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CONTENT Foreword
i
Preface
ii
CHAPTERS 1. Oscillation of partial difference equations with deviating arguments.
1
Patricia J. Y. Wong 2. Functional-analysis and partial difference equations.
49
Eugenia N. Petropoulou and Panayiotis D. Siafarikas 3. Partial difference equations and their application in system theory.
77
Jiri Gregor and Josef Hekrdla 4. Numerical schemes and difference equations.
111
Efstratios E. Tzirtzilakis and Nilkolaos G. Kafousalas Index
141
DEDICATION
In memory of Professor Panayiotis D. Siafarikas
i
Foreword
While “Mathematical Reviews” currently lists 1058 books containing “Partial Differential Equations” in their title and 128 books containing “Difference Equations” in their title, it only lists 3 books containing “Partial Difference Equations” in their title. On the other hand, 238 journal publications are listed containing “Partial Difference Equations” in their title, so research in this area is rather active and ongoing. This is due to the rich possibility of theoretical investigations and the numerous applications which partial difference equations enjoy. These facts illustrate that there is an urgent need to expand the availability of textbooks in the area of “Partial Difference Equations”. The book at hand, “Some Recent Advances in Partial Difference Equations”, as edited and presented by Professor Eugenia Petropoulou, is a welcome, timely, and excellent contribution filling the above described gap. Professor Petropoulou has done a terrific job in putting together this volume, offering four chapters on distinct topics of current interest in the area of partial difference equations. The first chapter covers oscillation theory of partial difference equations and is written by Professor Patricia Wong (Singapore), a world-wide leading expert in the area of differential, difference, and dynamic equations and in particular oscillation theory for these equations. Criteria for the nonexistence of positive solutions of certain partial difference equations with deviating arguments are presented and several examples are offered. The second chapter shows a connection between functional analysis and partial difference equations and is written by the late Professor Panayiotis Siafarikas (Greece), the internationally esteemed expert in this area of research, together with his student Professor Eugenia Petropoulou (Greece), who is the editor of this volume. A functional-analytic method to study partial difference equations is developed and illustrated by two fundamental examples. The third chapter discusses the connection of partial difference equations to systems theory and is written by Professor Jiˇr´ı Gregor and Professor Josef Hekrdla (Czech Republic). Existence and uniqueness results for initial value problems and boundary value problems involving linear partial difference equations are presented and extended to systems of linear partial difference equations. These results are applied to input–output relations of linear multidimensional systems. The fourth chapter offers some numerical schemes constituting partial difference equations and is written by Professor Efstratios Tzirtzilakis and Professor Nikolaos Kafoussias (Greece). Partial differential equations are discretized in order to obtain numerical schemes resulting in partial difference equations, and the connection of the solutions of these two equations is examined analytically and numerically. Of course these covered topics only scratch the surface of this exciting area of research. We look forward to future developments inspired by the publication of this volume. Martin Bohner, Rolla, Missouri (USA) October 20, 2010
ii
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Patricia J. Y. Wang
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10 2 F 102 F
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4 Some Recent Advances in Partial Difference Equations
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Some Recent Advances in Partial Difference Equations 5
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6 Some Recent Advances in Partial Difference Equations
Patricia J. Y. Wang
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Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 7
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8 Some Recent Advances in Partial Difference Equations
Patricia J. Y. Wang
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Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 9
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10 Some Recent Advances in Partial Difference Equations
B
Patricia J. Y. Wang
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Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 11
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12 Some Recent Advances in Partial Difference Equations
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Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 13
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14 Some Recent Advances in Partial Difference Equations
Patricia J. Y. Wang
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Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 15
1
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16 Some Recent Advances in Partial Difference Equations
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Patricia J. Y. Wang
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Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 17
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18 Some Recent Advances in Partial Difference Equations
Patricia J. Y. Wang
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Some Recent Advances in Partial Difference Equations 19
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20 Some Recent Advances in Partial Difference Equations
Patricia J. Y. Wang
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Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 21
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22 Some Recent Advances in Partial Difference Equations
1 E E 2 1 2 E
F 1 E E 2 E
Patricia J. Y. Wang
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Oscillation of Partial Differences Equations
$
Some Recent Advances in Partial Difference Equations 23
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24 Some Recent Advances in Partial Difference Equations
"
1 E 2
Patricia J. Y. Wang
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Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 25
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26 Some Recent Advances in Partial Difference Equations
Patricia J. Y. Wang
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Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 27
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28 Some Recent Advances in Partial Difference Equations
1 E 2 E 1 E 2 1 E E 2 1 2
Patricia J. Y. Wang
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1 42
Some Recent Advances in Partial Difference Equations 29
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21
1
21
2
1 33<2
1 E 2 1 3382 1 2
* 1 33;2 * 1 2 + 1 36 2 2
1
1
K 1 3382 1 36 2
E 2 1 2
2
1 E 1 2 F 1 2
1 2 ( " D 1 2 A * 12 F 1 2 * * 12 F * F +
E
1 36 2
! " $ * 12 + " " 1 36 2 1
+
2
1 362 1 2 +
" 1 362 1 342 " " "
1 E 2 1 2
1
21
2
1
2
+
1
2
1 33<2
30 Some Recent Advances in Partial Difference Equations
+
+
1
Patricia J. Y. Wang
2
1 2
2 1
2
1
1 E 2 1 2
1
2
1
2
1 36 2
1 2 + # " D
1 3382
1 362
+
" F + " 1
1
2
#"$
1 3642
2
& 3 " $ 1
2 1 2
1
" 1 3642 34 & F
2
1 E 2
3
1 2 + 1 3632
" + F 1 E 2 ( ( 3
( ( 3 " D & 3 & 3 33 &
1 2 F -
1
2E
( ( 3
-
3
1 3662 1 3652
Oscillation of Partial Differences Equations
1 42
G
Some Recent Advances in Partial Difference Equations 31
& 1 2 $ $ ( 1 3 2 " D
G 1 2E 1E E 2
1
" $
1 E 2E 1 E 2 1 2E
21
2E1 2
2 1 2 1 36;2
1
1 36;2 $ $ 1
2
2 1
2
B1 2 E C 1
2
1
1
2 1 2
" $" 1 3662 $ 1
2
F
-
.
1- .2 E 1 2
" . 1 - 2 ! 1 36<2 1 E 2
1
E 2
1- .2 E 1 2
1- .2 E 1 2
1 36<2
1- .2 E 1 2
1- .2 E 1 2
" 1 3 2 1 3682 1 35 2 " 1 42
1 3682
1 35 2
B 1 1 12 1222 1 1 12 1222C G G 1 2
32 Some Recent Advances in Partial Difference Equations
F
21
1
21
2 E 1 E 2 E 1 E 2 1 2 1 E E 2
1
1
1
Patricia J. Y. Wang
2E
2E
2 E 1 E 2 E 1 E 2 1 2 1 2
1- .2 E
1
2
1- .2 E 3 1 2
( " $
1
2E
1- .2 E 3
" 1 3652 36 & ( " 1 3662 $ 1 3652 D 1 $ $2 # 1 3662 ( 3 "
35 1 2 F
1 E 2
1 35 2 1 352
" F F ( ( 3 1 42 $ $ 1 2 =D 1 2 1 2 / 1 2 F 1 E E 2 ( 1
E 2 E 1 E 2 1 E E 2 E 1 E E 2 F 1 2 1 2 / 1 2 1 3542
Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 33
" $ 1 3 2 1 2
1
E 2 E 1 E 2 1 2 1 2 1 2 / 1 2 1 2
F
1
1
2
1
2 1
2 1
2
/
2
! $" 1 3542 $ $ 1
2
/
1 2
1
/
2
1 3532
F " 1 3532 $
/1 2
1
2
" 1 352
A / 1 2 " 1 3532 " = / F / 1 2 1 2 " 1 3532
/
"
/
/
* 12 F 1 2 $D
* 12 F *
/
E F 1 E 2
1 3562
1 3552
34 Some Recent Advances in Partial Difference Equations
( 1 3562
Patricia J. Y. Wang
1 E 2
" 1 352 3; 1 35 2
0
1 35;2
10 E 2
" 0 F F ( ( 3 0 1 2 $ $ 1 42 0 ( 35 " 1 3562 " " ) " "
/ /
/
/
F
/
/
! $" 1 3552 $
10 E 2 0
" 1 35;2 3< 1 35 2
1 E 2
1 35<2
" F F ( ( 3 + 1 42 $ $ ' 1 2 0 ( 35 " $ 1 3562 / 1 3562
/ /
Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 35
" $" 1 3552 $
1 E 2
# " 1 35<2
( * " 3 " " D D " * " " $ D 6 @ 1 21 E 42 1 E E 42 1 6 2 1 2 F 1 2 # F F 4 F F 4 @ 12 F " $ F A 1 1 E E 422 1 21 E 42 F 1 1 E E 422 1 2 1 1 E E 422 F 1 1 E E 422 " ) 1 21 E 42 1 2 F 1 2 F 1 2 F 1 2 F 1 2 ( 10 2 ' 1042 D
@ @ ( 1 82
11 E 2 21 E E 42 F 8F4 4 1 E 21 E 2 4 " 1F ; 652
36 Some Recent Advances in Partial Difference Equations
@ @ 3 9 D
1 2 F 3
Patricia J. Y. Wang
1 E 2
E F
4 E 3
1 4 2 D @ 4 @ 5 9 $ 3 1 21 E 42 F E F4 1 2 F 1 2 1 E 2 E F < 8 1 E E 2 # 1 452 D @ 3 ( 4 # F F 3 F ( 1 42 5 " @ 6 ( 44 9 1 482
1 21 E 42 1 2 1 21 E 42
F
1
2
F 4 E F 6
# " @ 3 5 ( 4 44 1 6 2 ! 1 6 2 $ 1 2 F 1 2
6 @ 1 21 E 21 21 E 32 1 E E 32 1 2 F 1 21 2
1 62
Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 37
( ) 12 F " $ F A "
1 21 E 21 21 E 32 1 2 F 1 2 F 1 2 F 1 2 F 1 21 2
@ 10 2 ' 1042 D ! ) @ 3 5 ( 4 44 D ( " 1 62 ! 1 62 $ 1 2 F 1 2
64 @ 1 2 F
1 E 2 1 E E 2E
1 E 2 1 E E 2 " 1 E 2 1 E 2 $ # F @ 12 F 12 F " $ F F & 2 F 1 2 F
1
1 E 2
1
2 F 1 2 F
1 642
1 E 2
1 2 F 1 2 F F ( " $ F F ! " $ 1 42 @ 1 482
¼
F
¼
B 1 2 1 2C
1 E
¼
¼
2 E 1 E 2
F 31 E 21 E 2 E # ( 4 44 1 642 ' ! 1 642 $ 1 2 F 1 2
38 Some Recent Advances in Partial Difference Equations
Patricia J. Y. Wang
63 @ 1 E 2 1 E E 2E 1 2 F
1 E 2 1 E E 2 1 632 " 1 E 2 1 E 2 $ 12 F 12 F 1 E 2 1 E 2 1 2 F 1 2 F 1 2 F 1 2 F 1 2 F 1 2 F F " ) ( 4 44 D ( 1 632 ! 1 632 $ 1 2 F 1 2 E
66 @ 1 2 F
1 21 E 21 E 2 1 E E 2 1 2
1 21 E 21 E 2 1 E E 2 1 21 2 " 1 E 2 1 E 2 $ 9 ) 12 F 12 F 1 21 E 21 E 2 1 2 F 1 2 F 1 2 1 21 E 21 E 2 1 2 F 1 2 F 1 21 2 1 2 F 1 2 F F ! " F F @ 1 42 D A 1 482 E
¼
¼
1 662
B 1 2 1 2C F 31 E 21 E 2 E
Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 39
9 ( 4 44 1 662 ! 1 662 $ 1 2 F 1 2
65 @ 1 E 2 1 E 2 1 2 F 1 21 2 B 1 E E 2C
1 652
" 1 42 ( ) 12 F " 2 F 1 2 F
1
1 E 2 1 E 2 1 21 2
2 F 1 2 F
1
( B 1 2 1 2C F ! ) ' @ ' 5 ( 4 44 D # 1 652 ! 1 652 $ ' $ 1 2 $ 1 2 F
6; @ 1 2 F 1 E 21 E 2 1 E E 2
1 6;2
& 12 F 1
2 F 1 2 F
1 E 21 E 2
1
2 F 1 2 F
B 1 2 1 2C F 1042 $ ! 1 6;2 $ $ $ 1 2 F 1 E 121 E 2 " 1
6< @ 1 E 2 1 2 G G 1 2 E 1 E 2 F
1 E 2 1 2 E
E
1 6<2
40 Some Recent Advances in Partial Difference Equations
Patricia J. Y. Wang
" $ I$ # F 12 F 12 F @ 12 F " $ F A 1 E 2 1 1 12 1222 F 1 1 12 1222 1 E 2 1 E 2 1 1 12 1222 F 1 1 12 1222 E " ) 1 E 2 1 E 2 1 2 F 1 2 F 1 2 F 1 2 F 1 E 2 E ( 1@ 2 1@42 D ! E 1 2 F 1 2 1 2 F E E
3 E F 3
@ ( 3 @ 1 3 82
E
E
(
3 EE 1 E 2 1 682
E E # 1 6<2 $ $ $ $ 1 682 *
1 2 F 14 2 1 2 ! I$ " D 1 6<2 $ 1 2 F 1 2
@ ( 3 4 9 F F " D 1 2 ( F 1 342 D ( ( 3 12 " @ 4 ( 3 3 9 $
1
2 F 31 E 21 E 2
Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 41
1 362 D # ( 3 12 I$ @ 3 ( 3 6
E 1 2 F E E
E E E
E F E
E
" 1 3 2 " F 9 F 3 1 3;2 31 E 21 E 2 6 " $ ( " @ 4 @ 6 ( 3 5 A @ 3 $ 1 3 2 " F ( 1 34 2 D " @ 4 @ 5 ( 3 ; & F (
1 E 2 1 2 F
E
1 3 42 " F 9 F 3
1
2 F 31 E 2
( 1 34 2 D ( 3 12 $ @ ; ( 3 < & F
1 E 2 1 2 F E
42 Some Recent Advances in Partial Difference Equations
Patricia J. Y. Wang
" D 1 3 62 " F 9 F 3 $ 1 3432 D ( ( 3 12 $ @ < ( 3 8 & F A @ 5 1 3 42 F ! " 1 3462 D 31 E 21 2 $ # " @ 5 @ 8 ( 3 & F A @ ; 1 3 62 F ( 1 3452 D " @ ; @ ( 3 ( 1 34;2 3 + 1 6 2 ( ( 3 12 D 1 6 2 1 2 F 1 32 13 2 1 62 16 2 @ ( 3 & F ( 1 33;2 3 " # " @ 5 @ ( 34 & F 0 1 3632 3 " ( " @ ; @ 4 ( 33 # " $ - F 4 1 3652 1 2 E - F 3 E 142 F 3 F 3 F
( " @ 4 @ 3 ( 35 3< 9 $ F 3 1 352 1 E 2½ 1 6 2 ½ 3 " $ ! ) 1 35;2 1 35<2 1 6 2 ( " @ 4
Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 43
68 @ 1 E 21 E 21 21 2 G G 1 2 F 1 2 61 E 21 E 2 41 E 21 E 21 21 2 E 1 2 61 E 21 E 2 E E E E 1 6 2 " F $ I$ ! * F 12 F 12 F F ( )
12 F 12 F " $ F F A " 1 E 21 E 21 21 2 1 2 F 1 2 F 61 E 21 E 2 41 E 21 E 21 21 2 1 2 F 1 2 F 61 E 21 E 2 1 2 F 1 2 F 1 2 F 1 2 F 1@ 2 1@42 D
1 2 F 6 E E
1
2F
1
2
/
1
/
E E E E 2
(
4 4 / 1 2 / 1 E E E E 2 (
@ ( 3 1 6 2 $ $ 1 $2
1 2 1 3 82 D 0 * 1 3 82 ' D " F F 4 F 3 F ! I$ F 1 6 2 $ 1 2 F 1 2
@ ( 3 9 D 1 34;2 1 6 2 " D * F F 3 F 4 F 6
44 Some Recent Advances in Partial Difference Equations
Patricia J. Y. Wang
@ 4 ( 3 4 3 3 33 35 3< ! ) $ D ( ( 3 12 I$
F
6 @
G G 1 2 F 1 E 2B1 2 CB1 2 C 1B1 2 C B1 2 C2
E E
1 6 42
" BC $ # F 12 F B1 2 C 12 F B1 2 C @ 12 F " ) 1
2 F 1 2 F
1
2 F 1 2 F 1 E 2B1 2 CB1 2 C
( 1 2 ! ) ( 3 3< D # 1 6 42 $ $ $ $ ! " 1 6 42 $ $
$ 1 2 F 6 @ G G 1 2 F 1 2B1 2 C 1 B1 2 C2 3 4 E B1 2 C1 2 1B1 2 C 2 3
E E E E
1 6 32
" BC F $ @ 12 F 12 F B1 2 C 12 F B1
2 C 12 F 9 ) 12 F 12 F 1 2 F 1 2 F 1 2 F 1 2 F 1 2B1 2 C 3 4 1 2 F 1 2 F B1 2 C1 2 1 2 F 1 2 F 3
Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 45
1 2 F 0 ( 3 33 35 3< ' D # " * 6 ! 1 6 32 $ $ $ 1 2 F
)
6 @ 1 21 2 1 2 G G 1 2 F 41 E 21 E 2 1 21 2 E 1 2 41 E 21 E 2
E E E E
1 6 62
" F $ # 12 F 12 F F A " 12 F 12 F 1 21 2 1 2 F 1 2 F 1 2 F 1 2 F 41 E 21 E 2 1 21 2 1 2 F 1 2 F 1 2 F 1 2 F 41 E 21 E 2 1 2 F ! ) ( 3 33 35 3< $ # " 1 6 62 $ $ 1 $2 ' 1 2 ! 1 6 62 $ $ $ $ 1 2 F
B C 7 0 "
. =)) 88 BC 7 0 " = 0 $
! " # $ %&&'( )
( " * 8 <
B4C 7 0 " . . . ( + $ + + 36 885
46 Some Recent Advances in Partial Difference Equations
Patricia J. Y. Wang
B3C 7 0 " ( + $ $ $ # @ ( . 0 6 > 0 885
B6C 7 0 " ( L M 9 + $ +
; ;< 88; B5C @ 9 ( 0 * ' # $ 84 4 884 B;C ! >N > & , -
$ @ +* 88 B
O 0@ . $ / -
$ 0 " M) 88 B8C P & O > & *0 1 # 2 , -
$ O " = 884 B C P & )) = ( -
$
# $ 0 " M) 8<< B C L L & @ @ M * $ # $ ; <3 883 B C L ( + $ <; ;8 883 B 4C 0 )$) M & . ) M 7 ) O " 0 884 B 3C ( + + $ <4 8 4 5 883 B 6C L M 9 7 0 " + * $ + + 54 <3 886 B 5C L M 9 7 0 " + 3 + $ $ 3< 4 <8 885 B ;C L M 9 7 0 " + ! 4 4 4838 6 ; 885
Oscillation of Partial Differences Equations
Some Recent Advances in Partial Difference Equations 47
B
B 8C L M 9 7 0 " $ + + 3 46 885 B C L M 9 7 0 " @ " $ + + 348 3< 885 B C L M 9 7 0 " + ' $ + $ 45 5 88< BC @ & Q > R S $ 12 + $ $ 3 + 556 ;8 886 B4C @ & Q > R S $ ' 12 A 3 + 5 ;; 8 886 B3C @ & Q > R S $ 1$2 A 3 + 544; 45 886 B6C @ & Q > R S $ 1$2 3 + ;<8 8; 885 B5C @ > R S $ ' 12 + 3 + 6;8 << 883 B;C @ A & M ( + + $ 45 84 88< B
48 Some Recent Advances in Partial Difference Equations
Patricia J. Y. Wang
B4 C L M 9 7 0 " + " + $ 46; <5 885
B4 C L M 9 7 0 " * 3 + $ $ 36 4 64 88; B4C L M 9 $ $ + $ 4646 6< 88< B44C L M 9 7 0 " + $ $ + 2 ;8 8 88< B43C L M 9 7 0 " 0 $ " $
$ + 8; 48 53 88< B46C L M 9 0 $ ' $ 6 78$ /!! &9 88< B45C > R ( & + $ + 3 65 ; 886 B4;C > R ( & + " $ Æ + $ 4546 3 88< B4 R ( & @ + 3 $ 6 5 886 B48C L @ )" !
5
9 " ) 8<8
49
Some Recent Advances in Partial Difference Equations, 2010, 49-76
CHAPTER 2
' !# ½
3# & 4 ¾ 3 ' & ( 3
- 5
5 ( * ( .+,, ( 6# 0 1 2 # # ¾ 3 5 ( 3
- - ( *
( .+,, ( 6# 0 2 # # ½
# $ % ! & ! At $ ( ! ! "#$% "#&$' "#&(( )*+## ,
1G 2 $ I $ ( " G " ½
Eugenia N. Petropoulou (Ed) All rights reserved - © 2010 Bentham Science Publishers Ltd.
Functional-analysis and Partial Difference
Some Recent Advances in Partial Difference Equations 50
1= 2 ' $ $ $ " = $ / 1" ) & A = 2 1 * B 6C2 " % = $ 1 2 $ 1 * B5
51 Some Recent Advances in Partial Difference Equations
Petropoulou and Siafarikas
, G " * $ 12 * * 142 * 132 * $ 1 $2 162 * 152 $ 1;2 " 1<2 $ 182 1 2 $ + $ $$ D* ! $ G " $
1 2
F
F
1 2
1 2
"
"
1 2
1 2
1 2
1 2
" V 1 2 10
% G $ $ B C2 ! $ $ D* G " $ ! 3 $ " 6 5 " * .$ 4 $ $ " $ G D 1 2 1 2 $
!
! $ " $ G " A D 1 2 ) ' " " " ) D*
Functional-analysis and Partial Difference
Some Recent Advances in Partial Difference Equations 52
+ T U " * ( $ $ $ A * " 1 )' 2 1 $ 2 0 %
G + G " ) " $ " T U 1 * ) 2 0 " 1 2 #"$ $ '
$ D D D " % ' A T U 0 ") ' B C & 7 B 4C " D " 0 " " 46 $ 0 " $ T0' A 0 ( 7 U !.0 P 0 8;3 " " ' " $ = % ! " " D* $ G ( " $ G " " 3 $ ( $ * G D 1 2 1 2 $ ! " 1 2 0 $ $ G 1 152 2 .$ $ " D " " 1 1;2 2 A " $ G " $ $ 1$ 2 " ( 142 1<2 1A $ 32
53 Some Recent Advances in Partial Difference Equations
Petropoulou and Siafarikas
( G $ # 12 2 12 2 " 2 2 ! " G $ " G + " G ' $ $ * $ ! ") R MR B6C 1 2 $ D* O ( ) D* $ $ $ $ $ G ! D* ") D " D " $ D* 0 B 3C $ * I $ A B 6 8C D $ D* O ' ?) D* B C ! " * $ $ ' $
" # $ 0 " 3 " G D 1 2 1 2 $ ( $ ) G $ " 2 6
7
½ ( ¾ # ! $ 1 2 1 2
Functional-analysis and Partial Difference
F
Some Recent Advances in Partial Difference Equations 54
1 "
" 1
&
&
F
2
3 "3"
3 2 "3"
" 4 " " 0 $ 1 2 1 2 1 2 F G $ 2 8
# ! $ # D $ 3 3 1 23 3 ( $
" " D A ?
% Æ 1 2 * A #? " 1 2 B4 ;5C
42
# 2 ! B3C " = G " $ A G 1 2 ( $
( +$ ( F 2 + $ = $ $ ( $ $ ' $ ( ' $ " " A Æ 1 2 "
32 !
3' # 2 ! = $ G ( 13 '2
55 Some Recent Advances in Partial Difference Equations
Petropoulou and Siafarikas
= 1 2 G 3 F 3 ' F ' " 5 E 6 E 0 .1 2 ! .1 2 F
14 2
! G G " 1 2 ( 1 .1 22 $ 14 2 D 2 ! B8C & * = " $ " % = 0 B8C " 1 2 ¾ F 1 2 62 6
9 : # ( # 1/ / 2 1 2 D 1
# / /
2F
/ /
1 2
( & / 7 1 1 22 1 2 ' D 7 1 1 22 F 8 1+ + 2 F 1 2+ +
/ " " * $ ' 1 * B C2 ( 1 2 $ $ / / + +
52 ;
# ! ) = $ 1 * 3 2 1 * 9 9 2 9 $ $ % $ 13 'H 9 9 2 " % $ D* A *
Functional-analysis and Partial Difference
Some Recent Advances in Partial Difference Equations 56
$ 13 'H 9 9 2 F 19 9 2 " ' 19 9 2
$ " ) " 1( 2 ( * 19 9 2 19 9 2 1
9 9
2F
1 2 1
2
9 9
142
" 1 2 Æ B6C ( *
19 92 142 $ 1 2 ;2
# ! $ D " <2 < # ( . %
F
1 2
1 2 F
! " " %
F
1 2
1 2 F
" $ 1 2 0 $ G G $$ ' * 1 162 2 ")
57 Some Recent Advances in Partial Difference Equations
Petropoulou and Siafarikas
%
! $ G " ( % $ ! " D B5C * 1 E 2 E 1 2 F B :12C 12
#"$ " 8<; " B;C ( B< 4 C " * ) B4 C " * ( % G " $ " $ B4C G B44C G " % G $ " $ B C 7 " $ B43C @ D 1 2 ( " " " " * " 1 1 2 1 22 F
1 21 2
1 2 1 2
13 2
# ( 13 2
1 2
¾ F
.$
1 2
: : #
F
1 2
"
12
" : F 3 : F 3 3 1 * * ; 38 B46C2 ! " $ # ! " D 1 2 " " " " *
Functional-analysis and Partial Difference
"
1 2
½
Some Recent Advances in Partial Difference Equations 58
F
1 2
1 2
132
& 2 # " ; 2 " $ F 1 ; 2; " F
1 ; 2
0 2
2 " 2 F
1 ; 2
1 ; 2 E (
=D " 2 ) ) " ) ;
F ; ) ; F ;
F
+ $ ) ) ) F ) F 2 ! ) 1 2 F 2 ; F 4
< )
) 1 2 F 2 ; F 4
< ) ;
1A * ; <1) 2 ! <1) 2 * 2 ) F ; ( )
1 ; 2; F ;
1 ; 2; F ;
1 ;2; E 1 ;2; E E 1 ; 2; E E
E1 ; 2; E 1 ; 2; E E 1 ; 2; E E F ; ) $ " ; F -2 $ ) ) " % "
59 Some Recent Advances in Partial Difference Equations
Petropoulou and Siafarikas
3 ( , ) ) ) ) D
) ; F ; ) ; F ;
F 4 F
) ;
F F 4
) ;
F
F
F
F
A $ 2 1) ; 2 F " F 1 ) ; 2 F ( $ 2 " F ) ; " $ ) ; F ) ; F F ! " $ ) ; F F 0
)
F 1 ; 2; E 1 ; 2; E E 1 ; 2; E E
E1 ; 2; E 1 ; 2; E E 1 ; 2; E E F $ " ; $ 1) ; 2 F 1 ; 2 1 ) ; 2 F 1 ; 2
1 ) ; ;
2 F ) ; F ; ! " $ ) ; F ; F
3 ( " " ) ) ) ) ) ) F 5 ) ) F ) ) F 5 ) ) F
" ,$
) F ) F ) F ) F
! *-
A D , ,$ ) * B46C B45C 0 ' $ G ' 2 2 D ( " "
Functional-analysis and Partial Difference
Some Recent Advances in Partial Difference Equations 60
34 ( : 2 D
1 2 F 1 ; 2 F 1 2
1342
:
2 2 " $
2
A D 1342 "'D !
1 2
¾ F
1 2 F
1 ; 2 F E
$ : 0 : ! 2 2 " :1 2 F :1 2 1 ; 2 F F ; 2 .$ 1 2 * 2 :1 2 F 1 2 ( $ F
1 2;
2
F
1 2 F 1 2
¾
E
A : $
:1 2
F ¾
1 2 F
1 ; 2 F
33 ( : 2 D 1 2 F 1 ; 2 F 1 2
:
2 ( 34
1332
61 Some Recent Advances in Partial Difference Equations
Petropoulou and Siafarikas
( D 1342 1332 1 2 ! = 1 2 0 =1 1 22
0 2 2
1 1 22 F 1 1 2 ; 2
=
A G $ B4 C " B44C 1 " 2 ! " $ $ " " 5
36 6 9 ( 1 E E 2 1) 2 1) 2 F 1) 21) 2 1 2 ) ) F ) )
4 1 E 2 1) 2 ) F )1) 2 3 1 E 2 ) 1) 2 F 1) 2 ) 6 11 2 1 2 >
" > >; F 11 2;
! 1 E E 2 F 1 ; 2 F 1 )) ; 2 F
F 11) 2 1) 2 ; 2 F 11) 2 1) 2 ; 2 ) ) !
1
2 F 1 1) 2 1) 2 ; 2 F F 1) ) ; 2 F 1) ) ; 2
2 F 1 ;
) ) 4 !
1
2 F 1 1) 2 ) ; 2 F F 11) 2 ) ; 2 F 1) 1) 2 ; 2
E 2 F 1 ;
) )
Functional-analysis and Partial Difference
3 !
Some Recent Advances in Partial Difference Equations 62
1 E 2 F 1 ; 2 F 1 )1) 2 ; 2 F
F 1) 1) 2 ; 2 F 11) 2 ) ; 2 ) ) 6 !
1 2 1 2 F 1 21 ; 2 F 1 1 2; 2 F 1 > ; 2 F 1> ; 2
35 6# 9 ( ' 1 2 D "
1 2F
1 ; 2 ; F
B 1 2C ;
2
'
1362
B 1 2C
A D 1362 " D
1 2 F
1 1 2 ; 2 F
2 .$
1 ;
2
1 2
1 2
1 1 2 ; 2 F
B 1 2C ; ; F B 1 2C
" 1 2 B 1 2C 0 " 5 D D* D 1362 AW 2 ( $ "
3; 6!; 0
9 ( ' 1 2 ' D 1362 AW 2
63 Some Recent Advances in Partial Difference Equations
Petropoulou and Siafarikas
1 2 B 1 2C 2 33 Æ " B 1 2C AW ! AW $ $ B 1 2C 1 2
1 1 22 1 2 F 4 F 1 2 1 2
4
( 4 $ ( 4 " 4 ½ F 1 2 1 2 ½ 1 2 ½ 1 2 ½
" 4 AW $ $ B 1 2C 1 2
B 1 2 E 1 2C B 1 2C 1 2 1 2 F B 1 2C 1 2 B 1 2 E 1 2C B1 1 2 2C 1 2 1 2 1 2
B 1 2 E 1 2C B 1 2C 1 2 1 2 F
1 2 ½
½
½
½
½
½
½
½
½
& ! 3 " "
$ 1 E E 2 F 1 1 2 E 1 2
16 2
1 2 1 2 ) " * 162 " 1 1 2 ) " * ( D $ 16 2'162 ( " $$ 16 2 0 34 36 ' ) " 1 2 2 1 E E 2 ) ) F ) ) 2 ! 2 1 2 1D 1 22 ( < ( 6 ( 1342 "
1) ) ; 2 F 11 ; 2 E 1 ; 2 1) ) ; 2 11 ; 2 1 ; 2 F
Functional-analysis and Partial Difference
Some Recent Advances in Partial Difference Equations 64
1) ) 1 ; 2 F
" ; F "
) )
2
$
1 F 1) ) 15 2 F
1642
1642 $ 16 2 2 ( ) ) ) 3 3 1642 1) 1) 2
F ) E
15 1)) 2 F )) E
;
;
E
" ;
1632
" " " 162 " ) 1632 " ; 1 ; 2 11) ) ; 2 F 1) ) ; 2 E 1; ; 2 E 1; ; 2 E E E" 1; ; 2 E " 1; ; 2 E " ; 2 " 1 ; 2 11) ) ; 2 F 1) ) ; 2 E "
1 ;2 F " 1 2 F "
! " $ ) 1632 " ; " F 1 2 " F 1 2 ! " ) 1632 " ; 1 ; 2 11) ) ; 2 F 1) ) ; 2 E 1; ; 2 E 1; ; 2 E E E" 1; ; 2 E " 1; ; 2 E " ; 2 " 1 ; 2 11) ) ; 2 F 1) ) ; 2 E
65 Some Recent Advances in Partial Difference Equations
Petropoulou and Siafarikas
1 ; 2 11) ) ;2 F 1) ) ;2 E 1 ; 2 11) ; 2 F 1) ;2 E 1 ; 2 11 ) ;2 F 1 ) ; 2 E 1 ;2 F 1 2 F
! " $ ) 1632 " ; F 1 42 F 1 E 2 ( 1632 ) 15 1) ) 2 F ) ) E
1 E 2; E
1 2;
1662
D ' 2 1662 " 2 ( 1 2 1 2 ( ( 6 1662 $ $ 16 2'162 2 " ") " 1662 & F 1) ) ( 1 0 " ) " 6 ! * # 2 ? " * 5 * $ " 15 * 2 * D 2 ? 1 * B46 ; '; C2 ( 1 5 $ D # ( ( 6 ( 1662
F 15 2
) )
E
1 E 2; E
1 2;
1652
1662 2 $ 1652 $ $ $ 16 2'162 .$ 1652 . "
15 2
) )
E
1 E 2; E
1 2
;
E ! "
Functional-analysis and Partial Difference
" F
1
;
& ''' 1 2 ' %(
Some Recent Advances in Partial Difference Equations 66
1 E 2; E
#$
;
"
F F 1 E 2
F F
F " F 1 2
F F
0
F (
F
1 E 2 E
F 1 E 2
1
E ) 1
´½µ ¾
1 2
E 1 2 ´½µ ¾
E 2 ´½µ E 1 2 ´½µ ¾
¾
= $ 1 342 " 1 2 ¾ F 1 1 2 ¾ E 1 E 2 ´½µ¾ E 1 2 ´½µ¾ A $ $ " $ 6 @ $ 16 2'162 1 2 1 2 " D 1 2 12 1 2 1 2 12 1 ( $ 16 2'162
16;2 1 2 ¾ 1 1 2 ¾ E 1 E 2 ´½µ¾ E 1 2 ´½µ¾
)
)
67 Some Recent Advances in Partial Difference Equations
Petropoulou and Siafarikas
64 1 2 1 2 1 2 F ( $ 16 2'162 $ 1%2 " 63 1 * * 11 2 ! F ) ) > " > >; F 11 2; F > F 11 2 ( 1 2 12 12 11 2 16<2
16;2 1 2 ¾ 11 2 1 2 ¾ E 1 E 2 ´½µ¾ E 1 2 ´½µ¾ 66 #"$ " 11 2 ? 16<2: ! 11 2 11 2 F 1682
)
" $ 16<2 ! 1662 15 2 F ) ) E
1 E 2; E
1 2;
16 2
> 1 16822 ) ) ( A $ 16 2 0 A $ 1 * B46 34C2 16 2 2
15 2 F 15 ) ) >2 F 16 2 $ 2 ( 16 2
F
1 ; 2; F
B1 ; 2; E 1 ; 2; E C
16 2
Æ 1 ; 2 16 2 " " " ( 6 . ) 16 2 "
Functional-analysis and Partial Difference
Some Recent Advances in Partial Difference Equations 68
; 1 ; 2 1) ) > ; 2 F 1 ; 2 F F
16 42
; 1 ; 2 1) ) > ; 2 F 1 ; 2 1) > ; 2 F
11 22 F 1 2 F F 11 22 F 1 21 2 F F 1 2 F F 1 2 F F ; ;
; ;
; ;
1
> ;
1
2 F
;
;
16 32
; 1 ; 2 1) ) > ; 2 F 1 ; 2 1) > ; 2 F
11 22 F 1 2 F F 11 22 F 1 21 2 F F 1 2 F F 1 2 F F ; ;
;
;
;
> ;
1
1 ; 2 F
;
;
16 62
" 1 ; 2 F F " 16 2 16 2 F 0 16 2 2 ( 1 2 12 ( 6 12 1682 16;2
1
) 2 E 1
E 2 ´½µ E 1 2 ´½µ ¾ ¾ 16 52 ! $ 1682 16<2 16 52 16;2 15 2 ) " D
1 2 15 2 ¾
¾
69 Some Recent Advances in Partial Difference Equations
Petropoulou and Siafarikas
0 * @ 1 $ 16 ;2 1 E E 2 F 1 2 5 1 2 F 1 2 F 16 <2 4 " 16 2'162 ( 1 2 F ( 4 5 1 2 F F 4 4 F $ 2
F 45
F 45
3
8
8
1 2 F 4
1 2 F 6 3;3
3
¾
" 6 " ) $ 16 ;2' 16 <2 " " TU $ 0 1 2 12 $ D 1 2 1F 0 12 D 5 8 F 8 F 8 F
1 2 F 4 8 8 < 3 F3 F3 F 1 2 F 3 4 3 ( 6 $ 16 ;2'16 <2
1 2 ¾ 5 1 E 2 ´½µ¾ E 1 2 ´½µ¾ 1 2 ¾ 65 < E 34 338 3 " " T U $
)
*
Functional-analysis and Partial Difference
Some Recent Advances in Partial Difference Equations 70
' & (
! 3 " "
$ 1 E E 2 11 2 1 2 F 1 2 E B 1 2C 15 2 1 2 1 2 ) " * 152 " 11 2 1 2 ) " * ( D $ 15 2'152 0 6 15 2'152 2 15 2 F ) ) E
1 E 2; E
1 2; E ) ) 1 2 1542
" F ) ) > > >; F 11 2; F 1 2 D 1362 ! 1 -2 $ 5 * 2 $ & 1542 )
F 15 2
B)
) E
1 E 2; E
1 2; E ) ) 1 2C F :1 2
1532 " $ D* 1" 2 $ 1532 2 = 1 2 AW D* ! B;C " D* # B4;C 5 ! @ @ AW $ $ * 1@ 2 @ D* @ " @ 1 @ @ @ * . 3 . 3 @ @ 2 ! 5 2 ? 1 <2 F 2 < < #"$ D* 1532 7 1532 ? 1 <2 ( <
: 1 2 & B E
1 E 2; E
1 2; E 1 2 C
+
,
71 Some Recent Advances in Partial Difference Equations
Petropoulou and Siafarikas
:1 2 & E E : 1 2 & 1 E 2 E &< 1562 " D 6 =D 1<2 F < &< " * F 3& < F ( < . < " & & 1 E 2 .
3& 1 E 2
1552
1562 $ :1 2 . E &< F < &< . E &< F < . < " 5 1532 1552 1532 2 < A $ $ " $ $ 5 @ $ 15 2'152 1 2 1 2 " D 1 2 12 1 2 1 2 12 11 2 F ( * & 3&
1
E 1 2 ´½µ 2 ½ E 1 E 2 ´½µ ½ ½
15;2
$ 15 2'152 < F & 54 @ $ 15 2'152 1 2 1 2 " D 1 2 12 1 2 1 2 12 11 2 ( 3
1
2 ½ E 1 E 2 ´½µ E 1 2 ´½µ ½ ½
1 11 2 2 15<2
$ 15 2'152 11 2 < F
Functional-analysis and Partial Difference
Some Recent Advances in Partial Difference Equations 72
53 @ $ 15 2'152 " '
11 2 * 1 1 2 1 2 " D 1 2 12 1 2 1 2 12 1
3 1
(
2 ½ E 1 E 2 ´½µ E 1 2 ´½µ ½ ½
1 1 2
1582
$ 1 < F 56 7 1552 15;2 15<2 1582 G ( D " * 0 4 % D " % G " $
) * $ " = $
0 1 2
' 1 2 12 " ' $ " T0 $ $ D* U 1 " 2
1 2 O @ N% T0 $ $ D* U L . 0 0 P 44; 6 3 < 12 @ N% T0 ' U 0 . P 4; 48; 44 ; 8
7
3 ' 1+G 2 " " 1 2 1+= 2 " " $ ! 1!;<2 1!<; 2 1!<;2
73 Some Recent Advances in Partial Difference Equations
Petropoulou and Siafarikas
$ += ! " $ $ " " $ * += 1 1( 2 1(22 ! ' = 1!;<2 O ! T0 * ' ' U L = P 8 <5 3 8;< 1!<; 2 O ! T0 U L . 0 0 P 3 448 4< 8<; 1!<;2 O ! T> " $ U L . 0 0 P 3 4< 3 8<; 1( 2 = ) (%% ) T0 T' % U += U L . 0 0 P 44 ;8 85 ; 1(2 = ) (%% ) T0 T' % U += !!U A 0 + P 4 1 6'52 5 4 54 8
B C 7 0 "
( . =)) 88 BC @ $ 4 $ + $ ( X A 4 B4C 9 > O 0 @ $ -
0 88 B3C 7 .) P 7 @ 88 B6C R M R 7
$ $ 3 + / $ # " @ " M) K0 ; B5C 7 .) # <
9 D 884
Functional-analysis and Partial Difference
Some Recent Advances in Partial Difference Equations 74
B;C O 9 . A ' /+$ 1 6$ ! $ 9 132 < 8;3 B O 5 $ 6 09 3 B8C = & * $ + + ;12;3 <3 856
B C 0 )$) M & . ) M 7 ) O " 0 884 B C = ) 0 ' $ 344; 3< 3 B C ! A /+$ 1 6$ ! $ 9 1328< 4 8;3 B 4C & 7 0 ' /+$ 1 6$ ! $ 9 132 5 ; 8;3 B 3C @ 7 . $ 3 + 4 12 4 46 B 6C LM 9 ( $ , $ + $ 3 64; 66; B 5C LM 9 7 0 " * $ " $ 3 $ 66 ; 63 888 B ;C LM 9 7 0 " . " + + 4588 ;6 B
75 Some Recent Advances in Partial Difference Equations
Petropoulou and Siafarikas
B C # 0 A* $ 5/$+ <5 ; 8 8;5
B C = > P & )) # 0 0 0 ' 8<< BC 7 9 &
& 7 9 . $ D* / 8 + 3 <5;4 5<< 8;8 B4C P P =$ 868 B3C & $ + 5 641 2 4 66 834 B6C 0 # / 0 9 ! 884 B5C O ! 1 E 2E 1 2 F B; :12C 12 3 + 1423 36 858 B;C O ! + $ "' " Æ W' ) 0 854 < 8<; B
Functional-analysis and Partial Difference
Some Recent Advances in Partial Difference Equations 76
B44C = ) ' ' $ + $ 65 /. $ /)9 368 6 843 4
B43C + $ 3 $ L B46C ! > > 1 )N 8< B45C O . + 2 0 D 8; B4;C @ L 7 # 0 D* *0 $ 6 5 + ( 1( %&=> 5 56
Some Recent Advances in Partial Difference Equations, 2010, 77-110
77
CHAPTER 3
" => 6 " ?4 3 5 ( ' ' ( @: ) * ( ) 4> ( .. .( @: A; 0 12 # # #:( 4 2 # # #:
&
)*+, ! +
$ $ )*+, ! - . )*+,! / 0 )*+,! 1 % *% . % $ $ $ $ ( % ! # . )*+ ! " 2 ! -, , . /0/1 Eugenia N. Petropoulou (Ed) All rights reserved - © 2010 Bentham Science Publishers Ltd.
78 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
* ( . 0 ! # 0 0 ( ! $ 0 % $ ( . 3$ 0 ! # $ $ ! - $ % 2 $
! / $ $ 4 5 06 7 8 ! ) 7 . 8! 9 ! & . . ! 1 ! & . ! - . )*+, 2 -, $ 1 : ;$ <%) ! # ( ! + 0 $ !! $ 3 3 % $ 3 !
% $
% % ! 1 ! 1 $ % ! # = ! # 0 $
0 ! - $ 7 8! " % % $ ! & $ :=;$ ! & )*+, ( $ ! & 0 .
0 " $ + ! -
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 79
$ 2 ! 9 )*+ ! " $ ! 1 )*+, Æ :>;! 1 0 ! " )*+, . !! :?$ @;$ :A;
B ! 1 2 B 7- . " " ) 8 C!
! ! #
E( , $ 0 % 7 8 0 4 4 E( ( 4 $ $ $ =$ ! # ( $ ! -! >! ! F ! 1 0 ! $ 5 ! # $ !
80 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
# $ ! $ $ (! - $ ! -! >! != 1 & $ 1 $ $
! # ! C % ! 5 $ $ 4 3
+ ,-
F, 0 )*+ 4 -! >!=! & 3 >!=! $ =$ $ , ! & / =! # >!=!= 4 4 4 ! &
3! # 0 : ; $ >!=!= : ; >!=!> 1 4 >!=! ! & 0 . )*+, Æ ( " ! # % Æ ¾½
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 81
" 0 ( 0 ! + >!=!?
>!=!@ $ $ 3 3 $ 3 3 3 3 0 " 0 ( $ ! & >!=!=$ . 0 $ 3 4 ! & / ( $ 3 3$ 3$ ! & 0
0 ! 0$ & ! & ! # 9 >! ! 3 >!=!=
A 4 > $ ) ' ) " $ D 1 > E ? 2 " $ " $
& ( $ 0 7( 8 ! & 0! 5 = = =! & ( $ $ $
82 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
>!=!A
& Æ Æ ! #
$ ! & 0 ! # )*+ Æ >!=!= !
# )*+,$ % )*+, ! # ! &
! & )*+, ! 6 >!=!> F >!=!= 2 . 0 G 1$ D $ D 0 $ 4 4 3 = ! & $ E$ $ $ 4 ! 1 4 4 >!=!=$ =! >!=!=
0 ! 9 :H$ I;! & >!=!> )*+, >!=!=! & & >!=!> ! 9 0 $ 4
% % /! & $ ! E $ B 7 8 ! & $ $
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 83
$ 4 3 4 3! 9 /! # 0 $ / ! - $ , 0 4 ! &
$ ! & % % ! 9 $ 0 ! & & >!=!> >!=!A 7 $ $ ! & >!=!> >!=!?$ >!=!@! 3 >!=!? / + >!=!= 4 $ $ B ( B 3$ =! " $ 4 3
4 3 ! " $ 9 >!=! &
A 4 T U
$ $ $ ! & ( ( 0! 1 0 3 ( 7 8 > ( ! # ( ! 9 $ 0 7$ 4 $ 0 !
84 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
8 9 >!=!@ & >!=!> $
Æ 2 $ . ! + Æ Æ 2 $ 3! # $ + >!=!?! & ( ! 9 $ % $ . $ ! & ! #, 7 8 >!=!=$ >!=!A >!=!> ! 9 )*+, >!=!H
4 Ñ Ñ Æ $ 4 4 % 0 $ !! = ! 6 >!=!A F >!=!H . 4 Ñ Ñ = & $ E$ $ $ 4 ! 1 4 4 & / >!=!H$ =!
0 ! 1 8 9 >!=!@ Æ $ ! & $ >!=!A ! 8 9 >!=!H # Æ ( $ :J;! & ! " !&
)*+,! 3 >!=!I )*+ Æ 4 Ñ Ñ % $ 3 3 3 3 3 3 3 3 4 >!=!I 4 $ 0 3 4 # 3 >!=!I 7 0 " ! &
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 85
# 3 >!=!I ! 9 :J;! # 3 3 ! # ! # >!=!I K K K K K K K K >!=!J K K K K K$ K$ $ ! & K K K $ . $ $ $ 0! 1 K K
K
K
$ $ $ $ $ 2 $ ( ! & $ $
! ! " 3 ! 9 !! : 3;! & >!=!J $ $ $ ( ( K $ :J;! & 0 & >!=!> >!=!A $ . >!=!A $ 0 ! 1 $ Æ & >!=!>$ >!=!A &$ / 7! & 7 8 >!=! & >!=!>$ &$ / ( $ >!=! $ $ 7 8! 1
4 # >!=!
! # $ Æ ! # :H;$ 0 &$ / 7$ >!=!= $
$ >!=!A!
86 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
?B
# . 7 8 !
. ! / $ )*+, 0 ! & $ !! #= $ = 4 # $ ( 3 3 3 3 ( ( 9 ! >!?! C $ 7 8 ( ! 3 >!=!J F % 4 $ 9 ! >!>! 9 $ L
1$ D$ L 1$ D$ L ! A
X B
C
A 44 #*
9 L% L 1% 1 D% D % 3 >!=! 3 F 1 L$ D L$ L $ & 9 ! >!> $ ! 5 , &" ' # $ ' 4 # $ ' 3 1 0 ' ! 1 &" # $ ' $ # $ $ ' 3 L' # $ '
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 87
F 0 ( 4 &" (L L ' # $ ' >!=!
!! #= $ = ' 3 >!=! = =$ $ $=$ 3$ ! C $ >!=! 3 )*+! & B ( 4 &" ! & $
$ ! " &" # $ ' >!=! $ ! " (&" # $ ' >!=! =$ B $ ) ( ! & 9 ! >!? ! & ( 9 (&"!
A 43 #* "
C $ >!=! 3 )*+ ! F * &" L ! & ) L 0 (1 (L L 3 $ (D (L (L $ ( (L (L 3 M >!=! 3! F 0 Æ Æ ) $ % % Æ ) $ $ Æ ! % % % % 3 >!=! > (&" & )*+ Æ $ ( 3 3 3 3 0 7 8 4 #= $ = ' 3 9 ! >!?! ' $ >!=! $ >!=! = ! # !! >!=! = #= 3 $3 ' 3 >!=! ? ( (
88 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
(K (K&" $ 9 ! >!@!
A 46 #* " !Y
# (K&" $ (K 4 &"
B (K1 (KL (KL 3$ (KD (KL (KL $ (K (KL (KL M B Æ )K $ % % Æ )K )*+ % % % >!=! @ % 3 )K (K! 5 ( 3 3 3 & >!=! > >!=! @ >!=! 3! & % % Æ ( % Æ (K Æ ( Æ (K! % % Æ ( Æ )K Æ ( Æ )K ( Æ )K = = ( Æ )K = = ! " >!=! > >!=! @ & >!=!> >!=! 3! + >!=!J )*+,! )*+, ! D )*+, ( ! !
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 89
>!=!= ! & # ! 9 0 &$ $ $ $ ! # & 0 $ 0 ! 9 $ ! & !
*
# >!=!= &$
$ 0 ( $ = $ 4 /$ Æ 2 $ !! $ 3 ! 9 $ $ $ 3! M 9 0 7 8 &
0! -! >!=! 3 & 3 & . >!=!= & 4 3! 8 9 >!=! #, *0 >!=! 3 . 0 2 4 3$ Æ $ & ! ! -! >!=! = ! / ! 9 F 4 & ! & 0 4 & 0 >!=!= 0 ( & ! # $ $ & 0 $ 7 0 8 3 >!=! A 3 + " 3$ 4 + 0 ! =! 7 4 ! F $ $ $ 4 & & ! F , 4 & &! # $ , &
90 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
& ! & 2 0 4 & 0 >!=!= &$ , >!=! H : & 0 >!=! A! 8 9 >!=! > # #$ ( # # $ # Ú % $ # Ú $ $ # 3 $ 3
# $ " 3 & >!=! H 2 $ : ;!
>!=! ? # $ >!=! A ( >!=! I $ !! &! 8 9
8 9 >!=! @ & 2 0 ( 3 & ! 9 : ;! 6 >!=! A # >!=! J " ¼ $ & $ $ >!=!=$ 2 >!=! H $ $ >!=! A 0 1! & $ ! -! >!=! H ! & 9 .
& $ & $ $ 3 3 ! =! & & >!=!= . & (% & >!=! J 0$ 0 &
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 91
& 3 , ! 6 >!=! I # >!=!=3 ¼ $ & $ 1 -% $ >!=!=$ 2 >!=! H $ $ >!=! A 0 ! F $ 0 0 ! 3 >!=! J F & H = > = = = = > > = / ? 3 3$ $ 3$ ! F Æ >!=!= >$ 3! & : H $ & 3 3 3 3 = 3 > 3 ? 3 ? 9 >!A! # 9 $ $
A 45 7 $
: ( ( $ ( ! # ( >!=! H ! F 0 , 4 & 4 & ! & , 0$ , ! 3 3 3 3 = > ? 3 ? 3$ 3 ? 3! & 4 3 3 $ 3 ?
92 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
>$ 3 = > 3 3$ ? 3! # 7 8 2 0 ( $ : 7 8 ( ! # $ ! & . ! & & >!=! A$ >!=! I! & 0 & >!=! J$ -% 0 = =$ $ =$ > (% 3 ?$ !
- 2 )*+,! -! >!=!=3 F Æ 0 $ 4 ! & $ >!=!= ! 4 ! Æ >!=!=
Æ 4 3 7 8 $ 0 4 Æ 3 Æ ! ! 6 >!=!= # Æ >!=!= >!=!=3 0 2 3 $ 0 ! 3 >!=!== ! # Æ $ ! 9 0 $ ! ! 2 & . >!=!= >!=!==! & / 0 3 ! & $ (% 0 $ -% ! $
2 & >!=! I!
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 93
# Æ $ 0 >!=!= >!=!== . 2 ! & ! 0! " : ; ! # >!=!= Æ $ 3 $ 7 8 ! ! $ ! Æ >!=!=>
/ ( $ : =;$ : >;$
/ ! >!=!=? =. 0/ 1/ 2 .. % $ / 2 % $ / / / 0 4 2 0 / % >!=!=@ # " >!=!=A ¼ 0/ 3 2 $ $ >!=!=? $ 0 >!=!=> 0 >!=!==$ >!=!=H ! 3
>!=!=I ¼ ! # 0/ 3 2 >!=!=? >!=!=H 0! # >!=!=@ 2
0! " 0 : >;!
'B
# & >!=! A$ >!=! I >!=!= >!=! H!
94 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
# & 0 $ 0
% ! M & >!=! A & >!=! I$ 2 ! & $ %
! # & 0 $ ! 9 : ;! -! >!=!== F Æ 0 $ 4 ! 9 $ 0
3 >!=!= & 0 ( &$ 3$ 3 &$ & 4 3 # >!=!= Æ $ $ & ! 6 >!=!=> + F
0 ! F >!=!= 1 2 >!=! H & ; 1 0 < >!=! A 0 ; & >!=! J >!=!=3
0! " 0 2 >!=!= ! 3 & &! M $ >!=!= 1 >!=! H $ ; 1 0 <$ >!=! A! ) $ : ;!
. ,-/
D ! 9 Æ >!=!= $ & ! 9 )*+, Æ
: ?;! & : @; ! " !& 2 !
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 95
-
# ! = =! 9 Æ 3 >! & B 4 $
?! & 7 0 @! & N >!>! $ 4 N A! 1 4 >!=!= >!=!A! # $ & >!=!> 1 = 0 ! & 1 =
>!=!= >!=!A $ >%@! # $ !! 4 $ 0 ! ! & ! # 0 4 & 3 $ 3 4 & ' & ' & ' &
5 ! ! $ &' & ' & : A; ! 6 >!>! ;6 3 < >!=!= 3 1 % A! # Æ >!=!= >!>!=
4 >!=!= 0 / / ( "
96 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
+ >!=!= >!>!= >!>!> 6 9 >!>!> "
2
N $ " >!>!? 9 0 " >!>!@ & >!>!@ &$ ! # >!>!? N F " & $ "$ "$ " "! 9 $ (. >!>!? $ "! " $ >!>!@ ! & >!>!@ ) ! " ) ! " " ! " 1 ?! 8 9 >!>!= & >!>!= Æ >!=!= . ! & ( ! # 7 4 8 4 0 87$ 8 4 0 >!=!= Æ K 7 # !! 7 #( & >!>! 4 6 >!>!> >!=!= 3 1 . % A! # Æ >!=!= >!>!A # #
¼
¼
¼
¼
4 >!=!= 0 / #% # ( / ( " #% #% #% $ # ½
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 97
1 4 >!=!= 8#( ! & 8 # 3! & 8 4 K 8 3 Æ K # ! D >!>!A Æ K 0 >!>!=! 9 8 4 & >!>! 8 ! 8 "$
! & )*+,! 6 >!>!? 9 4 $ : 4
9 9 >!>!H 2
! : >!>!I 1 %A! # Æ ! 4 $ 3 ! 3 9 : $ 3 9 : 55 2 D $ 9 3 9 " : ! " 3 9 : 9 3 9 " : ( F 9 3! & 9 3$ 9 3! " 9 3 $ 9 3! & >!>! ! !! >!>!H$ 9 3 ! 3 9 ! %! F 9 " : & 9 : 9 : " 9 : $ 9 :! & >!>! . 3 9 ! : 9 $ >!>!H$ >!>!I 9 : ! 5$ 9 : ! M $ . >!=!=! :
¼
¼
¼
¼
¼
¼
¼
¼
¼
¼
98 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
6 >!>!@ >!=!= 1
% A! *0
# ;
!
! 04 ) ! 2 ; ! ; ; " 2
;
! ) ! "
9 >!=!A 2 ! ; ! " $ 2 ;
) ! " ) ! " ) ! "!! & ) ! ) "! 2 ! ; ) ! " ! ! & !! " 0 ! 3 >!>!A E " + >!=!= " ! D >!=!? >!=!@ 3 3$ >!=!? Æ Æ $ & >!>!= & >!>! Æ ! & >!=!@ Æ $ 9 . >!>!= 0 $ ! 1 8 9 >!>!=$ ( 8 4 8 7 (! " 0 ( ! F + >!=!= O 8 4! 9 4 8 8 1 8 8 " " & Æ & >!>! $ $ 8 " 8 D ( O 2
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 99
3 3 %! " ( ! 1 B # " 3 '
# 8 >!>!= 0! 1 >!=!@ 8 K 8 K 8
8 " "$ K * K :* ; # K K * :* ; $ ' 6 0 # $
* * # 0 # > @ > ! " $ > @ ? 3 3 ½
½
½ ¾
& (
1 $ 0 ! " # " # 0 ! + " # $ ! & ! " # ! 4 " # " #! & ( % $ ! 0 0 $ 0
! * P% $ $ 9 $ !
$ $ % $ !! & Q % $ ! / $ ! 5 0 $
0 ! 5 $ 7 8 2
100 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
% % ! 9 % % ! ' " + 0 4 +
" 4 1 + + ! " # % ! 4 4 ! 1 ! 0 Æ ! ! & 2 $ $ ! # $ $ # ( !! P% 0 7 % 8 $ !! " ! $ 4 3 4 + % ( ! & " 2 1 0 $ $ ! 0 0 8 $
2 0 8 # ! # 4
$ !! ! & 2 ! C % ! 2 % ! & >!?!
! >!?! 3$ ! ! ! ! & 2 % ! )*+, $ ! & 0 4 -! >!?! F
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 101
3 + ! ! + 4 ! 9 + 0 3 9 4 + ! ! $
3 + ! +
4 +
3 6
5 $ 3+ + 4+ + ! # $ !! 4 3 9 + 7 8 4 4 3 ! & ! -! >!?!=
1 ! " #+ . . + ! 3 3 ! 4 + 4 + ! ! & . $ & ! + %! %! & ! ! +
!
!
%!
!
'
!
%!
' !
+ ! Æ 3 Æ! $ !Æ 4 + 4 + Æ! $ % % $ ! 3+
1 ! "# % %
9
" 3+ " + % ! & 0
$ ! & 9 >!H !
102 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
A 4; ( $
9 ! ! ! 0 < >!=!= % % Æ. ! & Æ ! 3 >!?!> F 4 3 Æ 3
3 3 3 3 $ 4 3 3! # 0 . 0 4 + Æ 3 . Æ ! * !! : H; Æ ! 6 >!?!? # >!?! 7 8
% ! # 0 2 %* 0 >!?! ! 1 % $ 0 ( 0 2 2 3 C ! & 0 0 4 1 - 9% 0
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 103
! # C 9% 9% 0 9% C 0 0 ! -! >!?!@ F 9% & 0 ! 4 4 & % 4 & 0 %
( , * 4 & $ !! % $ 1 & ! # 6 >!?!A F >!?! 72 8$ !! 3 # 9% $ & >!=!> 4 Æ >!?!=
3 4 & $ >!?! 4 " >!?! ! 5 & Æ 3 & 0 >!?! ! # . $ $ $ ! 9 $ D#D' $ $ $ P% $ $ P% ! D#D' 0 4 1 >!?! D#D' - !! 1 ! 6 >!?!H 1 D#D' $ !! (!
104 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
9 4 9 $
% $ % B 0! & $ ! D 4 E ! $ ! 9 ! C !! % F 9 ! & 2 % $ 0$ ! " ( %* $ ! # & >!=!A
% 0 )*+, Æ ! # Æ %
Æ ! 1 =%*
! & 0 % . )*+, )*+, 4 3 3 !! 0 ! ' % $ 4 ! # 9 %- % $ . 1 # $ =! & / , 2 3 - 3 -
- 5 1 - : $ : 1 - 5 !
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 105
>! 1 9 %- , . 0 3 & =%* 4 Æ :@;! # ! # 4 3 = 3 R 2 1 " = Æ ! & 4 3 3 1 = ! $ ! )*+
% $ ! ' : I;$ : J;$ :@;$ :?;! # =%* %* D#D' ! # 0 $ ! - $ & >!=!> $ >!=!= 0 0
:=3;$ :>;$ :A;! + >!=!= 0 0 >!=!= 0 0! & 0 0 >!=!= ! # 0 5 0 $ 5 ! C 1 ! / ! -! >!?!I & 0 >!=!= !!! 2 3 4 >!=!= 0 2 & 0 >!=!= !!! 2 3
4 >!=!= 0 2 # K K 0 0 ) ( $ ! 2 " ! 2 ! !
106 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
" ! 2 ! % % 0 2 0 ! C $ ! " $ ! & Æ $ ! >!?! Æ 08 9 8 8 8 8 8 8 0 >!?! ! & 0 $ !! S $ !! 08 % >!?! ! = >!?!J F ! 3 ! * 8 #/#. & !
! 2 & ! F 8 8 1 ! 8 & 8 ! ! !
2
½
¾
½
" ! 0 2 ! & ! & F >!?!J 708
!! 8 #/#. 1 ! & ! C F >!?!J $ !! & ! 0 Æ 3 # $ $ $ ! 1 !! 3
8 " 2 (! 5 ! 5 F >!?!J %*
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 107
%* $ ( %* $ ! 3 >!?! 3 = 3 3 4 4 3 1 & >!>! $ ! ! 3 C $ 2
% ! C
$ 0 & >!>! $ & >!>!@ 0! 9 D#D' 0
% >!=!A $ # $ !! & % # 4 6 >!?! F >!=!= 0 Æ 3 # " >!=!= !!! 5 % ! Æ ! - . 0 $ $ ! C Æ >!?!= ! / $ . ( " !1! * $ 2 := ;! & ! # 0 D#D' Æ >!?! $ $ ! & 0% 0 ! M 0 $ 0 )*+ ( 3 3 0 8 8 8 8 8 8 1 : I;!
¼
108 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
>!?! = & D#D' ! 8 8 3 8 8 =! 8 8 3 8 " %* )*+,$ 2 ! ( !
6
0
) ! # . 0 $ $
$ ! #
2 % $ . & ! )*+, )*+, ! & . ! 1 %* % )*+, ! 5
%* % 7 8! &
! ( (! & )*+, Æ ! $
% !
Partial Difference Equations and Their Application
Some Recent Advances in Partial Difference Equations 109
: ; /!)! 1 ! -, 6 ! - *(( #!$ JJ=! :=; T! 5( ! # % ,! + 9$ @ 4?AJ%?I3$ JIH! :>; M! ' ! - ! & & & $ =34 % H@$ JJ3! :?; C! =>4==?%=>J$ JIH! :I; T! E ! & ! & & $ @>4=?H%=A>$ JJI! :J; T! E ! " ! 2 $ ?4AH%I=$ JJ>! : 3; 9! 1! E ! 6 3! C U (4 $ J@J! : ; T! R ! D ! 2 $ H4 >% >?$ JJA! : =; ! D $ 5
! > & $ 4?3>%? @$ JIJ! : >; T! R ! 9 ! ? & $ I> 4@ %@J$ =33>! : ?; 9! E! D ! 1 Æ ! > & & $ =HA4H3=%H==$ =33=! : @; T! R ! " % ! 2 $ ?4>AJ%>I@$ JJH! : A; T! E ! & ! & $ H 4IA%JJ$ =33A! : H; T! E ! ! $ ?4=3@%= A$ JJ !
110 Some Recent Advances in Partial Difference Equations
Gregor and Hekrdla
: I; D! &! ', &! "! 5 ! " % . 0!
6.- - 2 0 6 : / $ JI ! : J; *! E ! " % % 0! 0 6 $ =?4=3 %=3H$ JHH! :=3; -! *! - U! & (
! 7 ! 1 )$ C U ($ JH@! := ; C!
Some Recent Advances in Partial Difference Equations, 2010, 111-140
111
CHAPTER 4
½ ! " #¾ $%& %! '%(! ! !%%! ! )% *%%+ %(! , ! !% '% ! + -. '% ! + "%%% % /! &( ! ¾ $%& %! '(% + 0%! && % ! + 1!% + .23 + "%%% %, ! / ( & ½
#
! 1 $ ( 0 2 9 =90 3 . 9 3 ! & 2 $ $ ! &
! & . $ ( ! & . $ $ ! , , Eugenia N. Petropoulou (Ed) All rights reserved - © 2010 Bentham Science Publishers Ltd.
112 Some Recent Advances in Partial Difference Equations
Tzirtzilakis and Kafoussias
& - ) $ , 9$ ! & )*+ . 9 * $ 0 0 $ S ! 1 $ . 4 >9 >9 & F > > 3 $ >9 & * >9 >) > >9 >9 & >) > &
$ ! & 0! &
0 $ Æ ( ! ' $ . B ! # $ $ $
! 1 . Æ ! ' $ $ 4%%!% %5! % % %5!$ ! * . 0 ! % (% % ! 1 ( 4 9 + - 9+- " - "- 9 R - 9R-
Numerical Schemes and Difference Equations
Some Recent Advances in Partial Difference Equations 113
9 * - 9*- 1 $ $ 0 . . ! " $ $ 0 . $
! 9 4 6!% % %! '%( !! $ 0$ ( 9! & 0 9! # 9
0 ! C 2 ! 1 ( Æ ! & 0 ! ' 0 % 0 % ! - $ $ ! D Æ 0 % $
$ ( Æ! & 0 %
2$ % ! 0&% '%( ( 9 ! M ( 0 0 $ ! 1 ( ! & Æ 9
0 ( ! & ! 6!% 7 % '%( 0 . ( ! 7 % '%( 7'8!
114 Some Recent Advances in Partial Difference Equations
Tzirtzilakis and Kafoussias
6!% $4%%!% '%( )*+,! # $
4 & ! & 0 2 ! 9 * # $ !!$ Æ96Æ Æ $ #0 ! &
B $ $ ( 0 0 0 $ ! 9 * ! # 0 ! & Æ 0 % ! & Æ 4%%!% %5, ! (% % % %5! ! # $ 0 Æ ( . %! # % ! $% %($ $ Æ$ %%
2 ! # 2 ! & . 4 $ ! !%! $ 2 ! 0 . ! #
! !% %!% $ 2 2 2 ! Æ 9 : %5 %!% (%%
Numerical Schemes and Difference Equations
Some Recent Advances in Partial Difference Equations 115
Æ ! 5 $ % ! #
$ Æ !
! " #
& 0 $ . $ . $ 0! & ; 4%%! ! & 0 . ! & %! $ . ( 0 ! 3 ) @
F 9V )$ 9 . ! & 0 . 9 9V 0 0 . Æ Æ) 2 ! 1 9 9V $ % $ $ ( % ! # Æ $ 0 2 9V ) 9 ) ( ! 9 $
! - ) & ;
& 0 % ! # ! # $ 0 9 ! & 9 ( !
116 Some Recent Advances in Partial Difference Equations
Tzirtzilakis and Kafoussias
& ; ; & 5 C !D E
& $ C ##$ 0 ',D $ 5 < J@ ! #
0 9 $ ) 3 7 8 ! & 9
102 !!$ W=(?.65 W( ?.65 W 5 0! $ 1 102 9 9 ! 9 ! # $ .6 5! # $ . 0 . $
!
1 "
1 # $ 9 * ! # Q 0 0 ! F $ $ 9 ) 0 ) 4 :3 ; :3 );! )
0 ) $ 0 ! F, 9 =90 3 9 3 ?!=! 9 ) =) ?!=!= >9 $ 9 >9 $ $ 9 (! !! 90 > >) > 9
! 900 > # 0 2 :3 ; :3 ) ; ) $ 9 ?! ! ' - 0 $ $ ) 0 ) ) ) ) ! & ) )!
Numerical Schemes and Difference Equations
Some Recent Advances in Partial Difference Equations 117
A 3 > D
& 9 ) ) 0 ! & 0 2 9 @ ! 9 9 ) 9 @ 9 9 ?!=!> 0 9 ) 9 9 ) 9 9 ) ) 9 9 ) ) 9 & 9 * & ! 9 9 ) >9 > 9 9 ) 9 ) =O > 0 > 0 >9 ?!=!? > 9 O > 0 O > 3 A ! 9 9 ) 9 ) >9 >9 =O > ?!=!@ > 0 0 9 ) @
¼ ¼
¼ ¼
¼ ¼
¼ ¼
¼
¼ ¼
118 Some Recent Advances in Partial Difference Equations
Tzirtzilakis and Kafoussias
M 2 $ ! " $ 9 @ 9 @ >9 4 9) = B = > 1 & + ?!=!@! & 90 , 90 0 @! & 7C8 0 9 @ 9 @ 9 @ 9 @ >9 C ?!=!A > C 4 9) = = 3 " 3 ?!=!A 79 * 8 9 @ 9 @! & ! & 90 ! $ $ & 9
@
)
9
=O >O 9 )
>9 > 0¼ ¼ > 9 > 0¼ ¼
>9 >
0 ¼ ¼
?!=!H
9 $ 90 7D ( * 8 9 @ 9 @ 9 @ 9 @ >9 C ?!=!I > - $ ?!=!H ?!=!? 0
90$
@ 9 @ C 9 @ 9 @ ?!=!J = = &
! & ! ' >9 >
9
Numerical Schemes and Difference Equations
Some Recent Advances in Partial Difference Equations 119
$ ! 9 $ 0 ?! ! 9 . >9 = >9 @ ?9= @ 9> @ C >
= >
?9- @ 9- = @ C 1 $ $ ( ! >9 >
9 - @
?!=! , 0 ! 9 7 8 9
90 ( ! 9 & 9 ) ) 9 ) ) >9 >9 =O >) >) 0 ) 0 9 ) 9 ) >9 >9 =O > 0 > 0 ?!=! $ 9 ) ) 9 ) 9 ) 9 ) = ) ) >9 > 9 =O >) 0 > 0 9 @ 9 @ 9 @ 9 @ = ) > 9 3 =O) >>)9 > 0 0 9 @ 9 @ 9 @ 9 @ = C ) C 3 ?!=! 3 ) & ) > 9 > 9 4 B =O >) 0 > 0 ¼ ¼
¼ ¼
¼ ¼
¼ ¼
¼ ¼
¼ ¼
¼ ¼
¼ ¼
¼ ¼
¼ ¼
120 Some Recent Advances in Partial Difference Equations
Tzirtzilakis and Kafoussias
C ) C ! 0 !%! 04 B 3 $ $ $ . $ 0$ !! 3 ) 3 ! 9 ?!=! 3$ C C )$
) :9 @ 9 @ ; 9 @ 9 @ = ) 9 9 ?!=! 9 9 = @ $ !! 9 @ 9 ! & 9 & D ( " 9&D" 2 ! 7 8 ( 9 @ ( 9 @! 1 2 $ ! & %
) 0 ! & % ! 5 $ ( 7 8 7 8! 9 $ % ! & ! & 9 %5 %!% (%% ! 1 $ $ $ !%! ! !%% ! Æ%! !! (% !% %!% !
+2
7" 8 ! 7 8$ 7 8 $ . $
Numerical Schemes and Difference Equations
Some Recent Advances in Partial Difference Equations 121
, ! & $ . $ ! 9 , $ ! 9 9 R C
# ! F 9V @% ! F 9 $ !! 0 ! # + @ 9 9 V + ?!>! 1 $ 0
) ?!>!= & 0 $ @ @ ! 0 >9 >9 ?!>!> >) > M $
$ 2 9V 9V 9V =V9 9V ?!>!? ) 0 9V 7 8 9 ! D ?!>! +
+
= = = = )
9
+
)
9
9
9
+
)
+
9
+
+
9
9
+
9
9
+
9
+ + +
?!>!? 0 9 . + 4
122 Some Recent Advances in Partial Difference Equations
Tzirtzilakis and Kafoussias
=
?!>!@ ) + + + ?!>!? ! 1 R C $ @ ?!>!@! ) 3! F B ! 1 $
9 $
+
B
+
0
= -
?!>!A
00 0 1 ! * ?!>!>$ ! $ ?!>!A $ 0$ $ = > - ! ( ?!>!@ 0 ) 3! * ?!>!>$ + + ) + ) ( 0 ?!>!H ! & ?!>!H 0 ! # $ R C
! 5 $ . ( ! R C ?!=! $ 2 $ !
!
½
1 $ 2 ?!=! 9 . & D ( " $ 9&D" C ) C ) 9 9 ?!>!I 9 9 = & $ . $
Numerical Schemes and Difference Equations
+
Some Recent Advances in Partial Difference Equations 123
+ = ) + +
< + ( 0 + ( 0 + ( 00 ) ( 0 ( 0 0 & ( 0 ( 0 = ( 0 ( 0 + +
?!>!J
) ( 0 ( 0 0 = & ( 0
= ) 0 =# 0 &
=# =( =# =#=( =# # )6 $ 0 E
, 4
: =# =#=( ; ?# & ?# I#
? I #
#
=(
D ( ?!>! 3 =( = = I# =# = & I# A# 3 # = $ 7 8 9&D" 3 # 9&D" !
124 Some Recent Advances in Partial Difference Equations
Tzirtzilakis and Kafoussias
!
D 2 ?!=!
9 & " 9&" C ) C ) 9 9 ?!>! 9 9 & ) + + ?!>! = + + & ) ( 0 0 ( 0 0 & ( 0 ( 0 ) ( 00 ( 0 0 ( 0 & ( 0 ( 0 + ) 0 0 # 0 0 + & $ 0 ( 0 # =( =( =# " , ! 5 $ ?#
9&" ! 1 $ 9&D" C C $ 0 !
!!
D 2 ?!=! 9 & 9 "
9&9" C ) C ) 9 9 ?!>! > 9 9 = &
Numerical Schemes and Difference Equations
Some Recent Advances in Partial Difference Equations 125
+ = ) + + & & ) ( 0 0 ( 0 & ( 0 ( 0 = ( 0 ) ( 0 0 ( 0 = & ( 0 ( 0 + ) 0 =# 0 & = + +
?!>! ?
=# =( =# =#=( =# & 0 : =# =#=( ; ?#
? =
?# I ? =# =( # # = $ ?# =# 9&9" ! # #
$ %!
9 $ 2 ! & (
F %9 9 7 8 9 ! & ( F 9 F99" $ C ) C ) 9 9 9 9 ? ?!>! @ & ) + + + + ? ?!>! A & ) ( 0 0 ( 0 & ( 0 ( 0 ?
126 Some Recent Advances in Partial Difference Equations
( 0 ( 0
Tzirtzilakis and Kafoussias
) ( 00 ( 0 & ? ( 0 + ( ? ) 0 + ( ( 0 ( 0
1
( ( (
( 0
( 0 ++
0 ?# 0 3 ? # 0 1 .
$ ! # " !
( ( ( (
& " # (
1 ?!>$ ! # ! 9 (! & ( 0 B 9 B 9 9 B 9 B 9 9 ?!?! B 9 B 9 9 B 9 B 9 9
! !
9&D" 4 ) 9 9 9 9 =#9 =#9 ?!?!= 9 9 = # )6 ! & ?!=! 9 3 ?!?!=4 9
Numerical Schemes and Difference Equations
Some Recent Advances in Partial Difference Equations 127
+ ?!?!= @ ! # ?!=!
0 )! & $ 0 ?!?! $ ?!?!= ( B 9 9 =#9 =#B 9 B 9 =# 9 =#B 9
=# =#B ! 5 $ $ 9
9 @ =# =#B 9 # 6= 4 = # =# B & 9 @ =# =# 9 @ =# B & =# @ = # 9 @ =# =# & 9 @ =@ ) ?!?!> 9 ) = ) # = = 9 @ B @ ?!?!? ?!?!> ?!?!? 9 @ =@ ) % = # " 3 ?!?!@ & ?!?!@ ?!=! ) 9 @ 0 0 = ) ) =) 9
128 Some Recent Advances in Partial Difference Equations
Tzirtzilakis and Kafoussias
!
M 9&" 9 9 # 9 9 9 9 #9 #9 # )6 ! & 0 @ ! D $ $ B 9 9 #B 9 #B 9 B 9 #B #B 9 9 #B #B ! 9 9 & $ ( 9 @ : # B B ; &
9 @
9 @
@
@
#
#
B B &
& =
B
B
@ # 9 @ =@ ) 9&D" ! 9 @
!!
?!?!A
9 9&9" ?!?!H 9 9 =# 9 9 9 =#9 =#9 & 0 @ 0 !
Numerical Schemes and Difference Equations
Some Recent Advances in Partial Difference Equations 129
M B 9 =# 9 =#B 9 & 9 @ =# =#B & 9 3 @ 4 9 5$ 9 @ =# =#B 9 # S= =# & 9 @ =# =#
9 @
=#
=#
9 @
@
@
=# B & =#
=# =# &
: = @ #; =@ )
9 @
?!?!I
9 # 0 = ) B B @ ?!?!J & ?!?!I ?!?!J # 9 @ =@ ) 1 $ ! 9 @
130 Some Recent Advances in Partial Difference Equations
Tzirtzilakis and Kafoussias
$ %!
D F %9 $ 9 9 ?# 9 9 ?!?! 3 @ 0 ! * B 9 B 9 ?# B 9 9 ?# B 9 9 B 9 B 9 B 9 9 ?# B B9 B9 ?# B B 9 &
?# B B @ 1 $ 0 Æ ( 9 $ @! " $ 0 $ ( 9 ) =) ! & $ $ 93 ) =)$ 4 9 @ @ @ =@ ) ?!?! D ( 9 @ ?# B B =@ ) & 9 @
? = & = & ? = = 9 @
9 @
#
#
B
B
B
@
B
)
@
)
@ ?#
=@ =@ ) 9 @ ) # ?!=! ! 9 @
)
Numerical Schemes and Difference Equations
Some Recent Advances in Partial Difference Equations 131
?!=! ! & 2 . $ 2 ?!?! 1 0 4 & ?!=! $ 9 ) =)$ ! & $ 2 $ . ! & ?!? 9 @ =@ ) 2 ! & $ $ ( 0 7* 8 ! & 2 ! & ! 9 ?!= 9&D" 2 ) 3 3
3 3=! # R C 2 9&D" # 6= = ) ! & $ ! & 9 ) ) 3$ 3!=$ 3!?$ 3!A$ 3!I !3! & ! 1 $ ! / 9&D" 9 ?!> ) 3 3 3 3 =@ R C ! # ! 9 ) 3 ? $ ) 3 A ! # ) ! & $ ! 9 ?!? 9&" ) 3 3 3 3=! # (
! # 9&D" ! 9 ?!@ ) 3 3 3 3 =@$ 9&D" 9 ?!>! & 9&D" ) 3 A !
132 Some Recent Advances in Partial Difference Equations
Tzirtzilakis and Kafoussias
9&" $ $ 9&D" ! 9 ?!A ) 3! ) 3 ) ) =! # $ 9&D" ! # 9&D" ) 33 ) 333! # 9&D" 9&" ) ! & 9&"$ $ ! & R C ) ! " $ $ 0 ! &
0 $ $ B 0 ) 3 0 $ 0
! 9 ?!H ) 3 3 3 3I! ) 3! & $ $ ) 3 3 3 3= ) 3 ? 9 ?!I! ) 3 3 3 3A 9 ?!J! ) 3 I ) 3 ! 9 0 ?!H$ ?!I ?!J ! 0 ) 3 3 3 3I! 1 $ ) 0 ) 0 ( ! & F %9 9 " F99" 2 ! 7 8 . 3 ) 3! 9 ?! 3 ?! ! # ( . $ )! # . 2 $ ,
Numerical Schemes and Difference Equations
Some Recent Advances in Partial Difference Equations 133
=@ ) - @ 1 ! $ 2 . $ $ @! & ?!=! 9 ) =) & $ . 9 ?! @ ) )! & 9 @ =@ )
R C ! - $
! " 2 $ $ . $ ! # . $
! $ $ !
9 @
134 Some Recent Advances in Partial Difference Equations
Tzirtzilakis and Kafoussias
A 3 7 A( G3 F G' F
A 34 7 A( G3 F 6 G' F
Numerical Schemes and Difference Equations
Some Recent Advances in Partial Difference Equations 135
A 33 7 A(@ G3 F G' F
A 36 7 A(@ G3 F 6 G' F
136 Some Recent Advances in Partial Difference Equations
Tzirtzilakis and Kafoussias
A 35 7 A(@ G3 F G' F
A 3; 7 A(A G3 F < G' F
Numerical Schemes and Difference Equations
Some Recent Advances in Partial Difference Equations 137
A 3< 7 A(A G3 F G' F
A 38 7 A(A G3 F 5 G' F
138 Some Recent Advances in Partial Difference Equations
Tzirtzilakis and Kafoussias
A 3 7 &AA G3 F G' F
A 3 7 &AA G3 F < G' F
Numerical Schemes and Difference Equations
Some Recent Advances in Partial Difference Equations 139
' 0
! ( 0 2 9 =90 3 9 3 ! D 2 $ $ !
$ ! # $ 2 $ $ ! 2 ! & $ $ ! & R C $ $ ! /
$ 0 $ ! 9 $ $ ! ' $ ! # , $ ! & 9 =90 3! & $ $
$ ! & ( $ ! 9 $ ! " $ $ 0 0 !
: ; /! )! 1 ! -, 6 ! - *(($ JJ=! :=; ! 9! 1 ! + 2 -, ! 1 )$ JHH! :>; T!*!T! 1 ! @ - ! -E %5 $ #$ C U (!
140 Some Recent Advances in Partial Difference Equations
Tzirtzilakis and Kafoussias
:?; T! )! D ! @ ! * ) $ #C$ - $ C U ($ =33 ! :@; ! $ -!U! 5 $ 1! X $ &! 1! P !
@ - ! " %R $ C U (! :A; &! T! ! @ - ! M )$ =33=! :H; !1!T! 9 ! 6 @ - ! " %R $ C U (! :I; )! *! F ! C ! & $ H==4H?%I?$ JA@! :J; /! +! - (! -, 6 ! R C / !$ JJ3! : 3; /! +! - (! & & + . @ -, ! " 0 ) ) F$ =33@! : ; E! ! ' D $ -! 1! 5 $ "! < ! 1 ! > 2 $ =J4==>%=@ $ J@3! : =; /! * / ; E! *! " ! + , @ -, ! ' M )$ JHI! : ?; 5!
6 @ A ! F " 0 & $ JJ@!
Some Recent Advances in Partial Difference Equations, 2010, 141-142
141
!" $ @ $ @> $ @ $ @>
$ A $ A $ A= 1 $ 3@ D#D' $ 3@$ 3H ( $ I $ I $ I $ IJ 0 + 5 $ H3 9 %- $ 3? 9 Y $ A= 9 $ AH % $ @H
$ @ 0 $ H
$ IA%II $ A@ F , $ ?$ =3 $ J@ $ JJ $ =
0 $ ? 0 $ > 0 $ > $ > $ ?$ =3 $ ? ( $ =3$ ==
$ =A $ => $ =?
$ =I $ =? $ =?
$ =J $ =@ % $ =@
$ >3 $ =A $ ?$ =3 ) + $ IJ 2$ IJ 0 ( $ IJ $ AA$ H $ H=$ I= $ I? $ 3@ $ 3@ $ H= / , $ 3? $ 3 ) +
Eugenia N. Petropoulou (Ed) All rights reserved - © 2010 Bentham Science Publishers Ltd.
142 Some Recent Advances in Partial Difference Equations
$ @=$ H= $ AA$ H $ H=$ J@$ JA $ ? $ ?$ 3$ =$ ?$ @$ =>$ =? $ ? $ ?$ 3$ =$ ?$ @$ =>$ =? $ J? $ J= $ JH$ JI $ H $ H= $ AA $ ? $ ?$ 3$ =$ ?$ @$ =>$ =? $ ? $ ?$ 3$ =$ ?$ @$ =>$ =? $ ? $ ? ) # $ ? $ J$ = %=>$ =@$ =I$ >3$ >=$ >? $ ? $ J$ = %=>$ =@$ =I$ >3$ >=$ >? $ ? $ J$ = %=>$ =@$ =I$ >3$ >=$ >? $ ? $ J$ = %=>$ =@$ =I$ >3$ >=$ >? $ ? $ ? " $ I3 $ JI $ 3 $ I C , $ A$ =
Eugenia N. Petropoulou
