Applied Mathematics and Computation 177 (2006) 844–851

www.elsevier.com/locate/amc

Approximate solution of complex differential equationsfor a rectangular domain with Taylor collocation method

Mehmet Sezer, Mustafa Gulsu *

Department of Mathematics, Faculty of Science, Mugla University, 48000 Mugla, Turkey

Abstract

In this study, Taylor collocation method is investigated for the approximate computation of high-order linear complexdifferential equations. Using the collocation points on any rectangular domain in the complex plane, the method trans-forms the given complex differential equation and the mixed conditions to matrix equation with unknown Taylor coeffi-cients. By means of the obtained matrix equations, the Taylor coefficients can be easily computed. Hence, the finite Taylorseries approach is obtained. Also, examples are presented and the results are discussed.� 2005 Elsevier Inc. All rights reserved.

Keywords: Taylor polynomials and series; Complex differential equations; Collocation methods

1. Introduction

Taylor and Chebyshev (matrix and collocation) methods to solve linear differential, integral, integro-differ-ential, difference, integro-difference and systems of integro-differential equations have been in many papers[1–10]. In this paper these methods are modified and developed for solving the linear complex differentialequation with variable coefficient

0096-3

doi:10

* CoE-m

Xm

k¼0

P kðzÞf ðkÞðzÞ ¼ gðzÞ; k P 0; k 2 Nþ ð1Þ

which is extended of the complex differential equations [11–14], under the mixed conditions

Xm�1

k¼0

XJ

j¼0

arkf ðkÞðnjÞ þ brkf ðkÞðz0Þ� �

¼ ½kr� r ¼ 0; 1; . . . ;m� 1. ð2Þ

003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

.1016/j.amc.2005.11.035

rresponding author.ail addresses: msezer@mu.edu.tr (M. Sezer), mgulsu@mu.edu.tr (M. Gulsu).

M. Sezer, M. Gulsu / Applied Mathematics and Computation 177 (2006) 844–851 845

Here the coefficients Pk(z) and the right-hand member g(z) are analytic functions in the square domainD = {z 2 C,z = x + iy,a 6 x 6 b,c 6 y 6 d,a,b,c,d 2 R}, ark, brk and kr are appropriate complex coefficients;nj, z0 2 D. We assume that the solution of Eq. (1) under the conditions (2) is expressed in the form

f ðzÞ ¼XN

n¼0

f ðnÞðz0Þn!

ðz� z0Þn; a 6 z; z0 6 b ð3Þ

which is a Taylor polynomial of degree N at z = z0, where f(n)(z0), n = 0,1, . . . ,N are the Taylor coefficients tobe determined. In addition, to compute the Taylor coefficients f(n)(z0), we use the collocation points

zpp ¼ xp þ iyp

so that

xp ¼ aþ b� aN

p; yp ¼ cþ d � cN

p; p ¼ 0; 1; . . . ;N . ð4Þ

2. Fundamental matrix relations

Let us consider the linear differential equation with variable coefficients (1) and find the truncated Taylorseries expansions of each term in the equation at z = z0 and their matrix representations. We first consider thedesired solution f(z) of the problem, defined by the series (3). Then we can put series (3) and its derivatives inthe matrix forms

f ðkÞðzÞ� �

¼ ZMkF; k ¼ 0; 1; . . . ;m 6 N ; ð5Þ

where

Z ¼ 1 ðz� z0Þ ðz� z0Þ2 . . . ðz� z0ÞN� �

;

Mk ¼

0 0 . . . 10!

0 . . . 0

0 0 . . . 0 11!

. . . 0

� � . . . �0 0 . . . 0 0 . . . 1

ðN�kÞ!

0 0 . . . 0 0 . . . 0

� � �0 0 . . . 0 0 . . . 0

26666666666664

37777777777775ðNþ1ÞxðNþ1Þ

;

F ¼ f ð0Þðz0Þ f ð1Þðz0Þ . . . f ðNÞðz0Þ� �T

.

For the collocation points z = zpp, the matrix relation (5) becomes

f ðkÞðzppÞ� �

¼ ZppMkF; p ¼ 0; 1; . . . ;N ; ð6Þ

where

Zpp ¼ 1 ðzpp � z0Þ ðzpp � z0Þ2 . . . ðzpp � z0ÞN� �

.

For p = 0,1, . . . ,N we can write the relation (6) in the form

f ðkÞðz00Þ� �

¼ Z00MkF

f ðkÞðz11Þ� �

¼ Z11MkF

. . .

f ðkÞðzNN Þ� �

¼ ZNN MkF

846 M. Sezer, M. Gulsu / Applied Mathematics and Computation 177 (2006) 844–851

or briefly

FðkÞ ¼

f ðkÞðz00Þf ðkÞðz11Þ���

f ðkÞðzNN Þ

266666664

377777775¼ HMkF; ð7Þ

H ¼

Z00

Z11

���

ZNN

2666666664

3777777775¼

1 ðz00 � z0Þ ðz00 � z0Þ2 � � � ðz00 � z0ÞN

1 ðz11 � z0Þ ðz11 � z0Þ2 � � � ðz11 � z0ÞN

� � � �� � � �� � � �1 ðzNN � z0Þ ðzNN � z0Þ2 � � � ðzNN � z0ÞN

26666666664

37777777775

.

On the other hand, substituting the collocation points z = zpp defined by (4) into Eq. (1), we have

Xm

k¼0

P kðzppÞf ðkÞðzppÞ ¼ gðzppÞ; p ¼ 0; 1; . . . ;N

or

Xm

k¼0

PkFðkÞ ¼ G. ð8Þ

Substituting the expression (7) into (8) we obtain the fundamental matrix equation

Xm

k¼0

P kHMkF ¼ G; ð9Þ

where

G ¼ gðz00Þ gðz11Þ . . . gðzNN Þ½ �T

Pk ¼

P kðz00Þ 0 � � � 0

0 P kðz11Þ � � � 0

� � �� � �� � �0 0 � � � P kðzNN Þ

2666666664

3777777775

.ð10Þ

We can obtain the corresponding matrix form for the conditions (2) as follows. By means of the relation (5),we have the matrix equation

Xm

k¼0

XJ

j¼0

½arkZðnjÞ þ brkZðz0Þ�Mk

( )F ¼ ½kr�; ð11Þ

where z0, nj 2 D, ark, brk 2 C, r = 0,1, . . . ,m � 1

ZðnjÞ ¼ 1 ðnj � z0Þ ðnj � z0Þ2 . . . ðnj � z0ÞNh i

; j ¼ 0; 1; . . . ; J ;

Zðz0Þ ¼ 1 0 0 . . . 0½ �.

Briefly, the matrix Eq. (11) are

UrF ¼ ½kr� or ½Ur; kr�; r ¼ 0; 1; . . . ;m� 1; ð12Þ

M. Sezer, M. Gulsu / Applied Mathematics and Computation 177 (2006) 844–851 847

where

Ur ¼Xm�1

k¼0

XJ

j¼0

d½arkZðnjÞ þ brkZðz0Þ�Mk � ur0 ur1 . . . urN½ �.

3. Method of solution

We now consider fundamental matrix Eq. (9) corresponding to Eq. (1). We can write Eq. (9) in the form

WF ¼ G; ð13Þ

where

W ¼ ½wpq� ¼Xm

k¼0

P kHMkF ¼ G; p; q ¼ 0; 1; . . . ;N

and G is defined in (10). The augmented matrix of Eq. (13) becomes

½W; G� ¼ ½wpq; gp�; p; q ¼ 0; 1; . . . ;N . ð14Þ

We now consider the matrix Eq. (12) corresponding to the conditions (2) and can write in the matrix forms

UiF ¼ ½ki�; i ¼ 0; 1; . . . ;m� 1

or the augmented matrix forms

½Ui; ki� ¼ ui0 ui0 . . . uiN ; ki½ �; ð15Þ

where

Ui ¼Xm�1

k¼0

XJ

j¼0

½arkZðnjÞ þ brkZðz0Þ�Mk.

Consequently, to find the unknown Taylor coefficients fn, n = 0,1, . . . ,N, related with the approximate solu-tion of the problem consisting of Eq. (1) and conditions (2), by replacing the m row matrices (15) by the last m

rows of the augmented matrix (14), we have new augmented matrix

½W�; G�� ¼

w00 w01 � � � w0N ; g0

w10 w11 � � � w1N ; g1

. . . . . . . . . . . .

wN�m;0 wN�m;1 � � � wN�m;N ; gN�m

u00 u01 � � � u0N ; k0

u10 u11 � � � u1N ; k1

. . . . . . . . . ; . . .

um�1;0 um�1;1 � � � um�1;N ; km�1

266666666666664

377777777777775

ð16Þ

or the corresponding matrix equation

W�F ¼ G�.

If det W* 5 0, we can write Eq. (16) as

F ¼ ðW�Þ�1G� ð17Þ

and the matrix F is uniquely determined. Thus the mth-order linear complex differential equation with variablecoefficients (1) under the conditions (2) has a unique solution. This solution is given by the truncated Taylorseries (3). Also we can easily check the accuracy of the obtained solutions as follows [2,4]: Since the Taylorpolynomial (3) is an approximate solution of Eq. (1), when the solutions f(z) and its derivatives are substitutedin Eq. (1), the resulting equation must be satisfied approximately; that is, for z = zi 2 D, i = 0,1,2, . . . ,

848 M. Sezer, M. Gulsu / Applied Mathematics and Computation 177 (2006) 844–851

EðziÞ ¼Xm

k¼0

P kðziÞf ðkÞðziÞ � gðziÞ�����

����� ffi 0

or

EðziÞ 6 10�ki ðki is any positive integerÞ.

If max ð10�kiÞ ¼ 10�k (k is any positive integer) is prescribed, then the truncation limit N is increased until thevalues E(zi) at each of the points zi becomes smaller than the prescribed 10�k.

4. Examples

Example 1. Let us first consider the linear second order complex differential equation

f 00ðzÞ þ zf 0ðzÞ þ zf ðzÞ ¼ ez þ 2zez; jzj 6 1

with f(0) = 1, f 0(0) = 1 and approximate the solution f(z) by the Taylor polynomial

f ðzÞ ¼X5

n¼0

f ðnÞðz0Þn!

ðz� z0Þn;

where a = 0, b = 1, c = 0, d = 1, z0 = 0, P0(z) = z, P1(z) = z, P2(z) = 1, g(z) = ez + 2zez. Then, for N = 5, thecollocation points are

z0 ¼ 0; z1 ¼1

5þ 1

5i; z2 ¼

2

5þ 2

5i; z3 ¼

3

5þ 3

5i; z4 ¼

4

5þ 4

5i; z5 ¼ 1þ i

and the fundamental matrix equation of problem is

ðP2HM2 þ P1HM1 þ P0HM0ÞF ¼ G;

where P0, P1, P2, H, M0, M1, M2 are matrices of order (6 · 6) defined by

P0 ¼ P1 ¼

0 0 0 0 0 0

0 1þi5

0 0 0 0

0 0 2þ2i5

0 0 0

0 0 0 3þ3i5

0 0

0 0 0 0 4þ4i5

0

0 0 0 0 0 1þ i

26666666664

37777777775; P2 ¼

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

26666666664

37777777775;

M0 ¼

1 0 0 0 0 0

0 1 0 0 0 0

0 0 12

0 0 0

0 0 0 16

0 0

0 0 0 0 124

0

0 0 0 0 0 1120

266666666664

377777777775; M1 ¼

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 12

0 0

0 0 0 0 16

0

0 0 0 0 0 124

0 0 0 0 0 0

26666666664

37777777775; M2 ¼

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 12

0

0 0 0 0 0 16

0 0 0 0 0 0

0 0 0 0 0 0

26666666664

37777777775;

H ¼

1 0 0 0 0 0

1 1þi5

225

i �2þ2i125

�4625

�4�4i3125

1 2þi5

825

i �16þ16i125

�64625

�128�128i3125

1 3þi5

1825

i �54þ54i125

�324625

�972�972i3125

1 4þi5

3225

i �128þ128i125

�1024625

�4096�4096i3125

1 1þ i 2i �2þ 2i �4 �4� 4i

26666666664

37777777775:

M. Sezer, M. Gulsu / Applied Mathematics and Computation 177 (2006) 844–851 849

The augmented matrix forms of the conditions for N = 5 are

TableError

z

0.0 + 00.1 + 00.2 + 00.3 + 00.4 + 00.5 + 00.6 + 00.7 + 00.8 + 00.9 + 01.0 + 1

TableError

z

0.0 + 00.1 + 00.2 + 00.3 + 00.4 + 00.5 + 00.6 + 00.7 + 00.8 + 00.9 + 01.0 + 1

½1 0 0 0 0 0 ; 1�;½0 1 0 0 0 0 ; 1�.

Taking N = 5, we obtain the following approximate solution. The solution is

f ðzÞ ¼ 1þ zþ 0:5z2 þ ð0:1656532806þ 0:0006418445835iÞz3 þ ð0:04247397905� 0:003715089246iÞz4

þ ð0:01074173967þ 0:003414311854iÞz5.

Taking N = 5, 6 the solutions obtained are compared with the exact solution y(x) = ez in Tables 1–3.

Example 2. Let us consider the second order complex differential equation

y00ðzÞ þ zy0ðzÞ þ 2zyðzÞ ¼ 2z sin zþ z cos z� sin z

with y(0) = 0, y 0(0) = 1 and approximate the solution f(z) by the truncated Taylor series in the form

f ðzÞ ¼X6

n¼0

yðnÞð0Þn!

zn

so that a = 0, b = 1, c = 0, P0(z) = 2z, P1(z) = z, P2(z) = 1 and g(z) = 2zsinz + zcosz � sinz. For N = 7, wehave the collocation points

1analysis of Example 1 for the Re(z) value

Exact solution (Reel) Present Met(z0 = 0)

N = 5 (Reel) Ne = 5 N = 6 (Reel) Ne = 6

.0i 1.00000000 1.00000000 0.000000 1.00000000 0.000000

.1i 1.099649667 1.099650127 0.460E�6 1.099649494 0.173E�6

.2i 1.197056021 1.197058066 0.204E�5 1.197055356 0.665E�6

.3i 1.289569374 1.289572683 0.330E�5 1.289568314 0.106E�5

.4i 1.374061539 1.374064758 0.321E�5 1.374060185 0.135E�5

.5i 1.446889037 1.446891796 0.275E�5 1.446887256 0.178E�5

.6i 1.503859541 1.503862872 0.333E�5 1.503857415 0.212E�5

.7i 1.540203025 1.540203452 0.427E�6 1.540201009 0.201E�5

.8i 1.550549297 1.550520219 0.290E�4 1.550545449 0.384E�5

.9i 1.528913812 1.528765906 0.147E�3 1.528893562 0.202E�4

.0i 1.468693940 1.468204123 0.489E�3 1.468605666 0.882E�4

2analysis of Example 1 for the Im(z) value

Exact solution (Im) Present Met(z0 = 0)

N = 5 (Im) Ne = 5 N = 6 (Im) Ne = 6

.0i 0.00000000 0.000000000 0.0000000 0.00000000 0.000000

.1i 0.110332988 0.110330942 0.204E�5 0.110333088 0.999E�7

.2i 0.242655268 0.242645839 0.942E�5 0.242655662 0.393E�6

.3i 0.398910553 0.398893389 0.171E�4 0.398911217 0.663E�6

.4i 0.580943900 0.580922057 0.218E�4 0.580944816 0.915E�6

.5i 0.790439083 0.790412124 0.269E�4 0.790440371 0.128E�5

.6i 1.028845666 1.028807744 0.379E�4 1.028847386 0.172E�5

.7i 1.297295112 1.297248987 0.461E�4 1.297297157 0.204E�5

.8i 1.596505341 1.596503892 0.144E�5 1.596508441 0.310E�5

.9i 1.926673304 1.926900527 0.227E�4 1.926682579 0.927E�5

.0i 2.287355287 2.288259023 0.903E�3 2.287388077 0.327E�4

Table 3The absolute error for the considered method at various points for Example 1

z Exact solution ði ¼ffiffiffiffiffiffiffi�1p

Þ Ne = 7 Ne = 8 Ne = 9

0.0 + 0.0i 0.000000000 + 0.000000000i 0.0000000 0.0000000 0.00000000.1 + 0.1i 0.110332988 + 0.110332988i 0.1470E�7 0.2559E�8 0.8000E�90.2 + 0.2i 0.242655268 + 0.242655268i 0.4977E�7 0.1985E�7 0.7810E�80.3 + 0.3i 0.398910553 + 0.398910553i 0.5623E�7 0.4632E�7 0.2376E�70.4 + 0.4i 0.580943900 + 0.580943900i 0.5747E�6 0.1568E�6 0.5459E�70.5 + 0.5i 0.790439083 + 0.790439083i 0.3019E�5 0.3058E�6 0.1077E�60.6 + 0.6i 1.028845666 + 1.028845666i 0.1109E�4 0.5281E�5 0.1852E�60.7 + 0.7i 1.297295112 + 1.297295112i 0.3287E�4 0.8382E�5 0.2948E�60.8 + 0.8i 1.596505341 + 1.596505341i 0.8398E�4 0.1250E�4 0.4405E�60.9 + 0.9i 1.926673304 + 1.926673304i 0.1923E�3 0.1783E�4 0.6264E�61.0 + 1.0i 2.287355287 + 2.287355287i 0.4053E�3 0.2479E�4 0.8500E�6

TableThe ab

z

0.0 + 00.1 + 00.2 + 00.3 + 00.4 + 00.5 + 00.6 + 00.7 + 00.8 + 00.9 + 01.0 + 1

850 M. Sezer, M. Gulsu / Applied Mathematics and Computation 177 (2006) 844–851

z0 ¼ 0; z1 ¼1þ i

7; z2 ¼

2þ 2i

7; z3 ¼

3þ 3i

7; z4 ¼

4þ 4i

7;

z5 ¼5þ 5i

7; z6 ¼

6þ 6i

7; z7 ¼ 1þ i

and the fundamental matrix equation

ðP2HM2 þ P1HM1 þ P0HM0ÞF ¼ G.

This equation has the solution

F ¼ ½0 1 0 � 1:000001540� 0:0000242272761i 0:000397069588þ 0:00035509611i 0:9938579500

þ 0:0003107030i 0:02785113649� 0:0304019089i� 0:9952329095þ 0:1359583529i�T.

Therefore, we find the solution

f ðzÞ ¼ z� ð0:1666669240þ 0:403801450110�5iÞz3 þ ð0:165478993810�4 þ 0:147938350010�4iÞz4

þ ð0:008282147276þ 0:259589083310�5Þz5 þ ð0:00003867977974� 0:00004223011000Þz6

� ð0:0001974652685 � 0:00002697650061iÞz7.

The values of this solution are compared with the exact solution f(z) = sinz for N = 7, 9, 11 in Table 4.

Example 3. Our last example is the linear complex differential equation with variable coefficients

f 0ðzÞ þ zf ðzÞ ¼ 2z2 � zþ 2

4solute error for the considered method at various points for Example 2

Exact solution ði ¼ffiffiffiffiffiffiffi�1p

Þ Ne = 7 Ne = 9 Ne = 11

.0i 0.0000000000 + 0.000000000i 0.000000000 0.000000000 0.000000000

.1i 0.1003329998 + 0.099666333i 0.510881E�8 0.735934E�9 0.78262E�10

.2i 0.2026559797 + 0.197322687i 0.168240E�7 0.123745E�7 0.150336E�8

.3i 0.3089186537 + 0.290919340i 0.259563E�7 0.631513E�7 0.817295E�8

.4i 0.4209894110 + 0.378327945i 0.358803E�7 0.200137E�6 0.267356E�7

.5i 0.5406126857 + 0.457304153i 0.476488E�7 0.488686E�6 0.665800E�7

.6i 0.6693640130 + 0.525452875i 0.568823E�7 0.101331E�5 0.139941E�6

.7i 0.8086020739 + 0.580196817i 0.735258E�7 0.187727E�5 0.261723E�6

.8i 0.9594171049 + 0.618749396i 0.826706E�7 0.320273E�5 0.449686E�6

.9i 1.122575130 + 0.6380930293i 0.275111E�6 0.513174E�5 0.724416E�6

.0i 1.298457581 + 0.6349639148i 0.250700E�5 0.782764E�5 0.110800E�5

M. Sezer, M. Gulsu / Applied Mathematics and Computation 177 (2006) 844–851 851

with the condition y(0) = � 1. For N = 3, the collocation points

z0 ¼ 0; z1 ¼1þ i

3; z2 ¼

2þ 2i

3; z3 ¼ 1þ i

and the matrix form of the problem is defined by

ðP1HM1 þ P0HM0ÞF ¼ G.

After the augmented matrices of the systems and conditions are computed, we obtain this solution

F ¼ �1 2 0 0 0½ �T.

Therefore, we find the exact solution

f ðzÞ ¼ 2z� 1.

5. Conclusions

High order linear complex differential equations are usually difficult to solve analytically. Then it is requiredto obtain the approximate solutions. For this reason, the present method has been proposed for approximatesolution and also analytical solution.

The method presented in this study is a method for computing the coefficients in the Taylor expansion ofthe solution of a linear complex differential equations, and is valid when the functions Pk(z) and g(z) aredefined in the rectangular domain D = {z 2 C,z = x + iy,a 6 x 6 b,c 6 y 6 d;a,b,c,d 2 R}.

The Taylor method is an effective method for the cases that the known functions have the Taylor seriesexpansion at z = z0. In this case, the Taylor polynomial solution f(z) and the values f(zj), zj 2 D can be easilyevaluated at low-computation effort. In addition, an interesting feature of this method is to find the analyticalsolutions if the equation has an exact solution that is a polynomial of degree N or less than N.

The method can also be extended to the system of linear complex differential equations with variable coef-ficients, but some modifications are required.

References

[1] R.P. Kanwal, K.C. Liu, A Taylor expansion approach for solving integral equations, Int. J. Math. Educ. Sci. Technol. 20 (3) (1989)411–414.

[2] S. Nas, S. Yalcınbas�, M. Sezer, A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equationsequations, Int. J. Math. Educ. Sci. Technol. 31 (2) (2000) 213–225.

[3] M. Sezer, A method for approximate solution of the second order linear differential equations in terms of Taylor polynomials, Int. J.Math. Educ. Sci. Technol. 27 (6) (1996) 821–834.

[4] S. Yalcınbas�, M. Sezer, The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms ofTaylor polynomials, Appl. Math. Comput. 112 (2002) 291–308.

[5] A. Akyuz, M. Sezer, Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coeficients,Appl. Math. Comput. 144 (2003) 237–247.

[6] M. Gulsu, M. Sezer, The approximate solution of high-order linear difference equation with variable coefficients in terms of Taylorpolynomials, Appl. Math. Comput. 168 (2005) 76–83.

[7] M. Sezer, M. Gulsu, A new polynomial approach for solving difference and Fredholm integro-difference equations with mixedargument, Appl. Math. Comput. 171 (2005) 332–344.

[8] A. Karamete, M. Sezer, A Taylor collocation method for the solution of linear integro-differential equations, Int. J. Comput. Math.79 (9) (2002) 987–1000.

[9] M. Sezer, A. Karamete, M. Gulsu, Taylor polynomial solutions of systems of linear differential equations with variable coefficients,Int. J. Comput. Math. 82 (6) (2005) 755–764.

[10] A. Akyuz, M. Sezer, A Chebyshev collocation method for the solution of linear differential equations, Int. J. Comput. Math. 72 (4)(1999) 491–507.

[11] M.R. Spiegel, Theory and Problems of Complex Variables, McGraw-Hill Inc., New York, 1972.[12] Y.M. Chiang, S. Wang, Oscillation results of certain higher-order linear differential equations with periodic coefficients in the complex

plane, J. Math. Anal. Appl. 215 (1997) 560–576.[13] J. Heittokangas, R. Korhonen, J. Rattya, Growth estimates for solutions of linear complex differential equations, Ann. Acad. Sci.

Fenn. Math. 29 (2004) 233–246.[14] L.V. Ahlfors, Complex Analysis, McGraw-Hill Inc., Tokyo, 1966.