2 ∂xn The large parentheses contain products of n – 1 different partial derivatives. Complete integral: Z n X w= Ck fk (xk ) dxk + Cn+1 , 2.

k=1

where the arbitrary constants C1 , . . . , Cn are related by the one constraint



C1 . . . Cn–1 + · · · + C1 . . . Ck–1 Ck+1 . . . Cn + · · · + C2 . . . Cn = 0.

     ∂w k1 ∂w k2 ∂w kn 4. ... = f1 (x1 )f2 (x2 ) . . . fn(xn)ϕ(w). ∂x1 ∂x2 ∂xn Complete integral: Z Z Z 1  –  1   1 k +···+k k n 1 1 dw = C1 dx1 + · · · + Cn ϕ(w) f1 (x1 ) fn (xn ) kn dxn + Cn+1 , where C1k1 . . . Cnkn = 1. 5.

f1 (x1 )



∂w

∂x1 Complete integral:

k1

+ · · · + fn(xn)

w = ϕ1 (x1 ) + · · · + ϕn (xn ),

where



∂w ∂xn

kn 

= w.

km – 1 ϕm (xm ) = km

Z

– k1m

fm

(xm ) dxm + Cm



km km –1

.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 167

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

168

Page 168

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES k1

∂w



∂w

kn

+ · · · + fn(xn) = g1 (x1 ) + · · · + gn(xn). ∂x1 ∂xn Complete integral:   Z  Z  g1 (x1 ) + C1 1/k1 gn (xn ) + Cn 1/kn w= dx1 + · · · + dxn + Cn+1 , f (x1 ) f (xn ) 6.

f1 (x1 )



where the arbitrary constants C1 , . . . , Cn are related by the one constraint C1 + · · · + Cn = 0. 7.

f1 (x1 )g1 (w)



∂w ∂x1

k

+ · · · + fn(xn)gn(w)



∂w ∂xn

k

= h(w).

Complete integral:  Z  k Z Z C1 g1(w)+· · · +Cnkgn(w) 1/k dx1 dxn dw = C1 +· · ·+C +Cn+1. n h(w) [f1(x1)]1/k [fn(xn)]1/k One of the constants C1 , . . . , Cn can be set equal to unity.     ∂w ∂w k ∂w k + f2 (x2 )g2 (w) + · · · + fn(xn)gn(w) = 0. ∂x1 ∂x2 ∂xn Complete integral: Z Z Z  k  1 dx2 dxn k k–1 x1 +C1 + C2 g2(w)+· · · +Cngn(w) dw = C2 +· · ·+Cn . 1/k [f2(x2)] [fn(xn)]1/k    n  X 1 ∂w mk ∂w 9. + n gk(xk) – hk(xk) = 0. P ∂x1 ∂xk fk(xk) k=2 8.

k=2

Complete integral:

1/mk n Z  X C1 fk (xk ) + hk (xk ) + Ck w = –C1 x1 + dxk + Cn+1 , gk (xk ) k=2

where the arbitrary constants C2 , . . . , Cn are related by the one constraint C2 + · · · + Cn = 0.

4.6.3. Equations Contain Arbitrary Functions of Two Variables 

     ∂w ∂w ∂w 1. f1 x1 , + f 2 x2 , + . . . + f n xn , = 0. ∂x1 ∂x2 ∂xn Complete integral: w = ϕ1 (x1 , C1 ) + ϕ2 (x2 , C2 ) + · · · + ϕn (xn , Cn ) + Cn+1 ,

where

Cn = –C1 – · · · – Cn–1 .

The functions ϕk = ϕk (xk , Ck ) are determined by the ordinary differential equations   dϕk fk xk , = Ck (k = 1, . . . , n). dxk

Solving these equations for the derivatives, one obtains linear separable equations, which are easy to integrate. Reference: E. Kamke (1965).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 168

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

169

4.6. EQUATIONS WITH ARBITRARY NUMBER OF VARIABLES CONTAINING ARBITRARY FUNCTIONS



     ∂w ∂w 2. f1 x1 , f 2 x2 , . . . f n xn , = a, ∂x1 ∂x2 ∂xn Complete integral: ∂w

w = ϕ1 (x1 , C1 ) + ϕ2 (x2 , C2 ) + · · · + ϕn (xn , Cn ) + Cn+1 ,

where

Page 169

a ≠ 0.

Cn =

a . C1 C2 . . . Cn–1

The functions ϕk = ϕk (xk , Ck ) are determined by the ordinary differential equations   dϕk = Ck fk xk , (k = 1, . . . , n). dxk Solving these equations for the derivatives, one obtains linear separable equations, which are easy to integrate. Reference: E. Kamke (1965).

3.

f1



∂w

+ a1 y1 ,

∂x1 Complete integral:

∂w ∂y1

+ a 1 x1



+ . . . + fn



∂w ∂xn

+ a nyn,

∂w ∂yn

+ a n xn



= 0.

w = –a1 x1 y1 – · · · – an xn yn + A1 x1 + · · · + An xn + B1 y1 + · · · + Bn yn , where the arbitrary constants Ak and Bm are related by the one constraint f1 (A1 , B1 ) + · · · + fn (An , Bn ) = 0.     ∂w ∂w ∂w ∂w 4. f1 + a1 y1 , + a 1 x1 . . . f n + a nyn, + anxn = b. ∂x1 ∂y1 ∂xn ∂yn The left-hand side of the equation consists of the product of n functions with complicated arguments. Complete integral: w = –a1 x1 y1 – · · · – an xn yn + A1 x1 + · · · + An xn + B1 y1 + · · · + Bn yn , where the arbitrary constants Ak and Bm are related by the one constraint f1 (A1 , B1 ) . . . fn (An , Bn ) = b.

4.6.4. Nonlinear Equations of General Form 1.

  ∂w ∂w ∂w + F x1 , , ..., = 0. ∂x1 ∂x2 ∂xn

Complete integral: w = C1 + C2 x2 + · · · + Cn xn – 

∂w

∂w

∂w

+ F x1 , , ..., ∂x1 ∂x2 ∂xn Complete integral: 2.

ax1

w = C1 e

ax1

+e



Z

F (x1 , C2 , . . . , Cn ) dx1 .

= aw.

ax1

(C2 x2 + · · · + Cn xn ) – e

Z

e–ax1 F (x1 , C2 eax1 , . . . , Cn eax1 ) dx1 .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 169

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

170 3.

Page 170

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES   ∂w ∂w ∂w + F x1 , , ..., = g(x1 )w. ∂x1 ∂x2 ∂xn

Complete integral: w = ϕ(x1 )(C1 + C2 x2 + · · · + Cn xn ) – ϕ(x1 )  Z where ϕ(x1 ) = exp g(x1 ) dx1 . 4.

∂w ∂x1



+ F x1 ,

∂w ∂x2

, ...,

∂w ∂xn



Z


dx1 F x1 , C2 ϕ(x1 ), . . . , Cn ϕ(x1 ) , ϕ(x1 )

= g(x1 )w + h2 (x1 )x2 + · · · + hn(x1 )xn.

Complete integral: w = x2 ϕ2 (x1 ) + · · · + xn ϕn (x1 ) + ψ(x1 ), where ϕk (x1 ) = Ck G(x1 ) + G(x1 ) ψ(x1 ) = C1 G(x1 ) – G(x1 )

5.

Z

Z

hk (x1 ) dx1 , G(x1 )

k = 2, . . . , n;

dx1 F (x1 , ϕ2 , . . . , ϕn ) , G(x1 )

G(x1 ) = exp

Z

 g(x1 ) dx1 .

  ∂w ∂w + wF x1 , , ..., ∂x1 ∂x2 ∂xn     n X ∂w ∂w ∂w ∂w + xk G k x1 , , ..., + H x1 , , ..., = 0. ∂x ∂x ∂x ∂x 2 n 2 n k=2 ∂w

Complete integral:

w = ϕ(x1 , C1 , . . . , Cn ) +

n X

xk ψk (x1 , C1 , . . . , Cn ),

k=2

where the functions ϕ, ψ2 , . . . , ψn are determined by solving the nonlinear system of ordinary differential equations ϕ′ + F (x1 , ψ2 , . . . , ψn )ϕ + H(x1 , ψ2 , . . . , ψn ) = 0, ψk′ + F (x1 , ψ2 , . . . , ψn )ψk + Gk (x1 , ψ2 , . . . , ψn ) = 0,

k = 2, . . . , n.

The prime denotes the derivative with respect to x1 . 6.

  ∂w ∂w ∂w + F x1 , ϕ2 (x2 ) + ψ2 (x2 ), . . . , ϕn(xn) + ψn(xn) = 0. ∂x1 ∂x2 ∂xn

Complete integral: Z Z Z C2 – ψ2 (x2 ) Cn – ψn (xn ) w= dx2 + · · · + dxn – F (x1 , C2 , . . . , Cn ) dx1 + C1 . ϕ2 (x2 ) ϕn (xn )

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 170

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.6. EQUATIONS WITH ARBITRARY NUMBER OF VARIABLES CONTAINING ARBITRARY FUNCTIONS

∂w

∂w

, ..., ∂x1 ∂xn Complete integral: 7.

F





Page 171

171

= 0. w = C1 x1 + . . . + Cn xn + Cn+1 .

The arbitrary constants C1 , . . . , Cn are related by the one constraint F (C1 , . . . , Cn ) = 0. Reference: E. Kamke (1965).



∂w



= 0. , ..., ∂x1 ∂xn Complete integral: w = C1 + C2 x2 + · · · + Cn xn + ϕ(x1 ),

8.

F x1 ,

∂w

where the function ϕ = ϕ(x1 ) is determined by the ordinary differential equation
F x1 , ϕ′ , C2 , . . . , Cn = 0. 9.



F a1 x1 + · · · + anxn,

∂w

∂x1

, ...,

Complete integral:

∂w

∂xn



w = C1 x1 + · · · + Cn xn + Cn+1 + ϕ(z),

= 0.

z = a1 x1 + · · · + an xn ,

where the function ϕ = ϕ(z) is determined by the ordinary differential equation
F z, a1 ϕ′z + C1 , . . . , an ϕ′z + Cn = 0. One of the constants C1 , . . . , Cn can be set equal to unity.   ∂w ∂w ∂w ∂w , ..., + · · · + xn = 0. 10. F + x1 ∂x1 ∂xn ∂x1 ∂xn Complete integral: w = ϕ(z) + Cn+1 ,

z = C1 x1 + · · · + Cn xn ,

where the function ϕ = ϕ(z) is determined by the ordinary differential equation
F (C1 ϕ′z , . . . , Cn ϕ′z + zϕ′z = 0.

One of the constants C1 , . . . , Cn can be set equal to ±1.   ∂w ∂w ∂w ∂w 11. F , ..., + xk+1 + · · · + xn = 0. ∂x1 ∂xn ∂xk+1 ∂xn Complete integral: w = C1 x1 + . . . + Ck xk + ϕ(z) + Cn+1 ,

z = Ck+1 xk+1 + · · · + Cn xn ,

where the function ϕ = ϕ(z) is determined by the ordinary differential equation
F (C1 , . . . , Ck , Ck+1 ϕ′z , . . . , Cn ϕ′z + zϕ′z = 0.

One of the constants Ck+1 , . . . , Cn can be set equal to ±1.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 171

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 172

172

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES   ∂w ∂w ∂w ∂w 12. F , ..., + x1 + · · · + xn = w. ∂x1 ∂xn ∂x1 ∂xn
Clairaut’s equation. Complete integral: w = C1 x1 + · · · + Cn xn + F C1 , . . . , Cn . References: E. Kamke (1965), A. M. Vinogradov and I. S. Krasilshchik (1997).



 ∂w ∂w ∂w ∂w 13. F x1 , , ..., + x2 + · · · + xn = w. ∂x1 ∂xn ∂x2 ∂xn Complete integral: w = C2 x2 +· · · +Cn xn +ϕ(x1 ). The function ϕ(x1 ) = ϕ(x
1 , C1 , . . . , Cn ) is determined by the ordinary differential equation F x1 , ϕ′ , C2 , . . . , Cn = ϕ.   ∂w ∂w ∂w ∂w F a 1 x1 + · · · + a n xn , , ..., + x1 + · · · + xn = w. ∂x1 ∂xn ∂x1 ∂xn Complete integral: 14.

w = C1 x1 + · · · + Cn xn + ϕ(z),

z = a1 x1 + · · · + an xn ,

where the function ϕ = ϕ(z) is determined by the ordinary differential equation
F z, a1 ϕ′z + C1 , . . . , an ϕ′z + Cn + zϕ′z = ϕ. 15.



F w,

∂w

∂x1 Complete integral:

, ...,

∂w

∂xn



= 0.

w = ϕ(z),

z = C1 x1 + · · · + Cn xn ,

where the function ϕ(z) is determined by the first-order autonomous ordinary differential equation
F ϕ, C1 ϕ′z , . . . , Cn ϕ′z = 0. Reference: E. Kamke (1965).



∂w

, ..., ∂x1 ∂xn ◦ 1 . Complete integral: 16.

F w,

∂w





= G x1

w = ϕ(z),

∂w ∂x1

+ x2

∂w ∂x2

+ · · · + xn

∂w ∂xn



.

z = C1 x1 + · · · + Cn xn ,

where the function ϕ(z) is determined by the first-order ordinary differential equation

F ϕ, C1 ϕ′z , . . . , Cn ϕ′z = G zϕ′z .

2◦ . If the left-hand side of the equation does not explicitly depend on w and is a homogeneous function of degree k in the derivatives, i.e., F (λp1 , . . . , λpn ) = λk F (p1 , . . . , pn ), and the function G is linear, G(ξ) = aξ, then a complete integral of the original equation is given by [Kamke (1965)]:  X k  1 n k–1 k–1 a w= Cm xm + Cn+1 . k F (C1 , . . . , Cn ) m=1

One of the constants C1 , . . . , Cn can be set equal to unity.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 172

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

4.6. EQUATIONS WITH ARBITRARY NUMBER OF VARIABLES CONTAINING ARBITRARY FUNCTIONS

17.



F a1 x1 + · · · + anxn + bw,

∂w ∂x1

, ...,

∂w ∂xn



Page 173

173

= 0.

For b = 0, see equation 4.6.4.14. For b ≠ 0, the substitution bu = a1 x1 + · · · + an xn + bw  an  ∂u a1 ∂u – , ..., – = 0. leads to an equation of the form 4.6.4.15: F bu, ∂x1 b ∂xn b
F ϕ1 (x1 , p1 ), ϕ2 (x2 , p2 ), . . . , ϕn(xn, pn) = 0,

∂w . ∂xk Here F is a composite function of the n functions ϕ1 , . . . , ϕn each of which depends on only one pair of Hamiltonian variables xk , pk (k = 1, . . . , n). Complete integral: 18.

pk =

w = w1 (x1 , C1 ) + w2 (x2 , C2 ) + · · · + wn (xn , Cn ) + Cn+1 . The arbitrary constants C1 , . . . , Cn are linked by the one constraint F (C1 , . . . , Cn ) = 0 and the functions wk = wk (xk , Ck ) are determined by the ordinary differential equations   dwk ϕk xk , = Ck ; k = 1, 2, . . . , n. dxk Solving these equations for the derivatives, one obtains linear separable equations, which are easy to integrate. References: E. Kamke (1965), A. P. Markeev (1990).


F ϕ1 (x1 , ψp1 ), ϕ2 (x2 , ψp2 ), . . . , ϕn(xn, ψpn) = 0, ∂w ∂w ψ = ψ(w), p1 = , . . . , pn = . ∂x1 ∂xn Z The substitution u = ψ(w) dw leads to an equation of the form 4.6.4.18: 19.


F ϕ1 (x1 , q1 ), ϕ2 (x2 , q2 ), . . . , ϕn (xn , qn ) = 0,

where

qk =

∂u . ∂xk

Reference: E. Kamke (1965).

20.

F1 (x1 , . . . , xk, p1 , . . . , pk) + F2 (xk+1 , . . . , xn, pk+1 , . . . , pn) = aw.

A complete integral can be represented as the sum of two functions, w = w1 (x1 , . . . , xk ) + w2 (xk+1 , . . . , xn ), which are determined by solving the two simpler equations  ∂w1    F1 (x1 , . . . , xk , q1 , . . . , qk ) – aw1 = C1 , ∂xm qm = ∂w2  F2 (xk+1 , . . . , xn , qk+1 , . . . , qn ) – aw2 = –C1 ,   ∂xm

for m = 1, . . . , k, for m = k + 1, . . . , n,

where C1 is an arbitrary constant.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 173

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

174 21.

Page 174

FIRST-ORDER NONLINEAR EQUATIONS WITH THREE OR MORE VARIABLES F1 (x1 , . . . , xk, p1 , . . . , pk) + eλw F2 (xk+1 , . . . , xn, pk+1 , . . . , pn) = 0.

A complete integral can be represented as the sum of two functions, w = w1 (x1 , . . . , xk ) + w2 (xk+1 , . . . , xn ), which are determined by solving the two simpler equations  ∂w1    e–λw1 F1 (x1 , . . . , xk , q1 , . . . , qk ) = C1 , ∂xm qm = ∂w2  eλw2 F2 (xk+1 , . . . , xn , qk+1 , . . . , qn ) = –C1 ,   ∂xm

for m = 1, . . . , k, for m = k + 1, . . . , n,

where C1 is an arbitrary constant. 22.


Fn . . . F3 (F2 (F1 (x1 , p1 ), x2 , p2 ), x3 , p3 ), . . . , xn, pn = 0,

pk =

∂w ∂xk

.

Complete integral:

w = w1 (x1 , C1 ) + w2 (x2 , C2 , C1 ) + · · · + wn–1 (xn–1 , Cn–1 , Cn–2 ) + wn (xn , Cn–1 ) + Cn . The functions wk are determined by the ordinary differential equations F1 (x1 , w1′ ) = C1 , Fk (Ck–1 , xk , wk′ ) = Ck , Fn (Cn–1 , xn , wn′ )

k = 2, . . . , n – 1,

= 0,

where wk′ stands for the derivative of wk with respect to xk . Solving these equations for the derivatives, one obtains linear separable equations, which are easy to integrate. References: A. P. Markeev (1990), V. V. Kozlov (1995).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 174

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 175

Chapter 5

Second-Order Parabolic Equations with One Space Variable 5.1. Equations with Power-Law Nonlinearities 2 5.1.1. Equations of the Form ∂w = a ∂ w2 + bw + cw 2

∂t

∂w

∂x

∂ 2w

=a – bw 2 . ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the function

1.

w1 = C12 w(C1 x + C2 , C12 t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Solutions: √ √ √ √ 6 )x2 + 12(4 – 6 )C1 x + 120(12 – 5 6 )at + 12(2 – 6 )C2 + 6C12 , √  2 x2 + C1 x + 10(3 – 6 )at + C2 √ √ √ √ a 12(4 + 6 )x2 + 12(4 + 6 )C1 x + 120(12 + 5 6 )at + 12(2 + 6 )C2 + 6C12 w= , √  2 b x2 + C1 x + 10(3 + 6 )at + C2 a 12(4 – w= b

where C1 and C2 are arbitrary constants.

3◦ . Traveling-wave solution (λ is an arbitrary constant): w = w(z),

z = x + λt,

where the function w(z) is determined by the autonomous ordinary differential equation ′′ awzz – λwz′ – bw2 = 0.

4◦ . Self-similar solution:

w = t–1 u(ξ),

ξ = xt–1/2 ,

where the function u(ξ) is determined by the ordinary differential equation au′′ξξ + 12 ξu′ξ + u – bu2 = 0 References for equation 5.1.1.1: T. Barannyk (2002), A. D. Polyanin and V. F. Zaitsev (2004, p. 1).

175

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 175

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

176

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 176

ONE SPACE VARIABLE

∂w ∂2w + aw(1 – w). = ∂t ∂x2 Fisher equation. This equation arises in heat and mass transfer, combustion theory, biology, and ecology. For example, it describes the mass transfer in a two-component medium at rest with a volume chemical reaction of quasi-first order. The kinetic function f (w) = aw(1 – w) models also an autocatalytic chain reaction in combustion theory. This is a special case of equation 5.1.3.2 with m = 2. 2.

1◦ . Suppose w(x, t) is a solution of the equation in question. Then the functions w1 = w(±x + C1 , t + C2 ), where C1 and C2 are arbitrary constants, are also solutions of the equation. 2◦ . Traveling-wave solutions (C is an arbitrary constant): √ 
–2 w(x, t) = 1 + C exp – 56 at ± 16 6a x , √ 
–2 5 1 w(x, t) = –1 + C exp – 6 at ± 6 6a x , √
1 + 2C exp – 56 at ± 16 –6a x w(x, t) =  √
2 . 1 + C exp – 56 at ± 16 –6a x

3◦ . Traveling-wave solutions:

w(x, t) = ±ξ 2 ϕ(ξ),

ξ = C1 exp

1 6

√


6a x + 56 at ,

where the function ϕ(ξ) is defined implicitly by Z dϕ ξ= – C2 , p ±(4ϕ3 – 1)

and C1 and C2 are arbitrary constants. For the upper sign, the inversion of this relation corresponds to the classical Weierstrass elliptic function, ϕ(ξ) = ℘(ξ + C3 , 0, 1). 4◦ . The substitution U = 1 – w leads to an equation of the similar form ∂U ∂2U = – aU (1 – U ). ∂t ∂x2 References for equation 5.1.1.2: R. A. Fisher (1937), M. J. Ablowitz and A. Zeppetella (1979), V. G. Danilov, V. P. Maslov, and K. A. Volosov (1995). 2 5.1.2. Equations of the Form ∂w = a ∂ w2 + b0 + b1 w + b2 w 2 + b3 w 3

∂t

∂x

∂ 2w

∂w =a – bw 3 . ∂t ∂x2 This is a special case of equation 5.1.2.5 with b0 = b1 = b2 = 0.

1.

1◦ . Suppose w(x, t) is a solution of the equation in question. Then the functions w1 = C1 w(±C1 x + C2 , C12 t + C3 ), where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 176

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

2◦ . Solutions: w(x, t) = ±

r

Page 177

177

2a 2C1 x + C2 . 2 b C1 x + C2 x + 6aC1 t + C3

3◦ . Traveling-wave solution (λ is an arbitrary constant): w = w(z),

z = x + λt,

where the function w(z) is determined by the autonomous ordinary differential equation ′′ – λwz′ – bw3 = 0. awzz

4◦ . Self-similar solution:

w = t–1/2 u(ξ),

ξ = xt–1/2 ,

where the function u(ξ) is determined by the ordinary differential equation au′′ξξ + 12 ξu′ξ + 12 u – bu3 = 0. 5◦ . Solution: w = xU (ζ),

ζ =t+

1 2 x , 6a

′′ –9abU 3 = 0. where the function U (ζ) is determined by the ordinary differential equation Uζζ

References for equation 5.1.2.1: P. A. Clarkson and E. L. Mansfield (1994), T. A. Barannyk and A. G. Nikitin (2004), G. Cicogna (2004).

∂w

∂2w

+ aw – bw 3 . ∂t ∂x2 This is a special case of equation 5.1.2.5 with b0 = b2 = 0. 2.

=

1◦ . Suppose w(x, t) is a solution of the equation in question. Then the functions w1 = ±w(±x + C1 , t + C2 ), where C1 and C2 are arbitrary constants, are also solutions of the equation (the signs are chosen arbitrarily). 2◦ . Solutions with a > 0 and b > 0: √ √

r C1 exp 12 2a x – C2 exp – 12 2a x a w= √ √


, b C1 exp 1 2a x + C2 exp – 1 2a x + C3 exp – 3 at 2 2 2 √ √

r   2C1 exp 2a x + C2 exp 12 2a x – 32 at a w= –1 , √ √

b C1 exp 2a x + C2 exp 1 2a x – 3 at + C3 2 2 where C1 , C2 , and C3 are arbitrary constants. 3◦ . Solution with a < 0 and b > 0: r |a| w= b cos


2|a| x + C1

, √ 1 3 2 2|a| x + C1 + C2 exp – 2 at sin

1 2

√

where C1 and C2 are arbitrary constants.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 177

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

178

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 178

ONE SPACE VARIABLE

4◦ . Solution with a > 0 (generalizes the first solution of Item 2◦ ): √ √ 

 w = C1 exp 12 2a x + 32 at – C2 exp – 12 2a x + 32 at U (z), √ √

z = C1 exp 12 2a x + 32 at + C2 exp – 12 2a x + 32 at + C3 ,

where C1 , C2 , and C3 are arbitrary constants, and the function U = U (z) is determined by ′′ = 2bU 3 (whose solution can be written the autonomous ordinary differential equation aUzz out in implicit form). 5◦ . Solution with a < 0 (generalizes the solution of Item 3◦ ): p

w = exp 32 at sin 12 2|a| x + C1 V (ξ), p

ξ = exp 32 at cos 12 2|a| x + C1 + C2 ,

where C1 and C2 are arbitrary constants, and the function V = V (ξ) is determined by the autonomous ordinary differential equation aVξξ′′ = –2bV 3 (whose solution can be written out in implicit form). 6◦ . See also equation 5.1.3.2 with m = 3. References for equation 5.1.2.2: F. Cariello and M. Tabor (1989), M. C. Nucci and P. A. Clarkson (1992), A. D. Polyanin and V. F. Zaitsev (2004, p. 3).

∂w ∂ 2w =a – bw 3 – cw 2 . ∂t ∂x2 This is a special case of equation 5.1.2.5 with b1 = b0 = 0. 3.

1◦ . Traveling-wave solutions: w(x, t) =



ct ±

r

b x+C 2a

–1

,

where C is an arbitrary constant. 2◦ . Solutions: w(x, t) = k

r

2a 1 ∂F , b F ∂x

where 

r

F = C1 x + kc

   2a kc c2 x + t + C3 , t + C2 exp – √ b 2b 2ab

k = ±1,

and C1 , C2 , and C3 are arbitrary constants. 3◦ . Solutions: w(x, t) = k

r

2a 1 ∂F c – , b F ∂x b

where 

     kc c2 kc c2 kc c2 F = C1 exp √ x – t + C2 √ x + t exp √ x – t + C3 , 2b b 2b 2ab 2ab 2ab

k = ±1.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 178

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 179

179

4◦ . The transformation b t = 2 τ, c

x=

√

2ab z, c

c w=– u b

leads to the equation ∂u 1 ∂ 2 u = + u2 (1 – u). ∂τ 2 ∂z 2 The Volosov’s solution of Eq. (1) in implicit form:

(1)

z = F (u, τ ), where the function F = F (u, τ ) is determined by the system of first-order partial differential equations ∂F 1 3uR 1 ∂F = + , = 2 , (2) ∂τ 2 2(u2 – u – R) ∂u u –u–R where the function R = R(u, τ ) is determined by the implicit formula u+R 3 1–u + ln = τ. R R 2

(3)

Relation (3) is the consistency condition for system (2). Remark. For overdetermined systems of two equations of the form (2) and their consistency condition, see Section 38.1. References for equation 5.1.2.3: A. D. Polyanin and V. F. Zaitsev (2004, pp. 3–4), K. A. Volosov (2007, http://eqworld.ipmnet.ru/eqarchive/view.php?id=56).

∂w

∂2w

– w(1 – w)(a – w). ∂t ∂x2 FitzHugh–Nagumo equation. This equation arises in population genetics and models the transmission of nerve impulses.

4.

=

1◦ . There are three stationary solutions: w = wk , where w1 = 0, w2 = 1, and w3 = a. The linear stability analysis shows that if –1 ≤ a < 0: if 0 < a < 1:

the solutions w = a, w = 1 are stable, the solutions w = 0, w = 1 are stable,

w = 0 is unstable; w = a is unstable.

There is a stationary nonhomogeneous solution that can be represented in implicit form (A and B are arbitrary constants): Z dw = ±x + B. q 1 4 1 1 3 2 4 w – 3 (a + 1)w + 2 aw + A 2◦ . Traveling-wave solutions (A is an arbitrary constant): w(x, t) =

1 + A exp



± 12

1 √ , 2 x + 12 (2a – 1)t

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 179

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

180

SECOND-ORDER PARABOLIC EQUATIONS w(x, t) = w(x, t) = w(x, t) = w(x, t) = w(x, t) = w(x, t) = w(x, t) = w(x, t) = w(x, t) = w(x, t) =

WITH

Page 180

ONE SPACE VARIABLE

a √ , 1 + A exp 2 a x + 12 a(2 – a)t  √  A exp ± 12 2 (1 – a)x + 12 (1 – a2 )t + a ,  √  A exp ± 12 2 (1 – a)x + 12 (1 – a2 )t + 1  1√  1 1 1 2 + 2 tanh ± 4 2 x + 4 (1 – 2a)t + A ,  1√  1 1 a + a tanh ± 4 2 ax + 14 a(a – 2)t + A , 2 2  1√  2 1 1 1 2 (1 + a) + 2 (1 – a) tanh ± 4 2 (1 – a)x + 4 (1 – a )t + A , 2a  1√ , (1 + a) – (1 – a) tanh ± 4 2 (1 – a)x + 14 (1 – a2 )t + A  1√  1 1 1 2 + 2 coth ± 4 2 x + 4 (1 – 2a)t + A ,  1√  1 1 a + a coth ± 4 2 ax + 14 a(a – 2)t + A , 2 2  1√  2 1 1 1 2 (1 + a) + 2 (1 – a) coth ± 4 2 (1 – a)x + 4 (1 – a )t + A , 2a  1√ . (1 + a) – (1 – a) coth ± 4 2 (1 – a)x + 14 (1 – a2 )t + A 

± 12

3◦ . “Two-phase” solution:

w(x, t) = z1 = ±

√ 2 2

x+

1 2

A exp(z1 ) + aB exp(z2 ) , A exp(z1 ) + B exp(z2 ) + C √

– a t, z2 = ± 22 ax + a 12 a – 1 t,

where A, B, and C are arbitrary constants.

4◦ . The solutions of Item 2◦ are special cases of the traveling-wave solution w(x, t) = w(ξ),

ξ = x + λt,

where λ is an arbitrary constant, and the function w(ξ) is determined by the autonomous ordinary differential equation ′′ wξξ – λwξ′ = w(1 – w)(a – w).

The substitution wξ′ = λy(w) leads to an Abel equation of the second kind:   ′ yyw – y = λ–2 aw – (a + 1)w2 + w3 .

The general solution of this equation with a = –1 and λ = ± √3 can be found in Polyanin 2 and Zaitsev (2003). 5◦ . Let us give two transformations that preserve the form of the original equation. The substitution u = 1 – w leads to an equation of the similar form with parameter a1 = 1 – a: ∂u ∂ 2 u = – u(1 – u)(1 – a – u). ∂t ∂x2

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 180

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 181

181

The transformation v(z, τ ) = 1 –

1 w(x, t), a

τ = a2 t,

z = ax

leads to an equation of the similar form with parameter a2 = 1 – a–1 :   1 ∂v ∂2v – v(1 – v) 1 – – v . = ∂τ a ∂z 2 Therefore, if w = w(x, t; a) is a solution of the equation in question, then the functions
w1 = 1 – w x, t; 1 – a ,
w2 = a – aw ax, a2 t; 1 – a–1

are also solutions of the equation. The aforesaid allows us to “multiply” exact solutions. References for equation 5.1.2.4: T. Kawahara and M. Tanaka (1983), M. C. Nucci and P. A. Clarkson (1992), N. H. Ibragimov (1994, p. 142), V. F. Zaitsev and A. D. Polyanin (1996, p. 392).

∂w ∂ 2w =a + b0 + b1 w + b2 w 2 + b3 w 3 . ∂t ∂x2 1◦ . Solutions are given by 5.

β ∂F w(x, t) = + λ, F ∂x

s

β=± –

2a , b3

(1)

where λ is any of the roots of the cubic equation b3 λ3 + b2 λ2 + b1 λ + b0 = 0

(2)

and the specific form of F = F (x, t) depends on the equation coefficients. Introduce the notation p1 = –3a,

p2 = β(b2 + 3b3 λ),

q1 = –

β (b2 + 3b3 λ), 2a

q2 = –

1 (3b3 λ2 + 2b2 λ + b1 ). (3) 2a

Four cases are possible. 1.1. For q2 ≠ 0 and q12 ≠ 4q2 , we have F (x, t) = C1 exp(k1 x + s1 t) + C2 exp(k2 x + s2 t) + C3 , q kn = – 12 q1 ± 12 q12 – 4q2 , sn = –kn2 p1 – kn p2 ,

(4)

where C1 , C2 , and C3 are arbitrary constants; n = 1, 2. 1.2. For q2 ≠ 0 and q12 = 4q2 , we have

F (x, t) = C1 exp(kx + s1 t) + C2 (kx + s2 t) exp(kx + s1 t) + C3 , k = – 12 q1 , s1 = – 14 p1 q12 + 12 p2 q1 , s2 = – 12 p1 q12 + 12 p2 q1 .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 181

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

182

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 182

ONE SPACE VARIABLE

1.3. For q2 = 0 and q1 ≠ 0, F (x, t) = C1 (x – p2 t) + C2 exp[–q1 x + q1 (p2 – p1 q1 )t] + C3 . 1.4. For q2 = q1 = 0, F (x, t) = C1 (x – p2 t)2 + C2 (x – p2 t) – 2C1 p1 t + C3 . Special case. Let a = 1,

b0 = 0,

b1 w + b2 w2 + b3 w3 = –bw(w – λ1 )(w – λ2 ).

By formulas (1)–(4) with λ = 0, one can obtain the solution w(x, t) = where

z1 = ± 12

C1 λ1 exp(z1 ) + C2 λ2 exp(z2 ) , C1 exp(z1 ) + C2 exp(z2 ) + C3 √

√ 1

z2 = ± 2

2b λ1 x + 12 bλ1 (λ1 – 2λ2 )t, 2b λ2 x + 12 bλ2 (λ2 – 2λ1 )t.

2◦ . There is a traveling-wave solution, w = w(x + γt). References for equation 5.1.2.5: V. G. Danilov and P. Yu. Sybochev (1991), N. A. Kudryashov (1993), P. A. Clarkson and E. L. Mansfield (1994). 2 5.1.3. Equations of the Form ∂w = a ∂ w2 + bw k + cw m + sw n

∂t

∂w

∂x

∂ 2w

=a + bw k. ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the functions

1.

w1 = C12 w(±C1k–1 x + C2 , C12k–2 t + C3 ), where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation. 2◦ . Traveling-wave solution: w = w(z),

z = x + λt,

where λ is an arbitrary constant and the function w(z) is determined by the autonomous ordinary differential equation ′′ – λwz′ + bwk = 0. awzz

3◦ . Self-similar solution:

x ξ= √ , t where the function u(ξ) is determined by the ordinary differential equation 1

w = t 1–k u(ξ),

au′′ξξ +

1 ′ 1 ξuξ + u + buk = 0. 2 k–1

4◦ . For k = 2 and k = 3, see equations 5.1.1.1 and 5.1.2.1, respectively, where other solutions can be found.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 182

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 183

183

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

∂w ∂2w + aw + bw m. = ∂t ∂x2 Kolmogorov–Petrovskii–Piskunov equation (a special case). This equation arises in heat and mass transfer, combustion theory, biology, and ecology.

2.

1◦ . Traveling-wave solutions:   2 w(x, t) = β + C exp(λt ± µx) 1–m ,   2 w(x, t) = –β + C exp(λt ± µx) 1–m ,

(1) (2)

where C is an arbitrary constant and the parameters λ, µ, and β are given by a(1 – m)(m + 3) λ= , 2(m + 1)

µ=

s

a(1 – m)2 , 2(m + 1)

β=

r

b – . a

2◦ . Solutions (1) and (2) are special cases of a wider class of solutions, the class of traveling-wave solutions: w = w(z), z = ±µx + λt. These are determined by the autonomous equation ′′ µ2 wzz – λwz′ + aw + bwm = 0.

For µ=

s

a(m + 3)2 , 2(m + 1)

λ = µ2

(3)

(m ≠ ±1, m ≠ –3)

the solution of equation (3) can be represented in parametric form as z=

m+3 ln f (ζ), m–1

where the function f (ζ) is given by Z  f (ζ) = ± C1 –

  2 w = ζ f (ζ) m–1 ,

4b ζ m+1 2 a(m – 1)

–1/2

dζ + C2 ,

and C1 and C2 are arbitrary constants. 3◦ . By the change of variable U (w) = µ2 λ–1 wz′ , equation (3) can be reduced to an Abel equation of the second kind: U Uw′ – U = a1 w + b1 wm ,

a1 = –aµ2 λ–2 ,

b1 = –bµ2 λ–2 .

The books by Polyanin and Zaitsev (2003) and Zaitsev and Polyanin (1994) present exact solutions of this equation for some values of m and a1 (b1 is any). References for equation 5.1.3.2: P. Kaliappan (1984), V. G. Danilov, V. P. Maslov, and K. A. Volosov (1995), V. F. Zaitsev and A. D. Polyanin (1996, p. 393).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 183

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

184

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 184

ONE SPACE VARIABLE

∂w ∂2w + aw + bw m + cw 2m–1 . = ∂t ∂x2 This equation arises in heat and mass transfer, combustion theory, biology, and ecology. For m = 2, see Subsection 5.1.2. 3.

1◦ . Traveling-wave solutions:   1 w(x, t) = β + C exp(λt + µx) 1–m ,

(1)

where C is an arbitrary constant and the parameters β, λ, and µ are determined by the system of algebraic equations aβ 2 + bβ + c = 0, µ2 – (1 – m)λ + a(1 – m)2 = 0,  µ2 – λ + (1 – m)[2a + (b/β) = 0.

(2) (3) (4)

The quadratic equation (2) for β can be solved independently. In the general case, system (2)–(4) gives four sets of the parameters, which generate four exact solutions of the original equation. 2◦ . Solution (1) is a special case of a wider class of traveling-wave solutions, w = w(z),

z = x + σt,

that are determined by the autonomous equation ′′ wzz – σwz′ + aw + bwm + cw2m–1 = 0.

(5)

The substitution U (w) = wz′ brings (5) to the Abel equation U Uw′ – σU + aw + bwm + cw2m–1 = 0, whose general solutions for some m (no constraints are imposed on a, b, and c) can be found in the books by Polyanin and Zaitsev (1995, 2003). 3◦ . The substitution

u = w1–m

leads to an equation with quadratic nonlinearity: ∂u ∂2u m  ∂u 2 u =u 2 + + a(1 – m)u2 + b(1 – m)u + c(1 – m). ∂t 1 – m ∂x ∂x

(6)

Solution (1) corresponds to a particular solution of (6) that has the form u = β + C exp(ωt + µx). For a = 0, equation (6) has also other traveling-wave solutions: r   c u(x, t) = (1 – m) bt ± – x + C. m Reference: V. F. Zaitsev and A. D. Polyanin (1996, pp. 393–394).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 184

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 185

185

∂w ∂2w + aw m–1 + bmw m – mb2 w 2m–1 . = ∂t ∂x2 Traveling-wave solution: w = w(z), z = x + λt, where λ is an arbitrary constant and the function w(z) is determined by the autonomous ordinary differential equation

4.

′′ – λwz′ + awm–1 + bmwm – mb2 w2m–1 = 0. (1) wzz For λ = 1, it can be shown that a one-parameter family of solutions to equation (1) satisfies the first-order equation a wz′ = w – bwm + . (2) mb Integrating (2) yields a solution in implicit form (A is any): Z dw 1 = z + A. (3) 2 m mb a + mbw – mb w In the special case a = 0, it follows from (3) that    1 w(z) = C exp (1 – m)z + b 1–m , where C is an arbitrary constant.

2 5.1.4. Equations of the Form ∂w = a ∂ w2 + f (x, t, w)

∂t

∂x

∂ 2w

∂w

+ s1 (bx + ct)k + s2 w n. =a ∂t ∂x2 This is a special case of equation 5.6.1.2 with f (z, w) = s1 z k + s2 wn . 1.

∂w ∂ 2w =a + s(w + bx + ct)k. ∂t ∂x2 This is a special case of equation 5.6.1.2 with f (z, w) = s(w + z)k . 2.

3.

∂w

=a

∂ 2w ∂x2

+ s(bx + ct)kw n.

∂t This is a special case of equation 5.6.1.2 with f (z, w) = sz k wn . ∂w

∂ 2w

+ btnxmw k. ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the function 4.

=a

w1 = C 2n+m+2 w(C k–1 x, C 2k–2 t), where C is an arbitrary constant, is also a solution of the equation. 2◦ . Self-similar solution: 2n+m+2 x w = t 2(1–k) u(ξ), ξ = √ , t where the function u = u(ξ) is determined by the ordinary differential equation 1 2n + m + 2 au′′ξξ + ξu′ξ + u + bξ m uk = 0. 2 2(k – 1)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 185

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

186

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 186

ONE SPACE VARIABLE

∂w ∂ 2w + sebx+ctw n. =a ∂t ∂x2 This is a special case of equation 5.6.1.2 with f (z, w) = sez wn .

5.

2 5.1.5. Equations of the Form ∂w = a ∂ w2 + bw k ∂w + f (w)

∂t

∂x

∂x

∂ 2w

∂w ∂w =a +b + cw + k1 w n1 + k2 w n2 . 2 ∂t ∂x ∂x This is a special case of equation 5.6.2.3 with f (t) = b. On passing from t, x to the new variables t, z = x + bt, one arrives at the simpler equation 1.

∂w ∂2w = a 2 + cw + k1 wn1 + k2 wn2 , ∂t ∂z special cases of which are discussed in Subsections 5.1.1 to 5.1.3. 2.

∂w

=

∂2w

+w

∂w

. ∂t ∂x Burgers equation. It is used for describing wave processes in gas dynamics, hydrodynamics, and acoustics. 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the function ∂x2

w1 = C1 w(C1 x + C1 C2 t + C3 , C12 t + C4 ) + C2 , where C1 , . . . , C4 are arbitrary constants, is also a solution of the equation. 2◦ . Solutions: A–x w(x, t) = , B+t 2 w(x, t) = λ + , x + λt + A 4x + 2A w(x, t) = 2 , x + Ax + 2t + B 6(x2 + 2t + A) w(x, t) = 3 , x + 6xt + 3Ax + B 2λ w(x, t) = , 1 + A exp(–λ2 t – λx)  exp A(x – λt)] – B   w(x, t) = –λ + A , exp A(x – λt) + B   w(x, t) = –λ + 2A tanh A(x – λt) + B ,     λ λx + B w(x, t) = 2 2 tanh – λx – B , λ t+A λ2 t + A   w(x, t) = –λ + 2A tan A(λt – x) + B , 2λ cos(λx + A) w(x, t) = , B exp(λ2 t) + sin(λx + A)     –1 2A (x + B)2 x+B w(x, t) = √ exp – A erf √ +C , 4(t + λ) π(t + λ) 2 t+λ

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 186

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

187

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

2 where A, B, C, and λ are arbitrary constants, and erf z ≡ √ π function (also called the probability integral).

Z

Page 187

z

exp(–ξ 2 ) dξ is the error

0

3◦ . Other solutions can be obtained using the following formula (Hopf–Cole transformation): 2 ∂u w(x, t) = , (1) u ∂x where u = u(x, t) is a solution of the linear heat equation ∂u ∂ 2 u = . ∂t ∂x2

(2)

For details about this equation, see the books Tikhonov and Samarskii (1990) and Polyanin (2002). 4◦ . Cauchy problem. Initial condition: w = f (x) at

t = 0,

Solution: w(x, t) = 2 where F (x, t) = √

–∞ < x < ∞.

∂ ln F (x, t), ∂x

  Z (x – ξ)2 1 ξ exp – – f (ξ ′ ) dξ ′ dξ. 4t 2 0 4πt –∞ Z

1

∞

5◦ . The Burgers equation is connected with the linear heat equation (2) by the B¨acklund transformation ∂u 1 – uw = 0, ∂x 2 ∂u 1 ∂(uw) – = 0. ∂t 2 ∂x Remark 1. The equation wt + wwx = wxx is often call the Burgers equation; it is obtained from the original equation by substituting –w for –w. Remark 2. The Hopf–Cole transformation (1) and equation 5.6.3.2, which is a generalized Burgers equation, were encountered much earlier in Fortsyth (1906). References for equation 5.1.5.2: E. Hopf (1950), J. Cole (1951), J. M. Burgers (1948), E. R. Benton and G. W. Platzman (1972), O. V. Rudenko and C. I. Soluyan (1975), N. H. Ibragimov (1994, pp. 180–182), V. F. Zaitsev and A. D. Polyanin (1996, pp. 395–396), M. Qin, F. Mei, and G. Fan (2007).

3.

∂w ∂t

=a

∂ 2w ∂x2

+ bw

∂w ∂x

.

Unnormalized Burgers equation. The scaling of the independent variables x = leads to an equation of the form 5.1.5.2:

a a z, t = 2 τ b b

∂w ∂ 2 w ∂w = +w . ∂τ ∂z ∂z 2

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 187

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

188

SECOND-ORDER PARABOLIC EQUATIONS

∂w ∂ 2w ∂w + bw =a + c. ∂t ∂x2 ∂x The transformation w = u(z, t) + ct,

WITH

Page 188

ONE SPACE VARIABLE

4.

z = x + 12 bct2 ,

leads to the Burgers equation 5.1.5.3: ∂u ∂u ∂2u = a 2 + bu . ∂t ∂z ∂z ∂w ∂2w ∂w + σw =a + bw. ∂t ∂x ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the functions

5.


w1 = ±w ±x – C1 σebt + C2 , t + C3 + Cbebt ,

where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation (one takes either the upper or the lower signs). 2◦ . Traveling-wave solution: w = w(z),

z = x + λt,

where λ is an arbitrary constant and the function w(z) is determined by the autonomous ordinary differential equation ′′ awzz – σwwz′ – λwz′ + bw = 0.

3◦ . Degenerate solution: w(x, t) = ∂w

∂w

=a

∂2w

+ b1 w + b0 . ∂t ∂x ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the function

6.

+ σw

b(x + C1 ) . σ(1 + C2 e–bt )


w1 = w x – C1 σeb1 t + C2 , t + C3 + Cb1 eb1 t ,

where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . The transformation w = u(z, t) –

b0 , b1

z = x+σ

b0 t, b1

leads to a simpler equation of the form 5.1.5.5: ∂u ∂u ∂2u + σu = a 2 + b1 u. ∂t ∂z ∂z

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 188

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 189

189

∂w ∂ 2w ∂w b2 + bw =a + w(w – k)(w + k). ∂t ∂x2 ∂x 9a Solution: bk k(–1 + C1 e4λx ) , λ= w= , 4λx 2λx+bkλt 12a 1 + C1 e + C2 e where C1 and C2 are arbitrary constants.

7.

Private communication: K. A. Volosov (2000).

∂w ∂2w ∂w + σw =a + b0 + b1 w + b2 w 2 + b3 w 3 . ∂t ∂x ∂x2 For b0 = 0, it is the Burgers–Huxley equation. Exact solutions of the equation are given by β ∂z + λ. w(x, t) = z ∂x Here, β and λ are any of the roots of the respective quadratic and cubic equations 8.

(1)

b3 β 2 + σβ + 2a = 0, b3 λ3 + b2 λ2 + b1 λ + b0 = 0, and the specific form of z = z(x, t) depends on the equation coefficients. 1◦ . Case b3 ≠ 0. Introduce the notation: p2 = λσ + βb2 + 3βλb3 , p1 = –βσ – 3a, q1 = –

βb2 + 3βλb3 , βσ + 2a

q2 = –

3b3 λ2 + 2b2 λ + b1 . βσ + 2a

Four cases are possible. 1.1. For q2 ≠ 0 and q12 ≠ 4q2 , we have z(x, t) = C1 exp(k1 x + s1 t) + C2 exp(k2 x + s2 t) + C3 , q 1 1 kn = – 2 q1 ± 2 q12 – 4q2 , sn = –kn2 p1 – kn p2 ,

where C1 , C2 , and C3 are arbitrary constants; n = 1, 2. 1.2. For q2 ≠ 0 and q12 = 4q2 , z(x, t) = C1 exp(kx + s1 t) + C2 (kx + s2 t) exp(kx + s1 t) + C3 , k = – 12 q1 , s1 = – 14 p1 q12 + 12 p2 q1 , s2 = – 12 p1 q12 + 12 p2 q1 . 1.3. For q2 = 0 and q1 ≠ 0, z(x, t) = C1 (x – p2 t) + C2 exp[–q1 x + q1 (p2 – p1 q1 )t] + C3 . 1.4. For q2 = q1 = 0, z(x, t) = C1 (x – p2 t)2 + C2 (x – p2 t) – 2C1 p1 t + C3 . 2◦ . Case b3 = 0 and b2 ≠ 0. Solutions are given by (1) with    2a b1 σ 2ab2 σ(b1 + 2b2 λ) β = – , z(x, t) = C1 + C2 exp Ax + A + t , A= , σ 2b2 σ 2ab2 where λ is a root of the quadratic equation b2 λ2 + b1 λ + b0 = 0. 3◦ . Case b3 = b2 = 0. See equations 5.1.5.4–5.1.5.6.

References for equation 5.1.5.8: P. G. Estevez and P. R. Gordoa (1990), N. A. Kudryashov (1993), P. G. Esteves (1994).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 189

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

190

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 190

ONE SPACE VARIABLE

∂w ∂ 2w m ∂w + bw =a . ∂t ∂x2 ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the function
w1 = C1 w C1m x + C2 , C12m t + C3 , 9.

where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation.

2◦ . Traveling-wave solutions:    –1/m λm b z + , z = x + λt; w(x, t) = C exp – a λ(m + 1)    1/m (m + 1)λ mλ w(x, t) = 1 + tanh z+C , z = x + λt, 2b 2a where C and λ are arbitrary constants. A wider family of traveling-wave solutions is presented in 5.6.3.8 for f (w) = bwm . 3◦ . There is a self-similar solution of the form 1 w(ξ, t) = |t|– 2m ϕ(ξ),

1 ξ = x|t|– 2 .

References for equation 5.1.5.9: A. D. Polyanin and V. F. Zaitsev (2004, p. 13), L. Zhang (2011).

5.1.6. Equations of the Form

∂w = a ∂ 2 w + f (x, t, w) ∂w + g(x, t, w) ∂t ∂x ∂x2 ∂w

∂ 2w

∂w

+ sw k. ∂t ∂x This is a special case of equation 5.6.2.1 with f (w) = swk . 1.

=a

∂x2

+ (bx + c)

1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = w(x + C1 e–bt , t + C2 ), where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . Generalized traveling-wave solution: w = w(z),

z = x + C1 e–bt ,

where the function w(z) is determined by the ordinary differential equation ′′ awzz + bzwz′ + swk = 0.

∂w

∂ 2w

=a

∂x2

+ btn

∂w

+ cw + k1 w m1 + k2 w m2 . ∂t ∂x This is a special case of equation 5.6.2.3 with f (t) = btn . On passing from t, x to the new b n+1 t , one arrives at the simpler equation variables t, z = x + n+1 2.

∂w ∂2w = a 2 + cw + k1 wm1 + k2 wm2 , ∂t ∂z special cases of which are discussed in Subsections 5.1.1 to 5.1.3.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 190

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 191

191

∂w k ∂w ∂ 2w . + w + bw =a ∂t t ∂x ∂x2 Modified Burgers equation. 3.

1◦ . Suppose w(x, t) is a solution of the equation in question. Then the functions w1 = C1 w(C1 x + C2 , C12 t), w2 = w(x – bC3 t1–k , t) + C3 (1 – k)t–k w3 = w(x – bC3 ln |t|, t) + C3 t

–1

if k ≠ 1, if k = 1,

where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation. 2◦ . Degenerate solution linear in x: (1 – k)x + C1 C2 tk + bt x + C1 w(x, t) = t(C2 + b ln |t|) w(x, t) =

if k ≠ 1, if k = 1,

where C1 and C2 are arbitrary constants. 3◦ . Self-similar solution: w(x, t) = u(z)t–1/2 ,

z = xt–1/2 ,

where the function u = u(z) is determined by the ordinary differential equation

au′′zz + 12 z – bu u′z + 12 – k u = 0. Reference for equation 5.1.6.3: A. D. Polyanin and V. F. Zaitsev (2002, p. 17).

∂w

∂w



1 ∂



∂w



 . 2

w

+ bw =a x – ∂t ∂x x ∂x ∂x x Cylindrical Burgers equation. The variable x plays the role of the radial coordinate. Solution: 2a 1 ∂θ w(x, t) = – , b θ ∂x where the function θ = θ(x, t) satisfies the linear heat equation with axial symmetry   ∂θ a ∂ ∂θ = x . ∂t x ∂x ∂x 4.

Reference: S. Nerney, E. J. Schmahl, and Z. E. Musielak (1996).

∂w ∂w ∂ 2w ∂w + bw =a + cxk + ckxk–1 w. 2 ∂t ∂x ∂x ∂x Solution: 2a 1 ∂θ w(x, t) = – , b θ ∂x where the function θ = θ(x, t) satisfies the linear equation

5.

∂θ ∂θ ∂2θ = a 2 + cxk . ∂t ∂x ∂x

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 191

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

192 6.

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 192

ONE SPACE VARIABLE

∂w ∂ 2w ∂w + bw =a + c(x + st)k. ∂t ∂x2 ∂x

This is a special case of equation 5.6.3.2 with f (x, t) = c(x + st)k . 7.

∂w ∂ 2w ∂w =a + cxk + stn. + bw ∂t ∂x ∂x2

This is a special case of equation 5.6.3.2 with f (x, t) = cxk + stn . 8.

∂w

=a

∂ 2w

+ (bx + cw k)

∂w

. ∂t ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the function ∂x2

w1 = w(x + C1 e–bt , t + C2 ), where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . Generalized traveling-wave solution: z = x + C1 e–bt ,

w = w(z),

where the function w(z) is determined by the ordinary differential equation ′′ + (bz + cwk )wz′ = 0. awzz

Reference for equation 5.1.6.8: A. D. Polyanin and V. F. Zaitsev (2004, p. 14).

9.

∂w

=a

∂ 2w

+ (bw m + ct + s)

∂w

. ∂t ∂x2 ∂x This is a special case of equation 5.6.3.12 with f (w) = bwm , g(t) = ct + s, and h(w) = 0. On passing from t, x to the new variables t, z = x + 12 ct2 + st, we obtain an equation of the form 5.1.5.9: ∂w ∂2w ∂w = a 2 + bwm . ∂t ∂z ∂z 10.

∂w ∂2w m k ∂w =a . + (bw + ct ) ∂t ∂x ∂x2

This is a special case of equation 5.6.3.12 with f (w) = bwm , g(t) = ctk , and h(w) = 0. c k+1 On passing from t, x to the new variables t, z = x + t , we obtain an equation k+1 of the form 5.1.5.9: ∂2w ∂w ∂w = a 2 + bwm . ∂t ∂z ∂z 11.

∂w ∂t

=a

∂2w ∂x2

+ s1 (bx + ct)kw n

∂w ∂x

+ s2 (bx + ct)pw q .

This is a special case of equation 5.6.3.14 with f (z, w) = s1 z k wn and g(z, w) = s2 z p wq .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 192

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

193

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES



2 5.1.7. Equations of the Form ∂w = a ∂ w2 + b ∂w

∂t

∂x

  ∂w 2 1. +b . =a ∂t ∂x2 ∂x 1◦ . Solutions: a w(x) = ln |Ax + B| + C, b w(x, t) = A2 bt ± Ax + B, ∂w

∂ 2w

∂x

Page 193

2

+ f (x, t, w)

(x + A)2 a w(x, t) = – – ln t + B, 4bt 2b a w(x, t) = ln |x2 + 2at + Ax + B| + C, b a w(x, t) = ln |x3 + 6axt + Ax + B| + C, b a w(x, t) = ln |x4 + 12ax2 t + 12a2 t2 + A| + B, b a2 λ2 a w(x, t) = – t + ln |cos(λx + A)| + B, b b where A, B, C, and λ are arbitrary constants. 2◦ . The substitution w(x, t) =

a ln |u(x, t)| b

leads to the linear heat equation ∂u ∂2u = a 2. ∂t ∂x For details about this equation, see the books by Tikhonov and Samarskii (1990) and Polyanin (2002).   ∂ 2w ∂w 2 ∂w 2. =a + b + c. ∂t ∂x2 ∂x The substitution a w(x, t) = ln |u(x, t)| + ct b leads to the linear heat equation ∂u ∂2u = a 2. ∂t ∂x   ∂w ∂ 2w ∂w 2 3. =a + b + cw + s. ∂t ∂x2 ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the functions w1 = w(±x + C1 , t + C2 ) + C3 ect , where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation. 2◦ . Generalized separable solution quadratic in x: w(x, t) =

c(x + C2 )2 2a ct s – e ln |C1 e–ct – 4b| + C3 ect – . –ct C1 e – 4b C1 c

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 193

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

194 4. 1◦ .

SECOND-ORDER PARABOLIC EQUATIONS ∂w ∂t

=

∂2w ∂x2

+



∂w ∂x

Solutions for a < 0:

2

WITH

Page 194

ONE SPACE VARIABLE

+ aw 2 .

√
w(x, t) = C1 exp –at ± x –a , √
C2 1 + w(x, t) = exp –at ± x –a , 2 C1 – at (C1 – at)

where C1 and C2 are arbitrary constants. The first formula represents a multiplicative separable solution and the second one, a generalized separable solution. 2◦ . Generalized separable solution for a < 0:  √
 √
w(x, t) = ϕ(t) + ψ(t) A exp x –a + B exp –x –a ,

where A and B are arbitrary constants, and the functions ϕ(t) and ψ(t) are determined by the autonomous system of first-order ordinary differential equations
ϕ′t = a ϕ2 + 4ABψ 2 , (1) ψt′ = a(2ϕ – 1)ψ.

(2)

Dividing equation (1) by (2) termwise yields the first-order equation (2ϕ – 1)ψϕ′ψ = ϕ2 + 4ABψ 2 . 3◦ . Generalized separable solution for a > 0:
√ w(x, t) = ϕ(t) + ψ(t) cos x a + C ,

where C is an arbitrary constant, and the functions ϕ(t) and ψ(t) are determined by the autonomous system of first-order ordinary differential equations
ϕ′t = a ϕ2 + ψ 2 , (3) ψt′ = a(2ϕ – 1)ψ.

(4)

Dividing equation (3) by (4) termwise yields a first-order equation. References for equation 5.1.7.4: V. A. Galaktionov and S. A. Posashkov (1989), V. F. Zaitsev and A. D. Polyanin (1996, p. 399).

5.

  ∂w 2 =a +b + bcw 2 + sw + k. ∂t ∂x2 ∂x

∂w

∂ 2w

1◦ . Generalized separable solution for c < 0:

√
w(x, t) = ϕ(t) + ψ(t) exp ±x –c ,

(1)

where the functions ϕ(t) and ψ(t) are determined by the autonomous system of first-order ordinary differential equations ϕ′t = bcϕ2 + sϕ + k, ψt′ = (2bcϕ + s – ac)ψ.

(2) (3)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 194

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 195

195

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

The solution of system (2), (3) is given by 2bcλ + s  , C1 exp –(2bcλ + s)t – bc   C1 C2 exp –(2bcλ + s + ac)t ψ(t) =    2 , C1 exp –(2bcλ + s)t – bc ϕ(t) = λ +



where C1 and C2 are arbitrary constants and λ = λ1 and λ = λ2 are roots of the quadratic equation bcλ2 + sλ + k = 0. 2◦ . For more complicated generalized separable solutions that involve hyperbolic and trigonometric functions of x, see equation 5.6.6.2 with f , g, h = const. References for equation 5.1.7.5: V. A. Galaktionov and S. A. Posashkov (1989), V. F. Zaitsev and A. D. Polyanin (1996, p. 399).

  ∂w ∂ 2w ∂w 2 6. =a +b + cw 2 + stnw + ktm. ∂t ∂x2 ∂x This is a special case of equation 5.6.6.2 with f = const, g = stn , and h = ktm .



2 5.1.8. Equations of the Form ∂w = a ∂ w2 + f x, t, w, ∂w

∂w



∂w

2

∂t

∂x

∂w

∂x

+ c. ∂t ∂x2 ∂x ∂x The substitution u = ew leads to the constant coefficient linear equation

1.

=a

∂ 2w



+a

+b

∂u ∂2u ∂u = a 2 +b + cu, ∂t ∂x ∂x which can be reduced to the linear heat equation [see Polyanin (2002, p. 93)]. ∂w

∂ 2w



∂w

2

3.

=a

+a

+ btn

∂w

+ ctm. ∂t ∂x2 ∂x ∂x This is a special case of equation 5.6.5.4 with f (x, t) = btn and g(x, t) = ctm . The substitution u = ew leads to the linear equation

2.

∂u ∂2u ∂u = a 2 + btn + ctm u. ∂t ∂x ∂x  2 ∂w ∂ 2w n ∂w =a + bt + ctmw + stk. ∂t ∂x2 ∂x

This is a special case of equation 5.6.6.1 with f (t) = btn , g(t) = ctm , and h(t) = stk . 4.

∂w ∂t

=a

∂ 2w ∂x2

λt

+ be



∂w ∂x

2

+ ceµtw + seνt.

This is a special case of equation 5.6.6.1 with f (t) = beλt , g(t) = ceµt , and h(t) = seνt .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 195

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

196 5. 1◦ .

SECOND-ORDER PARABOLIC EQUATIONS ∂w

=a

∂ 2w ∂x2

+ bw



∂w

2

WITH

Page 196

ONE SPACE VARIABLE

+ cw 3 + sw.

∂t ∂x Traveling-wave solutions with bc < 0:  r   c ac t , w = C exp ± – x + s – b b

where C is an arbitrary constant. Remark. The more general equation
wt = awxx + f (w) bwx2 + cw2 + sw,

where f (w) is an arbitrary function, has the same solution.

2◦ . Generalized separable solution with bc < 0 (generalizes the solutions of Item 1◦ ): q q

w = ϕ(t) exp x |c/b| + ψ(t) exp –x |c/b| ,

where the functions ϕ = ϕ(t) and ψ = ψ(t) are determined by the autonomous system of ordinary differential equations  ac  ′ ϕt = s – ϕ + 4cϕ2 ψ, b  ac  ′ ψt = s – ψ + 4cϕψ 2 . b Integrating this system, we get the multiplicative separable solution      q q

2(ac – bs) 4bcC1 C2 –1/2 w = C1 exp x |c/b| + C2 exp –x |c/b| C3 exp t + , b ac – bs

where C1 , C2 , and C3 are arbitrary constants. By setting C1 = 0 or C2 = 0, we get the solutions of Item 1◦ . 3◦ . Generalized separable solution with bc > 0: q q

w = ϕ(t) cos x c/b + ψ(t) sin x c/b ,

where the functions ϕ = ϕ(t) and ψ = ψ(t) are determined by the autonomous system of ordinary differential equations  ac  ′ ϕt = s – ϕ + c(ϕ2 + ψ 2 )ϕ, b  ac  ′ ψt = s – ψ + c(ϕ2 + ψ 2 )ψ. b

Integrating this system, we get the multiplicative separable solution      q q

bc(C12 + C22 ) –1/2 2(ac – bs) w = C1 cos x c/b + C2 sin x c/b C3 exp t + , b ac – bs where C1 , C2 , and C3 are arbitrary constants.

References for equation 5.1.8.5: A. D. Polyanin and V. F. Zaitsev (2002, p. 381; 2004, p. 699).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 196

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

6.

∂w ∂t

=

∂2w ∂x2

+

a w



∂w ∂x

2

Page 197

197

.

This is a special case of equation 5.6.6.8 with f (w) = a/w. The substitution ( 1 wa+1 if a ≠ –1, u= a+1 ln |w| if a = –1 leads to the constant coefficient linear equation ∂t u = ∂xx u. 7.

∂w ∂t

=

∂2w ∂x2

+ aw k



∂w ∂x

2

.

This is a special case of equation 5.6.6.8 with f (w) = awk . For k = 0, see equation 5.1.7.1, and for k = –1, see equation 5.1.8.6. The substitution   Z a wk+1 dw u = exp k+1 leads to the constant coefficient linear equation ∂t u = ∂xx u. 8.

∂w ∂t

=

∂2w ∂x2

+ aw



m

∂w ∂x

2

+ (bx + ct + s)

∂w ∂x

.

This is a special case of equation 5.6.6.10 with f (w) = awm , g(t) = b, and h(t) = ct + s.

  2 ∂w ∂w k∂ w = aw 5.1.9. Equations of the Form + f x, t, w, ∂t ∂x ∂x2 ∂w ∂ 2w = aw . ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the function

1.

w1 = C1–2 C2 w(C1 x + C3 , C2 t + C4 ), where C1 , . . . , C4 are arbitrary constants, is also a solution of the equation. 2◦ . Multiplicative separable solution: w(x, t) =

x2 + Ax + B , C – 2at

where A, B, and C are arbitrary constants. 3◦ . Traveling-wave solution in implicit form: Z dw 2 ak = kx + λt + C2 , λ ln |w| + C1 where C1 , C2 , k, and λ are arbitrary constants. 4◦ . For other exact solutions, see equation 5.1.9.26 with m = 1, Items 5◦ to 8◦ .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 197

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

198

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 198

ONE SPACE VARIABLE

∂w ∂ 2w + b. = aw ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the function

2.

w1 = C1–1 w(C1 x + C2 , C1 t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Solutions:

w(x, t) = Ax + B + bt,

x2 + Ax + B b – (C – 2at), C – 2at 4a where A, B, and C are arbitrary constants. The first solution is degenerate and the second one is a generalized separable solution. w(x, t) =

3◦ . Traveling-wave solution: w = w(z),

z = kx + λt,

where k and λ are arbitrary constants, and the function w(z) is determined by the autonomous ordinary differential equation ′′ ak2 wwzz – λwz′ + b = 0.

4◦ . Self-similar solution: w = tU (ξ),

ξ = x/t,

where the function U (ξ) is determined by the autonomous ordinary differential equation ′′ aU Uξξ + ξUξ′ – U + b = 0.

∂w

= aw

∂ 2w

+ bw + c. ∂t ∂x2 1◦ . Generalized separable solutions:

3.

c w(x, t) = Aebt x + Bebt – , b b(x + A)2 – Bce–bt – 2act + C w(x, t) = , Bbe–bt – 2a where A, B, and C are arbitrary constants (the first solution is degenerate). 2◦ . Traveling-wave solution: w = w(z),

z = kx + λt,

where k and λ are arbitrary constants, and the function w(z) is determined by the autonomous ordinary differential equation ′′ ak2 wwzz – λwz′ + bw + c = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 198

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 199

199

∂w ∂ 2w + cw 2 + kw + s. = aw ∂t ∂x2 This is a special case of equation 5.1.9.10 with b = 0.

4.

Reference: V. A. Galaktionov and S. A. Posashkov (1989).

5.

∂w

= aw

∂ 2w ∂x2

+ bw 2 + (ct + d)w + st + k.

∂t This is a special case of equation 5.6.9.3 with f (t) = ct + d and g(t) = st + k. ∂w

7.

∂w ∂t

= aw

∂ 2w

= aw

∂x2

∂ 2w ∂x2

+b

∂w

+ (ct + d)w + pt + k. ∂t ∂x This is a special case of equation 5.6.10.2 with f (t) ≡ 0, g(t) = b, h(t) = ct+d, and s(t) = pt+k. 6.

–

2 3

a



∂w ∂x

2

+ b.

Generalized separable solution: w(x, t) =

 1 3Ax3 + f2 (t)x2 + f1 (t)x + f0 (t) , a

where A is an arbitrary constant and the functions f2 (t), f1 (t), and f0 (t) are determined by the system of ordinary differential equations f2′ = 6Af1 – 23 f22 , f1′ = 18Af0 – 23 f1 f2 , f0′ = 2f0 f2 – 23 f12 + ab. The general solution of this system with A ≠ 0 has the form Z 2 1 1 f2 (t) = 3 ϕ(t) dt + 3B, f1 (t) = ϕ(t) dt + B + ϕ(t), A 2A Z 3 Z  1 1 1 f0 (t) = ϕ(t) dt + B + ϕ(t) ϕ(t) dt + B + ϕ′ (t), 2 2 9A 6A 36A2 t Z

where the function ϕ(t) is determined by the autonomous ordinary differential equation ϕ′′tt + 12ϕ2 – 36A2 ab = 0. The solution of the last equation is defined implicitly by Z (C1 + 72A2 abϕ – 8ϕ3 )–1/2 dϕ = ±t + C2 , and B, C1 , and C2 are arbitrary constants. Reference: J. R. King (1993), V. A. Galaktionov (1995), A. D. Polyanin and V. F. Zaitsev (2004, p. 20).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 199

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

200 8.

SECOND-ORDER PARABOLIC EQUATIONS ∂w ∂t

= aw

∂ 2w ∂x2

–a



∂w ∂x

2

WITH

Page 200

ONE SPACE VARIABLE

– b.

1◦ . Generalized separable solution quadratic in x: w=

(x + C1 )2 + C2 t – bt ln |t|, 2at

where C1 and C2 are arbitrary constants (one more arbitrary constant can be added by a translation in t). 2◦ . Generalized separable solution: w = ϕ(t) + ψ(t)(C1 eλx + C2 e–λx ), where C1 , C2 , and λ are arbitrary constants and the functions ϕ = ϕ(t) and ψ = ψ(t) are determined by the system of ordinary differential equations ϕ′t = 4aC1 C2 λ2 ψ 2 – b, ψt′ = aλ2 ϕψ. 3◦ . Generalized separable solution: w = ϕ(t) + ψ(t) sin(λx + C), where C and λ are arbitrary constants and the functions ϕ = ϕ(t) and ψ = ψ(t) are determined by the system of ordinary differential equations ϕ′t = –aλ2 ψ 2 – b, ψt′ = –aλ2 ϕψ. 4◦ . Generalized separable solution: w = C1 x + C2 –

(aC12

 x C2 + b)t + C3 t exp – – , aC1 t aC12 t β



β = 1–

b , aC12

where C1 , C2 , and C3 are arbitrary constants (C1 ≠ 0). Reference for equation 5.1.11.8: V. A. Galaktionov and S. R. Svirshchevskii (2007, pp. 37–38).

9.

  ∂w 2 ∂w = aw +b +c + pw + q. 2 ∂t ∂x ∂x ∂x

∂w

∂ 2w

This is a special case of equation 5.6.10.2 with f (t) = b, g(t) = c, h(t) = p, and s(t) = q.   ∂w ∂w 2 ∂2w = aw + b + cw 2 + kw + s. ∂t ∂x ∂x2 1◦ . Generalized separable solutions involving an exponential of x:

10.

w(x, t) = ϕ(t) + ψ(t) exp(±λx),

λ=



–c a+b

1/2

,

(1)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 200

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 201

201

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations ϕ′t = cϕ2 + kϕ + s,

(2)

2

(3)

ψt′ Integrating (2) yields

Z

= (aλ ϕ + 2cϕ + k)ψ.

dϕ = t + C1 . + kϕ + s On computing the integral, one can find ϕ = ϕ(t) in explicit form. The solution of equation (3) is expressed in terms of ϕ(t) as Z  2 (aλ ϕ + 2cϕ + k) dt , ψ(t) = C2 exp cϕ2

where C1 and C2 are arbitrary constants.

2◦ . There are also generalized separable solutions that involve hyperbolic and trigonometric functions (A is an arbitrary constant):   –c 1/2 w(x, t) = ϕ(t) + ψ(t) cosh(λx + A), λ = ; a+b   –c 1/2 w(x, t) = ϕ(t) + ψ(t) sinh(λx + A), λ = ; a+b  1/2 c w(x, t) = ϕ(t) + ψ(t) cos(λx + A), λ = . a+b

The functions ϕ = ϕ(t) and ψ = ψ(t) are determined by autonomous systems of first-order ordinary differential equations (these systems can be reduced to a single first-order equation each). For details about these solutions, see Items 2◦ to 4◦ of equation 5.6.10.1 with f (t) = k and g(t) = s. References for equation 5.1.9.10: V. A. Galaktionov and S. A. Posashkov (1989), A. D. Polyanin and V. F. Zaitsev (2002, p. 21).

11.

∂w ∂t

= aw

∂2w ∂x2

–

3 4

a



∂w ∂x

2

+ cw 2 + kw + s.

This is a special case of equation 5.1.9.10 with b = – 34 a. 1◦ . Solution with ac > 0: w = ϕ1 (t) + ϕ2 (t) cos(λx) + ϕ3 (t) sin(λx) + ϕ4 (t) cos(2λx) + ϕ5 (t) sin(2λx),

λ=

q

c/a,

where the functions ϕn (t) are determined by the autonomous system of first-order ordinary differential equations ϕ′1 = c[ϕ21 – 38 (ϕ22 + ϕ23 ) – 3(ϕ24 + ϕ25 )] + kϕ1 + s, ϕ′2 = c[ϕ1 ϕ2 – 3(ϕ2 ϕ4 + ϕ3 ϕ5 )] + kϕ2 , ϕ′3 = c[ϕ1 ϕ3 + 3(ϕ3 ϕ4 – ϕ2 ϕ5 )] + kϕ3 , ϕ′4 = c[–2ϕ1 ϕ4 + 38 (ϕ22 – ϕ23 )] + kϕ4 , ϕ′5 = c(–2ϕ1 ϕ5 + 34 ϕ2 ϕ3 ) + kϕ5 .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 201

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

202

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 202

ONE SPACE VARIABLE

2◦ . Solution with ac < 0: w = ϕ1 (t)+ϕ2 (t) cosh(λx)+ϕ3 (t) sinh(λx)+ϕ4 (t) cosh(2λx)+ϕ5 (t) sinh(2λx), λ =

q

–c/a,

where the functions ϕn (t) are determined by the autonomous system of first-order ordinary differential equations ϕ′1 = c[ϕ21 + 38 (ϕ23 – ϕ22 ) + 3(ϕ25 – ϕ24 )] + kϕ1 + s, ϕ′2 = c[ϕ1 ϕ2 + 3(ϕ3 ϕ5 – ϕ2 ϕ4 )] + kϕ2 , ϕ′3 = c[ϕ1 ϕ3 + 3(ϕ3 ϕ4 – ϕ2 ϕ5 )] + kϕ3 , ϕ′4 = c[–2ϕ1 ϕ4 + 38 (ϕ22 + ϕ23 )] + kϕ4 , ϕ′5 = c(–2ϕ1 ϕ5 + 34 ϕ2 ϕ3 ) + kϕ5 . 3◦ . Solution with c = 0: w = ϕ1 (t) + ϕ2 (t)x + ϕ3 (t)x2 + ϕ4 (t)x3 + ϕ5 (t)x4 , where the functions ϕn (t) are determined by the autonomous system of first-order ordinary differential equations ϕ′1 = a(2ϕ1 ϕ3 – 34 ϕ22 ) + kϕ1 + s, ϕ′2 = a(6ϕ1 ϕ4 – ϕ2 ϕ3 ) + kϕ2 , ϕ′3 = a(5ϕ23 + 32 ϕ2 ϕ4 ) + kϕ3 , ϕ′4 = a(6ϕ2 ϕ5 – ϕ3 ϕ4 ) + kϕ4 , ϕ′5 = a(2ϕ3 ϕ5 – 34 ϕ24 ) + kϕ5 . Remark. In the original equation, the parameters k and s can be replaced by arbitrary functions k = k(t) and s = s(t).

  ∂w 2 ∂w 12. =w –2 + bw + cw 2 + sw 3 . 2 ∂t ∂x ∂x ∂x This is a special case of equation 5.1.9.30 with m = 1. The substitution ∂w

∂ 2w

w = 1/f leads to the equation ∂f ∂2f ∂f = +b – cf – s. 2 ∂t ∂x ∂x This equation has a generalized separable solution of the form f

u = f (x)t + g(x), where the functions f = f (x) and g = g(x) are determined by the autonomous system of ordinary differential equations ′′ fxx + bfx′ – cf = f 2 , ′′ gxx + bgx′ – cg = f g + s.

Remark. In the original equation, the parameters b, c, and s can be replaced by arbitrary functions b = b(x), c = c(x), and s = s(x).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 202

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 203

203

∂w ∂ 2w . = aw 2 ∂t ∂x2 The substitution w = 1/v leads to an equation of the form 5.1.10.3:

13.

∂v ∂ =a ∂t ∂x



 1 ∂v . v 2 ∂x

Therefore the solutions of the original equation are expressed via solutions of the linear heat equation ∂u ∂2u =a 2 ∂t ∂y by the relations w=

∂u , ∂y

x = u.

The variable y should be eliminated to obtain w = w(x, t) in explicit form. Reference: N. H. Ibragimov (1985).

∂w ∂ 2w = aw 2 + bw 2 . 2 ∂t ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the function 14.

w1 = C12 w(C1–1 x + C2 , C12 t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . The transformation w = 1/u, τ = at leads to an equation of the form 5.1.11.2: ∂u ∂ = ∂t ∂x



1 ∂u u2 ∂x



–

b . a

∂w ∂ 2w = aw 2 + bw –1 . ∂t ∂x2 This is a special case of equation 5.1.9.27 with m = 2 and b = –1.

15.

1◦ . Suppose w(x, t) is a solution of the equation in question. Then the functions w1 = C1–1 w(±C12 x + C2 , C12 t + C3 ), where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation. 2◦ . Functional separable solutions:   b 1/2 2 w(x, t) = ± C1 (x + C2 ) + C3 exp(2aC1 t) – , aC1 where C1 , C2 , and C3 are arbitrary constants.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 203

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

204

SECOND-ORDER PARABOLIC EQUATIONS

∂w ∂ 2w + bw + cw –1 . = aw 2 ∂t ∂x2 Functional separable solutions:  Z 2bt 2 w(x, t) = ± bC1 e (x + C2 ) + C3 F (t) + 2cF (t)

WITH

Page 204

ONE SPACE VARIABLE

16.

dt F (t)

1/2

,

where C1 , C2 , and C3 are arbitrary constants. ∂w

2∂

2w

1



∂w

=w – w ∂t ∂x2 2 ∂x Multiplicative separable solution:

17.

2

w= q


F (t) = exp aC1 e2bt + 2bt ,

.

C2 x2 + C1 x + C0

,

(C12 – 4C0 C2 )t + C3

where C0 , C1 , C2 , and C3 are arbitrary constants.   2 1 ∂w ∂w 2 2∂ w 18. =w – w + aw + b. ∂t 2 ∂x ∂x2 Generalized separable solution w = ϕ2 (t)x2 + ϕ1 (t)x + ϕ0 (t), where the functions ϕn (t) are determined by the system of ordinary differential equations
ϕ′2 = ϕ2 2ϕ0 ϕ2 – 12 ϕ21 + aϕ2 ,
ϕ′1 = ϕ1 2ϕ0 ϕ2 – 12 ϕ21 + aϕ1 ,
ϕ′0 = ϕ0 2ϕ0 ϕ2 – 12 ϕ21 + aϕ0 + b.

Remark. In the original equation, the parameters a and b can be replaced by arbitrary functions a = a(t) and b = b(t). Reference: V. A. Galaktionov, S. A. Posashkov (1995).

  2 ∂w 1 ∂w 2 1 3 2∂ w 19. =w – w + w + aw + b. ∂t ∂x2 2 ∂x 2 Functional separable solution: w = ϕ1 (t) + ϕ2 (t) cos x + ϕ3 (t) sin x, where the functions ϕn = ϕn (t) are determined by the system of ordinary differential equations ϕ′1 = 12 ϕ1 (ϕ21 – ϕ22 – ϕ23 ) + aϕ1 + b, ϕ′2 = 12 ϕ2 (ϕ21 – ϕ22 – ϕ23 ) + aϕ2 , ϕ′3 = 12 ϕ3 (ϕ21 – ϕ22 – ϕ23 ) + aϕ3 . It follows from the last two equations that ϕ3 = Cϕ2 , where C is an arbitrary constant. Remark. The parameters of the equation can be arbitrary functions of time, a = a(t) and b = b(t). Reference: V. A. Galaktionov and S. R. Svirshchevskii (2007, p. 32).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 204

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

∂w

20.

∂t

=w

2∂

2w

∂x2

– 3w



∂w ∂x

2

+ bw 2

∂w ∂x

Page 205

205

+ cw 3 + sw 5 .

This is a special case of equation 5.1.9.30 with m = 2. The substitution w = f –1/2 leads to the equation f

∂f ∂f ∂2f +b = – 2cf – 2s. 2 ∂t ∂x ∂x

This equation has a generalized separable solution of the form u = f (x)t + g(x), where the functions f = f (x) and g = g(x) are determined by the autonomous system of ordinary differential equations ′′ fxx + bfx′ – 2cf = f 2 , ′′ gxx + bgx′ – 2cg = f g + 2s.

Remark. In the original equation, the parameters b, c, and s can be replaced by arbitrary functions b = b(x), c = c(x), and s = s(x).

∂w

21.

= aw 3

∂ 2w

. ∂t ∂x2 This is a special case of equation 5.1.9.20 with m = 3. Functional separable solution:  1/3 w(x, t) = a–1/3 3Ax3 + f2 (t)x2 + f1 (t)x + f0 (t) .

Here,

Z 2 1 1 f2 (t) = 3 ϕ(t) dt + 3B, f1 (t) = ϕ(t) dt + B + ϕ(t), A 2A Z  Z 3 1 1 1 f0 (t) = ϕ(t) dt + B + ϕ(t) ϕ(t) dt + B + ϕ′ (t), 2 2 9A 6A 36A2 t Z

where the function ϕ(t) is defined implicitly by Z

(C1 – 8ϕ3 )–1/2 dϕ = ±t + C2 ,

and A, B, C1 , and C2 are arbitrary constants. Setting C1 = 0 in the last relation, one obtains the function ϕ in explicit form: ϕ = – 12 (t + C2 )–2 . Reference: G. A. Rudykh and E. I. Semenov (1998).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 205

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

206

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 206

ONE SPACE VARIABLE

∂w ∂ 2w + bw –2 . = aw 3 ∂t ∂x2 This is a special case of equation 5.1.9.21 with m = 3 and b = –2. The substitution w = u1/3 leads to an equation of the form 5.1.9.7:  2 ∂2u 2 ∂u ∂u = au 2 – a + 3b. ∂t 3 ∂x ∂x 22.

Therefore the equation in question has a generalized separable solution of the form  1/3 w(x, t) = a–1/3 3Ax3 + f2 (t)x2 + f1 (t)x + f0 (t) . 23.

∂w

=w

3∂

2w

∂x2

– 4w

2



∂w

2

+ bw 3

∂w

+ cw 4 + sw 7 .

∂t ∂x ∂x This is a special case of equation 5.1.9.30 with m = 3. The substitution w = f –1/3 leads to the equation ∂f ∂2f ∂f = – 3cf – 3s. +b 2 ∂t ∂x ∂x This equation has a generalized separable solution of the form f

u = f (x)t + g(x), where the functions f = f (x) and g = g(x) are determined by the autonomous system of ordinary differential equations ′′ fxx + bfx′ – 3cf = f 2 , ′′ gxx + bgx′ – 3cg = f g + 3s.

Remark. In the original equation, the parameters b, c, and s can be replaced by arbitrary functions b = b(x), c = c(x), and s = s(x).

∂w

= aw 4

∂ 2w

+ bw + cw –1 . ∂t ∂x2 Functional separable solutions: q w(x, t) = ± ϕ(t)x2 + ψ(t)x + χ(t), 24.

where the functions ϕ(t), ψ(t), and χ(t) are determined by the system of first-order ordinary differential equations ϕ′t = 12 aϕ(4ϕχ – ψ 2 ) + 2bϕ, ψt′ = 12 aψ(4ϕχ – ψ 2 ) + 2bψ, χ′t = 12 aχ(4ϕχ – ψ 2 ) + 2bχ + 2c. It follows from the first two equations that ϕ = Cψ, where C is an arbitrary constant. Remark. In the original equation, the parameters a, b, and c can be replaced by arbitrary function s a = a(t), b = b(t), and c = c(t). Reference: V. A. Galaktionov, S. A. Posashkov, and S. R. Svirshchevskii (1995).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 206

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

25.

Page 207

207

∂w ∂ 2w + bxkw 5 . = aw 4 ∂t ∂x2

This is a special case of equation 5.6.11.4 with m = 4 and f (x) = bxk . ∂w ∂ 2w = aw m . ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the functions

26.

–2/m

w1 = C1

1/m

C2

w(±C1 x + C3 , C2 t + C4 ),

where C1 , . . . , C4 are arbitrary constants, are also solutions of the equation. 2◦ . Solutions: w(x) = Ax + B, w(x, t) = (±βx + βλt + A)1/m ,

β=



mλ , a(1 – m)

1/m m(x – A)2 w(x, t) = , 2a(2 – m)(B – t)   m (x + C)2 1/m m w(x, t) = A|t + B| m–2 + , 2a(m – 2) t + B   m(m–3) 1/m (x + A)2 2a(m – 2) m w(x, t) = + B(x + A) |ϕ(t)| 2 , ϕ(t) = C + t, ϕ(t) m where A, B, C, and λ are arbitrary constants (the first solution is degenerate). 3◦ . Traveling-wave solution in implicit form: Z dw βx + λt + C2 = , 1–m λw + C1 aβ 2 (1 – m) where C1 , C2 , β, and λ are arbitrary constants. To λ = 0 there corresponds a stationary solution, and to C1 = 0 there corresponds the second solution in Item 2◦ . 4◦ . Multiplicative separable solution: w(x, t) = (λt + A)–1/m f (x), where λ is an arbitrary constant, and the function f = f (x) is determined by the autonomous ′′ + λf 1–m = 0 (its solution can be written out in implicit ordinary differential equation amfxx form). 5◦ . Self-similar solution: w = w(z),

x z= √ , t

′′ + where the function w(z) is determined by the ordinary differential equation 2awm wzz ′ zwz = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 207

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

208

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 208

ONE SPACE VARIABLE

6◦ . Self-similar solution of a more general form: mβ+1 w = tβ U (ζ), ζ = xt– 2 , where β is an arbitrary constant, and the function U = U (ζ) is determined by the ordinary differential equation ′′ = βU – 12 (mβ + 1)ζUζ′ . aU m Uζζ This equation is generalized homogeneous, and, hence, its order can be reduced. 7◦ . Generalized self-similar solution: w = e–2λt ϕ(ξ), ξ = xeλmt , where λ is an arbitrary constant, and the function ϕ = ϕ(ξ) is determined by the ordinary differential equation aϕm ϕ′′ξξ = λmξϕ′ξ – 2λϕ.

This equation is generalized homogeneous, and, hence, its order can be reduced. 8◦ . Solution: w = (At + B)–1/m ψ(u), u = x + k ln(At + B), where A, B, and k are arbitrary constants, and the function ψ = ψ(u) is determined by the autonomous ordinary differential equation A ′′ = Akψu′ – ψ. aψ m ψuu m 1–m ◦ 9 . The substitution u = w leads to the equation   m ∂u ∂ ∂u =a u 1–m , ∂t ∂x ∂x which is considered in Subsection 5.1.10. ∂w

= aw m

∂ 2w

+ bw k. ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the functions

27.

w1 = C12 w(±C1k–m–1 x + C2 , C12k–2 t + C3 ), where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation. 2◦ . Traveling-wave solutions: w = w(z), z = kx + λt, where k and λ are arbitrary constants, and the function w(z) is determined by the autonomous ordinary differential equation ′′ – λwz′ + bwk = 0. awm wzz 3◦ . Self-similar solution for k ≠ 1: 1

k–m–1

w = t 1–k u(ξ), ξ = xt 2(1–k) , where the function u(ξ) is determined by the ordinary differential equation m–k+1 ′ 1 aum u′′ξξ + ξuξ + buk – u = 0. 2(1 – k) 1–k 4◦ . For m ≠ 1, the substitution u = w1–m leads to the equation   m ∂u k–m ∂u ∂ =a u 1–m + b(1 – m)u 1–m , ∂t ∂x ∂x which is considered in Subsection 5.1.11.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 208

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 209

209

5◦ . For k = 1, the transformation w(x, t) = ebt U (x, τ ),

τ=

1 bmt + const, e bm

leads to an equation of the form 5.1.9.26: ∂U ∂2U = aU m 2 . ∂τ ∂x 28.

∂w ∂ 2w ∂w = aw m + bxtn + ctkw. 2 ∂t ∂x ∂x

This is a special case of equation 5.6.11.5 with f (t) = btn and g(t) = ctk .

29.

 2 2 ∂w 2m ∂ w 2m–1 ∂w =w + (m – 1)w + aw m+1. 2 ∂t ∂x ∂x

The substitution w = u–1/m leads to an equation of the form 5.1.11.3: ∂u ∂ = ∂t ∂x



1 ∂u  – am, u2 ∂x

that can be linearized. 30.

2  2 ∂w ∂w m∂ w m–1 ∂w =w + bw m+1 + cw 2m+1 . – (m + 1)w + aw m 2 ∂t ∂x ∂x ∂x

For m ≠ 0, the substitution w = f –1/m leads to the equation f

∂f ∂2f ∂f = +a – bmf – cm. 2 ∂t ∂x ∂x

This equation has a generalized separable solution of the form u = f (x)t + g(x), where the functions f = f (x) and g = g(x) are determined by the autonomous system of ordinary differential equations ′′ fxx + afx′ – bmf = f 2 , ′′ gxx + agx′ – bmg = f g + cm.

Remark. In the original equation, the parameters a, b, and c can be replaced by arbitrary functions a = a(x), b = b(x), and c = c(x).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 209

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

210

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 210

ONE SPACE VARIABLE

  ∂w ∂ m ∂w 5.1.10. Equations of the Form =a w ∂t ∂x ∂x

⊲ Equations of this form admit traveling-wave solutions w = w(kx + λt).   ∂w ∂ ∂w 1. =a w . ∂t ∂x ∂x This is a special case of equation 5.1.10.8 with m = 1. It is a quadratic porous medium equation, which describes unsteady flow of groundwater with the presence of a free surface (J. Boussinesq, 1904), where w is the groundwater pressure. 1◦ . Solutions:

w(x, t) = C1 x + aC12 t + C2 ,

(x + C1 )2 C3 , + 6a(t + C2 ) |t + C2 |1/3 (x + C1 )2 w(x, t) = + C3 |x + C1 |1/2 |C2 – 6at|–5/8 , C2 – 6at where C1 , C2 , and C3 are arbitrary constants. w(x, t) = –

2◦ . Traveling-wave solution in implicit form: w – C2 ln |w + C2 | = C1 x + aC12 t + C3 . 3◦ . Solution in parametric form: x = (6at + C1 )ξ + C2 ξ 2 + C3 , w = –(6at + C1 )ξ 2 – 2C2 ξ 3 . 4◦ . Solution in parametric form: x = tf (ξ) + g(ξ), w = tfξ′ (ξ) + gξ′ (ξ), where the functions f = f (ξ) and g(ξ) are determined by the system of ordinary differential equations ′′ ′′′ (fξ′ )2 – f fξξ = afξξξ ,

(1)

fξ′ gξ′

(2)

–

′′ f gξξ

=

′′′ agξξξ .

The order of equation (1) can be reduced by two. Suppose a solution of equation (1) is known. Equation (2) is linear in g and has two linearly independent particular solutions g1 = 1,

g2 = f (ξ).

The second particular solution follows from the comparison of (1) and (2). The general solution of equation (1) can be represented in the form (see Polyanin and Zaitsev, 2003):  Z  Z g(ξ) = C1 + C2 f + C3 f ψ dξ – f ψ dξ ,   Z (3) 1 1 f = f (ξ), ψ = ′ 2 exp – f dξ . a (fξ )

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 210

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 211

211

It is not difficult to verify that equation (1) has the following particular solutions: f (ξ) = 6a(ξ + C)–1 , f (ξ) = Ceλξ – aλ,

(4)

where C and λ are arbitrary constants. One can see, taking into account (1) and (3), that the first solution in (4) leads to the solution of Item 3◦ . Substituting the second relation of (4) into (1), we obtain another solution. Remark. The above solution was obtained, with the help of the Mises transformation, from a solution of the hydrodynamic boundary layer equation (see 13.3.1.1, Items 5◦ and 7◦ ).

5◦ . For other solutions, see Items 4◦ to 9◦ of equation 5.1.10.8 with m = 1. References for equation 5.1.10.1: G. I. Barenblatt and Ya. B. Zel’dovich (1957), D. Zwillinger (1989, p. 311), A. D. Polyanin and V. F. Zaitsev (2002, pp. 22–23; 2004, p. 25).

∂w

∂



1 ∂w



=a . ∂t ∂x w ∂x Fast diffusion equation. This is a special case of equation 5.1.10.8 with m = –1. Solutions: 2.

w(x, t) = (C1 x – aC12 t + C2 )–1 , w(x, t) = (2at + C1 )(x + C2 )–2 , 2a(t + C1 ) w(x, t) = , (x + C2 )2 + C3 (t + C1 )2 C12 w(x, t) = , C2 + C3 exp(aC2 t – C1 x)     C12 C1 x C1 x –1 , w(x, t) = C3 exp – –1+ at + C2 at + C2 at + C2 2aC12 t + C2 w(x, t) = , sinh2 (C1 x + C3 ) C2 – 2aC12 t w(x, t) = , cosh2 (C1 x + C3 ) 2aC12 t + C2 w(x, t) = , cos2 (C1 x + C3 ) 2 sin(2at + C1 ) w(x, t) = , cos(2at + C1 ) – cos(2x + C2 ) 2 sinh(2at + C1 ) w(x, t) = , cosh(2x + C2 ) – cosh(2at + C1 ) 2 cosh(2at + C1 ) w(x, t) = , sinh(2x + C2 ) – sinh(2at + C1 ) 2 sinh(2at + C1 ) w(x, t) = – , cosh(2x + C2 ) + cosh(2at + C1 ) 2 sin(2at + C1 ) w(x, t) = , cosh(2x + C2 ) – cos(2at + C1 ) 2 sinh(2at + C1 ) w(x, t) = , cosh(2at + C1 ) – cos(2x + C2 )

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 211

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

212

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 212

ONE SPACE VARIABLE

where C1 , C2 , and C3 are arbitrary constants. References: O. V. Voinov (1994), V. V. Pukhnachov (1987), S. N. Aristov (1999), A. D. Polyanin and V. F. Zaitsev (2002, p. 23), R. O. Popovich, O. O. Vaneeva, and N. M. Ivanova (2007), N. M. Ivanova (2008).

3.

∂w ∂t

=a

∂ ∂x



w

–2 ∂w

∂x



.

Fujita–Storm equation. This is a special case of equation 5.1.10.8 with m = –2. 1◦ . Solutions: w(x, t) = ±(2C1 x – 2aC12 t + C2 )–1/2 , √   –1/2 2at C1 w(x, t) = ± ln 2 , x x t  –1/2 C1 (x + C2 )2 w(x, t) = ± + C3 exp(C1 t) , 2a ±1 , τ = at, w(x, t) = q  –τ 2 8τ f (τ ) 4 – e ce ± f (τ ) ±1 , τ = at, w(x, t) = q  –τ 2 –8τ f (–τ ) e ce ± f (τ ) – 4

f (τ ) =

f (τ ) =

p

c2 e–2τ + 2e–8τ + 2e–4τ x ,

p c2 e–2τ + 2e–8τ + 2e–4τ x ,

where C1 , C2 , C3 , and c are arbitrary constants. The first solution is of the traveling-wave type, the second is self-similar, and the third is a functional separable solution. In the last two equations, the plus or minus signs can be selected arbitrarily. ∂z 2◦ . Introduce a new unknown z = z(x, t) by w = and then integrate the resulting equation ∂x with respect to x to obtain  –2 2 ∂z ∂z ∂ z =a . (1) ∂t ∂x ∂x2 By the hodograph transformation x = u,

z = y,

(2)

equation (1) can be reduced to a linear heat equation for u = u(y, t): ∂u ∂2u =a 2. ∂t ∂y

(3)

Transformation (2) means that the dependent variable z is taken to be the independent variable, and the independent variable x, the dependent one. Solutions w = w(x, t) of the original equation are expressed via solutions u = u(y, t) of the linear equation (3) according to w=



∂u ∂y

–1

,

x = u(y, t).

(4)

The variable y should be eliminated from (4) to obtain w = w(x, t) in explicit form.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 212

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

213

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

3◦ . The transformation  Z x Z t ∂w x¯ = w(y, t) dy +a w–2 (x, τ ) (x, τ ) dτ , ∂x x0 t0 x=x0

¯t = t–t0 ,

Page 213

w(¯ ¯ x, ¯t) =

1 , (5) w(x, t)

where x0 and t0 are any numbers, leads to the linear equation ∂w ¯ ∂2w ¯ =a 2. ¯ ∂t ∂ x¯ The inversion of transformation (5) is given by  Z x¯ Z ¯t  ∂ w(¯ ¯ x , t′ ) ′ ¯ ′ x= w(x ¯ , t) dx + dt′ , ∂ x¯ ¯ t0 x ¯0 x ¯=¯ x0

t = ¯t – ¯t0 ,

w(x, t) =

1 . w(¯ ¯ x, ¯t)

4◦ . Suppose w = w(x, t) is a solution of the equation in question. Then the function w1 = w –

2twt + xwx + w at(wx – 12 w2 ) + 12 xw

is also a solution of the equation. References for equation 5.1.10.3: M. L. Storm (1951), G. W. Bluman and S. Kumei (1980), A. Munier, J. R. Burgan, J. Gutierres, E. Fijalkow, and M. R. Feix (1981), N. H. Ibragimov (1985), V. Tychynin, O. Petrova, and O. Tertyshnyk (2007), N. M. Ivanova (2008).

4.

  ∂w ∂ –4/3 ∂w =a w . ∂t ∂x ∂x

This is a special case of equation 5.1.10.8 with m = –4/3 (the equation admits more invariant solutions than for m ≠ –4/3). 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the function –3/4

w1 =

|A1 B2 – A2 B1 |3/2 C1 (A2 x + B2 )3



 A1 x + B1 w , C1 t + C2 , A2 x + B2

where A1 , A2 , B1 , B2 , C1 , and C2 are arbitrary constants (A1 B2 – A2 B1 ≠ 0), is also a solution of the equation. 2◦ . Solutions: w(x, t) = (±2C1 x – 3aC12 t + C2 )–3/4 , w(x, t) = (at + C1 )3/4 [(x + C2 )(C3 x + C2 C3 + 1)]–3/2 , w(x, t) = (±2C1 x3 + C2 x4 – 3aC12 x4 t)–3/4 ,  –3/4 (x + C1 )2 2 w(x, t) = + C3 (t + C2 ) , a(t + C2 )  –3/4 (x + C1 )2 2 4 w(x, t) = + C3 (t + C2 ) (x + C1 ) , a(t + C2 ) where C1 , C2 , and C3 are arbitrary constants. The first solution is of the traveling-wave type, the second is a solution in multiplicative separable form, and the others are functional separable solutions.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 213

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

214

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 214

ONE SPACE VARIABLE

3◦ . Functional separable solution:  –3/4 w(x, t) = ϕ4 (t)x4 + ϕ3 (t)x3 + ϕ2 (t)x2 + ϕ1 (t)x + ϕ0 (t) ,

where the functions ϕk = ϕk (t) are determined by the system of ordinary differential equations ϕ′0 = –a 34 ϕ21 + 2aϕ0 ϕ2 , ϕ′1 = –aϕ1 ϕ2 + 6aϕ0 ϕ3 , ϕ′2 = –aϕ22 + 32 aϕ1 ϕ3 + 12aϕ0 ϕ4 , ϕ′3 = –aϕ2 ϕ3 + 6aϕ1 ϕ4 , ϕ′4 = – 34 aϕ23 + 2aϕ2 ϕ4 . The prime denotes a derivative with respect to t. 4◦ . There are exact solutions of the following forms: w(x, t) = x–3 F (y), w(x, t) = x–3 G(z),

1 ; x tx2 z= . (x + 1)2 y =t–

5◦ . For other solutions, see equation 5.1.10.8 with m = –4/3. References for equation 5.1.10.4: L. V. Ovsiannikov (1959, 1982), J. R. King (1991), N. H. Ibragimov (1994, pp. 120, 128), V. A. Galaktionov (1995), G. A. Rudykh and E. I. Semenov (1998).

5.

∂w ∂t

=a

∂ ∂x



w

–2/3 ∂w

∂x



.

This is a special case of equation 5.1.10.8 with m = –2/3. 1◦ . Solution: 2◦ . The transformation

 –3/2 w = (C – 4at)3/2 (C – 4at)3/2 – x2 . t = τ,

where

∂v ∂ξ

x = v,

w = 1/u,

= u, leads to an equation of the form 5.1.10.4:   ∂ ∂u –4/3 ∂u =a u . ∂τ ∂ξ ∂ξ

References for equation 5.1.10.5: A. Munier, J. R. Burgan, J. Gutierres, E. Fijalkow, and M. R. Feix (1981), J. R. Burgan, A. Munier, M. R. Feix, and E. Fijalkow (1984), I. Sh. Akhatov, R. K. Gazizov, and N. H. Ibragimov (1989), J. R. King (1991), N. H. Ibragimov (1994, pp. 127–128).

6.

∂w ∂t

=a

∂ ∂x



w

–3/2 ∂w

∂x



.

This is a special case of equation 5.1.10.8 with m = –3/2.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 214

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 215

215

1◦ . Functional separable solution:  –2/3 , w(x, t) = a2/3 3Ax3 + f2 (t)x2 + f1 (t)x + f0 (t)

where A is an arbitrary constant and the functions f2 (t), f1 (t), and f0 (t) are determined by the system of ordinary differential equations f2′ = 6Af1 – 23 f22 , f1′ = 18Af0 – 23 f1 f2 , f0′ = 2f0 f2 – 23 f12 .

The general solution of this system with A ≠ 0 has the form Z 2 Z 1 1 f2 (t) = 3 ϕ(t) dt + 3B, f1 (t) = ϕ(t) dt + B + ϕ(t), A 2A Z  Z 3 1 1 1 ϕ(t) ϕ(t) dt + B + ϕ′ (t), f0 (t) = ϕ(t) dt + B + 2 2 9A 6A 36A2 t where the function ϕ(t) is determined by the autonomous ordinary differential equation ϕ′′tt + 12ϕ2 = 0. The solution of the last equation is given implicitly by Z (C1 – 8ϕ3 )–1/2 dϕ = ±t + C2 , and A, B, C1 , and C2 are arbitrary constants. Setting C1 = 0 in this relation, we find ϕ in explicit form: ϕ = – 12 (t + C2 )–2 . 2◦ . For other solutions, see equation 5.1.10.8 with m = –3/2. References for equation 5.1.10.6: J. R. King (1993), A. D. Polyanin and V. F. Zaitsev (2004, p. 28).

7.

∂w ∂t

=a

∂ ∂x



w

–1/2 ∂w

∂x



.

This is a special case of equation 5.1.10.8 with m = –1/2. Functional separable solution: w = [ϕ(x)t + ψ(x)]2 , where the functions ϕ = ϕ(x) and ψ = ψ(x) are determined by the system of ordinary differential equations ′′ aϕ′′xx = ϕ2 , aψxx = ϕψ. A particular solution of this system is ϕ=

6a , (x + C1 )2

ψ = C2 (x + C1 )3 +

C3 , (x + C1 )2

where C1 , C2 , and C3 are arbitrary constants. References: T. K. Amerov (1990), J. R. King (1992).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 215

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 216

216

SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE   ∂w ∂ m ∂w 8. =a w . ∂t ∂x ∂x This equation often occurs in nonlinear problems of heat and mass transfer, combustion theory, and flows in porous media. For example, it describes unsteady heat transfer in a quiescent medium with the heat diffusivity being a power-law function of temperature. For m = 1, –1, –2, –4/3, –2/3, –3/2, –1/2, see also equations 5.1.10.1 to 5.1.10.7. 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the function w1 = C1 w(C2 x + C3 , C1m C22 t + C4 ), where C1 , . . . , C4 are arbitrary constants, is also a solution of the equation. 2◦ . Solutions:* 1

w(x) = (Ax + B) m+1 , w(x, t) = (±kx + kλt + A)1/m , k = λm/a,  1 m m(x – A)2 w(x, t) = , 2a(m + 2)(B – t) 1  m m (x + C)2 m – w(x, t) = A|t + B| m+2 – , 2a(m + 2) t + B  1 m m m(x + A)2 – m(2m+3) 2 2(m+1) w(x, t) = + B|x + A| m+1 |ϕ(t)| , ϕ(t) = C – 2a(m + 2)t, ϕ(t) where A, B, C, and λ are arbitrary constants. The third solution for B > 0 and the fourth solution for B < 0 correspond to blow-up regimes (the solution increases without bound on a finite time interval). Special case. A solution satisfying the initial and boundary conditions w=0 w = kt is given by w(x, t) = where λ =

p



1/m

at

t=0

at

x = 0 (t > 0),

k(t – x/λ)1/m 0

(x > 0),

for 0 ≤ x ≤ λt, for x > λt,

akm /m.

3◦ . Traveling-wave solutions: w = w(z),

z = ±x + λt,

where the function w(z) is defined implicitly by Z wm dw a = C2 + z, λw + C1 and λ, C1 , and C2 are arbitrary constants. To λ = 0 there corresponds a stationary solution, and to C1 = 0 there corresponds the second solution in Item 2◦ . * For the sake of brevity, here and henceforth, exact solutions of nonlinear equations are given only for the domain of their spatial localization, where w ≡/ 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 216

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 217

217

4◦ . Multiplicative separable solution: w(x, t) = (λt + A)–1/m f (x),

(1)

where the function f = f (x) is defined implicitly by Z f m df 2λ , = ±x + C2 , b= p m+2 am(m + 2) C1 – bf and λ, C1 , and C2 are arbitrary constants. 5◦ . Self-similar solution: x z= √ (0 ≤ x < ∞), t where the function w(z) is determined by the ordinary differential equation
′ 2a wm wz′ z + zwz′ = 0. w = w(z),

(2)

Solutions of this sort usually describe situations where the unknown function assumes constant values at the initial and boundary conditions. To the particular solution of equation (2) with w(z) = k2 z 2/m there corresponds the third solution in Item 2◦ . Fujita (1952) obtained the general solution of equation (2) for m = –1 and m = –2; see also the book by Lykov (1967). With the boundary conditions w = 1 at

z = 0,

w=0

at z = ∞

the solution of equation (2) is localized and has the structure ( 1/m P (1 – Z, m) for 0 ≤ Z ≤ 1, w = (1 – Z) P (1, m) 0 for 1 ≤ Z < ∞, where ∞ X z 2a 2 Z = , z0 = , P (ξ, m) = bk ξ k , z0 mP (1, m) k=0

b0 = 1, b1 =

6◦ .

– 12 [m(m

+ 1)]–1 ,

. . . ; see Samarskii and Sobol’ (1963).

Self-similar solution: 1

w = t– m+2 F (ξ),

1

ξ = xt– m+2

(0 ≤ x < ∞).

Here, the function F = F (ξ) is determined by the first-order ordinary differential equation a(m + 2)F m Fξ′ + ξF = C,

(3)

where C is an arbitrary constant. To C = 0 in (3) there corresponds the fourth solution in Item 2◦ , which describes the propagation of a thermal wave coming from a plane source. For details, see the book by Zel’dovich and Raiser (1966). Performing the change of variable ϕ = F m in equation (3), one obtains ϕ′ξ = αϕ–1/m – βξ,

(4)

mC m where α = a(m+2) and β = a(m+2) . The books by Polyanin and Zaitsev (1995, 2003) present general solutions of equation (4) for m = –1 and m = 1.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 217

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

218

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 218

ONE SPACE VARIABLE

7◦ . Self-similar solution of a more general form: w = tβ g(ζ),

ζ = xt–

mβ+1 2 ,

β is any.

Here, the function g = g(ζ) is determined by the ordinary differential equation m 1 G′′ζζ = A1 ζG– m+1 G′ζ + A2 G m+1 ,

G = gm+1 ,

(5)

where A1 = –(mβ + 1)/(2a) and A2 = β(m + 1)/a. This equation is homogeneous, and, therefore, its order can be reduced (and then it can be transformed to an Abel equation of the second kind). Exact analytical solutions of equation (5) for various values of m can be found in Polyanin and Zaitsev (2003). 8◦ . Generalized self-similar solution: w = e–2λt ϕ(u),

u = xeλmt ,

λ is any,

where the function ϕ = ϕ(u) is determined by the ordinary differential equation a(ϕm ϕ′u )′u = λmuϕ′u – 2λϕ.

(6)

This equation is homogeneous, and, hence, its order can be reduced (and then it can be transformed to an Abel equation of the second kind). The substitution Φ = ϕm+1 brings (6) to an equation that coincides, up to notation, with (5). 9◦ . Solution: w = (t + A)–1/m ψ(u),

u = x + b ln(t + A),

A, b are any,

where the function ψ = ψ(u) is determined by the autonomous ordinary differential equation a(ψ m ψu′ )′u = bψu′ – ψ/m.

(7)

a Introduce the new dependent variable p(ψ) = ψ m ψu′ . Taking into account the identity b d b –m d = ψ p , we arrive at an Abel equation of the second kind: du a dψ pp′ψ = p – sψ m+1 ,

s = a/(mb2 ).

The general solutions of this equation with m = –3, –2, – 32 , –1 can be found in Polyanin and Zaitsev (2003). 10◦ . Unsteady point source solution with a = 1:   1/m  –1/(m+2) 2 x2 η0 – 2/(m+2) w(x, t) = At t  0

for |x| ≤ η0 t1/(m+2) , for |x| > η0 t1/(m+2) ,

where



m A= 2(m + 2)

1/m



Γ(1/m + 3/2) , η0 = E0 √ A π Γ(1/m + 1)

m/(m+2)

, Γ(z) =

Z

∞

e–ξ ξ z–1 dξ,

0

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 218

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 219

219

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

with Γ(z) being the gamma function. The above solution satisfies the initial condition w(x, 0) = E0 δ(x), where δ(x) is the Dirac delta function, and the condition of conservation of energy Z ∞ w(x, t) dx = E0 > 0. –∞

11◦ . The transformation  Z x Z t ∂w m et = t – t0 , x (x, τ ) e= w(y, t) dy + a w (x, τ ) dτ , ∂x x0 t0 x=x0

w e (e x, et) =

1 w(x, t)

takes a nonzero solution w(x, t) of the original equation to a solution w e (e x, et) of a similar equation   ∂w e ∂ ∂w e =a w e –m–2 . ∂e x ∂e x ∂et

References for equation 5.1.10.8: Ya. B. Zel’dovich and A. S. Kompaneets (1950), G. I. Barenblatt (1952), L. V. Ovsiannikov (1959, 1962, 1982), N. H. Ibragimov (1994, pp. 120–124), A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995), G. A. Rudykh and E. I. Semenov (1998), R. Z. Zhdanov and V. I. Lahno (1998).

  ∂ ∂w m ∂w =a w + bw k 5.1.11. Equations of the Form ∂t ∂x ∂x ⊲ Equations of this form admit traveling-wave solutions w = w(kx + λt).   ∂w ∂ ∂w 1. =a w + b. ∂t ∂x ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the functions w1 = C1–1 w(±C1 x + C2 , C1 t + C3 ), where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation. 2◦ . Generalized separable solutions linear and quadratic in x: w(x, t) = C1 x + (aC12 + b)t + C2 , w(x, t) = –

(x + C2 )2 3 + C3 |t + C1 |–1/3 + b(t + C1 ), 6a(t + C1 ) 4

where C1 , C2 , and C3 are arbitrary constants. The first solution is degenerate. 3◦ . Traveling-wave solution in implicit form: Z u du a = – ln | ± x + C1 t + C2 | + C3 , 2 au – C1 u + b

u=

w . ±x + C1 t + C2

4◦ . For other solutions, see equation 5.1.11.14 with m = 1 and k = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 219

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

220 2.

SECOND-ORDER PARABOLIC EQUATIONS   ∂w ∂ ∂w =a w + bw. ∂t ∂x ∂x

WITH

Page 220

ONE SPACE VARIABLE

This is a special case of equation 5.1.11.10 with m = 1. 1◦ . Additive separable solution: w=–

b (x + C1 )2 + C2 exp 6a

2 3 bt

where C1 and C2 are arbitrary constants.


,

2◦ . Generalized separable solution: w = –bebt (6aebt + C1 )–1 (x + C3 )2 + C2 ebt (6aebt + C1 )–1/3 , where C1 , C2 , and C3 are arbitrary constants. 3◦ . Generalized separable solution: w = –bebt (6aebt + C1 )–1 (x + C3 )2 + C2 ebt (6aebt + C1 )–5/8 (x + C3 )1/2 , where C1 , C2 , and C3 are arbitrary constants. 3.

∂w ∂t

=

∂ ∂x



w

–2 ∂w

∂x



+ b.

1◦ . Suppose w(x, t) is a solution of this equation. Then the functions w1 = C1 w(±C1 x + C2 , C1–1 t + C3 ), where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation. 2◦ . The transformation 2 ∂u , x=– bu ∂y

   b ∂ 1 ∂u –1 w(x, t) = – 2 ∂y u ∂y

(1)

leads to the equation   ∂ ∂Ψ Φ = 0, ∂y ∂y

where



∂ Φ= ∂y



1 ∂u u ∂y

–1

1 , Ψ= u



 ∂u ∂ 2 u – . ∂t ∂y 2

It follows that any solution u = u(x, t) of the linear heat equation ∂u ∂ 2 u – =0 ∂t ∂y 2

(2)

generates a solution (1) of the original nonlinear equation. References for equation 5.1.11.3: V. A. Dorodnitsyn and S. R. Svirshchevskii (1983), S. I. Svinolupov (1985).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 220

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

221

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

∂w

∂



–2 ∂w

Page 221



=a w – bw 3 . ∂t ∂x ∂x Functional separable solutions: 4.

 w(x, t) = ± C1 (x + C2 )2 + C3 exp(2aC1 t) –

b aC1

–1/2

,

where C1 , C2 , and C3 are arbitrary constants. 5.

∂w ∂t

=a

∂ ∂x



w

–1 ∂w

∂x



+ bw 2 .

1◦ . Generalized separable solution: w=



(x + C1 )2 + C2 t – bt ln |t| 2at

–1

,

where C1 and C2 are arbitrary constants (one more arbitrary constant can be added using a translation in t). 2◦ . Functional separable solution: w=

1 , ϕ(t) + ψ(t)(C1 eλx + C2 e–λx )

where C1 , C2 , and λ are arbitrary constants and the functions ϕ = ϕ(t) and ψ = ψ(t) are determined by the system of ordinary differential equations ϕ′t = 4aC1 C2 λ2 ψ 2 – b, ψt′ = aλ2 ϕψ. 3◦ . Functional separable solution: w=

1 , ϕ(t) + ψ(t) sin(λx + C)

where C and λ are arbitrary constants and the functions ϕ = ϕ(t) and ψ = ψ(t) are determined by the system of ordinary differential equations ϕ′t = –aλ2 ψ 2 – b, ψt′ = –aλ2 ϕψ. 4◦ . Functional separable solution:   –1 x C2 2 β w = C1 x + C2 – (aC1 + b)t + C3 t exp – – , aC1 t aC12 t

β =1–

b , aC12

where C1 , C2 , and C3 are arbitrary constants (C1 ≠ 0). Reference for equation 5.1.11.5: V. A. Galaktionov and S. R. Svirshchevskii (2007, pp. 37–38).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 221

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

222 6.

SECOND-ORDER PARABOLIC EQUATIONS   ∂w ∂ –4/3 ∂w =a w + bw –1/3 . ∂t ∂x ∂x

WITH

Page 222

ONE SPACE VARIABLE

1◦ . For ab > 0, the transformation w(x, t) = exp(±3λx)z(ξ, t),

1 ξ= exp(±2λx), 2λ

λ=



b 3a

1/2

,

leads to a simpler equation of the form 5.1.10.4:   ∂z ∂ –4/3 ∂z =a z . ∂t ∂ξ ∂ξ

(1)

2◦ . For ab < 0, the transformation z(ξ, t) , w(x, t) = cos3 (λx)

1 ξ = tan(λx), λ

λ=



b – 3a

1/2

,

also leads to simpler equation (1) (for its solutions, see equation 5.1.10.4). 3◦ . Multiplicative separable solution: w(x, t) = (t + C)3/4 u(x), where C is an arbitrary constant, and the function u = u(x) is determined by the autonomous ordinary differential equation a(u–4/3 u′x )′x + bu–1/3 – 34 u = 0. 4◦ . The substitution u = w–4/3 leads to an equation with quadratic nonlinearity of the form 5.1.9.11 with c = – 43 b and k = s = 0:  2 ∂u ∂2u 3 ∂u 4 = au 2 – a – bu2 . ∂t 4 ∂x 3 ∂x

(2)

For a = –b, equation (2) has a solution of the form u = ϕ1 (t) + ϕ2 (t) cos(kx) + ϕ3 (t) sin(kx) + ϕ4 (t) cos(2kx) + ϕ5 (t) sin(2kx),

k=

√2 , 3

where the functions ϕn = ϕn (t) are determined by the system of first-order ordinary differential equations (not written out here). For a = b, equation (2) has a solution of the form u = ϕ1 (t) + ϕ2 (t) cosh(kx) + ϕ3 (t) sinh(kx) + ϕ4 (t) cosh(2kx) + ϕ5 (t) sinh(2kx),

k=

√2 . 3

References for equation 5.1.11.6: N. H. Ibragimov (1994, pp. 134, 136), V. A. Galaktionov (1995), A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995), A. D. Polyanin and V. F. Zaitsev (2002, pp. 27–28).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 222

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

7.

∂w ∂t

=a

∂ ∂x



w

–3/2 ∂w

∂x



Page 223

223

+ bw 5/2 .

1◦ . Functional separable solution:  –2/3 . w(x, t) = a2/3 3Ax3 + f2 (t)x2 + f1 (t)x + f0 (t)

Here,

Z 2 1 1 f2 (t) = 3 ϕ(t) dt + 3B, f1 (t) = ϕ(t) dt + B + ϕ(t), A 2A Z 3 Z  1 1 1 ϕ(t) dt + B + ϕ(t) ϕ(t) dt + B + ϕ′ (t), f0 (t) = 2 2 9A 6A 36A2 t Z

where the function ϕ(t) is defined implicitly by Z (C1 – 108A2 abϕ – 8ϕ3 )–1/2 dϕ = t + C2 , and A, B, C1 , and C2 are arbitrary constants. 2◦ . For other solutions, see equations 5.1.11.12 with m = –3/2 and 5.1.11.14 with m = –3/2 and k = 5/2. 3◦ . The substitution w = u–2/3 leads to an equation of the form 5.1.9.7:

8.

∂w ∂t

=a

∂ ∂x

 2 ∂u ∂2u 2 ∂u 3 = au 2 – a – b. ∂t 3 ∂x 2 ∂x



w



1 ∂w √ w ∂x

–4/3 ∂w

∂x



+ cw 7/3 .

This is a special case of equation 5.1.12.5 with b = 0. See also equation 5.1.11.14 with m = –4/3 and k = 7/3. 9.

∂w ∂t

=a

∂ ∂x



√ + b w.

This is a special case of equation 5.1.11.8 with m = – 12 . This equation with a = 12 and b = –1 arises in plasma physics; it describes energy diffusion in a strong magnetic field in the presence of energy sinks due to plasma radiation. Generalized separable solution:  Z w = ϕ (x) C 2

dx –t ϕ2 (x)

2

,

where C is an arbitrary constant and the function ϕ = ϕ(x) is determined by the ordinary differential equation 2aϕ′′xx + bϕ + 2ϕ2 = 0. Reference: A. Oron and P. Rosenau (1986).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 223

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

224

SECOND-ORDER PARABOLIC EQUATIONS   ∂w ∂ m ∂w 10. =a w + bw. ∂t ∂x ∂x 1◦ . Solutions:

WITH

Page 224

ONE SPACE VARIABLE

1

w(x, t) = ebt (Ax + B) m+1 ,  1 m λm λ2 bmt bt w(x, t) = e ± x+ e +A , a ab  1 m bm2 (x – A)2 bt w(x, t) = e , bmt 2a(m + 2)(B – e )    1 2bmt bm2 (x + B)2 m – w(x, t) = A exp , m+2 2a(m + 2)  1 m bm2 (x + C)2 m – bt bmt + B| m+2 – , w(x, t) = e A|e 2a(m + 2) ebmt + B where A, B, C, and λ are arbitrary constants. 2◦ . By the transformation 1 bmt e + const bm the original equation can be reduced to an equation of the form 5.1.10.8:   ∂v ∂ m ∂v =a v . ∂τ ∂x ∂x w(x, t) = ebt v(x, τ ),

τ=

3◦ . See also equation 5.1.11.14 with k = 1.

References for equation 5.1.11.10: L. K. Martinson and K. B. Pavlov (1972), A. D. Polyanin and V. F. Zaitsev (2002, pp. 28–31).

∂w

∂



m ∂w



=a w + bw m+1 . ∂t ∂x ∂x 1◦ . Multiplicative separable solution (a = b = 1, m > 0):    2(m + 1) cos2 (πx/L) 1/m   m(m + 2) (t0 – t) w(x, t) =   0 11.

L , 2 (1) L for |x| > , 2 1/2 where L = 2π(m + 1) /m. Solution (1) describes a blow-up regime that exists on a limited time interval t ∈ [0, t0 ). The solution is localized in the interval |x| < L/2. for |x| ≤

2◦ . Multiplicative separable solution:  µx  Ae + Be–µx + D 1/m w(x, t) = , mλt + C s λ2 (m + 1)2 λ(m + 1) b B= 2 , D=– , µ=m – , 2 b(m + 2) a(m + 1) 4b A(m + 2) where A, C, and λ are arbitrary constants, ab(m + 1) < 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 224

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 225

225

3◦ . Multiplicative separable solution (C and λ are arbitrary constants): w(x, t) = (mλt + C)–1/m ϕ(x), where the function ϕ = ϕ(x) is determined by the ordinary differential equation a(ϕm ϕ′x )′x + bϕm+1 + λϕ = 0.

(2)

Equation (2) has the following solution in implicit form:  –1/2 Z 2λ b m m+2 2m+2 ϕ A– – dϕ = ±x + B, ϕ ϕ a(m + 2) a(m + 1) where A and B are arbitrary constants. 4◦ . Functional separable solutions [it is assumed that ab(m + 1) < 0]: s   –b 1/m w(x, t) = f (t) + g(t)eλx , λ = ±m , a(m + 1)

where the functions f = f (t) and g = g(t) are determined by the autonomous system of ordinary differential equations ft′ = bmf 2 ,

gt′ =

bm(m + 2) f g. m+1

Integrating yields f (t) = (C1 – bmt)–1 ,

m+2 g(t) = C2 (C1 – bmt)– m+1 ,

where C1 and C2 are arbitrary constants. 5◦ . Functional separable solution (A and B are arbitrary constants): s   –b λx –λx 1/m , λ=m w(x, t) = f (t) + g(t)(Ae + Be ) , a(m + 1)

(3)

where the functions f = f (t) and g = g(t) are determined by the autonomous system of ordinary differential equations ft′ = bmf 2 +

4bmAB 2 g , m+1

gt′ =

bm(m + 2) f g. m+1

(4)

On eliminating t from this system, one obtains a homogeneous first-order equation: fg′ =

m+1 f 4AB g + . m+2 g m+2 f

(5)

The substitution ζ = f /g leads to a separable equation. Integrating yields the solution of equation (5) in the form 2

f = ±g 4AB + C1 g– m+2


1

2,

C1 is any.

Substituting this expression into the second equation of system (4), one obtains a separable equation for g = g(t).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 225

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

226

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 226

ONE SPACE VARIABLE

6◦ . The functional separable solutions  1/m , w(x, t) = f (t) + g(t) cosh(λx)  1/m w(x, t) = f (t) + g(t) sinh(λx) are special cases of formula (3) with A = 12 , B =

1 2

and A = 12 , B = – 12 , respectively.

7◦ . Functional separable solution [it is assumed that ab(m + 1) > 0]: s  1/m b , λ=m , w(x, t) = f (t) + g(t) cos(λx + C) a(m + 1)

where the functions f = f (t) and g = g(t) are determined by the autonomous system of ordinary differential equations ft′ = bmf 2 +

bm 2 g , m+1

gt′ =

bm(m + 2) f g, m+1

which coincides with system (4) for AB = 14 . References for equation 5.1.11.11: N. V. Zmitrenko, S. P. Kurdyumov, A. P. Mikhailov, and A. A. Samarskii (1976), M. Bertsch, R. Kersner, and L. A. Peletier (1985), V. A. Galaktionov and S. A. Posashkov (1989), A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995), V. F. Zaitsev and A. D. Polyanin (1996, pp. 407–408).

  ∂w ∂ m ∂w 12. =a w + bw 1–m. ∂t ∂x ∂x This is a special case of equation 5.1.11.14 with k = 1 – m, where some solutions are also given. Functional separable solution for m ≠ –1 and m ≠ –2:  1/m m 1 bm2 – 2 m+2 w(x, t) = (x + A) + B|F | – F , F 4a(m + 1) 2a(m + 2) F = F (t) = C – t, m where A, B, and C are arbitrary constants. For m = –2 see also equation 5.1.11.4 and for m = –1 see equation 5.1.11.5. References: R. Kersner (1978 a, b).

∂w

∂



2n ∂w



=a w + bw 1–n. ∂t ∂x ∂x This is a special case of equation 5.1.11.14 with m = 2n and k = 1 – n. Generalized traveling-wave solution:  1/n x + C1 bn2 2a(n + 1) w(x, t) = ± √ – (C2 – kt) , k= , n C2 – kt 3a(n + 1) 13.

where C1 and C2 are arbitrary constants. Reference: A. D. Polyanin and V. F. Zaitsev (2004, p. 36).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 226

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

14.

∂w ∂t

=a

∂ ∂x



w



m ∂w

∂x

Page 227

227

+ bw k.

This is a special case of equation 5.6.15.2 with f (w) = awm and g(w) = bwk . For b = 0, see Subsection 5.1.10. 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the functions w1 = C12 w(±C1k–m–1 x + C2 , C12k–2 t + C3 ), where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation. 2◦ . A space-homogeneous solution and a stationary solution are given by (the latter is written out in implicit form): w(t) =  Z m w A–

(



(1 – k)bt + C

Cebt

2b a(m + k + 1)



1 1–k

if k ≠ 1,

if k = 1, –1/2 dw = ±x + B, wm+k+1

where A, B, and C are arbitrary constants. 3◦ . Traveling-wave solutions: w = w(z),

z = ±x + λt,

where the function w(z) is determined by the autonomous ordinary differential equation a(wm wz′ )′z – λwz′ + bwk = 0. The substitution u(w) =

(1)

a m ′ w wz λ

brings (1) to the Abel equation uu′w – u = –abλ–2 wm+k .

(2)

The book by Polyanin and Zaitsev (2003) presents exact solutions of equation (2) with m + k = –2, –1, – 12 , 0, 1. 4◦ . Self-similar solution for k ≠ 1: 1

w = t 1–k u(ξ),

k–m–1

ξ = xt 2(1–k) ,

where the function u(ξ) is determined by the ordinary differential equation a(um u′ξ )′ξ +

m–k+1 ′ 1 ξuξ + buk – u = 0. 2(1 – k) 1–k

Reference for equation 5.1.11.14: V. A. Dorodnitsyn (1982).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 227

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

228

SECOND-ORDER PARABOLIC EQUATIONS

5.1.12. Equations ofthe Form 

∂w = a ∂ w m ∂w ∂t ∂x ∂x

WITH

Page 228

ONE SPACE VARIABLE

+ bw + c1 w k1 + c2 w k2 + c3 w k3

⊲ Equations of this form admit traveling-wave solutions w = w(kx + λt).   ∂w ∂ ∂w 1. =a w + bw + c. ∂t ∂x ∂x For b = 0, see equation 5.1.11.1 and for c = 0, see equation 5.1.11.2. Generalized separable solutions linear and quadratic in x: aC12 2bt c e – , b b Z bebt (x + C2 )2 ebt cebt w(x, t) = + C3 1/3 + 1/3 e–bt ϕ1/3 (t)dt, ϕ(t) ϕ (t) ϕ (t)

w(x, t) = C1 xebt + C2 ebt +

ϕ(t) = C1 – 6aebt ,

where C1 , C2 , and C3 are arbitrary constants. The first solution is degenerate. Reference: A. D. Polyanin and V. F. Zaitsev (2004, p. 37).

∂w

∂



∂w



=a w + bw + cw 2 . ∂t ∂x ∂x This is a special case of equation 5.6.10.1. There are generalized separable solutions of the forms: 2.

w = f (t) + g(t)[a cosh(kx) + b sinh(kx)], w = f (t) + g(t)[a cos(kx) + b sin(kx)], where a and b are arbitrary constants, and the functions f (t) and g(t) and the constant k are determined by substituting these solutions into the original equation. References: W. I. Newman (1980), V. A. Galaktionov (1995), V. F. Zaitsev and A. D. Polyanin (1996, pp. 425–427).

 √ 1 ∂w + b + c w. √ ∂t ∂x w ∂x This is a special case of equation 5.1.12.12 with m = –1/2. Functional separable solution: 3.

∂w

=a

∂



w = [ϕ(x) + ψ(x)t]2 , where the functions ϕ = ϕ(x) and ψ = ψ(x) are determined by the system of ordinary differential equations aϕ′′xx – ϕψ + 12 cϕ + 12 b = 0, ′′ aψxx – ψ 2 + 12 cψ = 0.

This system can be easily integrated by starting from the latter equation. In particular, by setting ψ = const, one arrives at the special solution  2 1 b 2 w= ct – x + C1 x + C2 , 2 4a

where C1 and C2 are arbitrary constants.

Reference: V. A. Galaktionov and S. R. Svirshchevskii (2007, pp. 280–281).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 228

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

4.

∂w ∂t

=a

∂ ∂x



1 ∂w √ w ∂x



Page 229

229

√ + b + cw + d w.

Generalized separable solution:  w = f (x) exp

1 2 ct




2 + g(x) ,

where the functions f = f (x) and g = g(x) are determined by the system of ordinary differential equations ′′ 2afxx + cf g + df = 0, ′′ 2agxx + cg2 + dg + b = 0.

The latter equation has two constant special solutions: g = k1,2 = const. In both these cases, the former equation is a linear equation with constant coefficients, which is easily integrable and has solutions that are expressed in terms of exponentials or sines and cosines. Remark. The constants a, b, and d in the above equation can be replaced by arbitrary functions a = a(x), b = b(x), and d = d(x).

5.

∂w ∂t

=a

∂ ∂x



w

–2 ∂w

∂x



– bw – cw 3 .

Functional separable solutions for b ≠ 0:  Z 2bt 2 w(x, t) = ± bC1 e (x+C2 ) +C3 F (t)+2cF (t)

dt F (t)

–1/2

,

where C1 , C2 , and C3 are arbitrary constants.


F (t) = exp aC1 e2bt +2bt ,

Reference: A. D. Polyanin and V. F. Zaitsev (2004, p. 37).

6. 1◦ .

∂w ∂t

=a

∂ ∂x



w –4/3

∂w ∂x



+ bw –1/3 + cw.

Multiplicative separable solutions:
–3 w(x, t) = ect Aekx + Be–kx  –3 w(x, t) = ect A cos(kx) + B sin(kx)

if

b/(3a) = k2 > 0,

if

b/(3a) = –k2 < 0,

where A and B are arbitrary constants. 2◦ . The transformation

w = ect u(x, τ ),

τ =–

3 – 4 ct e 3 + const 4c

leads to a simpler equation of the form 5.1.11.6:   ∂u ∂ ∂u =a u–4/3 + bu–1/3 . ∂τ ∂x ∂x 3◦ . See also equation 5.1.12.10. Reference for equation 5.1.12.6: A. D. Polyanin and V. F. Zaitsev (2004, p. 38).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 229

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 230

230

SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE   ∂w ∂ –4/3 ∂w 7. =a w + bw + cw 5/3 . ∂t ∂x ∂x Functional separable solution:  –3/2 w(x, t) = ϕ(t)x2 + ψ(t)x + χ(t) ,

where the functions ϕ(t), ψ(t), and χ(t) are determined by the system of first-order ordinary differential equations ϕ′t = 12 aϕ(4ϕχ – ψ 2 ) – 23 bϕ, ψt′ = 12 aψ(4ϕχ – ψ 2 ) – 23 bψ, χ′t = 12 aχ(4ϕχ – ψ 2 ) – 23 bχ – 23 c. It follows from the first two equations that ϕ = Cψ, where C is an arbitrary constant.

Remark. In the original equation, the constants a, b, and c can be replaced by arbitrary functions a = a(t), b = b(t), and c = c(t). Reference: V. A. Galaktionov, S. A. Posashkov, and S. R. Svirshchevskii (1995).

  ∂w ∂ ∂w =a w –4/3 + bw + cw 7/3 . ∂t ∂x ∂x Functional separable solution:  –3/4 w(x, t) = ϕ4 (t)x4 + ϕ3 (t)x3 + ϕ2 (t)x2 + ϕ1 (t)x + ϕ0 (t) , 8.

where the functions ϕk = ϕk (t) are determined by the system of ordinary differential equations ϕ′0 = – 34 aϕ21 + 2aϕ0 ϕ2 – 43 bϕ0 – 43 c, ϕ′1 = –aϕ1 ϕ2 + 6aϕ0 ϕ3 – 43 bϕ1 , ϕ′2 = –aϕ22 + 32 aϕ1 ϕ3 + 12aϕ0 ϕ4 – 43 bϕ2 , ϕ′3 = –aϕ2 ϕ3 + 6aϕ1 ϕ4 – 43 bϕ3 , ϕ′4 = – 34 aϕ23 + 2aϕ2 ϕ4 – 43 bϕ4 .

The prime denotes a derivative with respect to t. Remark. In the original equation, the constants a, b, and c can be replaced by arbitrary functions a = a(t), b = b(t), and c = c(t). Reference: V. A. Galaktionov (1995).

9.

∂w

=

∂



w

–4/3 ∂w



∂t ∂x ∂x Functional separable solution:

–

3 4

w –1/3 + bw 5/3 + cw.

w = [ϕ1 (t) + ϕ2 (t) cos x + ϕ3 (t) sin x]–3/2 , where the functions ϕn = ϕn (t) are determined by the system of ordinary differential equations ϕ′1 = 12 ϕ1 (ϕ21 – ϕ22 – ϕ23 ) – 23 cϕ1 – 23 b, ϕ′2 = 12 ϕ2 (ϕ21 – ϕ22 – ϕ23 ) – 23 cϕ2 , ϕ′3 = 12 ϕ3 (ϕ21 – ϕ22 – ϕ23 ) – 23 cϕ3 . It follows from the last two equations that ϕ3 = Cϕ2 , where C is an arbitrary constant.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 230

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 231

231

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Remark. In the original equation, the constants b and c can be replaced by arbitrary functions b = b(t) and c = c(t). Reference: V. A. Galaktionov and S. R. Svirshchevskii (2007, p. 32).

10.

  ∂w ∂ –4/3 ∂w = w – aw –1/3 + bw 7/3 + cw. ∂t ∂x ∂x

The substitution u = w–4/3 leads to an equation with quadratic nonlinearity: ∂u ∂2u 3 =u 2 – ∂t 4 ∂x



∂u ∂x

2

+

1◦ . For a = 1, there is a solution of the form


4 au2 – cu – b . 3

u = ϕ1 (t) + ϕ2 (t) cos(kx) + ϕ3 (t) sin(kx) + ϕ4 (t) cos(2kx) + ϕ5 (t) sin(2kx),

k = 2 × 3–1/2 ,

where the functions ϕn = ϕn (t) are determined by the system of first-order ordinary differential equations (not written out here). 2◦ . For a = –1, there is a solution of the form u = ϕ1 (t)+ϕ2 (t) cosh(kx)+ϕ3 (t) sinh(kx)+ϕ4 (t) cosh(2kx)+ϕ5 (t) sinh(2kx),

k = 2×3–1/2 .

Reference for equation 5.1.12.10: V. A. Galaktionov (1995).

11.

∂w ∂t

=a

∂ ∂x



w

m ∂w

∂x



+ bw m+1 + cw.

1◦ . Multiplicative separable solutions:   1 w(x, t) = ect A cos(kx) + B sin(kx) m+1   1 w(x, t) = ect A exp(kx) + B exp(–kx) m+1

if b(m + 1)/a = k2 > 0, if b(m + 1)/a = –k2 < 0,

where A and B are arbitrary constants. 2◦ . The transformation

w = ect u(x, τ ),

τ=

1 cmt e + const cm

leads to a simpler equation of the form 5.1.11.11:   ∂u ∂ m ∂u =a u + bum+1 . ∂τ ∂x ∂x Special case. Multiplicative separable solution for m = –1:   b 2 w = A exp ct – x + Bx , 2a where A and B are arbitrary constants.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 231

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 232

232

SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE   ∂w ∂ m ∂w 12. =a w + b + cw –m. ∂t ∂x ∂x Functional separable solution:   1 m+1 b(m + 1) 2 , w = c(m + 1)t – x + C1 x + C2 2a where C1 and C2 are arbitrary constants.   ∂w ∂ m ∂w 13. =a w + bw + cw 1–m. ∂t ∂x ∂x 1◦ . Generalized traveling-wave solution:   a 2bmt c 1/m bmt bmt w(x, t) = C1 e x + e + C2 e – , b bm2 where C1 and C2 are arbitrary constants. 2◦ . For a more complicated solution, see 5.6.13.6 with f (t) = b and g(t) = c. Reference for equation 5.1.12.13: V. A. Galaktionov and S. A. Posashkov (1989).

∂w

∂



m ∂w



=a w + bw 1+m + cw + sw 1–m. ∂t ∂x ∂x This is a special case of equation 5.6.13.7 with f (t) = c and g(t) = s. The substitution u = wm leads to an equation of the form 5.1.9.10:   ∂u ∂2u a ∂u 2 = au 2 + + bmu2 + cmu + sm. ∂t m ∂x ∂x 14.

Reference: V. A. Galaktionov and S. A. Posashkov (1989).

  ∂w ∂ 2n ∂w =a w + bw + cw 1–n. 15. ∂t ∂x ∂x Generalized traveling-wave solutions:   Z dt 1/n w(x, t) = ϕ(t)(±x + C1 ) + cnϕ(t) , ϕ(t) where C1 and C2 are arbitrary constants.

  a(n + 1) –1/2 –2bnt ϕ(t) = C2 e – , bn2

Reference: A. D. Polyanin and V. F. Zaitsev (2004, p. 40).

∂w

∂



=a

wn

∂w



+ bw + ck + bkw n+1 + cw –n. ∂t ∂x ∂x Functional separable solutions:   1 √
√
 c n+1   w(x, t) = exp b(n + 1)t C1 cos x λ + C2 sin x λ – b 1  p
p
 c  n+1   w(x, t) = exp b(n + 1)t C1 cosh x |λ| + C2 sinh x |λ| – b bk where C1 and C2 are arbitrary constants and λ = (n + 1). a 16.

if λ > 0, if λ < 0,

Reference: V. A. Galaktionov (1994).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 232

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 233

233

h i ∂w ∂ ∂w 5.1.13. Equations of the Form = f (w) + g(w) ∂t ∂x ∂x ⊲ Equations of this form admit traveling-wave solutions w = w(kx + λt).   ∂w ∂ ∂w 2 1. = (aw + bw) . ∂t ∂x ∂x

This is a special case of equation 5.6.15.1 with f (w) = aw2 + bw and a special case of equation 5.1.13.4 with n=1. Solutions: q b w(x, t) = ± 2C1 x + 2aC12 t + C2 – , a x + C1 b , w(x, t) = ± √ – C2 – 4at 2a where C1 and C2 are arbitrary constants. The first solution is of the traveling-wave type and the second one is self-similar.   ∂w ∂ a ∂w 2. = . ∂t ∂x w 2 + b2 ∂x This is a special case of equation 5.6.15.1 with f (w) = a(w2 + b2 )–1 . 1◦ . Solutions (A and B are arbitrary constants): w(x) = b tan(Ax + B),
–1/2 w(x, t) = ±bx A – 2ab–2 t – x2 ,    –1/2 . w(x, t) = Ab exp(ab–2 t – x) 1 – A2 exp 2(ab–2 t – x)

2◦ . Traveling-wave solution in implicit form:   ak2 A w 1 2 2 λ(kx + λt) + B = 2 2 ln |w + A| – ln(w + b ) + arctan , 2 b b A +b where A, B, k, and λ are arbitrary constants. 3◦ . The substitution w= √ leads to the equation

bu 1 – u2

  2  2 ∂u a ∂u 2 ∂ u = 2 (1 – u ) 2 + u , ∂t ∂x b ∂x

(1)

(2)

which is a special case of 12.1.2.12 with F (t, ξ, η) = ab–2 (ξ – η). Equation (2) has multiplicative separable solutions Aeλx + Be–λx 2aλ2 u= √ , k= 2 ; b 4AB + Ce–kt (3) A sin(λx) + B cos(λx) 2aλ2 u= √ , k= 2 , b A2 + B 2 + Cekt where A, B, C, and λ are arbitrary constants. Formulas (1) and (3) provide two solutions of the original equation.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 233

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

234

SECOND-ORDER PARABOLIC EQUATIONS

4◦ . Solution:

WITH

Page 234

ONE SPACE VARIABLE

 a 1 w = b tan ± z ± 2 t + C , 2 b  
a 1 2 –2 z = x cos ± z – arctan ψ(z) ± 2 t + C , 2 b 

where C is an arbitrary constant and the function ψ = ψ(z) is determined by the ordinary differential equation   1 ψ ′ 2 ψz = (1 + ψ ) ±1 – . 2 z Here the function z = z(x, t) is defined implicitly. 5◦ . Solution:


C at w = b tan ϕ(z) + arctan ψ(z) + ln 2 , 2 b   2 2 b x C at z= cos–2 ϕ(z) + ln 2 , at 2 b 

where C is an arbitrary constant and the functions ϕ(z) and ψ(z) are determined by the system of ordinary differential equations ϕ′z

ψ = , 2z

ψz′

  1 C ψ ψ 2 = (1 + ψ ) – – . 2 2 2 z

Here the function z = z(x, t) is defined implicitly. References for equation 5.1.13.2: I. Sh. Akhatov, R. K. Gazizov, and N. H. Ibragimov (1989), N. H. Ibragimov (1994, pp. 125–126), P. W. Doyle and P. J. Vassiliou (1998).

3.

∂w ∂ = ∂t ∂x



 a ∂w . 1 – w 2 ∂x

Solutions: w = ±p

w

w w w

x + C1

, (x + C1 )2 + 2at + C2 eat–x =± √ , C 1 + C 2 e2(x–at) sinh(x + C1 ) = ±p , cosh2 (x + C1 ) – C2 e–2at cosh(x + C1 ) = ±p , sinh2 (x + C1 ) + C2 e–2at sin(x + C1 ) = ±p , 2at C2 e – cos2 (x + C1 )

where C1 and C2 are arbitrary constants.

Reference: P. W. Doyle and P. J. Vassiliou (1998).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 234

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

235

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

4.

Page 235

  2n n ∂w = (aw + bw ) . ∂t ∂x ∂x

∂w

∂

This is a special case of equation 5.6.15.1 with f (w) = aw2n + bwn . 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the function w1 = w(C1 x + C2 , C12 t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Traveling-wave solution:  q  b 1/n 2 w(x, t) = ± 2C1 nx + 2aC1 nt + C2 – , a where C1 and C2 are arbitrary constants. 3◦ . Self-similar solution: 1/n  x + C1 b w(x, t) = ± √ – , C2 – kt a(n + 1)

k=

2a(n + 1) . n

Reference for equation 5.1.13.4: A. D. Polyanin and V. F. Zaitsev (2004, pp. 41–42).

5.

  2n n ∂w = (aw + bw ) + cw 1–n. ∂t ∂x ∂x

∂w

∂

Generalized traveling-wave solution:  1/n b x + C1 cn2 , w(x, t) = ± √ – (C2 – kt) – a(n + 1) C2 – kt 3a(n + 1)

k=

2a(n + 1) , n

where C1 and C2 are arbitrary constants. Reference: A. D. Polyanin and V. F. Zaitsev (2004, p. 42).

6.

  2n n ∂w = (aw + bw ) + cw + sw 1–n. ∂t ∂x ∂x

∂w

∂

Generalized traveling-wave solutions:   Z Z b dt 1/n , w(x, t) = ϕ(t)(±x + C1 ) + ϕ(t) ϕ(t) dt + snϕ(t) n ϕ(t)   a(n + 1) –1/2 –2cnt ϕ(t) = C2 e – , cn2 where C1 and C2 are arbitrary constants.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 235

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

236

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 236

ONE SPACE VARIABLE

  ∂w ∂ m ∂w 5.1.14. Equations of the Form =a w + f (x, t, w) ∂t ∂x ∂x 1.

∂w ∂t

=a

∂

∂x



w

–4/3 ∂w

∂x



+ bxmw –1/3 .

For m = 0, see equation 5.1.11.6. For m ≠ 0, the original equation can be reduced to a simpler equation 5.1.10.4 that corresponds to the case b = 0 [see equation 5.6.13.1 with f (x) = bxm ]. 2.

∂w ∂t

=a

∂ ∂x



w

m ∂w

∂x



+ btnw.

This is a special case of equation 5.6.13.4 with f (t) = btn . 3.

∂w ∂t

=a

∂ ∂x



wm

∂w ∂x



+ beλtw.

This is a special case of equation 5.6.13.4 with f (t) = beλt . 4.

∂w ∂t

=a

∂ ∂x



wm

∂w ∂x



+ btnw 1–m.

This is a special case of equation 5.6.13.5 with f (t) = btn . 5.

∂w ∂t

=a

∂ ∂x



w

m ∂w

∂x



+ beλtw 1–m.

This is a special case of equation 5.6.13.5 with f (t) = beλt . 6.

∂w ∂t

=a

∂ ∂x



w

m ∂w

∂x



+ btnw + ctkw 1–m.

This is a special case of equation 5.6.13.6 with f (t) = btn and g(t) = ctk . 7.

  ∂ ∂w ∂w =a wm + beλtw + ceµtw 1–m. ∂t ∂x ∂x

This is a special case of equation 5.6.13.6 with f (t) = beλt and g(t) = ceµt . 8.

  ∂w ∂ ∂w =a wm + bw 1+m + ctnw + stkw 1–m. ∂t ∂x ∂x

This is a special case of equation 5.6.13.7 with f (t) = ctn and g(t) = stk . 9.

∂w ∂t

=a

∂ ∂x



w

m ∂w

∂x



+ bxnw 1+m.

This is a special case of equation 5.6.13.8 with f (x) = bxn .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 236

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 237

237

h i ∂w ∂ ∂w 5.1.15. Equations of the Form = f (w) + g(w) ∂w ∂t ∂x ∂x ∂x 1.

∂w ∂t

+ aw

∂w ∂x

=b

∂

∂x



w

∂w ∂x



.

This is a special case of equation 5.1.15.8 with n = 1. Degenerate solution linear in x: ax + b ln |t + C1 | + C2 , a2 (t + C1 )

w(x, t) =

where C1 and C2 are arbitrary constants. 2.

∂w ∂t

+ aw

∂w ∂x

=b

∂ ∂x



w

2 ∂w

∂x



.

This is a special case of equation 5.1.15.8 with n = 2. 1◦ . Traveling-wave solution in implicit form: 2b

Z

w2 dw = x + λt + C2 , aw2 + 2λw + C1

where C1 , C2 , and λ are arbitrary constants. 2◦ . Degenerate solution linear in x: w(x, t) = (x + C1 )f (t). Here, C1 is an arbitrary constant, and the function f = f (t) is determined by the ordinary differential equation ft′ + af 2 = 2bf 3 , whose solution can be represented in implicit form: 1 2b 2bf – a + ln = t + C2 . af a2 f 3.

∂w ∂t

=

∂

∂x



w

–2 ∂w

∂x



+ aw –2

∂w ∂x

.

Fokas–Yortsos equation. 1◦ . This equation can be linearized; see equation 5.6.15.5 (the special case in Item 2◦ ) and equation 5.6.13.3 with f (x) = a. 2◦ . Traveling-wave solution in implicit form: Z dw = x + λt + C2 , 2 w(λw + C1 w + a) where C1 , C2 , and λ are arbitrary constants (the integral can be calculated explicitly).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 237

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

238

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 238

ONE SPACE VARIABLE

3◦ . Multiplicative separable solution: p w(x, t) = f (x) C1 t + C2 ,

where C1 and C2 are arbitrary constants, and the function f = f (x) is determined by the autonomous ordinary differential equation (f –2 fx′ )′x + af –2 fx′ = 12 C1 f . 4◦ . Suppose w(x, t) is a solution of the equation in question. Then the function aw2 aw – wx

w1 = is also a solution of the equation.

References for equation 5.1.15.3: A. S. Fokas and Y. C. Yortsos (1982), V. Tychynin and O. Rasin (2004), N. M. Ivanova (2008).

4.

∂w ∂t

=

∂



∂w

a

∂x (w +

b)2

∂x



+c

∂w ∂x

.

The transformation u(z, t) = w(x, t) + b,

z = x + ct

leads to an equation of the form 5.1.10.3:   ∂ ∂u –2 ∂u =a u . ∂t ∂z ∂z 5.

  ∂w 2 + aw = (bw + cw) . ∂t ∂x ∂x ∂x

∂w

∂w

∂

1◦ . Degenerate solution linear in x: w(x, t) = f (t)x + g(t), where the functions f = f (t) and g = g(t) are determined by the system of ordinary differential equations ft′ + af 2 = 2bf 3 , gt′ + af g = 2bf 2 g + cf 2 . The solution of the first equation can be found in 5.1.15.2, Item 2◦ . The second equation is easy to integrate, since it is linear in g. 2◦ . Traveling-wave solution in implicit form: 2

Z

bw2 + cw dw = x + λt + C2 , aw2 + 2λw + C1

where C1 , C2 , and λ are arbitrary constants.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 238

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

6.

∂w ∂t

=a

∂ ∂x



w

m ∂w

∂x



+ b sign x

∂w ∂x

Page 239

239

.

This equation arises in biology. By changing from t and x to the new variables t and z = |x| + bt, we arrive at a simpler equation of the form 5.1.10.8:   ∂w ∂ m ∂w =a w . ∂t ∂z ∂z References: J. D. Murray (1989), N. Shigesada, K. Kawasaki, and E. Teramoto (1979).

7.

∂w ∂t

=a

∂ ∂x



wn

∂w ∂x



+b

∂w ∂x

.

1◦ . Suppose w(x, t) is a solution of this equation. Then the functions
w1 = Aw Bx + b(B – An B 2 )t + C1 , An B 2 t + C2 ,

where A, B, C1 , and C2 are arbitrary constants, are also solutions of the equation. 2◦ . The transformation w = U (z, t),

z = x + bt,

leads to a simpler equation of the form 5.1.10.8:   ∂ ∂U n ∂U =a U . ∂t ∂z ∂z 8.

  ∂w ∂w ∂ n ∂w + aw =b w . ∂t ∂x ∂x ∂x

1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = C1 w(C11–n x + C2 , C12–n t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Traveling-wave solution in implicit form: Z wn dw 2b = x + λt + C2 , aw2 + 2λw + C1 where C1 , C2 , and λ are arbitrary constants. 3◦ . Self-similar solution for n ≠ 2: w(x, t) = u(z)t1/(n–2) ,

z = xt–(n–1)/(n–2) ,

where the function u = u(z) is determined by the ordinary differential equation   n–1 1 n ′′ n–1 ′ 2 bu uzz + 2bnu (uz ) – au – z u′z – u = 0. n–2 n–2

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 239

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

240 9.

SECOND-ORDER PARABOLIC EQUATIONS   ∂w ∂ ∂w n ∂w =a w + (bw n + c) . ∂t ∂x ∂x ∂x

WITH

Page 240

ONE SPACE VARIABLE

Generalized traveling-wave solution: w(x, t) =



C2 – x a ln |t + C1 | c + – b(t + C1 ) b2 n(t + C1 ) b

1/n

,

where C1 and C2 are arbitrary constants. 10.

∂w ∂t

=a

∂ ∂x



w

2n ∂w

∂x



+ bw n

∂w ∂x

.

This is a special case of equation 5.1.15.11 with k = 2n and is a special case of equation 5.1.15.12 with b = s = 0 . 1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = C1 w(C1–n x + C2 , t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Traveling-wave solution in implicit form: Z

w2n dw

= x + C1 t + C3 b C1 w + C2 – n+1 wn+1 Z dw a = x + C1 t + C3 w2 (C1 w + C2 – b ln |x|) a

if

n ≠ –1,

if n = –1,

where C1 , C2 , and C3 are arbitrary constants. 3◦ . Multiplicative separable solution: w = (x + C1 )1/n ϕ(t), where C1 is an arbitrary constant and the function ϕ = ϕ(t) is determined by the separable first-order ordinary differential equation ϕ′t = a

n + 1 2n+1 b n+1 ϕ + ϕ . n n2

Integrating yields the general solution in implicit form –n

ϕ

  a(n + 1) a(n + 1) –n – ln ϕ + = –bt + C2 , bn bn

where C2 is an arbitrary constant. References for equation 5.1.15.10: C. M. Yung, K. Verburg, and P. Baveye (1994), A. D. Polyanin and V. F. Zaitsev (2004, p. 46), N. M. Ivanova (2008).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 240

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

241

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

11.

∂w ∂t

=a

∂ ∂x



w

k ∂w

∂x



+ bw n

∂w ∂x

Page 241

.

1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = C1 w(C1n–k x + C2 , C12n–k t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Traveling-wave solution in implicit form: a

Z

wk dw

= x + C1 t + C3 b C1 w + C2 – n+1 wn+1 Z wk dw a = x + C1 t + C3 C1 w + C2 – b ln |x|

if

n ≠ –1,

if n = –1,

where C1 , C2 , and C3 are arbitrary constants. 3◦ . Self-similar solution: 1

w = t k–2n U (z),

n–k

z = xt k–2n ,

where the function U = U (z) is determined by the ordinary differential equation  1  U + (n – k)zUz′ = a(U k Uz′ )′z + bU n Uz′ . k – 2n

Special case. For k = n, there is the generalized traveling-wave solution  1/n C2 – x a ln |t + C1 | w= + 2 , b(t + C1 ) b n(t + C1 )

where C1 and C2 are arbitrary constants (this solution is a special case of the one given in equation 5.1.15.12). References for equation 5.1.15.11: C. M. Yung, K. Verburg, and P. Baveye (1994), N. M. Ivanova (2008).

12.

∂w ∂ = ∂t ∂x



aw

2n

+ bw

n


∂w ∂x



+ (cw n + s)

∂w . ∂x

Generalized traveling-wave solution:

 1/n Z b w(x, t) = ϕ(t)x + (st + C1 )ϕ(t) + ϕ(t) ϕ(t) dt , n where C1 is an arbitrary constant and the function ϕ(t) is determined by the first-order separable ordinary differential equation ϕ′t =

a(n + 1) 3 ϕ + cϕ2 . n

Reference: A. D. Polyanin and V. F. Zaitsev (2004, p. 46).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 241

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

242

SECOND-ORDER PARABOLIC EQUATIONS

5.1.16. Equations hof the Form i 1.

∂w ∂t

WITH

Page 242

ONE SPACE VARIABLE

  ∂w = ∂ f (w) ∂w + g x, t, w, ∂w ∂t ∂x ∂x ∂x

=

∂

∂x



w

–1 ∂w

∂x



+ bx

∂w ∂x

.

This is a special case of equation 5.1.16.4 with n = –1. Solutions:
w(x, t) = C1 exp C2 xebt , 2b w(x, t) = , bt 2bxe – e2bt + C1 4be2bt , w(x, t) = 2 bt 4b (xe + C1 )2 + C2 e4bt C1 C22 exp(2bt + C2 xebt ) w(x, t) = , b[1 – C1 exp(C2 xebt )]2 C1 w(x, t) = , 2bC2 + C3 exp(C1 xebt + C1 C2 e2bt ) C12 e2bt , w(x, t) = b cos2 (C1 xebt + C2 ) where C1 , C2 , and C3 are arbitrary constants. Reference: N. M. Ivanova (2008).

2.

∂w ∂t

=a

∂ ∂x



w

–2 ∂w

∂x



+b

∂w ∂x

+ cw.

This is a special case of equation 5.6.13.10 with m = –2, f (t) = b, and g(t) = c. The transformation (A and B are arbitrary constants) w(x, t) = ect u(z, τ ),

z = x + bt + A,

τ =B–

1 –2ct e 2c

leads to an equation of the form 5.1.10.3:   ∂u ∂ ∂u =a u–2 . ∂τ ∂z ∂z Reference: V. A. Dorodnitsyn and S. R. Svirshchevskii (1983); the case b = 0 was treated.

3.

∂w ∂t

=a

∂ ∂x



w

–2 ∂w

∂x



+ (bx + c)

∂w ∂x

+ sw.

This is a special case of equation 5.6.13.12 with m = –2, f (t) = b, g(t) = c, and h(t) = s. 1◦ . For b = 0, see equation 5.1.16.2. 2◦ . Let b ≠ 0 and b ≠ s. The transformation w(x, t) = est u(z, τ ),

 c z = ebt x + , b

τ=

1 e2(b–s)t 2(b – s)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 242

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 243

243

leads to a simpler equation of the form 5.1.10.3:   ∂u ∂ ∂u =a u–2 . ∂τ ∂z ∂z 3◦ . Let b ≠ 0 and b = s. The transformation w(x, t) = ebt u(z, t),

 c z = ebt x + b

leads to a simpler equation of the form 5.1.10.3:   ∂u ∂ –2 ∂u =a u . ∂t ∂z ∂z

References for equation 5.1.16.3: S. I. Svinolupov (1985), V. F. Zaitsev and A. D. Polyanin (1996, pp. 434– 435).

4.

∂w ∂t

=a

∂ ∂x



w

n ∂w

∂x



– bx

∂w ∂x

.

This is a special case of equation 5.1.16.5 with c = s = 0. 1◦ . Suppose w(x, t) is a solution of this equation. Then the functions w1 = C1–2 w(±C1n x + C2 ebt , t + C3 ), where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation. 2◦ . Generalized traveling-wave solutions:
bt 2/n

w(x, t) = ±x + C1 e

  a(n + 2) –1/n 2bt , C2 e + bn

where C1 and C2 are arbitrary constants.

3◦ . Generalized traveling-wave solutions: w = w(z),

z = ±x + Cebt ,

where C is an arbitrary constant and the function w(z) is determined by the ordinary differential equation a(wn wz′ )′z – bzwz′ = 0. 4◦ . The transformation w(x, t) = u(z, τ ),

z = e–bt x,

τ =C–

1 –2bt e , 2b

where C is an arbitrary constant, leads to a simpler equation of the form 5.1.10.8:   ∂u ∂ n ∂u =a u . ∂τ ∂z ∂z If n = –1 or n = –2, see 5.1.10.2 or 5.1.10.3 for solutions of this equation.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 243

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 244

244

SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE   ∂w ∂ ∂w n ∂w 5. =a w + (bx + c) + sw. ∂t ∂x ∂x ∂x This is a special case of equation 5.6.13.12. The transformation  c 1 , τ= e(2b+sn)t , w(x, t) = est u(z, τ ), z = ebt x + b 2b + sn leads to a simpler equation of the form 5.1.10.8:   ∂u ∂ n ∂u =a u . ∂τ ∂z ∂z If n = –1 or n = –2, see 5.1.10.2 or 5.1.10.3 for solutions of this equation. Reference: V. F. Zaitsev and A. D. Polyanin (1996, pp. 434–435).

∂w

∂



n ∂w



n∂

2w

n–1



∂w

2

= w + aw + bw . ∂t ∂x ∂x ∂x2 ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the function 6.

w1 = C1 w(C2 x + C3 , C1n C22 t + C4 ), where C1 , . . . , C4 are arbitrary constants, is also a solution of the equation. 2◦ . Functional separable solution for b = 13 n(a – 2) – a – 1:  X 1/n 3 k w(x, t) = n ϕk (t)x . k=0

Here,

Z 2 1 1 ϕ3 (t) = A, ϕ2 (t) = ψ(t) dt + B, ϕ1 (t) = ψ(t) dt + B + ψ(t), 3A 2Aβn Z 3 Z  1 1 1 ϕ0 (t) = ψ(t) dt + B + ψ(t) ψ(t) dt + B + ψ ′ (t), 2 2 2 27A 6A βn 12A β 2 n2 t where the function ψ = ψ(t) is defined implicitly by Z
–1/2 C1 – 83 βnψ 3 dψ = C2 + t, Z

A, B, C1 , and C2 are arbitrary constants, β = a + 1; A ≠ 0, n ≠ 0, a > –1. 3◦ . Functional separable solution for b = 14 n(a – 3) – a – 1:  X 1/n 4 k w(x, t) = n ϕk (t)x . k=0

Here, the functions ϕk = ϕk (t) are determined by the system of ordinary differential equations ϕ′0 = – 34 βϕ21 + 2βϕ2 ϕ0 , ϕ′1 = –βϕ1 ϕ2 + 6βϕ3 ϕ0 , ϕ′2 = –βϕ22 + 32 βϕ1 ϕ3 + 12βϕ4 ϕ0 , ϕ′3 = –βϕ2 ϕ3 + 6βϕ1 ϕ4 , ϕ′4 = – 34 βϕ23 + 2βϕ2 ϕ4 , where β = n(a + 1); the prime denotes a derivative with respect to t.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 244

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 245

245

4◦ . There are exact solutions of the following forms: w(x, t) = F (z),

z = Ax + Bt;

w(x, t) = (At + B)–1/n G(x); w(x, t) = tβ H(ξ),

ξ = xt–

βn+1 2 ;

η = xent ;

w(x, t) = e–2t U (η),

w(x, t) = (At + B)–1/n V (ζ),

ζ = x + C ln(At + B),

where A, B, C, and β are arbitrary constants. The first solution is of the traveling-wave type, the second is a solution in multiplicative separable form, and the third is self-similar. Reference for equation 5.1.16.6: G. A. Rudykh and E. I. Semenov (1998).

5.1.17. Other Equations ∂w

∂2w



∂w

2

∂w

+ bw 2 . ∂t ∂x ∂x The substitution w = eu leads to the linear equation

1.

w

=w

∂x2

–

+ aw

∂u ∂ 2 u ∂u +a = + b. 2 ∂t ∂x ∂x Remark. In the original equation, the parameters a and b can be replaced by arbitrary functions a = a(x, t) and b = b(x, t).

  ∂w ∂w 2 ∂w ∂ 2w 2. w = aw +b + cw + sw 2 . 2 ∂t ∂x ∂x ∂x 1◦ . For a ≠ –b, the substitution a w = f a+b leads to the linear equation ∂f ∂2f ∂f s(a + b) =a 2 +c + f. ∂t ∂x a ∂x 2◦ . For a = –b, the substitution τ = at leads to an equation of the form 5.1.17.1. Remark. In the original equation, the parameters c and s can be replaced by arbitrary functions c = c(x, t) and s = s(x, t).

  ∂w ∂2w ∂w 2 ∂w 3. w =w + a + bw + cw 2 + sw 1–a. ∂t ∂x ∂x ∂x2 1◦ . For a ≠ –1, the substitution 1

w = f 1+a leads to the linear equation ∂f ∂2f ∂f = +b + c(a + 1)f + s(a + 1). 2 ∂t ∂x ∂x Remark. In the original equation, the parameters b, c, and s can be replaced by arbitrary functions b = b(x), c = c(x), and s = s(x).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 245

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

246 4.

SECOND-ORDER PARABOLIC EQUATIONS w

∂w ∂t

=w

∂2w ∂x2

+ (k – 1)



∂w ∂x

2

+ aw k+1

WITH

∂w ∂x

Page 246

ONE SPACE VARIABLE

.

The substitution w = u1/k leads to an unnormalized Burgers equation ∂u ∂ 2 u ∂u = + au , 2 ∂t ∂x ∂x that can be linearized (see equation 5.1.5.3). ∂w

= (aw 2 + bw 4 )

∂ 2w

. ∂t ∂x2 This is a special case of equation 5.6.16.3 with f (w) = aw2 + bw4 .

5.

1◦ . Self-similar solutions: w(x, t) = ±



a (x + C1 )2 – 2a(t + C2 ) b

1/2

,

where C1 and C2 are arbitrary constants. 2◦ . Functional separable solutions:   a 1/2 w(x, t) = ± ϕ(t)(x2 + C1 x + C2 ) – , b where C1 and C2 are arbitrary constants, and the function ϕ(t) is determined by the firstorder separable equation ϕ′t = –2aϕ2 + 12 b(4C2 – C12 )ϕ3 , whose solution can be written out in implicit form. 6.

∂w

= (aw 2 + bw 4 )

∂ 2w

∂t ∂x2 Functional separable solutions:

+ cw + kw –1 .

w(x, t) = ±

q

ϕ(t)x2 + ψ(t)x + χ(t),

where the functions ϕ(t), ψ(t), and χ(t) are determined by the system of first-order ordinary differential equations ϕ′t = 12 bϕ(4ϕχ – ψ 2 ) + 2cϕ, ψt′ = 12 bψ(4ϕχ – ψ 2 ) + 2cψ, χ′t = 12 (bχ + a)(4ϕχ – ψ 2 ) + 2cχ + 2k. It follows from the first two equations that ϕ = Cψ, where C is an arbitrary constant. Remark. The above remains true if the equation coefficients are arbitrary functions of time: a = a(t), b = b(t), c = c(t), and k = k(t).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 246

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

7.

Page 247

247

∂w ∂ 2w . = ax4–kw k ∂t ∂x2

This is a special case of equation 5.6.16.4 with f (u) = auk . The transformation w(x, t) = xu(z, t), z = 1/x leads to a simpler equation of the form 5.1.9.26: ∂u ∂2u = auk 2 . ∂t ∂z ∂w ∂ 2w = axnw k . ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the function

8.

w1 = C1 w(C2 x, C1k C22–n t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . The substitution u = w1–k leads to an equation of the form 5.1.17.10:   k ∂u ∂u n ∂ 1–k = ax u . ∂t ∂x ∂x 3◦ . The transformation w(x, t) = xu(z, t),

z = 1/x

leads to an equation of the similar form

9.

3m+4 ∂w = ax m+1 ∂t

∂u ∂2u = az 4–n–k uk 2 . ∂t ∂z   ∂ ∂w wm . ∂x ∂x

This is a special case of equation 5.1.17.10. The transformation 1

w(x, t) = x m+1 u(z, t),

z=

1 x

leads to a simpler equation of the form 5.1.10.8:   ∂u ∂ m ∂u =a u . ∂t ∂z ∂z 10.

∂w ∂t

= ax

n

∂ ∂x



w

m ∂w

∂x



.

This equation occurs in nonlinear problems of heat and mass transfer and is a special case of equation 5.6.17.16 with f (w) = awm . For n = 0, see equation 5.1.10.8.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 247

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

248

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 248

ONE SPACE VARIABLE

1◦ . Suppose w(x, t) is a solution of the equation in question. Then the function w1 = C1 w(C2 x, C1m C22–n t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Solutions: 1

w(x) = (Ax + B) m+1 , 

1 m mλ k= , a(n – 2)(2 + m – n – nm)  1 m
mβ 1 (1–n)β β 2–n w(x, t) = t xt +A , β= , a(2 – n) nm + n – m – 2  1 m m λ m+2 2 m+1 , w(x, t) = exp(–λt) (m + 1) x exp(λmt) + A , n= a m+1 1 2–n w(x, t) = k(λt + A)– m x m ,

where A, B, and λ are arbitrary constants. 3◦ . Multiplicative separable solution: w(x, t) = (λt + A)–1/m f (x), where the function f = f (x) is expressed via solutions to the Emden–Fowler equation ′′ + Fxx

1 λ(m + 1) –n x F m+1 = 0, am

F = f m+1 .

(1)

To the power-law particular solution of this equation there corresponds the second solution of the original equation in Item 2◦ . The order of equation (1) can be reduced; the equation is analyzed in detail in Polyanin and Zaitsev (2003), where its exact solutions for 26 different pairs of values of the parameters n and m are presented. 4◦ . Self-similar solution for n ≠ –2: w = w(z),

1

z = xt n–2 ,

where the function w(z) is determined by the ordinary differential equation
′ a(2 – n) wm wz′ z + z 1–n wz′ = 0.

(2)

The book by Zaitsev and Polyanin (1993) presents the general solution of equation (2) for m = –1 and any n. 5◦ . Self-similar solution: w = tα g(ζ),

ζ = xtβ ,

β=

mα + 1 , n–2

α is any,

where the function g(ζ) is determined by the ordinary differential equation
′ aζ n gm gζ′ ζ = βζgζ′ + αg.

(3)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 248

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 249

249

This equation is homogeneous, and, hence, its order can be reduced (thereafter it can be transformed to an Abel equation of the second kind). In the special case 1–n 1 α= , β= , nm + n – m – 2 nm + n – m – 2 a first integral of equation (3) is given by agm gζ′ = βζ 1–n g + C.

(4) 2◦ .

To C = 0 in (4) there corresponds the third solution in Item In the general case, the change of variable G = gm+1 brings (3) to the equation m 1 G′′ζζ = A1 ζ 1–n G– m+1 G′ζ + A2 ζ –n G m+1 ,

(5)

where A1 = β/a and A2 = α(m + 1)/a. Exact analytical solutions of equation (5) for various values of the parameters n and m can be found in the books by Polyanin and Zaitsev (1995, 2003). 6◦ . Generalized self-similar solution: w = eλ(n–2)t ϕ(u), u = xeλmt , λ is any, where the function ϕ(u) is determined by the ordinary differential equation aun (ϕm ϕ′u )′u = λmuϕ′u + λ(n – 2)ϕ. (6) This equation is homogeneous, so its order can be reduced (thereafter it can be transformed to an Abel equation of the second kind). m+2 In the special case n = , equation (6) has the first integral m+1 1 aϕm ϕ′ = λmu– m+1 ϕ + C. u

To C = 0 there corresponds the last solution in Item 2◦ . In the general case, the change of variable Φ = ϕm+1 brings (6) to an equation that coincides, up to notation, with (5). 7◦ . For n = 2, there are solutions of the form w = w(ξ), ξ = ln |x| – λt, which are defined implicitly by Z wm dw a(m + 1) = ξ + C2 , awm+1 – λ(m + 1)w + C1 where λ, C1 , and C2 are arbitrary constants. To the special case C1 = 0 there corresponds the solution    1 m m λ(m + 1) mλ + C|x| m+1 exp – t , w(x, t) = a m+1 where C is an arbitrary constant. 8◦ . The transformation 1 1 w(x, t) = x m+1 u(z, t), z = x leads to an equation of the similar form   4+3m–n–nm ∂ ∂u ∂u m+1 = az um . ∂t ∂z ∂z Reference for equation 5.1.17.10: V. F. Zaitsev and A. D. Polyanin (1996, pp. 410–412).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 249

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

250 11.

SECOND-ORDER PARABOLIC EQUATIONS ∂w ∂t

=

∂ ∂x



The substitution u =

ax + b cw + k

2

∂w ∂x

WITH

Page 250

ONE SPACE VARIABLE

 .

cw + k (c ≠ 0) leads to an equation of the form 5.1.10.3: ax + b   ∂u ∂ –2 ∂u = u . ∂t ∂x ∂x

Reference: A. Munier, J. R. Burgan, J. Gutierres, E. Fijalkow, and M. R. Feix (1981).

12.

  ∂w ∂ n m ∂w =a x w . ∂t ∂x ∂x

This is a special case of equation 5.6.17.5 with f (x) = axn . 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the function w1 = C1 w(C2 x, C1m C22–n t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Let m ≠ –1 and 2m – 2n – nm + 3 ≠ 0. The transformation 1–n

w(x, t) = x m+1 u(ξ, t),

ξ=x

2m–2n–nm+3 m+1

leads to an equation of the similar form   3m–3n–2nm+4 ∂u ∂ ∂u m =A ξ 2m–2n–nm+3 u , ∂t ∂ξ ∂ξ

(1)

  2m – 2n – nm + 3 2 . where A = a m+1 3m + 4 3◦ . In the special case n = , the transformed equation is very simple and coincides, 2m + 3 up to notation, with equation 5.1.10.8:   ∂u ∂ m ∂u =A u . ∂t ∂ξ ∂ξ 4◦ . In the special case of n = 2 and m = –2, the transformed equation becomes   ∂ ∂u –2 ∂u =A u , ∂t ∂ξ ∂ξ so it coincides with equation 5.1.10.3 (which can further be reduced to the linear heat equation). Reference for equation 5.1.17.12: V. F. Zaitsev and A. D. Polyanin (1996, pp. 412–413).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 250

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

∂w

a ∂



n

m ∂w

Page 251

251



= n x w . ∂t x ∂x ∂x This equation occurs in nonlinear problems of heat and mass transfer and other applications. For n = 0, see equation 5.1.10.8. To n = 1 there correspond two-dimensional problems with axial symmetry, and to n = 2 there correspond three-dimensional spherically symmetric problems. Equation with n = 5 are encountered in the theory of static turbulence. To m > 0 there corresponds slow diffusion and to m < 0, fast diffusion.

13.

1◦ . Suppose w(x, t) is a solution of the equation in question. Then the function w1 = C1 w(C2 x, C1m C22 t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Solutions:
1 w(x) = Ax1–n + B m+1 ,  1 mx2 m w(x, t) = , k = 2a(nm + m + 2), A – kt  1 m(n+1) m mx2 – w(x, t) = A|kt + B| nm+m+2 – , k = 2a(nm + m + 2), kt + B    1 m 4aλ m+2 2 w(x, t) = A exp – t + λx , n=– , m m where A, B, and λ are arbitrary constants. 3◦ . Solutions with m = –

2 : n+1

 n+1
– 2 λ 2 –2aλt C+x e w(x, t) = e , n+1  2  n+1 1+n 1+n ξ ξ – 2 w(x, t) = t 1–n ln , ξ = xt 2(1–n) , n–1 C –aλ(n+1)t



where C and λ are arbitrary constants. 1 2 n–3 4◦ . Some other solutions with m = –1, m = – , m = – , and m = can be found 2 n+1 2 in King (1990 c) and Ivanova (2008). References for equation 5.1.17.13: Ya. B. Zel’dovich and A. S. Kompaneets (1950), G. I. Barenblatt (1952, 1989), R. E. Pattle (1959), Ya. B. Zel’dovich and Yu. P. Raiser (1966), J. R. King (1990 c), L. I. Sedov (1993), A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995), V. F. Zaitsev and A. D. Polyanin (1996, p. 413), N. M. Ivanova (2008).

14.

∂w

= k(ax2 + bx + c)mw 4–2m

∂ 2w

. ∂t ∂x2 This is a special case of equation 5.6.16.5 with f (u) = ku–2m .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 251

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

252

SECOND-ORDER PARABOLIC EQUATIONS

WITH

1◦ . The transformation p w(x, t) = u(z, t) ax2 + bx + c,

z=

Z

Page 252

ONE SPACE VARIABLE

ax2

dx + bx + c

(1)

leads to the equation

∂u ∂2u = ku4–2m 2 + k(ac – 14 b2 )u5–2m , ∂t ∂z

(2)

which has a traveling-wave solution u = u(z + λt) and a multiplicative separable solution u = f (t)g(z). Using the change of variable ϕ = u2m–3 , one obtains from (2) an equation of the form 5.1.11.11,   ∂ϕ ∂ n ∂ϕ =k ϕ + pϕn+1 , ∂t ∂z ∂z 4 – 2m n= , p = k(2m – 3)(ac – 14 b2 ), 2m – 3 which admits a wide class of exact solutions. 2◦ . By the transformation   1 w(x, t) = v(ξ, t) 2m+3 ,

ξ=

Z

(ax2

dx , + bx + c)m

the original equation can be reduced to equation 5.6.17.5:   4–2m ∂v ∂ ∂v = F (ξ)v 2m–3 , ∂t ∂ξ ∂ξ where the function F (ξ) is defined parametrically by Z k dx F (ξ) = , ξ= . 2 m 2 (ax + bx + c) (ax + bx + c)m

(3)

(4)

(5)

Note some special cases of equation (4) where the function F = F (ξ) can be written out in explicit form:   ∂v ∂ cos2 ξ ∂v =k , m = 1, a = 1, b = 0, c = 1; ∂t ∂ξ v 2 ∂ξ   ∂v ∂ cosh2 ξ ∂v =k , m = 1, a = –1, b = 0, c = 1; ∂t ∂ξ v 2 ∂ξ   ∂v ∂ v –3/2 ∂v 1 =k , m = , a = –1, b = 0, c = 1. ∂t ∂ξ cos ξ ∂ξ 2 Reference for equation 5.1.17.14: V. F. Zaitsev and A. D. Polyanin (1996, pp. 413–414).

15.

∂w

= aeλx

∂



w –1

∂w



+b

∂w

. ∂t ∂x ∂x ∂x For exact solutions to this equations, see Ivanova (2008).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 252

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES

Page 253

253

  k ∂ 5 p ∂f (t)w ∂f (t)w f (t)(1 – w) 16. = 4 x . ∂t x ∂x ∂x Closed von K´arm´an–Howarth equation. It is used for the statistical description of decaying isotropic turbulence in the limit of large Reynolds numbers (see von K´arm´an and Howarth (1938) and Oberlack and Peters (1993)). Here w is the normalized double correlation function and f (t) = u′2 (t) is the turbulence intensity, and BLL = f (t)w is the longitudinal correlation function. 1◦ . Self-similar solution for f (t) = t–2(σ+1)/(σ+3) (σ is an arbitrary constant): w = w(ξ),

ξ = xt–2/(σ+3)

where the function w(ξ) is determined by the ordinary differential equation   √ k d dw 2 dw σ+1 5 ξ 1–w + ξ +2 w=0 dξ σ + 3 dξ σ+3 ξ 4 dξ subject to the boundary conditions w(ξ) = 1 at

ξ = 0,

w(ξ) → 0

as ξ → ∞.

Special case. For Loitsianskii’s decay law, which corresponds to σ = 4, we have a solution in implicit form √     √ 7 1+ 1–w √ ξ = k ln –2 1–w . 2 1– 1–w √ Geometric interpretation: in the new variables x = ξ/(7k), y = w this solution is transformed to the tractrix equation p   p 1 + 1 – y2 1 p x = – 1 – y 2 + ln , 2 1 – 1 – y2 arising in the differential geometry. The curve x = x(y) is the element of Beltarmi surface. Reference: V. N. Grebenev and M. Oberlack (2009).

2◦ . The substitution u = f (t)(1 – w) leads to the simpler equation   √ ∂u ∂u k ∂ = 4 x5 u + ft′ (t). ∂t ∂x x ∂x

(1)

3◦ . For a linear function f (t) = at + b, equation (1) admits a stationary solution 2/3  3a –4 u = C1 x + C2 – x , 10k where C1 and C2 are arbitrary constants. The corresponding nonstationary solution of the original equation is expressed as w =1–

2/3 1  3a C1 x–4 + C2 – x . at + b 10k

(2)

4◦ . For a suitable f (t), equation (1) admits a generalized separable solution of the form u = [ϕ(t)x + ψ(t)]2 .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 253

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

254

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 254

ONE SPACE VARIABLE

5.2. Equations with Exponential Nonlinearities 2 5.2.1. Equations of the Form ∂w = a ∂ w2 + b0 + b1 eλw + b2 e2λw

∂t

1.

∂w

=a

∂ 2w ∂x2

∂x

+ beλw .

∂t 1◦ . Suppose w(x, t) is a solution of this equation. Then the functions w1 = w(±C1λ x + C2 , C12λ t + C3 ) + 2 ln |C1 |,

where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation. 2◦ . Traveling-wave solution (k and β are arbitrary constants): w = w(z),

z = kx + βt,

where the function w(z) is determined by the autonomous ordinary differential equation ′′ ak2 wzz – βwz′ + beλw = 0.

3◦ . Solution: w = u(ξ) –

1 ln t, λ

x ξ= √ , t

where the function u(ξ) is determined by the ordinary differential equation au′′ξξ +

1 ′ 1 ξu + + beλu = 0. 2 ξ λ

Reference for equation 5.2.1.1: N. H. Ibragimov (1994, p. 135), A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995).

∂w

∂2w

+ a + beλw . ∂t ∂x2 This equation occurs in heat and mass transfer and combustion theory. 2.

=

1◦ . Traveling-wave solutions:
 2  ln β + C exp ±µx – 12 aλt , λ
 2  w(x, t) = – ln –β + C exp ±µx – 12 aλt , λ w(x, t) = –

β=

r

b – , µ= a

r

aλ , 2

where C is an arbitrary constant.

2◦ . The solutions of Item 1◦ are special cases of the traveling-wave solutions w = w(z),

z = ±µx + σt,

that satisfy the autonomous equation ′′ µ2 wzz – σwz′ + a + beλw = 0.

(1)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 254

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.2. EQUATIONS WITH EXPONENTIAL NONLINEARITIES

For µ=

q

1 2 aλ,

Page 255

255

σ = µ2 ,

the general solution of equation (1) can be written out in parametric form as z=2

Z

f ′ (τ ) dτ τ  + C1 , f (τ ) λτ f (τ ) + 2

w=

2 ln |f (τ )|, λ

where the function f (τ ) is defined by

√ C2 – 2 ln τ + τ 2 + k f (τ ) = , √ λ τ2 + k

k=

4b , aλ2

and C1 and C2 are arbitrary constants. Reference for equation 5.2.1.2: V. F. Zaitsev and A. D. Polyanin (1996, p. 414).

3.

∂w ∂2w = + a + beλw + ce2λw . ∂t ∂x2

Equations of this form are encountered in problems of heat and mass transfer and combustion theory. 1◦ . Traveling-wave solutions for a ≠ 0: w(x, t) = –

 1  ln β + C exp(±µx – aλt) , λ

µ=

1√ –cλ, β

(1)

where C is an arbitrary constant and the parameter β is determined by solving the quadratic equation aβ 2 + bβ + c = 0. 2◦ . Traveling-wave solutions for a = 0: w(x, t) = –

√
1 ln ± –cλ x – bλt + C . λ

(2)

3◦ . The substitution u = e–λw leads to an equation with quadratic nonlinearity: u

∂u ∂2u =u 2 – ∂t ∂x



∂u ∂x

2

– aλu2 – bλu – cλ.

The particular solution u = β + C exp(λt + µx) of this equation generates a solution (1). 4◦ . Solutions (1) and (2) are special cases of a wider class of traveling-wave solutions w = w(x + σt). Reference for equation 5.2.1.3: V. F. Zaitsev and A. D. Polyanin (1996, p. 415).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 255

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

256

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 256

ONE SPACE VARIABLE

  ∂w ∂ λw ∂w 5.2.2. Equations of the Form =a e + f (w) ∂t ∂x ∂x ⊲ Equations of this form admit traveling-wave solutions w = w(kx + λt). 1.

∂w

=a

∂



eλw

∂w



. ∂t ∂x ∂x This equation governs unsteady heat transfer in a quiescent medium in the case where the thermal diffusivity is exponentially dependent on temperature. 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the function w1 = w(C1 x + C2 , C3 t + C4 ) +

1 C3 ln 2 , λ C1

where C1 , . . . , C4 are arbitrary constants, is also a solution of the equation. 2◦ . Solutions:   2 ±x + A , w(x, t) = ln √ λ B – 2at
1
1 w(x, t) = – ln C – 2aλµt + ln λµx2 + Ax + B , λ λ

where A, B, C, and µ are arbitrary constants. The first solution is self-similar and the second one is an additive separable solution. 3◦ . Traveling-wave solution in implicit form: x + βt + C1 = a 4◦ . Self-similar solution: w = w(y),

Z

eλw dw . βw + C2

√ y = x/ t,

where the function w(y) is determined by the ordinary differential equation a(eλw wy′ )′y + 12 ywy′ = 0. 5◦ . Solution:

w(x, t) = U (ξ) + 2kt,

ξ = xe–kλt ,

where k is an arbitrary constant, and the function U = U (ξ) is determined by the ordinary differential equation 2k – kλξUξ′ = a(eλU Uξ′ )′ξ . 6◦ . Solution:

1 ln t, ζ = x + β ln t, λ where β is an arbitrary constant, and the function F = F (ζ) is determined by the first-order ordinary differential equation (C is an arbitrary constant) w(x, t) = F (ζ) –

–ζ + βλF = aλeλF Fζ′ + C.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 256

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 257

257

5.2. EQUATIONS WITH EXPONENTIAL NONLINEARITIES

7◦ . Solution:

2b + 1 ln t, θ = xtb , λ where b is an arbitrary constant, and the function G = G(θ) is determined by the ordinary differential equation 2b + 1 – + bθG′θ = (aeλG G′θ )′θ . λ w(x, t) = G(θ) –

8◦ . The substitution ϕ = eλw leads to an equation of the form 5.1.9.1: ∂ϕ ∂2ϕ = aϕ 2 . ∂t ∂x References for equation 5.2.2.1: L. V. Ovsiannikov (1959, 1982), N. H. Ibragimov (1994, pp. 110, 119), A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995).

  ∂ ∂w w ∂w =a e + b. 2. ∂t ∂x ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = w(C1 x + C2 , t + C3 ) – 2 ln |C1 |, where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Additive separable solutions: w(x, t) = ln |C1 x + C2 | + bt + C3 ,   2a w(x, t) = 2 ln | ± x + C1 | – ln C2 e–bt – . b The first solution is degenerate. 3◦ . The transformation w = bt + u(x, τ ),

τ=

1 bt e + const b

leads to an equation of the form 5.2.2.1:   ∂u ∂ ∂u =a eu . ∂τ ∂x ∂x Reference for equation 5.2.2.2: A. D. Polyanin and V. F. Zaitsev (2002, p. 41).

  ∂w ∂ w ∂w 3. = e – a2 ew. ∂t ∂x ∂x Solutions: ±A exp[2(±ax + B)] + 2 exp(±ax + B) + D ax – B, w(x, t) = ln 2a2 (t + C) ±

where A, B, C, and D are arbitrary constants (either the upper or the lower signs must be taken).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 257

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 258

258

SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE   ∂w ∂ w ∂w 4. =a e – bew . ∂t ∂x ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the functions w1 = w(±x + C1 , C2 t + C3 ) + ln C2 , where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation. 2◦ . Additive separable solution: w(x, t) = u(x) – ln(aC1 t + C2 ), where C1 and C2 are arbitrary constants, and the function u = u(x) is determined by the ordinary differential equation u′′xx + (u′x )2 + C1 e–u –

b = 0. a

Integrating yields the general solution in implicit form: Z 

–2u

C3 e

–u

– 2C1 e

b + a

–1/2

du = ±x + C4 .

The integral is computable, so the solution can be rewritten in explicit form (if a = 1 and b > 0, see 5.2.2.3 for a solution). 3◦ . The substitution u = ew leads to an equation of the form 5.1.9.10: ∂u ∂2u = au 2 – bu2 . ∂t ∂x 5.

∂w ∂t

=

∂ ∂x



w ∂w

e

∂x



+ aew + b,

a ≠ 0,

b ≠ 0.

1◦ . Additive separable solution for a = k2 > 0:   w(x, t) = ln C1 cos(kx) + C2 sin(kx) + bt + C3 , where C1 , C2 , and C3 are arbitrary constants.

2◦ . Additive separable solution for a = –k2 < 0:   w(x, t) = ln C1 cosh(kx) + C2 sinh(kx) + bt + C3 . 3◦ . The transformation

w = bt + u(x, τ ),

τ=

1 bt e + const b

leads to an equation of the form 5.2.2.4:   ∂u ∂ u ∂u = e + aeu . ∂τ ∂x ∂x Reference for equation 5.2.2.5: A. D. Polyanin and V. F. Zaitsev (2002, pp. 41–42).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 258

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.2. EQUATIONS WITH EXPONENTIAL NONLINEARITIES

6.

∂w ∂t

=a

∂ ∂x



w ∂w

e

∂x



Page 259

259

+ beλw .

1◦ . Suppose w(x, t) is a solution of this equation. Then the functions w1 = w(±C1λ–1 x + C2 , C12λ t + C3 ) + 2 ln |C1 |, where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation. 2◦ . Solution for λ ≠ 0: w(x, t) = u(z) –

1 ln t, λ

z = 2 ln |x| +

1–λ ln t, λ

where the function u = u(z) is determined by the ordinary differential equation

7.

∂w ∂t

=a

∂ ∂x



  2aλe–z 2(eu u′z )′z – eu u′z + bλeλu = (1 – λ)u′z – 1. λw ∂w

e

∂x



+ b + ce–λw.

Functional separable solution:   1 bλ 2 w = ln cλt – x + C1 x + C2 , λ 2a where C1 and C2 are arbitrary constants. 8.

  ∂w ∂w ∂ =a eλw + beλw + c + se–λw . ∂t ∂x ∂x

This is a special case of equation 5.6.14.4 with f (t) = c and g(t) = s. Functional separable solutions: o p p  1 n αt  ln e C1 cos(x β ) + C2 sin(x β ) + γ if abλ > 0, λ n o p p   1 w = ln eαt C1 cosh(x –β ) + C2 sinh(x –β ) + γ if abλ < 0. λ

w=

Here, C1 and C2 are arbitrary constants and

α = λ(bγ + c),

β = bλ/a,

where γ = γ1,2 are roots of the quadratic equation bγ 2 + cγ + s = 0. Reference: V. A. Galaktionov and S. A. Posashkov (1989).

i f (w) ∂w + g(w) ∂x ∂x

5.2.3. Equations of the Form ∂w = ∂

∂t

h

⊲ Equations of this form admit traveling-wave solutions w = w(kx + λt).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 259

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 260

260

SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE   ∂w ∂ λw ∂w 1. =a we . ∂t ∂x ∂x Traveling-wave solution:   1 a w(x, t) = ln C1 x + C12 t + C2 , λ λ where C1 and C2 are arbitrary constants. Reference: A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995).

  ∂w ∂ λw ∂w 2. =a we + b. ∂t ∂x ∂x Generalized traveling-wave solution:   aC12 2bλt 1 bλt bλt w(x, t) = ln C1 e x + e + C2 e , λ bλ2 where C1 and C2 are arbitrary constants. Reference: A. D. Polyanin and V. F. Zaitsev (2004, p. 56).

  ∂w ∂ w ∂w 3. =a we + b(w + 2). ∂t ∂x ∂x Functional separable solution:   b 2bt 2 w(x, t) = ln C1 e – (x + C2 ) , 2a where C1 and C2 are arbitrary constants.   ∂w ∂ λw ∂w 4. =a we + be–λw . ∂t ∂x ∂x Traveling-wave solution:   2   aC1 1 w(x, t) = ln C1 x + + bλ t + C2 , λ λ where C1 and C2 are arbitrary constants.   ∂w ∂ λw ∂w 5. =a we + b + ce–λw. ∂t ∂x ∂x Generalized traveling-wave solution:   aC12 2bλt 1 c bλt bλt w(x, t) = ln C1 e x + e + C2 e – , λ b bλ2 where C1 and C2 are arbitrary constants. Reference: A. D. Polyanin and V. F. Zaitsev (2004, p. 57).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 260

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

261

5.2. EQUATIONS WITH EXPONENTIAL NONLINEARITIES

6.

∂w ∂t

=

∂ ∂x



λw

awe

λw

+ be


∂w ∂x



Page 261

+ b + ce–λw .

The substitution w = u – b/a leads to an equation of the form 5.2.3.5:   ∂u ∂ ∂u = ae–bλ/a ueλu + b + cebλ/a e–λu . ∂t ∂x ∂x  
∂ ∂w 2λw λw ∂w 7. = ae + bwe . ∂t ∂x ∂x Self-similar solutions:   1 b ±x + C1 w(x, t) = ln √ – , λ C2 – 2at aλ where C1 and C2 are arbitrary constants.  
∂ ∂w 2λw λw ∂w = ae + bwe + c. 8. ∂t ∂x ∂x Generalized traveling-wave solutions:   Z 1 b w(x, t) = ln ±ϕ(t)x + C1 ϕ(t) + ϕ(t) ϕ(t) dt , λ λ

ϕ(t) =



–2cλt

C2 e

a – cλ

–1/2

,

where C1 and C2 are arbitrary constants.  
∂ ∂w n n ∂w 9. =a w exp λw + bw 1–n. ∂t ∂x ∂x Generalized traveling-wave solutions: 1/n  aC 2 1 ln z , z = C1 ebnλt x + 2 1 2 e2bnλt + C2 ebnλt , w(x, t) = λ bn λ where C1 and C2 are arbitrary constants.   ∂w ∂ –2 1/w ∂w 10. =a w e . ∂t ∂x ∂x Solution: w= where

z2 + c , 2at

x

z= ln

x2 +c 2e2 at

+

√ 2 c x

ln

x2 +c 2e2 at

+

√ 2 –c x

z=

x √ c

+

b x

√ x– –c √ x+ –c

+

b x

arctan x ln

if c > 0, if c < 0,

where b and c are arbitrary constants. References: I. Tsyfra, A. Messina, A. Napoli, and V. Tretynyk (2004), N. M. Ivanova (2008).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 261

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

262

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 262

ONE SPACE VARIABLE

5.2.4. Other Equations Explicitly Independent of x and t ∂2w ∂w ∂w = +β + a + beλw + ce2λw . 2 ∂t ∂x ∂x On passing from t, x to the new variables t, z = x + βt, one arrives at a simpler equation of the form 5.2.1.3: ∂w ∂ 2 w + a + beλw + ce2λw . = ∂t ∂z 2 1.

2.

∂w ∂t

=a

∂ 2w ∂x2

+ beλw

∂w ∂x

.

This is a special case of equation 5.6.3.8 with f (w) = beλw . 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the function w1 = w(C1 x + C2 , C12 t + C3 ) +

1 ln C1 , λ

where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Apart from the traveling wave w = w(x + λt), there is also an exact solution of the form w = ϕ(ξ) –

3.

1 ln t, 2λ

x ξ= √ . t

2  ∂2w ∂w λw ∂w = + ae . ∂t ∂x2 ∂x

This is a special case of equation 5.6.6.8 with f (w) = aeλw . The substitution   Z a λw u = exp e dw λ leads to the linear heat equation ∂t u = ∂xx u. ∂w ∂ 2w = aeλw . ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the function

4.

w1 = w(C1 x + C2 , C3 t + C4 ) +

1 C3 ln 2 , λ C1

where C1 , . . . , C4 are arbitrary constants, is also a solution of the equation. 2◦ . Traveling-wave solution:     1 C1 β w(x, t) = ln C2 exp z + , λ C1 ak2

z = kx + βt,

where C1 , C2 , k, and β are arbitrary constants.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 262

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.2. EQUATIONS WITH EXPONENTIAL NONLINEARITIES

Page 263

263

3◦ . Additive separable solutions:   2 1 cos (C2 x + C3 ) , w(x, t) = ln λ 2C22 (at + C1 )   1 sinh2 (C2 x + C3 ) w(x, t) = ln , λ 2C22 (at + C1 )   1 cosh2 (C2 x + C3 ) w(x, t) = ln , λ 2C22 (C1 – at) where C1 , C2 , and C3 are arbitrary constants; note that ln(A/B) = ln |A| – ln |B| for AB > 0. 4◦ . Self-similar solution: w = w(y),

√ y = x/ t,

where the function w(y) is determined by the ordinary differential equation ′′ aeλw wyy + 12 ywy′ = 0.

5◦ . Solution: w(x, t) = U (ξ) + 2kt,

ξ = xe–kλt ,

where k is an arbitrary constant, and the function U = U (ξ) is determined by the ordinary differential equation ′′ . 2k – kλξUξ′ = aeλU Uξξ 6◦ . Solution:

1 ln t, ζ = x + β ln t, λ where β is an arbitrary constant, and the function F = F (ζ) is determined by the autonomous ordinary differential equation w(x, t) = F (ζ) –

′′ . βλFζ′ – 1 = aλeλF Fζζ

7◦ . Solution:

2b + 1 ln t, θ = xtb , λ where b is an arbitrary constant, and the function G = G(θ) is determined by the ordinary differential equation 2b + 1 – + bθG′θ = aeλG G′′θθ . λ w(x, t) = G(θ) –

∂w

= aew

∂ 2w

+ beλw . ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the functions 5.

w1 = w(±C1λ–1 x + C2 , C12λ t + C3 ) + 2 ln |C1 |, where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 263

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

264

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 264

ONE SPACE VARIABLE

2◦ . Traveling-wave solution: w = w(ξ),

ξ = kx + βt,

where k and β are arbitrary constants, and the function w(ξ) is determined by the autonomous ordinary differential equation ′′ ak2 ew wξξ – βwξ′ + beλw = 0.

3◦ . Solution for λ ≠ 0: 1 1–λ ln t, z = 2 ln x + ln t, λ λ where the function u = u(z) is determined by the ordinary differential equation
2aλeu–z 2u′′zz – u′z + bλeλu = (1 – λ)u′z – 1. w(x, t) = u(z) –

4◦ . Additive separable solution for λ = 1:

w(x, t) = – ln(kt + C) + ϕ(x), where the function ϕ(x) is determined by the autonomous ordinary differential equation aϕ′′xx + b + ke–ϕ = 0. 5◦ . Additive separable solutions for λ = 0:   b cos2 (C2 x + C3 ) w(x, t) = ln , 2C22 a – C1 e–bt   b sinh2 (C2 x + C3 ) w(x, t) = ln , a – C1 e–bt 2C22   b cosh2 (C2 x + C3 ) w(x, t) = ln , C1 e–bt – a 2C22 where C1 , C2 , and C3 are arbitrary constants; note that ln(A/B) = ln |A| – ln |B| for AB > 0.   ∂w ∂w ∂ λw ∂w 6. + aw =b e . ∂t ∂x ∂x ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the function   1 a w1 = w eC1 x + C1 eC1 t + C2 , eC1 t + C3 – C1 , λ λ where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Traveling-wave solution in implicit form: Z eλw dw 2b = x + βt + C2 , aw2 + 2βw + C1 where C1 , C2 , and β are arbitrary constants. 3◦ . Solution:

1 x a ln t, z = – ln t, λ t λ where the function u = u(z) is determined by the ordinary differential equation   a ′ 1 au – z – u + = b(eλ uu′z )′z . λ z λ w(x, t) = u(z) +

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 264

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.2. EQUATIONS WITH EXPONENTIAL NONLINEARITIES

7.

∂w

=a

∂



λw ∂w

e



+ beβw

Page 265

265

∂w

. ∂t ∂x ∂x ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the functions
w1 = w e(β–λ)k x + A, e(2β–λ)k t + B + k,

where A, B, and k are arbitrary constants, are also solutions of the equation. 2◦ . Traveling-wave solution in implicit form: Z eλw dw = x + C1 t + C3 , a C1 w + C2 – (b/β)eβw where C1 , C2 , and C3 are arbitrary constants. 3◦ . Solution with λ ≠ 2β: w = U (ζ) +

1 ln t, λ – 2β

β–λ

ζ = xt λ–2β ,

where the function U = U (ζ) is determined by the ordinary differential equation  1  (β – λ)ζUζ′ + 1 = a(eλU Uζ′ )′ζ + beβU Uζ′ . λ – 2β

4◦ . Additive separable solution with λ = 2β:

w = ϕ(t) +

1 ln x, β

where the function ϕ = ϕ(t) is determined by the separable first-order ordinary differential equation βϕ′t = ae2βϕ + beβϕ . References for equation 5.2.4.7: M. P. Edwards (1994), N. M. Ivanova (2008).



=

∂

2w

ae

w

+ ce


∂w




∂w + kew + s . ∂t ∂x ∂x ∂x This is a special case of equation 5.2.4.10 with b = 0 and λ = 1.  

∂w ∂w ∂ 2w w ∂w = ae + bwe + kew + s . 9. ∂t ∂x ∂x ∂x This is a special case of equation 5.2.4.10 with c = 0 and λ = 1. Generalized traveling-wave solution:   Z w(x, t) = ln ϕ(t)x + (st + C1 )ϕ(t) + bϕ(t) ϕ(t) dt , 8.

∂w

where C1 is an arbitrary constant, and the function ϕ(t) is determined by the first-order separable ordinary differential equation ϕ′t = aϕ3 + kϕ2 ,

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 265

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

266

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 266

ONE SPACE VARIABLE

whose general solution can be written out in implicit form 1 a aϕ + k – ln + + C2 = t. kϕ k2 ϕ

Here C2 is an arbitrary constant and k ≠ 0. In special cases, we have

ϕ(t) = (C2 – 2at)–1/2 –1

if k = 0, if a = 0.

ϕ(t) = (C2 – kt)

Reference: A. D. Polyanin and V. F. Zaitsev (2004, p. 60).

10.

∂w ∂ = ∂t ∂x



2λw

ae

λw

+ bwe

λw

+ ce

Generalized traveling-wave solution:


∂w ∂x




∂w + keλw + s . ∂x

  Z 1 b w = ln ϕ(t)x + (st + C1 )ϕ(t) + ϕ(t) ϕ(t) dt , λ λ where C1 is an arbitrary constant, and the function ϕ = ϕ(t) is determined by the first-order separable ordinary differential equation ϕ′t = aϕ3 + kϕ2 . For solutions to this equation, see the previous equation 5.2.4.9.

5.2.5. Equations Explicitly Dependent on x and/or t 1.

∂w ∂t

=a

∂ 2w ∂x2

+ ceλw+bx+ct.

This is a special case of equation 5.6.1.2 with f (z, w) = cez+λw . 2.

  ∂w ∂ 2w ∂w 2 =a + aλ + beβx+µt–λw . ∂t ∂x ∂x2

This is a special case of equation 5.6.4.9 with f (x, t) ≡ 0 and g(x, t) = beβx+µt . 3.

∂w ∂t

=a

∂ ∂x



λw ∂w

e

∂x



+ btn.



+ beµt.

This is a special case of equation 5.6.14.1 with f (t) = btn . 4.

∂w ∂t

=a

∂ ∂x



λw ∂w

e

∂x

This is a special case of equation 5.6.14.1 with f (t) = beµt .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 266

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.2. EQUATIONS WITH EXPONENTIAL NONLINEARITIES

5.

∂w ∂t

=a

∂ ∂x



λw ∂w

e

∂x



Page 267

267

+ beλw + ceµt.

This is a special case of equation 5.6.14.4 with f (t) = ceµt and g(t) = 0.   ∂w ∂ λw ∂w 6. =a e + btne–λw . ∂t ∂x ∂x This is a special case of equation 5.6.14.2 with f (t) = 0 and g(t) = btn .   ∂w ∂ λw ∂w 7. =a e + be–λw+µt. ∂t ∂x ∂x This is a special case of equation 5.6.14.2 with f (t) = 0 and g(t) = beµt .   ∂w ∂ λw ∂w =a e + beµt + ce–λw+νt. 8. ∂t ∂x ∂x

This is a special case of equation 5.6.14.2 with f (t) = beµt and g(t) = ceνt .   ∂w ∂ λw ∂w 9. =a e + (bx + c)eλw. ∂t ∂x ∂x Additive separable solution: w=–

1 ln(λt + C) + ϕ(x), λ

where C is an arbitrary constant and the function ϕ(x) is determined by the second-order linear ordinary differential equation ′′ aψxx + λ(bx + c)ψ + λ = 0,   ∂w ∂w ∂ 10. =a eλw + beλw+µx . ∂t ∂x ∂x Additive separable solution:

w=–

ψ = eλϕ .

1 ln(λt + C) + ϕ(x), λ

where C is an arbitrary constant and the function ϕ(x) is determined by the second-order linear ordinary differential equation ′′ aψxx + λbeµx ψ + λ = 0, ψ = eλϕ .   ∂w ∂w ∂ 11. =a xneλw . ∂t ∂x ∂x This is a special case of equation 5.6.17.12 with f (x) = axn .   ∂w ∂ λw+µx ∂w 12. =a e . ∂t ∂x ∂x This is a special case of equation 5.6.17.12 with f (x) = aeµx .

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 267

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

268

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 268

ONE SPACE VARIABLE

5.3. Equations with Hyperbolic Nonlinearities 5.3.1. Equations Involving Hyperbolic Cosine 1.

∂w ∂t

=a

∂ 2w ∂x2

+ b coshk(λw).

This is a special case of equation 5.6.1.1 with f (w) = b coshk (λw). 2.

∂w ∂t

=a

∂ 2w ∂x2

+ β coshk(λw + bx + ct).

This is a special case of equation 5.6.1.2 with f (z, w) = β coshk (z + λw). 3.

∂w ∂t

=a

∂ 2w ∂x2

+ (bx + c)

∂w ∂x

+ s coshk(λw).

This is a special case of equation 5.6.2.1 with f (w) = s coshk (λw). 4.

  ∂w ∂2w ∂w 2 k = + b cosh (λw) . ∂t ∂x ∂x2

This is a special case of equation 5.6.6.8 with f (w) = b coshk (λw). 5.

∂w ∂t

=

∂2w ∂x2

k

+ b cosh (λw)



∂w ∂x

2

+ c coshk(βt)

∂w ∂x

.

This is a special case of equation 5.6.6.10 with f (w) = b coshk (λw), g(t) = 0, and h(t) = c coshk (βt). 6.

∂w ∂t

=a

∂ ∂x



λw ∂w

e

∂x



+ 2b cosh(λw) + c coshk(βt).

This is a special case of equation 5.6.14.4 with f (t) = c coshk (βt) and g(t) = b. 7.

  ∂w 2 =a cosh (βw) . ∂t ∂x ∂x

∂w

∂

This is a special case of equation 5.6.15.1 with f (w) = a cosh2 (βw). Self-similar solutions:   1 ±x + C1 w(x, t) = arcsinh √ , β C2 – 2at where C1 and C2 are arbitrary constants. 8.

∂w ∂t

=a

  ∂w coshk(βw) . ∂x ∂x ∂

This is a special case of equation 5.6.15.1 with f (w) = a coshk (βw).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 268

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.3. EQUATIONS WITH HYPERBOLIC NONLINEARITIES

Page 269

269

5.3.2. Equations Involving Hyperbolic Sine 1.

∂w ∂t

=a

∂ 2w ∂x2

+ b sinhk(λw).

This is a special case of equation 5.6.1.1 with f (w) = b sinhk (λw). 2.

∂w ∂t

=a

∂ 2w ∂x2

+ β sinhk(λw + bx + ct).

This is a special case of equation 5.6.1.2 with f (z, w) = β sinhk (z + λw). 3.

∂w ∂t

=a

∂ 2w ∂x2

+ (bx + c)

∂w ∂x

+ s sinhk(λw).

This is a special case of equation 5.6.2.1 with f (w) = s sinhk (λw). 4.

  ∂w 2 ∂w ∂2w k + b sinh (λw) . = ∂t ∂x2 ∂x

This is a special case of equation 5.6.6.8 with f (w) = b sinhk (λw). 5.

  ∂w ∂2w ∂w 2 ∂w k k = + b sinh (λw) + c sinh (βt) . ∂t ∂x2 ∂x ∂x

This is a special case of equation 5.6.6.10 with f (w) = b sinhk (λw), g(t) = 0, and h(t) = c sinhk (βt). 6.

  ∂w ∂ λw ∂w =a e + 2b sinh(λw) + c sinhk(βt). ∂t ∂x ∂x

This is a special case of equation 5.6.14.4 with f (t) = c sinhk (βt) and g(t) = –b. 7.

  ∂w 2 =a sinh (βw) . ∂t ∂x ∂x

∂w

∂

This is a special case of equation 5.6.15.1 with f (w) = a sinh2 (βw). Self-similar solutions:   1 ±x + C1 w(x, t) = arccosh √ , β C2 – 2at where C1 and C2 are arbitrary constants. 8.

  ∂w ∂ ∂w k =a sinh (βw) . ∂t ∂x ∂x

This is a special case of equation 5.6.15.1 with f (w) = a sinhk (βw).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 269

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

270

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 270

ONE SPACE VARIABLE

5.3.3. Equations Involving Hyperbolic Tangent ∂ 2w ∂w =a + b tanhk(λw). ∂t ∂x2 This is a special case of equation 5.6.1.1 with f (w) = b tanhk (λw). 1.

∂w

∂ 2w

+ β tanhk(λw + bx + ct). ∂t ∂x2 This is a special case of equation 5.6.1.2 with f (z, w) = β tanhk (z + λw). 2.

=a

∂w ∂ 2w ∂w + (bx + c) =a + s tanhk(λw). 2 ∂t ∂x ∂x This is a special case of equation 5.6.2.1 with f (w) = s tanhk (λw). 3.

4.

∂w ∂t

=

∂2w ∂x2

k

+ b tanh (λw)



∂w ∂x

2

.

This is a special case of equation 5.6.6.8 with f (w) = b tanhk (λw). 5.

∂w ∂t

=

∂2w ∂x2

+ b tanhk(λw)



∂w ∂x

2

+ c tanhk(βt)

∂w ∂x

.

This is a special case of equation 5.6.6.10 with f (w) = b tanhk (λw), g(t) = 0, and h(t) = c tanhk (βt).   ∂w ∂w ∂ k 6. =a tanh (βw) . ∂t ∂x ∂x This is a special case of equation 5.6.15.1 with f (w) = a tanhk (βw).

5.3.4. Equations Involving Hyperbolic Cotangent ∂w ∂ 2w =a + b cothk(λw). ∂t ∂x2 This is a special case of equation 5.6.1.1 with f (w) = b cothk (λw). 1.

∂w

∂ 2w

+ β cothk(λw + bx + ct). ∂t ∂x2 This is a special case of equation 5.6.1.2 with f (z, w) = β cothk (z + λw).

2.

=a

∂w ∂ 2w ∂w =a + (bx + c) + s cothk(λw). 2 ∂t ∂x ∂x This is a special case of equation 5.6.2.1 with f (w) = s cothk (λw). 3.

4.

  ∂w ∂2w ∂w 2 k = + b coth (λw) . ∂t ∂x ∂x2

This is a special case of equation 5.6.6.8 with f (w) = b cothk (λw).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 270

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.4. EQUATIONS WITH LOGARITHMIC NONLINEARITIES

5.

∂w ∂t

=

∂2w ∂x2

k

+ b coth (λw)



∂w ∂x

2

+ c cothk(βt)

∂w ∂x

Page 271

271

.

This is a special case of equation 5.6.6.10 with f (w) = b cothk (λw), g(t) = 0, and h(t) = c cothk (βt).   ∂w ∂ ∂w k 6. =a coth (βw) . ∂t ∂x ∂x This is a special case of equation 5.6.15.1 with f (w) = a cothk (βw).

5.4. Equations with Logarithmic Nonlinearities 2 5.4.1. Equations of the Form ∂w = a ∂ w2 + f (x, t, w)

∂t

∂x

∂ 2w

∂w

+ b ln w. =a ∂t ∂x2 This is a special case of equation 5.6.1.1 with f (w) = b ln w. 1.

∂w

∂2w

+ aw ln w. ∂t ∂x2 This is a special case of equation 5.6.1.1 with f (w) = aw ln w. 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the functions w1 = exp(C1 eat )w(±x + C2 , t + C3 ), where C1 , C2 , and C3 are arbitrary constants, are also solutions of the equation. 2◦ . Functional separable solutions:   A2 2at at at w(x, t) = exp Ae x + e + Be , a   w(x, t) = exp 12 – 14 a(x + A)2 + Beat ,   a(x + A)2 1 at –at at ln(1 + Be ) + Ce w(x, t) = exp – + e , 4(1 + Be–at ) 2B where A, B, and C are arbitrary constants. 3◦ . Multiplicative separable solution:   w(x, t) = exp Aeat + f (x) , where the function f (x) is defined implicitly by Z
–1/2 Be–2f – af + 12 a df = ±x + C.

2.

=

(1) (2)

Relations (1) and (2) involve three arbitrary constants, A, B, and C. 4◦ . There are more complicated solutions of the form   w(x, t) = exp Aeat + f (x + bt) , where the function f (ξ) is determined by the autonomous ordinary differential equation ′′ fξξ + (fξ′ )2 – bfξ′ + af = 0.

References for equation 5.4.1.2: V. A. Dorodnitsyn (1979, 1982), A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995), A. D. Polyanin and V. F. Zaitsev (2002, p. 46).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 271

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

272 3.

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 272

ONE SPACE VARIABLE

∂w ∂2w + aw ln w + bw. = ∂t ∂x2

The substitution w = e–b/a u leads to an equation of the form 5.4.1.2: ∂u ∂ 2 u = + au ln u. ∂t ∂x2 ∂w ∂2w = + aw ln w + (bx + c)w. ∂t ∂x2 This is a special case of equations 5.6.1.5 and 5.6.1.7.

4.

∂2w ∂w + aw ln w + (bx + ct + k)w. = ∂t ∂x2 This is a special case of equation 5.6.1.7. 5.

∂w ∂2w = + aw ln w + (bx2 + cx + k)w. 2 ∂t ∂x This is a special case of equation 5.6.1.9. 6.

∂w

∂2w

+ a(w + b) ln2 (w + b). ∂t ∂x2 1◦ . The substitution w = eu – b leads to an equation of the form 5.1.7.4:  2 ∂u ∂ 2 u ∂u = + + au2 . ∂t ∂x ∂x2

7.

=

(1)

2◦ . Solutions of equation (1) for a < 0: √
u(x, t) = C1 exp –at ± x –a , √
1 C2 u(x, t) = + exp –at ± x –a , 2 C1 – at (C1 – at)

where C1 and C2 are arbitrary constants. The first solution is a traveling-wave solution and the second one is a generalized separable solution. 3◦ . Equation (1) has also generalized separable solutions of the following forms:   √ √ u(x, t) = ϕ(t) + ψ(t) A exp(x –a ) + B exp(x –a ) if a < 0,  √ √  u(x, t) = ϕ(t) + ψ(t) A sin(x a ) + B cos(x a ) if a > 0.

For details, see 5.1.7.4.

References for equation 5.4.1.7: V. A. Galaktionov and S. A. Posashkov (1989), A. D. Polyanin and V. F. Zaitsev (2002, p. 47).

8.

∂w

=

∂2w

  + (1 + kw) a ln2 (1 + kw) + b ln(1 + kw) + c .

∂t ∂x2 This is a special case of equation 5.6.1.10.

Reference: V. A. Galaktionov and S. A. Posashkov (1989).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 272

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.4. EQUATIONS WITH LOGARITHMIC NONLINEARITIES

Page 273

273

5.4.2. Other Equations ∂w

∂2w

∂w

=

∂ 2w

∂w

+ cw ln w. ∂t ∂x2 ∂x This is a special case of equation 5.6.2.6 with f (t) = 0, g(t) = bt, h(t) = c, and p(t) = s(t) = 0.

2.

∂w

=a

+a

∂w

+ bw ln w. ∂t ∂x2 ∂x On passing from t, x to the new variables t, z = x + at, one obtains a simpler solution of the form 5.4.1.2: ∂w ∂ 2 w + bw ln w. = ∂t ∂z 2 1.

∂ 2w

+ bx

∂w

+ cw ln w. ∂t ∂x This is a special case of equation 5.6.2.6 with f (t) = b, g(t) = 0, h(t) = c, and p(t) = s(t) = 0. 3.

4.

∂w ∂t

=a

+ bt

=

∂x2

a ∂ xk ∂x



x

k ∂w

∂x



+ bw ln w.

The values k = 1 and k = 2 correspond to problems with axial and central symmetry, respectively. 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the functions w1 = exp(C1 ebt ) w(±x, t + C2 ), where C1 and C2 are arbitrary constants, are also solutions of the equation. 2◦ . Functional separable solution:  w(x, t) = exp –

 1 bx2 bt bt –bt + Be + (k + 1)e ln(1 + Ae ) , 2A 4a(1 + Ae–bt )

where A and B are arbitrary constants. 3◦ . Multiplicative separable solution:
w(x, t) = exp Aebt θ(x),

where A is an arbitrary constant and the function θ(x) is determined by the second-order ordinary differential equation   a d k dθ x + bθ ln θ = 0. dx xk dx References for equation 5.4.2.4: A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995), A. D. Polyanin and V. F. Zaitsev (2002, pp. 47–48).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 273

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

274

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 274

ONE SPACE VARIABLE

∂w ∂ 2w ∂w + (b ln w + c) =a . ∂t ∂x2 ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the function 5.

w1 = eC1 w(x + bC1 t + C2 , t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Traveling-wave solution:

    b c w(x, t) = exp C1 exp – x + b2 C2 t + 1 – aC2 – , a b

where C1 and C2 are arbitrary constants.

3◦ . Generalized traveling-wave solution:   C1 – x a ln |t + C2 | c w(x, t) = exp + – . b(t + C2 ) b2 t + C2 b 4◦ . Solution:

w(x, t) = eλt u(z), z = x + 12 bλt2 + kt, where k and λ are arbitrary constants, and the function u(z) is determined by the autonomous ordinary differential equation au′′zz + (b ln u + c – k)u′z – λu = 0. To λ = 0, there corresponds a traveling-wave solution. Reference for equation 5.4.2.5: A. D. Polyanin and V. F. Zaitsev (2004, p. 67).

6.

∂w ∂t

=

∂2w ∂x2

k

+ a ln (bw)



∂w ∂x

2

.

This is a special case of equation 5.6.6.8 with f (w) = a lnk (bw).   ∂w ∂w ∂ ∂w 7. =a wn + (b ln w + c) . ∂t ∂x ∂x ∂x 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the function w1 = Aw(A–n x + bA–n ln A t + B, A–n t + C), where A, B, and C are arbitrary constants, is also a solution of the equation. 2◦ . Traveling-wave solution in implicit form: Z wn dw a = –x + C1 t + C3 , bw ln w + (c – b + C1 )w + C2 where C1 , C2 , and C3 are arbitrary constants. 3◦ . Solution:

x b + ln t, t n where the function U = U (z) is determined by the ordinary differential equation  b ′ 1 a(U n Uz′ )′z + b ln U + z + c – U – U = 0. n z n w = t1/n U (z),

z=

References for equation 5.4.2.7: C. M. Yung, K. Verburg, and P. Baveye (1994), N. M. Ivanova (2008).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 274

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

275

5.4. EQUATIONS WITH LOGARITHMIC NONLINEARITIES

8.

Page 275

  ∂w = (a ln w + b) . ∂t ∂x ∂x

∂w

∂

This is a special case of equation 5.6.15.1 with f (w) = a ln w + b. 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the function w1 = w(C1 x + C2 , C12 t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Solutions:

 q  b 2 w(x, t) = exp ± 2C1 x + 2aC1 t + C2 – , a   C2 ± x a+b w(x, t) = exp √ , – a C1 – 2at

where C1 and C2 are arbitrary constants. The first solution represents a traveling wave and the second one is self-similar.   ∂w ∂ ∂w 9. = (a ln w + b) + cw. ∂t ∂x ∂x Generalized traveling-wave solution:   C2 ± x a+b c w(x, t) = exp √ – (C1 – 2at) – , a C1 – 2at 3a where C1 and C2 are arbitrary constants. Reference: A. D. Polyanin and V. F. Zaitsev (2004, p. 68).

  ∂w 10. = (a ln w + b) + cw ln w + sw. ∂t ∂x ∂x Generalized traveling-wave solution:  Z Z w(x, t) = exp ϕ(t)(C1 ± x) + (a + b)ϕ(t) ϕ(t) dt + sϕ(t) ∂w

∂

ϕ(t) =



–2ct

C2 e

a – c

–1/2

 dt , ϕ(t)

,

where C1 and C2 are arbitrary constants. ∂w

∂ 2w



∂w

2

∂w

+ bw 2 + cw 2 ln w. ∂t ∂x2 ∂x ∂x The substitution w = eu leads to the linear equation 11.

w

=w

–

+ aw

∂u ∂ 2 u ∂u = +a + cu + b. 2 ∂t ∂x ∂x Remark. In the original equation, the parameters a, b, and c can be replaced by arbitrary functions a = a(x), b = b(x), and c = c(x).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 275

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

276

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 276

ONE SPACE VARIABLE

5.5. Equations with Trigonometric Nonlinearities 5.5.1. Equations Involving Cosine ∂w

∂ 2w

+ b cosk(λw). ∂t ∂x2 This is a special case of equation 5.6.1.1 with f (w) = b cosk (λw).

1.

=a

∂w ∂ 2w + β cosk(λw + bx + ct). =a ∂t ∂x2 This is a special case of equation 5.6.1.2 with f (z, w) = β cosk (z + λw). 2.

∂w

∂ 2w

∂w

+ s cosk(λw). ∂t ∂x This is a special case of equation 5.6.2.1 with f (w) = s cosk (λw).

3.

4.

∂w ∂t

=a

=

∂x2

∂2w ∂x2

+ (bx + c)

+ b cosk(λw)



∂w ∂x

2

.

This is a special case of equation 5.6.6.8 with f (w) = b cosk (λw). 5.

∂w ∂t

=

∂2w ∂x2

k

+ b cos (λw)



∂w ∂x

2

+ c cosk(βt)

∂w ∂x

.

This is a special case of equation 5.6.6.10 with f (w) = b cosk (λw), g(t) = 0, and h(t) = c cosk (βt). ∂w

2

∂2w

= a cos (λw + β) . ∂t ∂x2 The substitution u = tan(λw + β) leads to an equation of the form 5.1.13.2:   ∂u ∂ 1 ∂u =a . ∂t ∂x u2 + 1 ∂x 6.

7.

  ∂w ∂ ∂w 2 =a cos (βw) . ∂t ∂x ∂x

This is a special case of equation 5.6.15.1 with f (w) = a cos2 (βw). Self-similar solutions: w(x, t) =

1 ±x + C1 arcsin √ , β 2at + C2

where C1 and C2 are arbitrary constants. 8.

  ∂w k =a cos (βw) . ∂t ∂x ∂x

∂w

∂

This is a special case of equation 5.6.15.1 with f (w) = a cosk (βw).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 276

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.5. EQUATIONS WITH TRIGONOMETRIC NONLINEARITIES

Page 277

277

5.5.2. Equations Involving Sine 1.

∂ 2w ∂w =a + b sink(λw). ∂t ∂x2

This is a special case of equation 5.6.1.1 with f (w) = b sink (λw). 2.

∂w ∂t

=a

∂ 2w ∂x2

+ β sink(λw + bx + ct).

This is a special case of equation 5.6.1.2 with f (z, w) = β sink (z + λw). 3.

∂w ∂ 2w ∂w + (bx + c) =a + s sink(λw). 2 ∂t ∂x ∂x

This is a special case of equation 5.6.2.1 with f (w) = s sink (λw). 4.

∂w ∂t

=

∂2w ∂x2

k

+ b sin (λw)



∂w ∂x

2

.

This is a special case of equation 5.6.6.8 with f (w) = b sink (λw). 5.

∂w ∂t

=

∂2w ∂x2

k

+ b sin (λw)



∂w ∂x

2

+ c sink(βt)

∂w ∂x

.

This is a special case of equation 5.6.6.10 with f (w) = b sink (λw), g(t) = 0, and h(t) = c sink (βt). 6.

∂w

= a sin2 (λw)

∂2w

. ∂t ∂x2 The substitution u = cot(λw) leads to an equation of the form 5.1.13.2: ∂u ∂ =a ∂t ∂x 7.

∂w ∂t

=a



 1 ∂u . u2 + 1 ∂x

  ∂w sin2 (βw) . ∂x ∂x ∂

This is a special case of equation 5.6.15.1 with f (w) = a sin2 (βw). Self-similar solutions: w(x, t) =

1 ±x + C1 arccos √ , β 2at + C2

where C1 and C2 are arbitrary constants. 8.

  ∂w k =a sin (βw) . ∂t ∂x ∂x

∂w

∂

This is a special case of equation 5.6.15.1 with f (w) = a sink (βw).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 277

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

278

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 278

ONE SPACE VARIABLE

5.5.3. Equations Involving Tangent ∂ 2w ∂w =a + b tank(λw). ∂t ∂x2 This is a special case of equation 5.6.1.1 with f (w) = b tank (λw). 1.

∂w

∂ 2w

+ β tank(λw + bx + ct). ∂t ∂x2 This is a special case of equation 5.6.1.2 with f (z, w) = β tank (z + λw).

2.

=a

∂w ∂ 2w ∂w + (bx + c) =a + s tank(λw). 2 ∂t ∂x ∂x This is a special case of equation 5.6.2.1 with f (w) = s tank (λw). 3.

4.

  ∂w ∂w 2 ∂2w k = + b tan (λw) . ∂t ∂x ∂x2

This is a special case of equation 5.6.6.8 with f (w) = b tank (λw). 5.

∂w ∂t

=

∂2w ∂x2

k

+ b tan (λw)



∂w ∂x

2

+ c tank(βt)

∂w ∂x

.

This is a special case of equation 5.6.6.10 with f (w) = b tank (λw), g(t) = 0, and h(t) = c tank (βt).   ∂w ∂w ∂ k =a tan (βw) . 6. ∂t ∂x ∂x This is a special case of equation 5.6.15.1 with f (w) = a tank (βw).

5.5.4. Equations Involving Cotangent ∂w ∂ 2w =a + b cotk(λw). ∂t ∂x2 This is a special case of equation 5.6.1.1 with f (w) = b cotk (λw).

1.

∂w

∂ 2w

+ β cotk(λw + bx + ct). ∂t ∂x2 This is a special case of equation 5.6.1.2 with f (z, w) = β cotk (z + λw). 2.

=a

∂w ∂ 2w ∂w =a + (bx + c) + s cotk(λw). 2 ∂t ∂x ∂x This is a special case of equation 5.6.2.1 with f (w) = s cotk (λw). 3.

4.

  ∂w ∂2w ∂w 2 k = + b cot (λw) . ∂t ∂x ∂x2

This is a special case of equation 5.6.6.8 with f (w) = b cotk (λw).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 278

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

279

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

5.

∂w ∂t

=

∂2w ∂x2

k

+ b cot (λw)



∂w ∂x

2

+ c cotk(βt)

∂w ∂x

Page 279

.

This is a special case of equation 5.6.6.10 with f (w) = b cotk (λw), g(t) = 0, and h(t) = c cotk (βt).   ∂ ∂w ∂w k 6. =a cot (βw) . ∂t ∂x ∂x This is a special case of equation 5.6.15.1 with f (w) = a cotk (βw).

5.5.5. Equations Involving Inverse Trigonometric Functions 1.

∂w

∂



eλ arctan w ∂w



= . ∂t ∂x 1 + w 2 ∂x Solution in parametric form:  
1 ′ w = tan ϕ(z) + arctan 2zϕz – ln t , λ   1 x2 = z cos2 ϕ(z) + ln t , λ

where z is the parameter and the function ϕ = ϕ(z) is determined by the first-order ordinary differential equation (C is an arbitrary constant)   1 C –z 1 1 ′ ϕz = tan ln –ϕ – . 2z λ 2 λ(C – z) Reference: I. Sh. Akhatov, R. K. Gazizov, and N. H. Ibragimov (1989, pp. 29–30).

2.

∂w

∂t Solution:

=



∂w ∂x

2

 –1    2 ∂w ∂ w +1 exp k arctan , ∂x ∂x2

k ≠ 0.

  w2 = u(z) – x2 , z = t exp –k arctan(x/w) , where the function u(z) is determined by the ordinary differential equation 
 2 2k2 z 2 uu′′zz – k2 z(3zu′z – 2u)u′z – 4u2 – 12 (k2 z 2 u′z + 4u2 )u′z exp k arctan 12 kzu–1 u′z = 0. Reference: I. Sh. Akhatov, R. K. Gazizov, and N. H. Ibragimov (1989, pp. 29–33).

5.6. Equations Involving Arbitrary Functions 2 5.6.1. Equations of the Form ∂w = a ∂ w2 + f (x, t, w)

∂t

∂w

∂x

∂ 2w

=a + f (w). ∂t ∂x2 Kolmogorov–Petrovskii–Piskunov equation. Equations of this form are often encountered in various problems of heat and mass transfer (with f being the rate of a volume chemical reaction), combustion theory, biology, and ecology. For f = f (w) having power-law, exponential, or logarithmic form, see Subsections 5.1.1 to 5.1.3, equations 5.2.1.1 to 5.2.1.3, or equations 5.4.1.2, 5.4.1.3, 5.4.1.7, and 5.4.1.8, respectively. 1.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 279

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

280

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 280

ONE SPACE VARIABLE

1◦ . The equation has an implicit nonstationary solution independent of the space variable: Z dw = t + C, C is an arbitrary constant. f (w)

2◦ . Stationary solution in implicit form: –1/2 Z  Z 2 C1 – f (w) dw dw = C2 ± x. a 3◦ . Traveling-wave solutions: w = w(z),

z = ±x + λt,

where λ is an arbitrary constant. The function w = w(z) is determined by the autonomous ordinary differential equation ′′ – λwz′ + f (w) = 0. awzz

(1)

The transformation ξ = (λ/a)z,

U (w) = wξ′

brings (1) to the Abel equation U Uw′ – U + aλ–2 f (w) = 0.

(2)

The book by Polyanin and Zaitsev (2003) presents a considerable number of solutions to equation (2) for various f = f (w). References for equation 5.6.1.1: A. N. Kolmogorov, I. G. Petrovskii, and I. S. Piskunov (1937), Ya. B. Zel’dovich, G. I. Barenblatt, V. B. Librovich, and G. M. Makhviladze (1985).

∂w ∂ 2w =a + f (bx + ct, w). ∂t ∂x2 Solution: w = w(ξ), 2.

ξ = bx + ct,

where the function w(ξ) is determined by the ordinary differential equation ′′ – cwξ′ + f (ξ, w) = 0. ab2 wξξ

 x 3. =a + f √ ,w . ∂t ∂x2 t t ◦ 1 . Suppose w(x, t) is a solution of this equation. Then the function ∂w

∂ 2w

1



w1 = w(Cx, C 2 t), where C is an arbitrary constant, is also a solution of the equation. 2◦ . The transformation τ = ln t,

x ξ= √ t

leads to the equation ∂w ∂ 2 w 1 ∂w =a 2 + ξ + f (ξ, w), ∂τ 2 ∂ξ ∂ξ which admits exact solutions of the form w = w(ξ).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 280

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

Page 281

281

∂w ∂ 2w + bw ln w + f (t)w. =a ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the functions 4.



w1 = exp C1 ebt w ±x + C2 , t ,

where C1 and C2 are arbitrary constants, are also solutions of the equation. 2◦ . Generalized traveling-wave solution:   Z a 2 2bt bt bt bt –bt w(x, t) = exp Ae x + Be + A e + e e f (t) dt , b where A and B are arbitrary constants. 3◦ . Functional separable solution:   w(x, t) = exp ϕ(t)x2 + ψ(t) .

Here, the functions ϕ(t) and ψ(t) are given by bebt ϕ(t) = , A – 4aebt

bt

bt

ψ(t) = Be + e

where A and B are arbitrary constants.

Z

  e–bt 2aϕ(t) + f (t) dt,

4◦ . There are also functional separable solutions of the more general form   w(x, t) = exp ϕ2 (t)x2 + ϕ1 (t)x + ϕ0 (t) ,

where the functions ϕ2 (t), ϕ1 (t), and ϕ0 (t) are determined by a system of ordinary differential equations (see equation 5.6.1.9) that can be integrated. 5◦ . Solution:

  Z bt bt –bt w(x, t) = exp Ae + e e f (t) dt + Φ(x + λt) ,

where A and λ are arbitrary constants, and the function Φ = Φ(z) is determined by the autonomous ordinary differential equation aΦ′′zz + a(Φ′z )2 – λΦ′z + bΦ = 0, the order of which can be reduced by one. 6◦ . The substitution

 Z  bt –bt w(x, t) = exp e e f (t) dt u(x, t)

leads to a simpler equation of the form 5.4.1.2: ∂u ∂2u = a 2 + bu ln u. ∂t ∂x

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 281

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

282

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 282

ONE SPACE VARIABLE

  ∂w ∂ 2w + bw ln w + f (x) + g(t) w. =a ∂t ∂x2 1◦ . Multiplicative separable solution:   Z bt bt –bt e g(t) dt ϕ(x), w(x, t) = exp Ce + e 5.

where C is an arbitrary constant, and the function ϕ(t) is determined by the ordinary differential equation aϕ′′xx + bϕ ln ϕ + f (x)ϕ = 0. 2◦ . With the substitution

 Z  bt –bt w(x, t) = exp e e g(t) dt u(x, t)

one arrives at the simpler equation ∂u ∂2u = a 2 + bu ln u + f (x)u. ∂t ∂x ∂w ∂ 2w + f (t)w ln w + g(t)w. =a ∂t ∂x2 1◦ . Generalized traveling-wave solution:   w(x, t) = exp Φ(t)x + Ψ(t) , 6.

where the functions Φ(t) and Ψ(t) are given by Z F F F Φ(t) = Ae , Ψ(t) = Be + e e–F (aA2 e2F + g) dt,

F =

Z

f dt,

and A and B are arbitrary constants. 2◦ . Functional separable solution:   w(x, t) = exp ϕ(t)x2 + ψ(t) ,

where the functions ϕ(t) and ψ(t) are given by  –1 Z Z F F ϕ(t) = e A – 4a e dt , F = f dt, Z ψ(t) = BeF + eF e–F (2aϕ + g) dt, and A and B are arbitrary constants. 3◦ . There are also functional separable solutions of the more general form   w(x, t) = exp ϕ2 (t)x2 + ϕ1 (t)x + ϕ0 (t) ,

where the functions ϕ2 (t), ϕ1 (t), and ϕ0 (t) are determined by a system of ordinary differential equations (see equation 5.6.1.9), which can be integrated. Reference for equation 5.6.1.6: V. F. Zaitsev and A. D. Polyanin (1996, pp. 419–420).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 282

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

Page 283

283

  ∂w ∂ 2w + f (t)w ln w + g(t)x + h(t) w. =a ∂t ∂x2 1◦ . Generalized traveling-wave solution:   w(x, t) = exp ϕ(t)x + ψ(t) , where the functions ϕ(t) and ψ(t) are given by Z Z F F –F ϕ(t) = Ae + e e g dt, F = f dt, Z ψ(t) = BeF + eF e–F (aϕ2 + h) dt, 7.

and A and B are arbitrary constants. 2◦ . There are also functional separable solutions of the form   w(x, t) = exp ϕ2 (t)x2 + ϕ1 (t)x + ϕ0 (t) , where the functions ϕ2 (t), ϕ1 (t), and ϕ0 (t) are determined by a system of ordinary differential equations (see equation 5.6.1.9), which can be integrated. Reference for equation 5.6.1.7: V. F. Zaitsev and A. D. Polyanin (1996, p. 420).

8.

∂w

=a

∂ 2w

  + f (x)w ln w + bf (x)t + g(x) w.

∂t ∂x2 Multiplicative separable solution:

  w(x, t) = e–bt exp ϕ(x) , where the function ϕ(x) is determined by the ordinary differential equation aϕ′′xx + a(ϕ′x )2 + f (x)ϕ + g(x) + b = 0.

For f , g = const, this equation can be reduced by the substitution u(ϕ) = (ϕ′x )2 to a first-order linear equation. Reference: V. F. Zaitsev and A. D. Polyanin (1996, pp. 420–421).

9.

∂w

=a

∂ 2w

  + f (t)w ln w + g(t)x2 + h(t)x + s(t) w.

∂t ∂x2 Functional separable solution:

  w(x, t) = exp ϕ2 (t)x2 + ϕ1 (t)x + ϕ0 (t) , where the functions ϕn (t) (n = 1, 2, 3) are determined by the system of first-order ordinary differential equations with variable coefficients ϕ′2 = 4aϕ22 + f ϕ2 + g, ϕ′1 = 4aϕ2 ϕ1 + f ϕ1 + h,

(1) (2)

ϕ′0 = f ϕ0 + aϕ21 + 2aϕ2 + s. (3) Here, the arguments of the functions f , g, h, and s are not specified, and the prime denotes a derivative with respect to t. Equation (1) for ϕ2 = ϕ2 (t) is a Riccati equation, so it can be reduced to a second-order linear equation. The books by Kamke (1977), Polyanin and Zaitsev (2003) present a large number of solutions to this equation for various f and g. Given a solution of equation (1), the solutions of equations (2) and (3) can be constructed successively, due to the linearity of each of them in the unknown. Reference: V. F. Zaitsev and A. D. Polyanin (1996, p. 421).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 283

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

284

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 284

ONE SPACE VARIABLE

  ∂w ∂2w 2 + (bw + c) k ln (bw + c) + f (t) ln(bw + c) + g(t) . =a ∂t ∂x2 The substitution bw + c = exp u, u = u(x, t) leads to an equation of the form 5.6.6.2:  2 ∂u ∂u ∂2u + bku2 + bf (t)u + bg(t), = a 2 +a ∂t ∂x ∂x which has exponential and sinusoidal solutions with respect to x. 10.

2 5.6.2. Equations of the Form ∂w = a ∂ w2 + f (x, t) ∂w + g(x, t, w)

∂t

∂w

∂x

∂x

∂w

+ f (w). ∂t ∂x2 ∂x This equation governs unsteady mass transfer with a volume chemical reaction in an inhomogeneous fluid flow. 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the function

1.

=a

∂ 2w

+ (bx + c)

w1 = w(x + C1 e–bt , t + C2 ), where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . Generalized traveling-wave solution: w = w(z), z = x + C1 e–bt , where the function w(z) is determined by the ordinary differential equation ′′ + (bz + c)wz′ + f (w) = 0. awzz Reference for equation 5.6.2.1: A. D. Polyanin and V. F. Zaitsev (2002, p. 56).

  ∂w a ∂ n ∂w 2. = n x + f (t)w ln w. ∂t x ∂x ∂x This equation can be rewritten as ∂w ∂ 2 w an ∂w =a 2 + + f (t)w ln w. ∂t x ∂x ∂x Functional separable solution:   w(x, t) = exp ϕ(t)x2 + ψ(t) , where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations with variable coefficients ϕ′t = 4aϕ2 + f ϕ, ψt′ = 2a(n + 1)ϕ + f ψ; the arguments of the functions f and g are not specified. Integrating the first equation and then the second, we obtain  –1 Z Z F F ϕ(t) = e A – 4a e dt , F = f dt, Z F F ψ(t) = Be + 2a(n + 1)e ϕe–F dt,

where A and B are arbitrary constants.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 284

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

3.

∂w ∂ 2w ∂w + f (t) =a + g(w). ∂t ∂x2 ∂x

On passing from t, x to the new variables t, z = x+

Z

Page 285

285

f (t) dt, one obtains a simpler equation

∂w ∂2w = a 2 + g(w), ∂t ∂z which has a traveling-wave solution w = w(kz + λt). 4.

∂ 2w ∂w ∂w =a + g(t, w). + f (t) 2 ∂t ∂x ∂x

On passing from t, x to the new variables t, z = x+

Z

f (t) dt, one obtains a simpler equation

∂w ∂2w = a 2 + g(t, w). ∂t ∂z   ∂ 2w ∂w ∂w =a + bw ln w + g(x) + h(t) w. + f (x) 2 ∂t ∂x ∂x Multiplicative separable solution:   Z w(x, t) = exp Cebt + ebt e–bt h(t) dt ϕ(x), 5.

where C is an arbitrary constant, and the function ϕ(t) is determined by the ordinary differential equation aϕ′′xx + f (x)ϕ′x + bϕ ln ϕ + g(x)ϕ = 0. ∂ 2w

    ∂w + xf (t) + g(t) + h(t)w ln w + xp(t) + s(t) w. ∂t ∂x 1◦ . Generalized traveling-wave solution:   w(x, t) = exp xϕ(t) + ψ(t) , 6.

∂w

=a

∂x2

where the functions ϕ = ϕ(t) and ψ = ψ(t) are determined by the system of first-order ordinary differential equations with variable coefficients   ϕ′t = f (t) + h(t) ϕ + p(t), (1) ψt′ = h(t)ψ + aϕ2 + g(t)ϕ + s(t).

(2)

Integrating (1) and then (2), we obtain (C1 and C2 are arbitrary constants) Z  Z Z p(t) dt, E(t) = exp f (t) dt + h(t) dt , ϕ(t) = C1 E(t) + E(t) E(t) Z  Z aϕ2 (t) + g(t)ϕ(t) + s(t) ψ(t) = C2 H(t) + H(t) dt, H(t) = exp h(t) dt . H(t) 2◦ . See equation 5.6.2.7 with r(t) = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 285

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

286

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 286

ONE SPACE VARIABLE

 ∂w   2 ∂w ∂ 2w  + xf (t) + g(t) r(t) + xp(t) + s(t) w. =a + h(t)w ln w + x ∂t ∂x2 ∂x Functional separable solution:   w(x, t) = exp x2 ϕ(t) + xψ(t) + χ(t) , 7.

where the functions ϕ = ϕ(t), ψ = ψ(t), and χ = χ(t) are determined by the system of first-order ordinary differential equations with variable coefficients ϕ′t = 4aϕ2 + (2f + h)ϕ + r, ψt′ = (4aϕ + f + h)ψ + 2gϕ + p,

(1) (2)

χ′t = hχ + 2aϕ + aψ 2 + gψ + s.

(3)

For r ≡ 0, equation (1) is a Bernoulli equation, so it is easy to integrate. In the general case, equation (1) for ϕ = ϕ(t) is a Riccati equation, so it can be reduced to a second-order linear equation. The books by Kamke (1977) and Polyanin and Zaitsev (2003) present a large number of solutions to the Riccati equation for various f , h, and r. With equation (1) solved, the solutions of equations (2) and (3) can be obtained with ease, since these are linear in their unknowns ψ = ψ(t) and χ = χ(t).    2  g(t) ∂w ∂w ∂ 2w 8. + xf (t) + p(t) + s(t) w. =a + h(t)w ln w + x ∂t ∂x2 x ∂x Functional separable solution:   w(x, t) = exp ϕ(t)x2 + ψ(t) , where the functions ϕ = ϕ(t) and ψ = ψ(t) are determined by the system of first-order ordinary differential equations with variable coefficients ϕ′t = 4aϕ2 + (2f + h)ϕ + p, ψt′ = hψ + 2(a + g)ϕ + s.

(1) (2)

For p ≡ 0, equation (1) is a Bernoulli equation, so it is easy to integrate. In the general case, equation (1) for ϕ = ϕ(t) is a Riccati equation, so it can be reduced to a second-order linear equation. The books by Kamke (1977) and Polyanin and Zaitsev (2003) present a large number of solutions to the Riccati equation for various f , h, and r. With equation (1) solved, the solution of the linear equation (2) can be obtained with ease. Reference: A. D. Polyanin and V. F. Zaitsev (2002, p. 55).

5.6.3. Equations of the Form

∂w = a ∂ 2 w + f (x, t, w) ∂w + g(x, t, w) ∂t ∂x ∂x2 ∂w ∂ 2w ∂w + bw =a + f (t). 2 ∂t ∂x ∂x The transformation Z t w = u(z, t) + f (τ ) dτ , 1.

t0

z =x+b

Z

t

(t – τ )f (τ ) dτ , t0

where t0 is any number, leads to the Burgers equation 5.1.5.3: ∂u ∂2u ∂u = a 2 + bu . ∂t ∂x ∂x

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 286

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

Page 287

287

∂w ∂ 2w ∂w + bw =a + f (x, t). ∂t ∂x2 ∂x This is a special case of equation 5.6.3.4. Let us substitute ∂u w= ∂x and then integrate the resulting equation with respect to x to arrive at an equation of the form 5.6.4.3:   ∂ 2 u b ∂u 2 ∂u =a 2 + + F (x, t), ∂t 2 ∂x ∂x Z where F (x, t) = f (x, t) dx + g(t) with g(t) being an arbitrary function. 2.

Reference: A. R. Fortsyth (1906).

3.

∂w

=a

∂ 2w

+ bw

∂w

+

∂ h

i f (x, t)w .

∂t ∂x2 ∂x ∂x This is a special case of equation 5.6.3.4. Solution: 2a 1 ∂θ w(x, t) = , b θ ∂x where the function θ = θ(x, t) satisfies the linear equation ∂θ ∂2θ ∂θ = a 2 + f (x, t) . ∂t ∂x ∂x 4.

∂w

=a

∂ 2w

+ bw

∂w

∂t ∂x2 ∂x The B¨acklund transformation

+

∂ h

∂x

i f (x, t)w + g(x, t).

  Z ∂Z b ∂w 1 2 = Z a + bw + f (x, t)w + g(x, t) dx , ∂t 2a ∂x 2 ∂Z b = Zw, ∂x 2a links the equation in question with the linear equation Z ∂Z ∂2Z ∂Z b = a 2 + f (x, t) + Z g(x, t) dx. ∂t ∂x 2a ∂x ∂ 2w

  ∂w + bw + f (t) + g(t). ∂t ∂x2 ∂x The transformation Z t Z t Z t g(τ ) dτ , z = x + f (τ ) dτ + b (t – τ )g(τ ) dτ , w = u(z, t) +

5.

∂w

=a

t0

t0

t0

where t0 is any number, leads to the Burgers equation 5.1.5.3: ∂u ∂2u ∂u = a 2 + bu . ∂t ∂x ∂x

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 287

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

288

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 288

ONE SPACE VARIABLE

∂w ∂w ∂ 2w + g(t)w + h(t). + f (t)w =a ∂t ∂x ∂x2 Suppose w(x, t) is a solution of this equation. Then the function Z  Z
w1 = w x + ψ(t), t + ϕ(t), ϕ(t) = C exp g(t) dt , ψ(t) = – f (t)ϕ(t) dt, 6.

where C is an arbitrary constant, is also a solution of the equation.

Remark. This remains true if the equation coefficient a is an arbitrary function of time, a = a(t).

∂ 2w

  ∂w + f (t) ln w + g(t) . ∂t ∂x2 ∂x Generalized traveling-wave solution: 7.

∂w

=a

  Z   w(x, t) = exp ϕ(t)(x + C1 ) + ϕ(t) aϕ(t) + g(t) dt ,

Z –1 ϕ(t) = – f (t) dt + C2 ,

where C1 and C2 are arbitrary constants. 8.

∂w

=a

∂ 2w ∂x2

+ f (w)

∂w

∂t ∂x Generalized Burgers equation.

.

1◦ . Traveling-wave solution: w = w(z),

z = x + λt

are defined implicitly by Z dw a = z + B, λw – F (w) + A

F (w) =

Z

f (w) dw,

where A and B are arbitrary constants. 2◦ . For the special forms of the function f (w) = bw, f (w) = bwm , f (w) = beλw , and f (w) = b ln w + c, see equations 5.1.5.3, 5.1.5.9, 5.2.4.2, and 5.4.2.5, respectively, where other solutions are also given. ∂w ∂ 2w ∂w =a + f (w) + g(w). ∂t ∂x ∂x2 Traveling-wave solution: w = w(z), 9.

z = x + λt,

where the function w = w(z) is determined by the second-order autonomous ordinary differential equation   ′′ awzz + f (w) – λ wz′ + g(w) = 0,

which can be reduced with the change of variable wz′ = u(w) to a first-order equation. For exact solutions of the above ordinary differential equation with various f (w) and g(w), see Polyanin and Zaitsev (2003).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 288

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

Page 289

289

 ∂w ∂w ∂2w  + f (w) + bx =a + g(w). ∂t ∂x2 ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the function 10.

w1 = w(x + C1 e–bt , t + C2 ), where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . Generalized traveling-wave solution: w = w(z), z = x + C1 e–bt , where the function w(z) is determined by the ordinary differential equation   ′′ + f (w) + bz wz′ + g(w) = 0. awzz Reference for equation 5.6.3.10: A. D. Polyanin and V. F. Zaitsev (2002, p. 56).

11.

 ∂w ∂2w  ∂w =a + f (w) + g(t) . 2 ∂t ∂x ∂x

On passing from t, x to the new variables t, z = x + the form 5.6.3.8:

12.

∂w ∂t

=a

∂2w ∂x2

g(t) dt, one obtains an equation of

∂w ∂2w ∂w = a 2 + f (w) . ∂t ∂z ∂z   ∂w + h(w). + f (w) + g(t) ∂x Z

On passing from t, x to the new variables t, z = x+ of the form 5.6.3.9:

Z

g(t) dt, one obtains a simpler equation

∂w ∂2w ∂w = a 2 + f (w) + h(w). ∂t ∂z ∂z

 ∂w ∂w ∂2w  =a + f (w) + g(t) + bx + h(w). 2 ∂t ∂x ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the function 13.

w1 = w(x + Ce–bt , t), where C is an arbitrary constant, is also a solution of the equation. 2◦ . Generalized traveling-wave solution: Z –bt –bt w = w(z), z = x + Ce + e ebt g(t) dt,

where the function w(z) is determined by the ordinary differential equation   ′′ awzz + f (w) + bz wz′ + h(w) = 0. 14.

∂w

∂t Solution:

=a

∂2w ∂x2

+ f (bx + ct, w)

∂w ∂x

+ g(bx + ct, w).

w = w(ξ), ξ = bx + ct, where the function w(ξ) is determined by the ordinary differential equation   ′′ ab2 wξξ + bf (ξ, w) – c wξ′ + g(ξ, w) = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 289

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

290

SECOND-ORDER PARABOLIC EQUATIONS

ONE SPACE VARIABLE

WITH



2 5.6.4. Equations of the Form ∂w = a ∂ w2 + b ∂w

∂t

1.

∂x

  ∂ 2w ∂w ∂w 2 =a +b + f (x) + g(t). ∂t ∂x ∂x2

Page 290

∂x

2

+ f (x, t, w)

This is a special case of equation 5.6.4.3. Additive separable solution: w(x, t) = At + B +

Z

g(t) dt + ϕ(x).

Here, A and B are arbitrary constants, and the function ϕ(x) is determined by the ordinary differential equation aϕ′′xx + b ϕ′x )2 + f (x) – A = 0, which can be reduced, with the change of variable ϕ′x = equation:

2.

a ψx′ , to a second-order linear b ψ

  ′′ ψxx + ba–2 f (x) – A ψ = 0.

  ∂w 2 +b + f (t)x2 + g(t)x + h(t). =a ∂t ∂x2 ∂x

∂w

∂ 2w

This is a special case of equation 5.6.4.3. Generalized separable solution quadratic in x: w(x, t) = ϕ(t)x2 + ψ(t)x + χ(t), where the functions ϕ(t), ψ(t), and χ(t) are determined by the system of first-order ordinary differential equations with variable coefficients ϕ′t = 4bϕ2 + f , ψt′ = 4bϕψ + g,

(1) (2)

χ′t = 2aϕ + bψ 2 + h.

(3)

Equation (1) for ϕ is a Riccati equation. In the special case f = const, it can be easily integrated by separation of variables. Having determined ϕ, one finds the solutions of equation (2) and then (3), which are linear in the unknowns ψ and χ, respectively.   ∂ 2w ∂w 2 ∂w =a + b + f (x, t). ∂t ∂x ∂x2   b The substitution u = exp w leads to a linear equation for u = u(x, t): a 3.

∂u ∂2u b = a 2 + f (x, t)u. ∂t a ∂x

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 290

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

291

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

4.

Page 291

  ∂w 2 =a +b + cw + f (t). ∂t ∂x2 ∂x

∂w

∂ 2w

1◦ . Suppose w(x, t) is a solution of this equation. Then the functions w1 = w(±x + C1 , t) + C2 ect , where C1 and C2 are arbitrary constants, are also solutions of the equation. 2◦ . Generalized separable solution quadratic in x: c(x + C2 )2 2a ct w(x, t) = – e ln |C1 e–ct – 4b| + C3 ect + ect –ct C1 e – 4b C1

Z

e–ct f (t) dt.

3◦ . Solution: ct

ct

w(x, t) = Ae + e

Z

e–ct f (t) dt + Θ(ξ),

ξ = x + λt,

where A and λ are arbitrary constants, and the function Θ(ξ) is determined by the autonomous ordinary differential equation aΘ′′ξξ + b Θ′ξ 4◦ . The substitution


2

– λΘ′ξ + cΘ = 0. ct

w = U (x, t) + e

Z

e–ct f (t) dt

leads to the simpler equation   ∂U ∂2U ∂U 2 =a 2 +b + cU . ∂t ∂x ∂x Reference for equation 5.6.4.4: A. D. Polyanin and V. F. Zaitsev (2002, p. 57).

5.

  ∂w 2 =a +b + cw + f (x). ∂t ∂x2 ∂x

∂w

∂ 2w

1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = w(x, t + C1 ) + C2 ect , where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . Additive separable solution: w(x, t) = Aect + ϕ(x), where A is an arbitrary constant and the function ϕ(x) is determined by the ordinary differential equation aϕ′′xx + b(ϕ′x )2 + cϕ + f (x) = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 291

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

292 6.

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 292

ONE SPACE VARIABLE

  ∂w 2 =a +b + cw + f (x) + g(t). ∂t ∂x2 ∂x ∂ 2w

∂w

Additive separable solution: ct

ct

w(x, t) = ϕ(x) + Ae + e

Z

e–ct g(t) dt,

where A is an arbitrary constant and the function ϕ(x) is determined by the ordinary differential equation aϕ′′xx + b(ϕ′x )2 + cϕ + f (x) = 0. 7.

∂w ∂t

=a

  ∂w 2 + b + g(t)w + h(t). ∂x2 ∂x

∂ 2w

This is a special case of equation 5.6.6.1 with f (t) = b. 8.

  ∂w 2 =a +b + cw 2 + f (t)w + g(t). ∂t ∂x2 ∂x

∂w

∂ 2w

This is a special case of equation 5.6.6.2. 9.

  ∂w 2 =a + aλ + f (x, t) + g(x, t)e–λw. ∂t ∂x2 ∂x

∂w

∂ 2w

The substitution u = exp(λw) leads to the linear equation ∂u ∂2u = a 2 + λf (x, t)u + λg(x, t). ∂t ∂x 10.

    ∂w 2 1 x =a +b + f √ ,w . ∂t ∂x2 ∂x t t

∂w

∂2w

1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = w(Cx, C 2 t), where C is an arbitrary constant, is also a solution of the equation. 2◦ . The transformation τ = ln t, ξ = xt–1/2 leads to the equation  2 ∂w ∂2w ∂w 1 ∂w =a 2 +b + ξ + f (ξ, w), ∂τ ∂ξ 2 ∂ξ ∂ξ which admits an exact solution of the form w = w(ξ). 3◦ . In the special case f = f (ξ), there is also a exact solution of the form w = Cτ + ϕ(ξ), where C is an arbitrary constant, and the function ϕ(ξ) is determined by the ordinary differential equation aϕ′′ξξ + b(ϕ′ξ )2 + 12 ξϕ′ξ + f (ξ) – C = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 292

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

Page 293

293

5.6.5. Equations of the Form

1.

  ∂w = a ∂ 2 w + b ∂w 2 + f (x, t, w) ∂w + g(x, t, w) ∂t ∂x ∂x ∂x2

  ∂w 2 ∂w +b + f (x) =a + kw + g(x) + h(t). 2 ∂t ∂x ∂x ∂x

∂w

∂ 2w

1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = w(x, t) + Cekt , where C is an arbitrary constant, is also a solution of the equation. 2◦ . Additive separable solution: kt

kt

w(x, t) = ϕ(x) + Ce + e

Z

e–kt h(t) dt,

where the function ϕ(x) is determined by the ordinary differential equation aϕ′′xx + b(ϕ′x )2 + f (x)ϕ′x + kϕ + g(x) = 0. 2.

  ∂w 2 ∂w =a +b + f (t) + cw 2 + g(t)w + h(t). 2 ∂t ∂x ∂x ∂x

∂w

∂ 2w

This is a special case of equation 5.6.6.5.   ∂w ∂ 2w ∂w 2 ∂w =a + kw 2 + f (t)w + g(t). + b + cw ∂t ∂x ∂x ∂x2 There are generalized separable solutions of the form 3.

w(x, t) = ϕ(t) + ψ(t) exp(λx), where λ = λ1,2 are roots of the quadratic equation bλ2 + cλ + k = 0.   ∂w 2 ∂w ∂w ∂ 2w =a +b + f (x, t) + g(x, t). 4. 2 ∂t ∂x ∂x ∂x   b The substitution u = exp w leads to the linear equation a

5.

∂u ∂2u ∂u b = a 2 + f (x, t) + g(x, t)u. ∂t ∂x a ∂x   ∂w ∂ 2w ∂w 2 ∂w =a + aλ + f (x, t) + g(x, t) + h(x, t)e–λw. 2 ∂t ∂x ∂x ∂x

The substitution u = exp(λw) leads to the linear equation

∂u ∂2u ∂u = a 2 + f (x, t) + λg(x, t)u + λh(x, t). ∂t ∂x ∂x

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 293

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

294

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 294

ONE SPACE VARIABLE

5.6.6. Equations of the Form

  ∂w = a ∂ 2 w + f (x, t, w) ∂w 2 + g(x, t, w) ∂w + h(x, t, w) ∂t ∂x ∂x ∂x2

  ∂ 2w ∂w ∂w 2 1. =a + f (t) + g(t)w + h(t). ∂t ∂x ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the functions Z  w1 = w(±x + C1 , t) + C2 exp g(t) dt ,

where C1 and C2 are arbitrary constants, are also solutions of the equation. 2◦ . Generalized separable solution quadratic in x: w(x, t) = ϕ(t)x2 + ψ(t)x + χ(t), where the functions ϕ(t), ψ(t), and χ(t) are determined by the system of first-order ordinary differential equations with variable coefficients ϕ′t = 4f ϕ2 + gϕ, ψt′ = (4f ϕ + g)ψ,

(1) (2)

χ′t = gχ + 2aϕ + f ψ 2 + h.

(3)

Equation (1) is a Bernoulli equation for ϕ, so it can be readily integrated. Having determined ϕ, we can obtain the solutions of equation (2) and then (3), which are linear in the unknowns ψ and χ, respectively. Finally, we have  –1 Z Z G G ϕ = e A1 – 4 e f dt , G = g dt, Z  (4) ψ = A2 exp (4f ϕ + g) dt , G

G

χ = A3 e + e

Z

e–G (2aϕ + f ψ 2 + h) dt,

where A1 , A2 , and A3 are arbitrary constants. To the limit passage A1 → ∞ in (4) there corresponds a degenerate solution with ϕ ≡ 0. Reference for equation 5.6.6.1: V. F. Zaitsev and A. D. Polyanin (1996, p. 422).

∂w

∂ 2w



∂w

2

=a + f (t) + bf (t)w 2 + g(t)w + h(t). ∂t ∂x2 ∂x 1◦ . Generalized separable solutions: √
w(x, t) = ϕ(t) + ψ(t) exp ±x –b , b < 0,

2.

(1)

where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations with variable coefficients (the arguments of f , g, and h are not indicated) ϕ′t = bf ϕ2 + gϕ + h, ψt′ = (2bf ϕ + g – ab)ψ.

(2) (3)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 294

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

Page 295

295

Equation (2) is a Riccati equation for ϕ = ϕ(t), so it can be reduced to a second-order linear equation. The books by Kamke (1977) and Polyanin and Zaitsev (2003) present a large number of solutions to this equation for various f , g, and h. Given a solution of (2), one can solve the linear equation (3) for ψ = ψ(t) to obtain   Z ψ(t) = C exp –abt + (2bf ϕ + g) dt , (4)

where C is an arbitrary constant. Note two special integrable cases of equation (2). Solution of equation (2) with h ≡ 0:  –1 Z ϕ(t) = eG C1 – b f eG dt ,

G=

Z

g dt,

where C1 is an arbitrary constant. If the functions f , g, and h are proportional, g = αf , h = βf (α, β = const), then the solution of (2) is given by Z Z dϕ = f dt + C2 , (5) bϕ2 + αϕ + β where C2 is an arbitrary constant. On integrating the left-hand side of equation (5) and solving for ϕ, one can find ϕ = ϕ(t) in explicit form. 2◦ . Generalized separable solution of a more general form: √
√
  w(x, t) = ϕ(t) + ψ(t) A exp x –b + B exp –x –b , b < 0, (6) where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations with variable coefficients
ϕ′t = bf ϕ2 + 4ABψ 2 + gϕ + h, (7) ψt′ = 2bf ϕψ + gψ – abψ. (8) Solving (8) for ϕ to express it in terms of ψ and then substituting the resulting expression into (7), one arrives at a second-order nonlinear equation for ψ (if f , g, h = const, this equation is autonomous and, hence, its order can be reduced). Note two special cases of solution (6) where the exponentials combine to form hyperbolic functions: √
w(x, t) = ϕ(t) + ψ(t) cosh x –b , A = 12 , B = 12 , √
w(x, t) = ϕ(t) + ψ(t) sinh x –b , A = 12 , B = – 12 . 3◦ . Generalized separable solution (c is an arbitrary constant): √
w(x, t) = ϕ(t) + ψ(t) cos x b + c , b > 0, (9) where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations with variable coefficients
ϕ′t = bf ϕ2 + ψ 2 + gϕ + h, (10) ψt′ = 2bf ϕψ + gψ – abψ. (11) Solving (11) for ϕ to express it in terms of ψ and then substituting the resulting expression into (10), one arrives at a second-order nonlinear equation for ψ (if f , g, h = const, this equation is autonomous and, hence, its order can be reduced).

References for equation 5.6.6.2: V. A. Galaktionov and S. A. Posashkov (1989, the case f , g, h = const was considered), V. F. Zaitsev and A. D. Polyanin (1996, pp. 422–423).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 295

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

296 3.

SECOND-ORDER PARABOLIC EQUATIONS ∂w

=a

∂ 2w ∂x2

+ f (t)



∂w

2

+ bf (t)w

∂w

WITH

Page 296

ONE SPACE VARIABLE

+ cf (t)w 2 + g(t)w + h(t).

∂t ∂x ∂x There are generalized separable solutions of the form

w(x, t) = ϕ(t) + ψ(t) exp(λx), where λ = λ1,2 are roots of the quadratic equation λ2 + bλ + c = 0.   ∂ 2w ∂w ∂w 2 ∂w 4. =a + f (x) + g(x) + bw + h(x) + p(t). 2 ∂t ∂x ∂x ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = w(x, t) + Cebt , where C is an arbitrary constant, is also a solution of the equation. 2◦ . Additive separable solution: bt

bt

w(x, t) = ϕ(x) + Ce + e

Z

e–bt p(t) dt,

where the function ϕ(x) is determined by the ordinary differential equation aϕ′′xx + f (x)(ϕ′x )2 + g(x)ϕ′x + bϕ + h(x) = 0. ∂w

∂ 2w



∂w

2

∂w

+ bf (t)w 2 + h(t)w + p(t). Z On passing from t, x to the new variables t, z = x + g(t) dt, one arrives at an equation of 5.

∂t

=a

∂x2

+ f (t)

∂x

+ g(t)

∂x

the form 5.6.6.2:

  ∂w ∂2w ∂w 2 = a 2 + f (t) + bf (t)w2 + h(t)w + p(t). ∂t ∂z ∂z

   ∂w ∂2w ∂w ∂w 2  6. =a +h(t)w+p(t)x2 +q(t)x+s(t). +f (t) + g (t)x+g (t) 1 0 2 ∂t ∂x ∂x ∂x Generalized separable solution quadratic in x: w(x, t) = ϕ(t)x2 + ψ(t)x + χ(t), where the functions ϕ(t), ψ(t), and χ(t) are determined by the system of first-order ordinary differential equations with variable coefficients ϕ′t = 4f ϕ2 + (2g1 + h)ϕ + p, ψt′ = (4f ϕ + g1 + h)ψ + 2g0 ϕ + q,

(1) (2)

χ′t = hχ + 2aϕ + f ψ 2 + g0 ψ + s.

(3)

Equation (1) is a Riccati equation for ϕ = ϕ(t) and, hence, can be reduced to a secondorder linear equation. For solution of such equations, see Kamke (1977) and Polyanin and Zaitsev (2003). In the special case p ≡ 0, (1) is a Bernoulli equation, so it can be readily integrated. Given a solution of (1), equations (2) and (3) can be easily solved, since these are linear in their respective unknowns ψ and χ.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 296

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

7.

∂w ∂t

=a

∂ 2w ∂x2

– ak

The substitution u = w

1 w

1–k



∂w ∂x

2

+ f (x, t)

∂w ∂x

Page 297

297

+ g(x, t)w + h(x, t)w k.

leads to the linear equation

∂u ∂2u ∂u = a 2 + f (x, t) + (1 – k)g(x, t)u + (1 – k)h(x, t). ∂t ∂x ∂x 8.

∂w ∂t

=

∂2w ∂x2

+ f (w)



∂w ∂x

2

.

The substitution u=

Z

F (w) dw,

where

Z  F (w) = exp f (w) dw ,

leads to the linear heat equation for u = u(x, t): ∂u ∂ 2 u = . ∂t ∂x2 9.

∂w ∂t

=

∂2w ∂x2

+ f (w)



∂w ∂x

2

+ g(x)

∂w ∂x

.

The substitution u=

Z

F (w) dw,

where

Z  F (w) = exp f (w) dw ,

leads to a linear equation for u = u(x, t): ∂u ∂ 2 u ∂u = + g(x) . ∂t ∂x ∂x2 Some exact solutions of this equation, for arbitrary g, can be found in Polyanin (2002). 10.

   ∂w ∂w 2  ∂w ∂ 2w = + f (w) + xg(t) + h(t) . ∂t ∂x ∂x ∂x2

The substitution

u=

Z

F (w) dw,

where

Z  F (w) = exp f (w) dw ,

leads to a linear equation for u = u(x, t):  ∂u ∂u ∂ 2 u  = + xg(t) + h(t) . 2 ∂t ∂x ∂x

This equation can be reduced to the linear heat equation; see Polyanin (2002, p. 133).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 297

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

298 11.

SECOND-ORDER PARABOLIC EQUATIONS ∂w ∂t

=

∂ 2w ∂x2

+ f (w)



∂w ∂x

2

+ g(x, t)

∂w ∂x

WITH

Page 298

ONE SPACE VARIABLE

.

The substitution u=

Z

F (w) dw,

Z  F (w) = exp f (w) dw ,

where

leads to the linear equation ∂u ∂ 2 u ∂u + g(x, t) . = 2 ∂t ∂x ∂x 12.

  ∂2w ∂w ∂w 2 ∂w =a + f (w) + g(w) + h(w). 2 ∂t ∂x ∂x ∂x

Traveling-wave solution: w = w(z),

z = x + λt,

where the function w = w(z) is determined by the autonomous ordinary differential equation ′′ awzz + f (w)(wz′ )2 + [g(w) – λ]wz′ + h(w) = 0.

(1)

The substitution wz′ = u(w) leads to the first-order equation auu′w + f (w)u2 + [g(w) – λ]u + h(w) = 0.

(2)

For exact solutions of the ordinary differential equations (1) and (2) for various f (w), g(w), and h(w), see the book by Polyanin and Zaitsev (2003). Note that in the special case h ≡ 0, equation (2) becomes linear and, hence, can be readily integrated.

  2 ∂w ∂ w ∂w 5.6.7. Equations of the Form =a + f x, t, w, ∂t ∂x ∂x2 1.

  ∂w ∂w k ∂w ∂ 2w =a + f (x) + g(x) + bw + h(x) + p(t). ∂t ∂x ∂x ∂x2

1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = w(x, t) + Cebt , where C is an arbitrary constant, is also a solution of the equation. 2◦ . Additive separable solution: bt

bt

w(x, t) = ϕ(x) + Ce + e

Z

e–bt p(t) dt,

where the function ϕ(x) is determined by the second-order ordinary differential equation aϕ′′xx + f (x)(ϕ′x )k + g(x)ϕ′x + bϕ + h(x) = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 298

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

299

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

2.

Page 299

  ∂w ∂ 2w ∂w + f x, =a + bw + g(t). ∂t ∂x2 ∂x

1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = w(x, t) + Cebt , where C is an arbitrary constant, is also a solution of the equation. 2◦ . Additive separable solution: bt

bt

w(x, t) = ϕ(x) + Ce + e

Z

e–bt g(t) dt,

where the function ϕ(x) is determined by the second-order ordinary differential equation aϕ′′xx + f (x, ϕ′x ) + bϕ = 0. 3.

  1 ∂w ∂ 2w ∂w + wf t, =a . ∂t ∂x2 w ∂x

1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = C1 w(x + C2 , t), where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . Multiplicative separable solution:   Z 2 w(x, t) = A exp λx + aλ t + f (t, λ) dt , where A and λ are arbitrary constants.



2 5.6.8. Equations of the Form ∂w = f (x, t) ∂ w2 + g x, t, w, ∂w

∂t

∂x

∂ 2w ∂w = (ax2 + b) + ax + f (w). 2 ∂t ∂x ∂x Z dx The substitution z = leads to an equation of the form 5.6.1.1: √ ax2 + b 1.

∂w

∂x



∂w ∂ 2 w = + f (w). ∂t ∂z 2   ∂w f (t) ∂ n ∂w 2. = x + g(t)w ln w. ∂t xn ∂x ∂x This equation can be rewritten in the form ∂w ∂ 2 w nf (t) ∂w = f (t) 2 + + g(t)w ln w. ∂t x ∂x ∂x

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 299

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

300

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 300

ONE SPACE VARIABLE

Functional separable solution:

  w(x, t) = exp ϕ(t)x2 + ψ(t) ,

where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations with variable coefficients ϕ′t = 4f ϕ2 + gϕ, ψt′ = 2(n + 1)f ϕ + gψ; the arguments of f and g are omitted. Successively integrating, one obtains  –1 Z Z G G ϕ(t) = e A – 4 f e dt , G = g dt, Z G G ψ(t) = Be + 2(n + 1)e f ϕe–G dt,

where A and B are arbitrary constants.    2  ∂w ∂ 2w h(t) ∂w 3. = f (t) + xg(t) + + s(t)w ln w + x p(t) + q(t) w. ∂t ∂x2 x ∂x Functional separable solution:   w(x, t) = exp ϕ(t)x2 + ψ(t) ,

where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations with variable coefficients ϕ′t = 4f ϕ2 + (2g + s)ϕ + p, ψt′ = sψ + 2(f + h)ϕ + q.

(1) (2)

For p ≡ 0, equation (1) is a Bernoulli equation and, hence, can be easily integrated. In the general case, (1) is a Riccati equation for ϕ = ϕ(t), so it can be reduced to a second-order linear equation. The books by Kamke (1977) and Polyanin and Zaitsev (2003) present a considerable number of solutions to this equation for various f , g, s, and p. Having solved equation (1), one can find ψ = ψ(t) from the linear equation (2).   ∂w ∂ ∂w 4. = f (t) eλx + g(t)w ln w + h(t)w. ∂t ∂x ∂x Functional separable solution:   w(x, t) = exp ϕ(t)e–λx + ψ(t) ,

where the functions ϕ(t) and ψ(t) are determined by the ordinary differential equations ϕ′t = λ2 f (t)ϕ2 + g(t)ϕ, ψt′ = g(t)ψ + h(t). Integrating yields  –1 Z 2 ϕ(t) = G(t) A – λ f (t)G(t) dt , Z h(t) ψ(t) = BG(t) + G(t) dt, G(t) where A and B are arbitrary constants.

G(t) = exp

Z



g(t) dt ,

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 300

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

Page 301

301

  ∂w 5. = f (x) + aw ln w. ∂t ∂x ∂x This equation can be rewritten in the form ∂w

∂

∂w ∂2w ∂w = f (x) 2 + fx′ (x) + aw ln w. ∂t ∂x ∂x 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the function w1 = exp(C1 eat ) w(x, t + C2 ), where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . Multiplicative separable solution: w(x, t) = exp(Ceat ) ϕ(x), where C is an arbitrary constant, and the function ϕ(t) is determined by the ordinary differential equation (f ϕ′x )′x + aϕ ln ϕ = 0.     ∂ ∂w ∂w 6. = f (x) + aw ln w + g(x) + h(t) w. ∂t ∂x ∂x ◦ 1 . Suppose w(x, t) is a solution of this equation. Then the function w1 = exp(Ceat )w(x, t), where C is an arbitrary constant, is also a solution of the equation. 2◦ . Multiplicative separable solution:   Z at at –at e h(t) dt ϕ(x), w(x, t) = exp Ce + e where the function ϕ(x) is determined by the ordinary differential equation (f ϕ′x )′x + aϕ ln ϕ + g(x)ϕ = 0.   ∂w ∂ 2w ∂w = f (x) + g(x) + aw ln w + h(x) + s(t) w. 2 ∂t ∂x ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the function 7.

w1 = exp(Ceat )w(x, t), where C is an arbitrary constant, is also a solution of the equation. 2◦ . Multiplicative separable solution:   Z w(x, t) = exp Ceat + eat e–at s(t) dt ϕ(x), where the function ϕ(x) is determined by the ordinary differential equation f (x)ϕ′′xx + g(x)ϕ′x + aϕ ln ϕ + h(x)ϕ = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 301

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

302 8.

SECOND-ORDER PARABOLIC EQUATIONS ∂w ∂t

= f (x)

∂ 2w ∂x2

+ g(x)



∂w ∂x

2

+ h(x)

∂w ∂x

WITH

Page 302

ONE SPACE VARIABLE

+ aw + p(x) + q(t).

1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = w(x, t) + Ceat , where C is an arbitrary constant, is also a solution of the equation. 2◦ . Additive separable solution: w(x, t) = ϕ(x) + Ceat + eat

Z

e–at q(t) dt,

where the function ϕ(x) is determined by the ordinary differential equation f (x)ϕ′′xx + g(x)(ϕ′x )2 + h(x)ϕ′x + aϕ + p(x) = 0. 9.

∂w ∂t

= f (x)

∂ 2w ∂x2

+ g(x)



∂w ∂x

k

+ h(x)

∂w ∂x

+ aw + p(x) + q(t).

1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = w(x, t) + Ceat , where C is an arbitrary constant, is also a solution of the equation. 2◦ . Additive separable solution: at

at

w(x, t) = ϕ(x) + Ce + e

Z

e–at q(t) dt,

where the function ϕ(x) is determined by the ordinary differential equation f (x)ϕ′′xx + g(x)(ϕ′x )k + h(x)ϕ′x + aϕ + p(x) = 0. 10.

∂w ∂t

= f (x)

∂ 2w ∂x2



+ g x,

∂w ∂x



+ aw + h(t).

1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = w(x, t) + Ceat , where C is an arbitrary constant, is also a solution of the equation. 2◦ . Additive separable solution: at

at

w(x, t) = ϕ(x) + Ce + e

Z

e–at h(t) dt,

where the function ϕ(x) is determined by the ordinary differential equation f (x)ϕ′′xx + g(x, ϕ′x ) + aϕ = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 302

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 303

303

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

5.6.9. Equations of the Form

∂w = aw ∂ 2 w + f (x, t, w) ∂w + g(x, t, w) ∂t ∂x ∂x2 ∂w

∂ 2w

+ f (x)w + bx + c. ∂t ∂x2 Generalized separable solution:

1.

= aw

1 w(x, t) = (bx + c)t + Ax + B – a

Z

x

(x – ξ)f (ξ) dξ, x0

where A, B, and x0 are arbitrary constants. ∂w

= aw

∂ 2w

+ f (t)w + g(t). ∂t ∂x2 1◦ . Degenerate solution linear in x: 2.

w(x, t) = F (t)(Ax + B) + F (t)

Z

g(t) dt, F (t)

F (t) = exp

Z

 f (t) dt ,

where A and B are arbitrary constants. 2◦ . Generalized separable solution quadratic in x: 2

g(t) dt, ϕ(t) Z  F (t) = exp f (t) dt ,

w(x, t) = ϕ(t)(x + Ax + B) + ϕ(t)  –1 Z ϕ(t) = F (t) C – 2a F (t) dt ,

Z

where A, B, and C are arbitrary constants. ∂w ∂ 2w = aw + cw 2 + f (t)w + g(t). ∂t ∂x2 This is a special case of equation 5.6.10.1 with b = 0. 3.

∂w ∂ 2w = aw – ak2 w 2 + f (x)w + b1 sinh(kx) + b2 cosh(kx). 2 ∂t ∂x Generalized separable solution:   w(x, t) = t b1 sinh(kx) + b2 cosh(kx) + ϕ(x). 4.

Here, the function ϕ(x) is determined by the linear nonhomogeneous ordinary differential equation with constant coefficients aϕ′′xx – ak2 ϕ + f (x) = 0,

whose general solution is given by ϕ(x) = C1 sinh(kx) + C2 cosh(kx) – where A, B, and x0 are arbitrary constants.

1 ak

Z

x

x0

  f (ξ) sinh k(x – ξ) dξ,

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 303

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

304

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 304

ONE SPACE VARIABLE

∂w ∂ 2w + ak2 w 2 + f (x)w + b1 sin(kx) + b2 cos(kx). = aw ∂t ∂x2 Generalized separable solution:   w(x, t) = t b1 sin(kx) + b2 cos(kx) + ϕ(x). 5.

Here, the function ϕ(x) is determined by the linear nonhomogeneous ordinary differential equation with constant coefficients aϕ′′xx + ak2 ϕ + f (x) = 0,

whose general solution is given by 1 ϕ(x) = C1 sin(kx) + C2 cos(kx) – ak

Z

where A, B, and x0 are arbitrary constants.

x x0

  f (ξ) sin k(x – ξ) dξ,

∂w ∂w ∂ 2w + f (t) = aw + g(t)w. ∂t ∂x2 ∂x The transformation Z Z w(x, t) = G(t)u(z, τ ), z = x + f (t) dt, τ = G(t) dt, 6.

G(t) = exp

Z

g(t) dt



leads to a simpler equation of the form 5.1.9.1: ∂u ∂2u = au 2 . ∂τ ∂z ∂w

8.

∂w

= aw

∂ 2w

+ f (t)

∂w

+ g(t)w + h(t). ∂t ∂x This is a special case of equation 5.6.10.5. 7.

= aw

∂x2

∂ 2w ∂x2

∂t The transformation

+ xf (t)

∂w ∂x

+ g(t)w.

w(x, t) = G(t)u(z, τ ),

z = xF (t),

τ=

Z

F 2 (t)G(t) dt,

where the functions F (t) and G(t) are given by Z  Z  F (t) = exp f (t) dt , G(t) = exp g(t) dt , leads to a simpler equation of the form 5.1.9.1: ∂u ∂2u = au 2 . ∂τ ∂z

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 304

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

305

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

 ∂w ∂w ∂ 2w  + xf (t) + g(t) = aw + h(t)w. ∂t ∂x2 ∂x The transformation Z w(x, t) = H(t)u(z, τ ), z = xF (t) + g(t)F (t) dt,

Page 305

9.

τ=

Z

F 2 (t)H(t) dt,

where the functions F (t) and H(t) are given by Z  Z  F (t) = exp f (t) dt , H(t) = exp h(t) dt , leads to a simpler equation of the form 5.1.9.1: ∂2u ∂u = au 2 . ∂τ ∂z

10.

∂w

= aw

∂2w ∂x2

+ f (x)w

∂t Generalized separable solution:

∂w ∂x

+ g(t)w + h(t).

w(x, t) = ϕ(t)Θ(x) + ψ(t), where the functions ϕ(t), ψ(t), and Θ(x) are determined by the system of ordinary differential equations ϕ′t = Cϕ2 + g(t)ϕ,   ψt′ = Cϕ + g(t) ψ + h(t), aΘ′′xx + f (x)Θ′x = C, where C is an arbitrary constant. Integrating successively, one obtains –1 Z   Z ϕ(t) = G(t) A1 – C G(t) dt , G(t) = exp g(t) dt , Z h(t) ψ(t) = A2 ϕ(t) + ϕ(t) dt, ϕ(t)   Z  Z Z Z dx C dx 1 Θ(x) = B1 + B2 + F (x) dx , F (x) = exp f (x) dx , F (x) a F (x) a

where A1 , A2 , B1 , and B2 are arbitrary constants. ∂w

∂2w

∂w

+ g(x)w 2 + h(t)w. ∂t ∂x Multiplicative separable solution:  –1 Z  Z w(x, t) = ϕ(x)H(t) A – B H(t) dt , H(t) = exp h(t) dt . 11.

= aw

∂x2

+ f (x)w

Here, A and B are arbitrary constants, and the function ϕ(x) is determined by the secondorder linear ordinary differential equation aϕ′′xx + f (x)ϕ′x + g(x)ϕ = B. For exact solutions of this equation with various f (x) and g(x), see Kamke (1977) and Polyanin and Zaitsev (2003).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 305

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

306

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 306

ONE SPACE VARIABLE

5.6.10. Equations of the Form

∂w = ∂t

1.

 2 2 ∂w ∂ w ∂w (aw+b) 2 +f (x, t, w) ∂x +g(x, t, w) ∂x +h(x, t, w) ∂x

  ∂w 2 = aw +b + cw 2 + f (t)w + g(t). ∂t ∂x2 ∂x

∂w

∂ 2w

1◦ . Generalized separable solutions involving an exponential of x: w(x, t) = ϕ(t) + ψ(t) exp(±λx),

λ=



–c a+b

1/2

,

(1)

where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations with variable coefficients (the arguments of f and g are not indicated) ϕ′t = cϕ2 + f ϕ + g,

(2)

2

(3)

ψt′

= (aλ ϕ + 2cϕ + f )ψ.

Equation (2) is a Riccati equation for ϕ = ϕ(t), so it can be reduced to a second-order linear equation. The books by Kamke (1977) and Polyanin and Zaitsev (2003) present a large number of solutions to this equation for various f and g. In particular, for g ≡ 0, equation (2) is a Bernoulli equation, which is easy to integrate. In another special case, f , g = const, a particular solution of (2) is a number, ϕ = ϕ0 , which is a root of the quadratic equation cϕ20 + f ϕ0 + g = 0. The substitution u = ϕ – ϕ0 leads to a Bernoulli equation. Given a solution of (2), the solution of equation (3) can be obtained in the form ψ(t) = C exp

Z

 (aλ ϕ + 2cϕ + f ) dt , 2

(4)

where C is an arbitrary constant. 2◦ . Generalized separable solution involving hyperbolic cosine (A is an arbitrary constant): w(x, t) = ϕ(t) + ψ(t) cosh(λx + A),

λ=



–c a+b

1/2

,

(5)

where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations with variable coefficients (the arguments of f and g are not specified) ϕ′t = cϕ2 – bλ2 ψ 2 + f ϕ + g, ψt′

2

= (aλ ϕ + 2cϕ + f )ψ.

(6) (7)

Solving equation (7) for ϕ to express it in terms of ψ and then substituting the resulting expression into (6), one arrives at a second-order nonlinear equation for ψ, which is autonomous if f , g = const and, hence, its order can be reduced.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 306

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 307

307

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

3◦ . Generalized separable solution involving hyperbolic sine (A is an arbitrary constant): w(x, t) = ϕ(t) + ψ(t) sinh(λx + A),

λ=



–c a+b

1/2

,

where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations ϕ′t = cϕ2 + bλ2 ψ 2 + f ϕ + g, ψt′ = (aλ2 ϕ + 2cϕ + f )ψ. 4◦ . Generalized separable solution involving a trigonometric function (A is an arbitrary constant):  1/2 c w(x, t) = ϕ(t) + ψ(t) cos(λx + A), λ = , (8) a+b where the functions ϕ(t) and ψ(t) are determined by the system of first-order ordinary differential equations ϕ′t = cϕ2 + bλ2 ψ 2 + f ϕ + g, ψt′

2

= (–aλ ϕ + 2cϕ + f )ψ.

(9) (10)

Solving equation (10) for ϕ to express it in terms of ψ and substituting the resulting expression into (9), one arrives at a second-order nonlinear equation for ψ, which is autonomous if f , g = const and, hence, its order can be reduced. References for equation 5.6.10.1: V. A. Galaktionov (1995), V. F. Zaitsev and A. D. Polyanin (1996, pp. 425–427).

2.

∂w ∂t

= aw

∂ 2w ∂x2

+ f (t)



∂w ∂x

2

+ g(t)

∂w ∂x

+ h(t)w + s(t).

Generalized separable solution quadratic in x: w(x, t) = ϕ(t)x2 + ψ(t)x + χ(t), where the functions ϕ(t), ψ(t), and χ(t) are determined by the system of first-order ordinary differential equations with variable coefficients (the arguments of f , g, h, and s are not specified) ϕ′t = 2(2f + a)ϕ2 + hϕ, ψt′ = (4f ϕ + 2aϕ + h)ψ + 2gϕ,

(1) (2)

χ′t = (2aϕ + h)χ + f ψ 2 + gψ + s.

(3)

Equation (1) is a Bernoulli equation for ϕ = ϕ(t), so it is easy to integrate. After that, equation (2) and then (3) can be solved with ease, since both are linear in their respective unknowns ψ and χ. References: V. A. Galaktionov (1995), V. F. Zaitsev and A. D. Polyanin (1996, p. 427).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 307

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

308 3.

SECOND-ORDER PARABOLIC EQUATIONS ∂w

= aw

∂ 2w

+ f (x)

∂x2



∂w

∂t ∂x Multiplicative separable solution:

2

+ g(x)w

∂w ∂x

WITH

Page 308

ONE SPACE VARIABLE

+ h(x)w 2 + p(t)w.

w(x, t) = ϕ(x)ψ(t), where ϕ(x) and ψ(t) are determined by the following system of ordinary differential equations (C is an arbitrary constant): aϕϕ′′xx + f (x)(ϕ′x )2 + g(x)ϕϕ′x + h(x)ϕ2 = Cϕ, ψt′

2

= Cψ + p(t)ψ.

(1) (2)

The general solution of equation (2) is given by Z   –1 Z p(t) dt , ψ(t) = P (t) A – C P (t) dt , P (t) = exp where A is an arbitrary constant. In the special case f ≡ 0, equation (1) can be reduced, on dividing it by ϕ, to a second-order linear equation; for exact solutions of this equation with various g(x) and h(x), see Kamke (1997) and Polyanin and Zaitsev (2003). 4.

∂w

= (aw + b)

∂t The transformation

∂ 2w ∂x2

+c



∂w ∂x

2

+ f (t)

∂w ∂x

+ kw 2 + g(t)w + h(t).

b u(z, t) = w(x, t) + , z = x + a leads to an equation of the form 5.6.10.1 for u = u(z, t). 5.

∂w ∂t

= (aw + b)

∂ 2w ∂x2

+ f (t)



∂w ∂x

2

Z

f (t) dt

  ∂w + g1 (t)x + g0 (x) ∂x + h(t)w + p2 (t)x2 + p1 (t)x + p0 (t).

There is a generalized separable solution quadratic in x: w(x, t) = ϕ(t)x2 + ψ(t)x + χ(t), where the functions ϕ(t), ψ(t), and χ(t) are determined by a system of first-order ordinary differential equations with variable coefficients, which is not written out here.

5.6.11. Equations of the Form

∂w = aw m ∂ 2 w + f (x, t) ∂w + g(x, t, w) ∂t ∂x ∂x2 ∂w ∂ 2w 2 ∂ = w2 [f (x)w]. + w ∂t ∂x2 ∂x 1◦ . Consider the nonlinear equation 1.

 ∂x 2 ∂x ∂ 2 x = + f (x) , ∂t ∂z ∂z 2

(1)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 308

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 309

309

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

which is reduced to the linear heat equation (see Equation 5.6.6.8). Dividing the equation by xz , differentiating with respect to z, changing to the von Mises variables t, z, x = x(t, z)

=⇒

t, x, w = w(t, x),

and taking into account the formula ∂ x  t

∂z xz

≡

where

w=

∂x , ∂z

xz xtz – xt xzz wt , = 2 w xz

we arrive at the original equation.

2◦ . The original equation is related to equation (1) by the B¨acklund transformation xt = wz + f (x)w2 , xz = w. ∂w

= aw 4

∂2w

+ f (x)w 5. ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the function 2.

w1 = C1 w(x, C14 t + C2 ), where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . Suppose u = u(x) is a nontrivial solution of the second-order linear ordinary differential equation au′′xx + f (x)u = 0. (1) Then the transformation Z dx w ξ= , z= 2 u u simplifies the original equation bringing it to the form ∂z ∂2z = az 4 2 . ∂t ∂ξ Using the change of variable v = z –3 , we obtain an equation of the form 5.1.10.4:   ∂v ∂ ∂v =a v –4/3 . ∂t ∂ξ ∂ξ 3◦ . Multiplicative separable solution:

w(x, t) = (4λt + C)–1/4 g(x), where C and λ are arbitrary constants, and the function g = g(x) is determined by Yermakov’s equation ′′ agxx + f (x)g + λg–3 = 0. (2) Given a particular solution, u = u(x), of the linear equation (1), the general solution of the nonlinear equation (2) can be expressed as (e.g., see Polyanin and Zaitsev, 2003)   Z λ 2 dx 2 2 2 Ag = – u + u B + A , a u2 where A and B are arbitrary constants (A ≠ 0).

Reference for equation 5.6.11.2: V. F. Zaitsev and A. D. Polyanin (1996, pp. 427–428).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 309

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

310

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 310

ONE SPACE VARIABLE

∂w ∂ 2w + f (t)w. = aw m ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the functions 3.

w1 = C1–2 w(±C1m x + C2 , t), where C1 and C2 are arbitrary constants, are also solutions of the equation. 2◦ . The transformation w(x, t) = F (t)u(x, τ ),

τ=

Z

m

F (t) dt,

F (t) = exp

Z

 f (t) dt ,

leads to a simpler equation of the form 5.1.9.20: ∂2u ∂u = aum 2 . ∂τ ∂x ∂w

= aw m

∂ 2w

+ f (x)w m+1. ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the function

4.

w1 = C1 w(x, C1m t + C2 ), where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . Multiplicative separable solution (C and λ are arbitrary constants): w(x, t) = (mλt + C)–1/m ϕ(x), where the function ϕ = ϕ(x) is determined by the ordinary differential equation aϕm ϕ′′xx + f (x)ϕm+1 + λϕ = 0. 5.

∂w

= aw m

∂ 2w ∂x2

∂t The transformation

+ xf (t)

∂w ∂x

w(x, t) = u(z, τ )G(t),

+ g(t)w.

z = xF (t),

τ=

Z

F 2 (t)Gm (t) dt,

where the functions F (t) and G(t) are given by Z  Z  F (t) = exp f (t) dt , G(t) = exp g(t) dt , leads to a simpler equation of the form 5.1.9.26: ∂u ∂2u = aum 2 . ∂τ ∂z Reference: V. F. Zaitsev and A. D. Polyanin (1996, p. 428).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 310

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

311

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

 ∂w ∂w ∂ 2w  + f (t)x + g(t) = aw m + h(t)w. ∂t ∂x2 ∂x The transformation Z w(x, t) = u(z, τ )H(t), z = xF (t) + g(t)F (t) dt,

Page 311

6.

τ=

Z

F 2 (t)H m (t) dt,

where the functions F (t) and H(t) are given by Z  Z  F (t) = exp f (t) dt , H(t) = exp h(t) dt , leads to a simpler equation of the form 5.1.9.26:

∂u ∂2u = aum 2 . ∂τ ∂z

5.6.12. Equations of  the Form 

∂w = a ∂ w ∂w +f (x, t) ∂w +g(x, t, w) ∂t ∂x ∂x ∂x

  ∂w ∂ ∂w 1. =a w + f (t). ∂t ∂x ∂x Generalized separable solutions linear and quadratic in x: Z 2 w(x, t) = C1 x + aC1 t + C2 + f (t) dt,

Z (x + C2 )2 –1/3 –1/3 w(x, t) = – + C3 (t + C1 ) + (t + C1 ) (t + C1 )1/3 f (t) dt, 6a(t + C1 ) where C1 , C2 , and C3 are arbitrary constants. The first solution is degenerate.   ∂w ∂ ∂w 2. =a w + f (t)w + g(t). ∂t ∂x ∂x This is a special case of equation 5.6.13.6 with m = 1.   ∂w ∂ ∂w =a w + bw 2 + f (t)w + g(t). 3. ∂t ∂x ∂x This is a special case of equation 5.6.13.7 with m = 1.   ∂w ∂ ∂w 4. =a w + f (x)w 2 . ∂t ∂x ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = C1 w(x, C1 t + C2 ), where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . Multiplicative separable solution: w = (λt + C)–1 ϕ(x), where λ and C are arbitrary constants, and the function ϕ(x) is determined by the ordinary differential equation a(ϕϕ′x )′x + f (x)ϕ2 + λϕ = 0.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 311

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

312 5.

SECOND-ORDER PARABOLIC EQUATIONS   ∂w ∂ ∂w ∂w =a w + f (t) + g(t)w. ∂t ∂x ∂x ∂x

WITH

Page 312

ONE SPACE VARIABLE

This is a special case of equation 5.6.13.8 with m = 1. The transformation Z Z w(x, t) = G(t)u(z, τ ), z = x + f (t) dt, τ = G(t) dt,

G(t) = exp

Z



g(t) dt ,

leads to a simpler equation of the form 5.1.10.1:   ∂u ∂ ∂u =a u . ∂t ∂z ∂z 6.

∂w ∂t

=a

∂ ∂x



w

∂w ∂x



+ xf (t)

∂w ∂x

+ g(t)w.

The transformation w(t, x) = u(z, τ )G(t),

z = xF (t),

τ=

Z

F 2 (t)G(t) dt,

where the functions F (t) and G(t) are given by F (t) = exp

Z

 f (t) dt ,

G(t) = exp

Z



g(t) dt ,

leads to a simpler equation of the form 5.1.10.1:   ∂u ∂ ∂u =a u . ∂τ ∂z ∂z 7.

∂w ∂t

=a

∂ ∂x



w

∂w ∂x



  ∂w + xf (t) + g(t) + h(t)w. ∂x

This is a special case of equation 5.6.13.10 with m = 1.

5.6.13. Equations of the Form 

∂w = a ∂ w m ∂w +f (x, t) ∂w +g(x, t, w) ∂t ∂x ∂x ∂x

1.

  ∂w ∂ ∂w =a w –4/3 + f (x)w –1/3. ∂t ∂x ∂x

1◦ . Suppose w(x, t) is a solution of this equation. Then the function –4/3

w1 = C1 w(x, C1

t + C2 ),

where C1 and C2 are arbitrary constants, is also a solution of the equation.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 312

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 313

313

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

2◦ . The substitution w = v –3 leads to an equation of the form 5.6.11.2: ∂v ∂2v = av 4 2 – 13 f (x)v 5 . ∂t ∂x 3◦ . Suppose u = u(x) is any nontrivial particular solution of the second-order linear ordinary differential equation au′′xx – 13 f (x)u = 0. The transformation ξ=±

Z

dx , u2

z = wu3

simplifies the original equation, bringing it to equation 5.1.10.4:   ∂ ∂z –4/3 ∂z =a z . ∂t ∂ξ ∂ξ Reference for equation 5.6.13.1: V. F. Zaitsev and A. D. Polyanin (1996).

2.

∂w ∂t

=a

∂ ∂x



1 ∂w w 2 ∂x



+ f (x).

This equation can be reduced to the generalized Burgers equation (see Example 26.36). 3.

∂w ∂t

=

∂ ∂x



1 ∂w w 2 ∂x



∂

–

∂x



f (x) w



.

The substitution w = 1/u leads to an equation of the form 5.6.11.1: ∂u ∂2u ∂ = u2 2 + u2 [f (x)u]. ∂t ∂x ∂x Therefor the original equation can be reduced to a second-order linear equation. 4.

∂w ∂t

=a

∂ ∂x



w

m ∂w

∂x



+ f (t)w.

The transformation w(x, t) = u(x, τ )F (t),

τ=

Z

m

F (t) dt,

F (t) = exp

Z

 f (t) dt ,

leads to a simpler equation of the form 5.1.10.8:   ∂u ∂ m ∂u =a u . ∂τ ∂x ∂x If m = –1 or m = –2, see 5.1.10.2 or 5.1.10.3 for solutions of this equation.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 313

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 314

314

SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE   ∂w ∂ m ∂w 5. =a w + f (t)w 1–m. ∂t ∂x ∂x The substitution u = wm leads to an equation of the form 5.6.10.2:   ∂u ∂2u a ∂u 2 = au 2 + + mf (t), ∂t m ∂x ∂x

which admits a generalized separable solution of the form u = ϕ(t)x2 + ψ(t)x + χ(t).   ∂w ∂ ∂w 6. =a wm + f (t)w + g(t)w 1–m. ∂t ∂x ∂x The substitution u = wm leads to an equation of the form 5.6.10.2:   ∂u ∂2u a ∂u 2 = au 2 + + mf (t)u + mg(t), ∂t m ∂x ∂x

which admits a generalized separable solution of the form u = ϕ(t)x2 + ψ(t)x + χ(t). References: V. A. Galaktionov (1995), V. F. Zaitsev and A. D. Polyanin (1996, p. 429).

  ∂w ∂ m ∂w =a w + bw 1+m + f (t)w + g(t)w 1–m. 7. ∂t ∂x ∂x For b = 0, see equation 5.6.13.6. The substitution u = wm leads to an equation of the form 5.6.10.1:   ∂u ∂2u a ∂u 2 = au 2 + + bmu2 + mf (t)u + mg(t), ∂t m ∂x ∂x which admits generalized separable solutions of the following forms: u(x, t) = ϕ(t) + ψ(t) exp(±λx), u(x, t) = ϕ(t) + ψ(t) cosh(λx + C), u(x, t) = ϕ(t) + ψ(t) sinh(λx + C), u(x, t) = ϕ(t) + ψ(t) cos(λx + C), where the functions ϕ(t) and ψ(t) are determined by systems of appropriate first-order ordinary differential equations, the parameter λ is a root of a quadratic equation, and C is an arbitrary constant. References: V. A. Galaktionov (1995), V. F. Zaitsev and A. D. Polyanin (1996, p. 429).

∂w

∂



wm

∂w



+ f (x)w 1+m. ∂t ∂x ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = C1 w(x, C1m t + C2 ), where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . Multiplicative separable solution: 8.

=a

w = (λmt + C)–1/m ϕ(x), where λ and C are arbitrary constants, and the function ϕ(x) is determined by the equation 1

′′ aψxx + (m + 1)f (x)ψ + λ(m + 1)ψ m+1 = 0, ψ = ϕm+1 . The book by Polyanin and Zaitsev (2003) presents exact solutions of this equation for various f (x).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 314

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

315

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

∂w

∂



m ∂w

Page 315



=a w + g(x)w m+1 + f (t)w. ∂t ∂x ∂x Multiplicative separable solution: 9.

w = ϕ(x)ψ(t), where the functions ϕ = ϕ(x) and ψ = ψ(t) are determined by the ordinary differential equations a(ϕm ϕ′x )′x + g(x)ϕm+1 + C1 ϕ = 0, ψt′ – f (t)ψ + C1 ψ m+1 = 0, and C1 is an arbitrary constant. The general solution of the second equation for ψ is given by –1/m  Z Z F mF ψ(t) = e C2 + mC1 e dt , F = f (t) dt, where C2 is an arbitrary constant.   ∂w ∂ ∂w m ∂w 10. =a w + f (t) + g(t)w. ∂t ∂x ∂x ∂x The transformation Z Z w(x, t) = u(z, τ )G(t), z = x + f (t) dt, τ = Gm (t) dt,

Z  G(t) = exp g(t)dt ,

leads to a simpler equation of the form 5.1.10.8:   ∂u ∂ m ∂u =a u . ∂τ ∂z ∂z

  ∂w ∂ ∂w m ∂w =a w + xf (t) + g(t)w. 11. ∂t ∂x ∂x ∂x The transformation w(t, x) = u(z, τ )G(t),

z = xF (t),

τ=

Z

F 2 (t)Gm (t) dt,

where the functions F (t) and G(t) are given by Z  Z  F (t) = exp f (t) dt , G(t) = exp g(t) dt , leads to a simpler equation of the form 5.1.10.8:   ∂u ∂ m ∂u =a u . ∂τ ∂z ∂z In the special case m = –2, this equation can be transformed to the linear heat equation (see 5.1.10.3). Reference: V. F. Zaitsev and A. D. Polyanin (1996, p. 430).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 315

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 316

316

SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE     ∂w ∂w ∂ m ∂w 12. =a w + xf (t) + g(t) + h(t)w. ∂t ∂x ∂x ∂x The transformation Z Z w(x, t) = u(z, τ )H(t), z = xF (t) + g(t)F (t) dt, τ = F 2 (t)H m (t) dt, where the functions F (t) and H(t) are given by Z  Z  F (t) = exp f (t) dt , H(t) = exp h(t) dt , leads to a simpler equation of the form 5.1.10.8:   ∂u ∂ m ∂u =a u . ∂τ ∂z ∂z If m = –1 or m = –2, see 5.1.10.2 or 5.1.10.3 for solutions of this equation. Reference: V. F. Zaitsev and A. D. Polyanin (1996, pp. 434–435).

  ∂w ∂ λw ∂w =a e + f (x, t, w) 5.6.14. Equations of the Form ∂t ∂x ∂x   ∂w ∂ λw ∂w =a e + f (t). 1. ∂t ∂x ∂x 1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = w(C1 x + C2 , t) –

2 ln |C1 |, λ

where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . The transformation w(x, t) = u(x, τ ) + F (t),

τ=

Z

  exp λF (t) dt,

F (t) =

Z

f (t) dt,

leads to a simpler equation of the form 5.2.2.1:   ∂u ∂ λu ∂u =a e . ∂τ ∂x ∂x

Reference for equation 5.6.14.1: A. D. Polyanin and V. F. Zaitsev (2002, p. 74).

2.

∂w ∂t

=a

∂ ∂x



eλw

∂w ∂x



+ f (t) + g(t)e–λw.

The substitution u = eλw leads to an equation of the form 5.6.9.2: ∂u ∂2u = au 2 + λf (t)u + λg(t). ∂t ∂x This equation admits a generalized separable solution of the form u = ϕ(t)x2 + ψ(t)x + χ(t).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 316

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

3.

∂w ∂t

=a

∂ ∂x



λw ∂w

e

∂x



Page 317

317

+ f (x) + (bx + c)e–λw .

The substitution u = eλw leads to an equation of the form 5.6.9.1: ∂u ∂2u = au 2 + λf (x)u + λ(bx + c). ∂t ∂x This equation admits a generalized separable solution of the form u = λ(bx + c)t + ϕ(x). 4.

∂w ∂t

=a

∂ ∂x



λw ∂w

e

∂x



+ beλw + f (t) + g(t)e–λw.

For b = 0, see equation 5.6.14.2. The substitution u = eλw leads to an equation of the form 5.6.10.1: ∂u ∂2u = au 2 + bu2 + λf (t)u + λg(t). ∂t ∂x This equation admits generalized separable solutions of the following forms: u(x, t) = ϕ(t) + ψ(t) exp(±µx), u(x, t) = ϕ(t) + ψ(t) cosh(µx + C), u(x, t) = ϕ(t) + ψ(t) sinh(µx + C), u(x, t) = ϕ(t) + ψ(t) cos(µx + C), where the functions ϕ(t) and ψ(t) are determined by systems of appropriate first-order ordinary differential equations, µ is a root of a quadratic equation, and C is an arbitrary constant. Reference: V. F. Zaitsev and A. D. Polyanin (1996, pp. 430–431).

5. 1◦ .

∂w ∂t

=a

∂ ∂x



eλw

∂w ∂x



+ f (x)eλw.

Suppose w(x, t) is a solution of this equation. Then the function w1 = w(x, C1 t + C2 ) +

1 ln C1 , λ

where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . Additive separable solution: w=–

1 ln(λt + C) + ϕ(x), λ

where λ and C are arbitrary constants, and the function ϕ(x) is determined by the secondorder linear ordinary differential equation ′′ aψxx + λf (x)ψ + λ = 0,

ψ = eλϕ .

Reference for equation 5.6.14.5: V. F. Zaitsev and A. D. Polyanin (1996, p. 431).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 317

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 318

318

SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE   ∂w ∂ λw ∂w 6. =a e + g(x)eλw + f (t). ∂t ∂x ∂x Additive separable solution: w = ϕ(x) + ψ(t), where the functions ϕ = ϕ(x) and ψ = ψ(t) are determined by the ordinary differential equations a(eλϕ ϕ′x )′x + g(x)eλϕ + C1 = 0, ψt′

λψ

– f (t) + C1 e

= 0,

(1) (2)

and C1 is an arbitrary constant. Equation (1) can be reduced, with the change of variable ′′ + λg(x)U + λC = 0. The general solution of equation U = eλϕ , to the linear equation aUxx 1 (2) is given by   Z Z 1 λF ψ(t) = F – ln C2 + λC1 e dt , F = f (t) dt, λ where C2 is an arbitrary constant.

5.6.15. Equations hof the Form i

  ∂w = ∂ f (w) ∂w + g x, t, w, ∂w ∂t ∂x ∂x ∂x

  ∂w 1. = f (w) . ∂t ∂x ∂x This equation is frequently encountered in nonlinear problems of heat and mass transfer (with f being the thermal diffusivity or diffusion coefficient) and the theory of flows through porous media. For f (w) = awm , see Subsection 5.1.10; for f (w) = eλw , see equation 5.2.2.1; and for f (w) = a ln w + b, see equation 5.4.2.8. ∂w

∂

1◦ . Suppose w(x, t) is a solution of the equation in question. Then the function w1 = w(C1 x + C2 , C12 t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Traveling-wave solution in implicit form: Z f (w) dw 2 k = kx + λt + C2 , λw + C1

(1)

where C1 , C2 , k, and λ are arbitrary constants. To λ = 0 there corresponds a stationary solution. 3◦ . Self-similar solution: w = w(z),

x z= √ t

(0 ≤ x < ∞),

where the function w(z) is determined by the ordinary differential equation  ′ f (w)wz′ z + 12 zwz′ = 0.

(2)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 318

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 319

319

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

Solutions of this form usually correspond to constant w in the initial and boundary conditions for the original partial differential equation (w0 , w1 = const): (initial condition), w = w0 at t = 0 w = w1 at x = 0 (boundary condition), w → w0 as x → ∞ (boundary condition). Then the boundary conditions for the ordinary differential equation (2) are as follows: w → w0 as z → ∞. (3) w = w1 at z = 0, –1 –2 2 –1 For f (w) = aw , f (w) = aw , and f (w) = (αw + βw + γ) , the general solutions of (2) were obtained by Fujita (1952); see also the book by Lykov (1967). 4◦ . We now describe a simple method for finding an f (w) such that equation (2) admits an exact solution. To this end, we integrate equation (2) with respect to z and then apply the hodograph transformation (where w is regarded as the independent variable and z as the dependent one) to obtain Z  1 ′ z dw + A , A is an arbitrary constant. (4) f (w) = – zw 2 Substituting a specific expression z = z(w) for z on the right-hand side of relation (4), one obtains a one-parameter family of functions f (w) for which z = z(w) solves equation (2). The explicit form of w = w(z) is obtained by the inversion of z = z(w). The method just outlined was devised by Philip (1960); he obtained a large number of exact solutions to the original equation for various f = f (w). Some of his results, those corresponding to a problem with the initial and boundary conditions of (3) with w0 = 0 and w1 = 1, are listed below in Table 5.1. All solutions are written out in implicit form, z = z(w), and are valid within the range of their spatial localization 0 ≤ w ≤ 1. TABLE 5.1 Solutions of equation 5.6.15.1 for various f = f (w), where z = xt–1/2 Function f = f (w)

No 1 2 3 4 5 6 7 8 9 10 11

1 2

n n n w – w2n 2 2(n + 1)   n (1 – w)n–1 – (1 – w)2n 2(n + 1) n n w–2n – w–n 2(1 – n) 2
1 sin2 12 πw 2   1 sin(πw) πw + sin(πw) 8   1 sin2 (πw) 5 + cos(πw) 16

 cos 12 πw cos 12 πw + 21 πw – 1 w arccos w + 1 1 √ – 2 2 1 – w2 π – 2(1 – w) arcsin(1 – w) 1 √ – 2 4 2w – w2 w arcsin w 1 2 √ + w 4 4 1 – w2 1 (1 2

– ln w)

Solution z = z(w)

Conditions

1 – wn

n>0

(1 – w)n

n>0

w–n – 1

0
cos

1 πw 2

cos2

1 πw 2

cos3

1 πw 2





1 – sin( 12 πw) arccos w arcsin(1 – w) √

1 – w2

– ln w

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 319

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

320

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 320

ONE SPACE VARIABLE

5◦ . There is another way to find an f (w) for which equation (2) admits exact solutions. By direct substitution, one can verify that equation (2) is satisfied by w = φ′z ,

f (w) =

s + φ – zφ′z , 2φ′′zz

(5)

where φ = φ(z) is an arbitrary function, and s is an arbitrary constant. Expressions (5) define a parametric representation of f = f (w); the explicit representation is obtained by eliminating z. For example, assuming in (5) that 1 (w0 – w1 )e–λz λ

φ(z) = w0 z +

(λ > 0, w1 > w0 ),

and eliminating z, one obtains A + B + C ln(w – w0 ), w = w0 + (w1 – w0 )e–λz , w – w0   where A = – 12 sλ–1 , B = 12 λ–2 1 + ln(w1 – w0 ) , and C = – 12 λ–2 . Note that this solution satisfies the boundary conditions of (3). Likewise, one can construct other f (w). f (w) =

6◦ . Here is one more method for constructing an f (w) for which equation (2) admits exact solutions. Suppose w ¯ = w(z) ¯ is a solution of equation (2) with ¯ = w(z) ¯ is ′ an f (w). Then w also a solution of the more complicated equation [F (w)wz′ z + 12 zwz′ = 0 with F (w) = f (w) + Ag(w)

(A is an arbitrary constant),

(6)

where the function g = g(w) is defined parametrically by g(w) =

1 , w ¯ ′z

w = w(z). ¯

(7)

For example, the function w ¯ = bz 2/m , where b is some constant, is a particular solution of equation (2) if f (w) is a power-law function, f (w) = awm . It follows from (6) and (7) m–2

that w ¯ is also a solution of equation (2) with f (w) = awm + Aw 2 . For the first solution presented in Table 5.1, the method outlined gives the following one-parameter family of functions: f (w) =

n n n w – w2n + Awn–1 , 2 2(n + 1)

for which z = 1 – wn is a solution of the equation. 7◦ . The transformation 1 ¯t = t – t0 , w(¯ , ¯ x, ¯t) = w(x, t)   Z x Z t
∂w x¯ = w(y, t) dy + f w(x0 , τ ) (x, τ ) dτ ∂x x0 t0 x=x0

(8)

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 320

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

Page 321

321

takes a nonzero solution w(x, t) of the original equation to a solution w(¯ ¯ x, ¯t) of a similar equation,     ∂ ¯ ∂w ¯ 1 1 ∂w ¯ ¯ = f (w) ¯ , f (w) = 2 f . (9) ¯ ∂t ∂ x¯ ∂ x¯ w w

In the special case of power-law dependence, f (w) = awm , transformation (8) leads to equation (9) where f¯(w) = aw–m–2 . 8◦ . The equation in question is represented in conservative form, i.e., in the form of a conservation law. Another conservation law:   ∂ ∂w ∂ (xw) + F (w) – xf (w) = 0, ∂t ∂x ∂x Z where F (w) = f (w) dw. 9◦ . For f (w) = a(w2 + b)–1 , see equation 5.1.13.2 and Subsection 30.3.2 (Example 30.9). 10◦ . For f (w) = aw2n + bwn , see equation 5.1.13.3. References for equation 5.6.15.1: L. V. Ovsiannikov (1959, 1962, 1982), V. A. Dorodnitsyn and S. R. Svirshchevskii (1983), W. Strampp (1982), J. R. Burgan, A. Munier, M. R. Feix, and E. Fijalkow (1984), J. R. King (1990 b), N. H. Ibragimov (1994, pp. 110, 118, 119), A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov (1995), V. F. Zaitsev and A. D. Polyanin (1996, pp. 435–438), P. W. Doyle and P. J. Vassiliou (1998).

  ∂w ∂ ∂w 2. = f (w) + g(w). ∂t ∂x ∂x This equation governs unsteady heat conduction in a quiescent medium in the case where the thermal diffusivity and the rate of reaction are arbitrary functions of temperature. 1◦ . Suppose w(x, t) is a solution of this equation. Then the functions w1 = w(±x + C1 , t + C2 ), where C1 and C2 are arbitrary constants, are also solutions of the equation. 2◦ . Traveling-wave solutions: w = w(z),

z = ±x + λt,

where the function w(z) is determined by the autonomous ordinary differential equation [f (w)wz′ ]′z – λwz′ + g(w) = 0.

(1)

The substitution

1 f (w)wz′ λ brings (1) to an Abel equation of the second kind: y(w) =

′ yyw – y = ϕ(w),

where

ϕ(w) = –λ–2 f (w)g(w).

(2)

The book by Polyanin and Zaitsev (2003) presents a considerable number of solutions to equation (2) for various ϕ = ϕ(w).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 321

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

322

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 322

ONE SPACE VARIABLE

3◦ . Let the function f = f (w) be arbitrary and let g = g(w) be defined by g(w) =

A + B, f (w)

where A and B are some numbers. In this case, there is a functional separable solution, which is defined implicitly by Z

f (w) dw = At –

1 Bx2 + C1 x + C2 , 2

where C1 and C2 are arbitrary constants. 4◦ . Let now g = g(w) be arbitrary and let f = f (w) be defined by A1 A2 w + B A2 A3 f (w) = + g(w) g(w) Z dw , Z = –A2 g(w)

Z

Z dw,

(3) (4)

where A1 , A2 , and A3 are some numbers. Then there are generalized traveling-wave solutions of the form w = w(Z),

Z= √

A1 A2 ±x + C2 – – (2A3 t + C1 ), 2A3 t + C1 A3 3A3

where the function w(Z) is determined by the inversion of (4), and C1 and C2 are arbitrary constants. 5◦ . Let g = g(w) be arbitrary and let f = f (w) be defined by     Z Z 1 dw A1 w + A3 Z dw exp –A4 , g(w) g(w)   Z dw A2 1 – Z= exp –A4 , A4 g(w) A4

f (w) =

(5) (6)

where A1 , A2 , A3 , and A4 are some numbers (A4 ≠ 0). In this case, there are generalized traveling-wave solutions of the form w = w(Z),

Z = ϕ(t)x + ψ(t),

where the function w(Z) is determined by the inversion of (6), 

2A4 t

ϕ(t) = ± C1 e

A3 – A4

–1/2

,

 Z Z ψ(t) = –ϕ(t) A1 ϕ(t) dt + A2

 dt + C2 , ϕ(t)

and C1 and C2 are arbitrary constants.

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 322

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

Page 323

323

6◦ . Let f = f (w) be arbitrary and let g = g(w) be defined by  Z  a g(w) = +b f (w) dw + c , f (w) where a, b, and c are arbitrary constants. In this case, there are generalized separable solutions defined implicitly by Z f (w) dw + c = eat z(x), where

p
p
if b < 0, z(x) = C1 exp x |b| + C2 exp –x |b| √
√
z(x) = C1 cos x b + C2 sin x b if b > 0, if b = 0,

z(x) = C1 x + C2

and C1 and C2 are arbitrary constants. 7◦ . Let f (w) and g(w) be as follows: f (w) =

wϕ′w (w),

  ϕ(w) g(w) = a w + 2 ′ , ϕw (w)

where ϕ(w) is an arbitrary function and a is any number. Then there are functional separable solutions defined implicitly by ϕ(w) = C1 e2at – 12 a(x + C2 )2 , where C1 and C2 are arbitrary constants. 8◦ . Let f (w) and g(w) be as follows: f (w) = A

V (z) , Vz′ (z)

  g(w) = B 2z –1/2 Vz′ (z) + z –3/2 V (z) ,

where A and B are arbitrary constants (AB ≠ 0), V (z) is an arbitrary function, and the function z = z(w) is determined implicitly by the formula Z w = z –1/2 Vz′ (z) dz + C1 ; (7) C1 is an arbitrary constant. Then there exists a functional separable solution of the form (7) where (x + C3 )2 BC2 z=– + 2Bt + , 4At + C2 2A and C2 and C3 are arbitrary constants. 9◦ . Group classification of solutions to the equation in question was carried out by Dorodnitsyn (1979, 1982); see also Dorodnitsyn and Svirshchevskii (1983), Galaktionov, Dorodnitsyn, Elenin, Kurdyumov, and Samarskii (1986), and Ibragimov (1994). As a result, only a limited number of equations were extracted that possess symmetries other than translations. This means that symmetry analysis does not allow one to obtain the solutions given in Items 3◦–8◦ above (for equations with functional arbitrariness).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 323

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

324

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 324

ONE SPACE VARIABLE

10◦ . If f = dF (w)/dw and g = aF (w) + bw + c, where F (w) is an arbitrary function, and a, b, and c are arbitrary constants, then there is a conservation law  –bt     e p(x)w t + e–bt p(x)x F (w) – p(x)(F (w))x + ϕ(x) x = 0. Here,

 √ √  C1 sin( ax) + C√2 cos( ax) √ p(x) = C1 e –ax + C2 e– –ax  C1 x + C2 ′ where ϕx = cp(x); C1 and C2 are arbitrary constants.

if a > 0, if a < 0, if a = 0,

11◦ . For specific equations of this form, see Subsections 5.1.1 to 5.1.3, 5.1.11 to 5.1.13, 5.2.1 to 5.2.3, and 5.4.1. References for equation 5.6.15.2: V. A. Dorodnitsyn (1979), V. A. Galaktionov (1994), A. D. Polyanin and V. F. Zaitsev (2004, pp. 107–108), A. D. Polyanin and E. A. Vyazmina (2005).

  ∂w g(t) 3. = f (w) + + h(x). ∂t ∂x ∂x f (w) Functional separable solution in implicit form: Z Z Z x f (w) dw = g(t) dt – (x – ξ)h(ξ) dξ + C1 x + C2 , ∂w

∂

x0

where C1 and C2 are arbitrary constants, and x0 is any number.   ∂w ∂ ∂w ∂w 4. = f (w) – (ax + b) . ∂t ∂x ∂x ∂x This equation governs unsteady heat and mass transfer in an inhomogeneous fluid flow in the cases where the thermal diffusivity is arbitrarily dependent on temperature. 1◦ . Suppose w(x, t) is a solution of the equation in question. Then the function w1 = w(x + C1 eat , t + C2 ), where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . Generalized traveling-wave solution: w = w(z),

z = x + C1 eat ,

where the function w(z) is determined by the ordinary differential equation [f (w)wz′ ]′z – (az + b)wz′ = 0. 3◦ . Functional separable solution in implicit form: Z f (w) dw = C1 e–at (ax + b) + C2 .

4◦ . On passing from t, x to the new variables  
1 b –2at –at 1–e , τ= , ζ =e x+ 2a a

one obtains a simpler equation of the form 5.6.15.1 for w(ζ, τ ):   ∂w ∂ ∂w = f (w) . ∂τ ∂ζ ∂ζ

Reference for equation 5.6.15.4: A. D. Polyanin and V. F. Zaitsev (2004, p. 109).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 324

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

325

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

  ∂w ∂w 5. = f (w) + g(w) . ∂t ∂x ∂x ∂x 1◦ . Traveling-wave solution in implicit form: Z f (w) dw 2 k = kx + λt + C2 , λw – kG(w) + C1 ∂w

Page 325

∂

G(w) =

where C1 , C2 , k, and λ are arbitrary constants. 2◦ . The transformation   dz = w dx + f (w)wx + G(w) dt,

dτ = dt,

u = 1/w

Z

g(w) dw,

dz = zx dx + zt dt

leads to an equation of the similar form   ∂ ∂u ∂u ∂u = Φ(u) + Ψ(u) , ∂τ ∂z ∂z ∂z

where 1 Φ(u) = 2 f u

  1 , u

    1 1 1 Ψ(u) = g –G , u u u

G(w) =

Z




g(w) dw.

Special case. For f (w) = a and g(w) = bw, the original equation is an unnormalized Burgers equation 5.1.5.3. The above transformation brings it to the solvable equation   ∂u ∂ a ∂u b ∂u = + , ∂τ ∂z u2 ∂z 2u2 ∂z which was studied by Fokas and Yortsos (1982).

3◦ . Let f (w) and g(w) be defined as   Z ′ f (w) = Zw A1 w + A3 Z dw ,

g(w) = A2 + A4 Z,

where Z = Z(w)

(1)

is a prescribed function (chosen arbitrarily). Then the original equation has the following generalized traveling-wave solution: Z w = w(Z), Z = ϕ(t)x + (A2 t + C1 )ϕ(t) + A1 ϕ(t) ϕ(t) dt,

where C1 is an arbitrary constant, the function w(Z) is determined by the inversion of (1), and the function ϕ(t) is determined by the first-order separable ordinary differential equation ϕ′t = A3 ϕ3 + A4 ϕ2 ,

(2)

whose general solution can be written out in implicit form. In special cases, solutions of equation (2) are given by ϕ(t) = (C2 – 2A3 t)–1/2 –1

ϕ(t) = (C2 – A4 t) 4◦ .

if A4 = 0, if A3 = 0.

Conservation law: 

 Dt (w) + Dx –f (w)wx – G(w) = 0,

G(w) =

Z

g(w) dw.

Reference for equation 5.6.15.5: A. D. Polyanin and V. F. Zaitsev (2004, pp. 109–110).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 325

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 326

326

SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE   ∂w ∂ ∂w ∂w 6. = f (w) + g(t) . ∂t ∂x ∂x ∂x This equation governs unsteady heat conduction in a moving medium in the case where the thermal diffusivity is arbitrarily dependent on temperature.Z On passing from t, x to the new variables t, z = x + g(t) dt, one obtains a simpler

equation of the form 5.6.15.1:

  ∂w ∂ ∂w = f (w) . ∂t ∂z ∂z

  ∂w ∂w f (w) + xg(t) . ∂t ∂x ∂x ∂x On passing from t, x to the new variables (A and B are arbitrary constants) Z  Z 2 τ = G (t) dt + A, z = xG(t), where G(t) = B exp g(t) dt , 7.

∂w

=

∂

one obtains a simpler equation of the form 5.6.15.1 for w(τ , z):   ∂w ∂ ∂w = f (w) . ∂τ ∂z ∂z     ∂w ∂w ∂ ∂w 8. = f (w) + xg(t) + h(t) . ∂t ∂x ∂x ∂x The transformation Z Z w = U (z, τ ), z = xG(t) + h(t)G(t) dt, τ = G2 (t) dt,

Z  G(t) = exp g(t) dt ,

leads to a simpler equation of the form 5.6.15.1:   ∂U ∂ ∂U = f (U ) . ∂τ ∂z ∂z Reference: V. F. Zaitsev and A. D. Polyanin (1996, p. 441).

  ∂w ∂w 9. = f (w) + g(w) + h(w). ∂t ∂x ∂x ∂x For g ≡ const, this equation governs unsteady heat conduction in a medium moving at a constant velocity in the case where the thermal diffusivity and the reaction rate are arbitrary functions of temperature. Traveling-wave solution: ∂w

∂

w = w(z),

z = x + λt,

where the function w(z) is determined by the autonomous ordinary differential equation [f (w)wz′ ]′z + [g(w) – λ]wz′ + h(w) = 0.

(1)

The substitution y(w) = f (w)wz′ brings (1) to the Abel equation ′ yyw + [g(w) – λ]y + f (w)h(w) = 0.

(2)

The books by Polyanin and Zaitsev (1995, 2003) present a large number of exact solutions to equation (2) for various f (w), g(w), and h(w).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 326

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

10.

Page 327

327

    ∂w ∂w = f (w) + ax + g(w) + h(w). ∂t ∂x ∂x ∂x

∂w

∂

1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = w(x + C1 e–at , t + C2 ), where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . Generalized traveling-wave solution: z = x + C1 e–at ,

w = w(z),

where the function w(z) is determined by the ordinary differential equation [f (w)wz′ ]′z + [az + g(w)]wz′ + h(w) = 0. Reference for equation 5.6.15.10: A. D. Polyanin and V. F. Zaitsev (2004, p. 111). 2 5.6.16. Equations of the Form ∂w = f (x, w) ∂ w2

∂t

∂w

∂x

∂ 2w

= f (x)w m . ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the function

1.

w1 = C1 w(x, C1m t + C2 ), where C1 and C2 are arbitrary constants, is also a solution of the equation. 2◦ . Multiplicative separable solution: w(x, t) = (mλt + C)–1/m ϕ(x), where C and λ are arbitrary constants, and the function ϕ = ϕ(x) is determined by the generalized Emden–Fowler equation  –1 ϕ′′xx + λ f (x) ϕ1–m = 0.

(1)

For m = 1, a solution of equation (1) is given by Z x (x – ξ) ϕ(x) = –λ dξ + Ax + B, x0 f (ξ) where A, B, and x0 are arbitrary constants. The books by Polyanin and Zaitsev (1995, 2003) present a large number of solutions to equation (1) for various f (x). 3◦ . The transformation u = w/x, ξ = 1/x leads to an equation of the similar form ∂u ∂2u = F (ξ)um 2 , ∂t ∂ξ

F (ξ) = ξ 4–m f (1/ξ).

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 327

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

328

SECOND-ORDER PARABOLIC EQUATIONS

WITH

Page 328

ONE SPACE VARIABLE

∂w f (x) ∂ 2 w . = ∂t aw + b ∂x2 Generalized separable solution linear in t:  1 ϕ(x)t + ψ(x) – b , w(x, t) = a where the functions ϕ(x) and ψ(x) are determined by the system of ordinary differential equations f (x)ϕ′′xx – ϕ2 = 0, ′′ – ϕψ = 0. f (x)ψxx The first equation can be treated independently from the second. The second equation has a particular solution ψ(x) = ϕ(x), so its general solution is given by Z dx ψ(x) = C1 ϕ(x) + C2 ϕ(x) , ϕ2 (x) where C1 and C2 are arbitrary constants. 2.

3.

∂w

= f (w)

∂2w

. ∂t ∂x2 1◦ . Suppose w(x, t) is a solution of this equation. Then the function w1 = w(C1 x + C2 , C12 t + C3 ), where C1 , C2 , and C3 are arbitrary constants, is also a solution of the equation. 2◦ . Traveling-wave solution in implicit form: Z Z dw dw 2 k = kx + λt + C2 , F (w) = , λF (w) + C1 f (w) where C1 , C2 , k, and λ are arbitrary constants. 3◦ . Self-similar solution: x + C1 w = U (z), z = √ , C2 t + C3 where C1 , C2 , and C3 are arbitrary constants, and the function U (z) is determined by the ordinary differential equation ′′ + 12 C2 zUz′ = 0. f (U )Uzz dw 4◦ . The substitution u = leads to an equation of the form 5.6.15.1: f (w)   ∂u ∂ ∂u = F (u) , ∂t ∂x ∂x where the function F is defined parametrically as Z dw F (u) = f (w), u = . f (w) To obtain F = F (u) in explicit form, one should eliminate w from the two relations. 5◦ . Conservation laws: Dt (u) + Dx (–wx ) = 0, Dt (xu) + Dx (w – xwx ) = 0,

Z

where Dt = in Item 4◦ .

∂ ∂t , Dx

=

∂ ∂x ;

wx is the partial derivative of w with respect to x; and u is defined

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

Page 328

C8723: A.D.Polyanin and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, Second Edition, 9781420087239, Chapman & Hall/CRC, 2011 (v.3, Nov 2011)

329

5.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS

4.

∂w = x4 f ∂t



w x



Page 329

∂ 2w . ∂x2

The transformation u = w/x, ξ = 1/x leads to a simpler equation of the form 5.6.16.3: ∂u ∂2u = f (u) 2 . ∂t ∂ξ 



∂2w . √ 2 ∂t ax2 + bx + c ∂x With the transformation p w(x, t) = u(z, t) ax2 + bx + c, 5.

∂w

4

=w f

w

Z

z=

ax2

dx + bx + c

one arrives at the simpler equation

∂u ∂2u = u4 f (u) 2 + (ac – 14 b2 )u5 f (u), ∂t ∂z which has a traveling-wave solution u = u(z + λt). Reference: V. F. Zaitsev and A. D. Polyanin (1996, p. 432).

5.6.17. Equations of the Form

  ∂w = f (x, t, w) ∂ 2 w + g x, t, w, ∂w ∂t ∂x ∂x2

  ∂w ∂ 2w ∂w 2 1. = f (t) +w – aw 3 . ∂t ∂x2 ∂x 1◦ . Multiplicative separable solutions for a > 0:   Z √ w(x, t) = C exp ±x a + a f (t) dt , where C is an arbitrary constant. 2◦ . Multiplicative separable solution for a > 0: √ x a

w(x, t) = C1 e

√
–x a F

+ C2 e

e