Applied Mathematics and Computation 160 (2005) 401–410

www.elsevier.com/locate/amc

Approximate calculation of eigenvalueswith the method of weightedresiduals–collocation method

_Ibrahim C� elikFaculty of Arts and Sciences, Department of Mathematics, Pammukkale University,

Denizli 20017, Turkey

Abstract

In this study, the collocation method of the weight residual methods are investigated

for the approximate computation of higher Sturm–Liouville eigenvalues, where trial

solution is accepted as the Chebyshev series. The obtained approximate eigenvalues are

compared with the previous computational results [ANZIAM J. 42 (2000) C96, Numer.

Math. 47 (1985) 289, Numer. Math. 59 (1991) 243, Appl. Numer. Math. 23 (1997) 311,

Computing 26 (1981) 123].

� 2003 Elsevier Inc. All rights reserved.

Keywords: Eigenvalue problem; Collocation method; Chebyshev series

1. Introduction

The concept of an eigenvalue problem is rather important both in pure and

applied mathematics, a physical system, such as a pendulum, a vibrating and

rotating shaft. The physical system connected with eigenpairs of the system.

The Sturm–Liouville systems arise from vibration problems in continuum

mechanics. In physic, they describe boundary value problems corresponding to

simply harmonic standing waves. It is commonly assumed in physics that wavemotion can be resolved into simply harmonic standing waves, each of which

oscillates with its proper frequency.

E-mail addresses: i.celik@pamukkale.edu.tr, ibrahimcelik35@hotmail.com (_I. C� elik).

0096-3003/$ - see front matter � 2003 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2003.11.011

402 _I. C�elik / Appl. Math. Comput. 160 (2005) 401–410

A general Sturm–Liouville problem can be written as the following differ-

ential equation

d

dxpðxÞ du

dx

� �þ ðrðxÞk� q1ðxÞÞu ¼ 0:

It can be reduced to a study of the canonical Liouville normal form

u00 þ ðk� qðxÞÞu ¼ 0:

In this study, the Liouville normal form was investigated. If it is difficult tosolve the Sturm–Liouville problems or there are no exact solutions of Sturm–

Liouville problems, they can be solved by various approximate methods. The

weighted residuals methods, the variational methods and finite difference

methods are the most common methods for these calculations. For all these

approximate methods, the differential eigenvalue problem is reduced to an

algebraic system which gives approximate eigenvalues of the problem.

For the solution of the eigenvalue problem, some studies have been worried

out. Fox and Parker [5] used Chebyshev series to solve differential eigenvalueproblems. Paine et al. [7] showed that in the case of the second order centered

finite difference method with uniform mesh, the error, when qðxÞ is constant (inthe Liouville normal form), has the same asymptotic form (for k ! 1) as the

error for general qðxÞ. Thus, they gave corrected finite difference approxima-

tion.

Fix [4] investigated the following Sturm–Liouville problem for general qðxÞ.

u00 þ ðk� qðxÞÞu ¼ 0 06 x6 p;u0ð0Þ � auð0Þ ¼ u0ðpÞ � buðpÞ ¼ 0;oruð0Þ ¼ uðpÞ ¼ 0:

9>>=>>; ð1Þ

In order to study the asymptotic behavior of eigenvalues for large k, he used

the function Uðx; kÞ, the modifier Pr€ufer phase, which is defined for any given

solution uðx; kÞ of Eq. (1) by the equation

tanðUÞ ¼ ðk� qðxÞÞ12u0

u:

He found recurrence formulas to obtain eigenvalues of Eq. (1) for general qðxÞ.Andrew and Paine [2] improved the results of Numerov�s method with

asymptotic correction technique. Vanden Berghe and De Meyer [3] have

developed special tow step methods producing very accurate results. Ghe-

lardnoi [6] investigated the approximations of Sturm–Liouville eigenvaluesusing some linear multistep methods, called boundary value methods and

correction technique of Andrew–Paine and Paine et al. is extended to these

methods.

_I C�elik / Appl. Math. Comput. 160 (2005) 401–410 403

By using the following Liouville normal form

u00 þ ðk� qðxÞÞu ¼ 0;

uð0Þ ¼ uðpÞ ¼ 0;

the asymptotic correction of [2,7] can be outlined as follows.

Since the exact eigenvalues kk ¼ k2, k ¼ 1; 2; . . . , is known for qðxÞ � 0, the

closed form of the errors can be calculate as

enk ¼ dðnÞk � kk; k ¼ 1; 2; . . . ;

where dðnÞk indicates algebraic eigenvalues obtained by the finite difference

methods.

For qðxÞ 6� 0, kðnÞk eigenvalues of more general problem can be solved by

using the finite difference techniques. Hence the corrected eigenvalues was

obtained as

~kðnÞk ¼ kðnÞk � eðnÞk :

The aim of the present paper is to study the collocation method for

approximate computation of eigenvalues, where trial solution is taken as the

Chebyshev polynomials, and the asymptotic correction technique use for col-

location method.

2. Approximate calculation of eigenvalues with the method of weighted residuals–collocation method

In these methods we assume a trial solution of the form

uðxÞ ¼ a02þXn�1

r¼1

arTrðxÞ; �16 x6 1; ð2Þ

where TrðxÞ are the Chebyshev polynomials and ar are the undetermined

parameters. For any finite range, a6 x6 b, can be transformed to the basic

range �16 x6 1. The trial function (2) is substituted into the Liouville normal

form

LðuÞ þ ku ¼ 0;

uð�1Þ ¼ uð1Þ ¼ 0;

where L ¼ d2

dx2 � qðxÞ, to form the residuals;

R ¼ d2

dx2a02

"þXn�1

r¼1

arTrðxÞ#þ ½k� qðxÞ� a0

2

"þXn�1

r¼1

arTrðxÞ#: ð3Þ

404 _I. C�elik / Appl. Math. Comput. 160 (2005) 401–410

As the number of TrðxÞ is increased, the residuals will be smaller. We take

xi ¼ cosðn� i� 1Þp

ðn� 1Þ ; i ¼ 1; 2; 3; . . . ; ðn� 2Þ; ð4Þ

which has called the turning points of Chebyshev polynomials, as the collo-

cation points and residual Eq. (3) is vanished at the collocation points, then we

obtain (n� 2) equations as follows:

La02

"þXn�1

r¼1

arTrðxiÞ#þ k

a02

"þXn�1

r¼1

arTrðxiÞ#¼ 0;

where

TrðxiÞ ¼ cosðn� i� 1Þrp

ðn� 1Þ ; i ¼ 1; 2; 3; . . . ; ðn� 2Þ:

Using the boundary conditions gives

uð�1Þ ¼ a02� a1 þ a2 � a3 þ � � � þ ð�1Þn�1an�1 ¼ 0;

uð1Þ ¼ a02þ a1 þ a2 þ a3 þ � � � þ an�1 ¼ 0:

�ð5Þ

Thus, the following algebraic equation system is obtained as;

a02� a1 þ a2 � a3 þ � � � þ ð�1Þn�1an�1 ¼ 0;

a02þ a1 þ a2 þ a3 þ � � � þ an�1 ¼ 0;

L a02þPn�1

r¼1 arTrðxiÞh i

þ k a02þPn�1

r¼1 arTrðxiÞh i

¼ 0;

i ¼ 1; 2; . . . ; ðn� 2Þ:

8>>><>>>:

ð6Þ

This is a set of n homogenous linear equations for the undetermined param-eters ar. This set of equations has a nontrivial solution only if the determinant

of the coefficients matrix vanishes. This gives an equation of degree (n� 2) in kand has (n� 2) roots which are the first (n� 2) approximate eigenvalues of the

original problem.

3. Applications

The collocation method has been used to solve the following eigenvalue

problems of Liouville normal form.

u00 þ ðk� qðxÞÞu ¼ 0; 06 x6 p;

uð0Þ ¼ uðpÞ ¼ 0;ð7Þ

one choose the trial solution as

uðxÞ ¼ a02þXn�1

r¼1

arTþr ðxÞ;

_I C�elik / Appl. Math. Comput. 160 (2005) 401–410 405

then

u00ðxÞ ¼ b02þXn�3

r¼1

brTþr ðxÞ:

Using the boundary conditions, the following equations can be obtained.

uð0Þ ¼ a02� a1 þ a2 � a3 þ a4 � a5 þ a6 � a7 þ � � � þ ð�1Þn�1an�1 ¼ 0;

uðpÞ ¼ a02þ a1 þ a2 þ a3 þ a4 þ a5 þ a6 þ a7 þ � � � þ an�1 ¼ 0:

If uðxÞ and u00ðxÞ are substituted into the Sturm–Liouville equation, then the

following residual equation is obtained.

R ¼ b02þXn�3

r¼1

brTþr ðxÞ þ ðk� qðxÞÞ a0

2

þXn�1

r¼1

arTþr ðxÞ

!;

where

br ¼16

p2ððr þ 1Þðr þ 2Þarþ2 þ 2ðr þ 2Þðr þ 4Þarþ4 þ 3ðr þ 3Þðr þ 6Þarþ6

þ . . .Þ:

If one takes

xi ¼ cosn� 1� in� 1

� �; i ¼ 1; 2; . . . ; n� 2

as the collocation points. At the end, at collocation points, the value of TrðxÞ is

TrðxiÞ ¼ cosrðn� 1� iÞ

n� 1

� �; r ¼ 1; 2; 3 . . . ; n� 1; i ¼ 1; 2; 3; . . . ; n� 2:

The residual equation vanishes at the collocation points, thus consideration of

the boundary conditions and the residuals equation gives the n� n algebraic

homogeny equation system.

If qðxÞ � 0, then one can calculate the closed form errors

enk ¼ dcollðnÞk � kq¼0k ; k ¼ 1; 2; . . . ;

where kq¼0k is exact eigenvalues and dcollðnÞk is obtain in the n� n system of

algebraic equations for qðxÞ � 0.

For qðxÞ 6� 0, kcollðnÞk can be obtained in the n� n algebraic equation system.Hence the corrected eigenvalue can be presented as

~kcollðnÞk ¼ kcollðnÞk � eðnÞk :

Table 1

Corrected collocation solution for qðxÞ ¼ ex

k kk kcollð39Þk eð39Þk~kcollð39Þk ¼ kcollð39Þk þ eð39Þk

1 4.8966694 4.8966694 0.0000000 4.8966694

2 10.045190 10.045190 0.000000 10.045190

3 16.019267 16.019267 0.000000 16.019267

4 23.266271 23.266271 0.000000 23.266271

5 32.263707 32.263707 0.000000 32.263707

6 43.220020 43.220020 0.000000 43.220020

7 56.181594 56.181594 0.000000 56.181594

8 71.152998 71.152998 0.000000 71.152998

9 88.132119 88.132119 0.000000 88.132119

10 107.11668 107.11668 0.00000 107.11668

11 128.10502 128.10502 0.00000 128.10502

12 151.09604 151.09604 0.00000 151.09604

13 176.08900 176.08900 0.00000 176.08900

14 203.08337 203.08337 0.00000 203.08337

15 232.07881 232.07881 0.00000 232.07881

16 263.07507 263.07507 0.00000 263.07507

17 296.07196 296.07198 )0.00002 296.07196

18 331.06934 331.06940 )0.00005 331.06935

19 368.06713 368.06769 )0.00052 368.06717

20 407.06524 407.04923 0.01502 407.06425

Table 2

Corrected collocation solution for qðxÞ ¼ ðxþ 0:1Þ�2

k kk kcollð39Þk eð39Þk~kcollð39Þk ¼ kcollð39Þk þ eð39Þk

1 1.5198658 1.5198658 0.0000000 1.5198658

2 4.9433098 4.9433098 0.000000 4.9433098

3 10.284663 10.284663 0.000000 10.284663

4 17.559958 17.559958 0.000000 17.559958

5 26.782863 26.782863 0.000000 26.782863

6 37.964426 37.964426 0.000000 37.964426

7 51.113358 51.113358 0.000000 51.113358

8 66.236448 66.236448 0.000000 66.236448

9 83.338962 83.338963 0.000000 83.338963

10 102.42499 102.42499 0.00000 102.42499

11 123.49771 123.49771 0.00000 123.49771

12 146.55961 146.55961 0.00000 146.55961

13 171.61264 171.61265 0.00000 171.61265

14 198.65837 198.65838 0.00000 198.65838

15 227.69803 227.69803 0.00000 227.69803

16 258.73262 258.73262 0.00000 258.73262

17 291.76293 291.76296 )0.00002 291.76294

18 326.78963 326.78968 )0.00005 326.78963

19 363.81325 363.81390 )0.00052 363.81338

20 402.83424 402.81735 0.01502 402.83237

406 _I. C�elik / Appl. Math. Comput. 160 (2005) 401–410

Table 3

For qðxÞ ¼ ex error table

kk US

ð~KðhÞk �

kkÞ103

CN

ð~KðhÞk �

kkÞ103

IO2

ð~KðhÞk �

kkÞ103

VD1

ðKðhÞk �

kkÞ103

UT

ð~KðhÞk �

kkÞ103

MT

ð~KðhÞk �

kkÞ103

VD2

ð~KðhÞk �

kkÞ103

Andrew [1]

ðkk�~Kð40Þk Þ103

Collocation

ðkk�~kcollð39Þk Þ103

1 4.8966694 )0.0139 )0.0026 )0.0000 0.0037 0.0000 )0.0000 0.0000 0.00225 0.0000

2 10.045190 )0.0753 )0.0326 )0.0001 0.0328 0.0009 )0.0001 0.0000 0.0287 0.000

3 16.019267 )0.2940 )0.1111 0.0454 0.000

4 23.266271 )0.4927 )0.2318 )0.0011 0.3320 0.0096 )0.0000 0.0000 0.0935 0.000

5 32.263707 )1.1286 )0.3878 0.476 0.000

6 43.220020 )1.6781 )0.5823 )0.0033 0.6885 0.0930 )0.0005 0.0027 0.976 0.000

7 56.181594 )0.7448 )0.8158 1.53 0.000

8 71.152998 )5.7267 )1.0917 0.0106 1.2511 0.8407 )0.0015 )0.0138 2.15 0.000

9 88.132119 )5.7250 )1.4106 2.86 0.000

10 107.11668 )2.2789 )1.7816 0.0829 1.9832 4.4872 )0.0078 )0.0290 3.65 0.00

11 128.10502 )23.286 )2.1949 4.54 0.00

12 151.09604 )21.656 )2.6672 0.3047 2.9440 15.635 )0.0058 )0.0673 5.51 0.00

13 176.08900 )10.082 )3.2113 6.58 0.00

14 203.08337 )68.490 )3.8143 0.7551 4.2989 38.375 )0.0110 )0.0103 7.74 0.00

15 232.07881 )82.532 )4.4995 9.01 0.00

16 263.07507 )26.684 )5.2801 1.5548 5.6635 66.140 0.0113 )0.1343 10.4 0.00

17 296.07196 )117.02 )6.1622 11.8 0.00

18 331.06934 )254.16 )7.1575 2.8152 6.5856 68.815 0.1348 )0.1434 13.4 )0.0119 368.06713 )118.14 )8.3085 15.1 )0.0420 407.06524 )51.722 )9.6295 4.5852 9.5445 11.307 0.4650 )0.0538 16.9 0.99

_IC�elik

/Appl.Math.Comput.160(2005)401–410

407

Table 4

For qðxÞ ¼ ðxþ 0:1Þ�2error table

k kk US

ð~KðhÞk � kkÞ103

CN

ð~KðhÞk � kkÞ103

IO2

ð~KðhÞk � kkÞ103

VD1

ðKðhÞk � kkÞ103

Collocation

ðkk � ~kcollð39Þk Þ103

1 1.5198658 )0.0868 )0.0462 0.0106 )0.023 0.0000

2 4.9433098 )0.5548 )0.2931 0.0664 )0.166 0.0000

3 10.284663 )1.7289 )0.9037 0.000

4 17.559958 )3.8803 )2.0115 0.4330 )1.381 0.000

5 26.782863 )7.3116 )3.7173 0.000

6 37.964426 )12.144 )6.0952 1.2207 )3.056 0.000

7 51.113358 )18.296 )9.1986 0.000

8 66.236448 )27.192 )13.071 2.3659 )8.718 0.000

9 83.338962 )36.928 )17.753 0.001

10 102.42499 )47.755 )23.289 3.6638 )15.518 0.00

11 123.49771 )66.529 )29.720 0.00

12 146.55961 )81.467 )37.101 4.8035 )23.058 0.00

13 171.61264 )96.132 )45.486 )0.0114 198.65837 )135.13 )54.968 5.3975 )30.017 )0.0115 227.69803 )161.78 )65.627 0.00

16 258.73262 )170.50 )77.573 4.9420 )34.243 )0.0017 291.76293 )243.50 )90.918 )0.0118 326.78963 )334.06 )105.82 2.9121 )33.587 0.00

19 363.81325 )337.51 )122.45 )0.1320 402.83424 )367.43 )141.02 )1.3146 )25.122 1.87

408

_I.C�elik

/Appl.Math.Comput.160(2005)401–410

_I C�elik / Appl. Math. Comput. 160 (2005) 401–410 409

For qðxÞ ¼ ex and qðxÞ ¼ ðxþ 0:1Þ�2, the approximate eigenvalues, the

corrected eigenvalues and errors of Eq. (7) with the collocation method aregiven, respectively in Tables 1 and 2 for n ¼ 40.

4. Conclusions

The collocation method has been used to compute eigenvalues of the Sturm–

Liouville problem for n ¼ 40. The result of the computed approximate eigen-

values have been compared with the result of corrected Numerov method(CN), the uniderivative Simpson method (US), implicit Obrechkoff with

derivatives up to y00 (IO2), the method of Van den Berghe and De Meyer

(VD1), uniderivative top method (UT), multiderivative top method (MT), the

method of Van den Berghe and De Meyer (VD2) and with the study of Andrew

[1]. The errors of Sturm–Liouville problems are given in Tables 3 and 4 for the

CN, US, IO2, VD1, UT, MT and VD2 methods. The results are taken from [6]

for n ¼ 40, qðxÞ ¼ ex and qðxÞ ¼ ðxþ 0:1Þ�2. The errors for different solutions

could be seen in Tables 3 and 4 for qðxÞ ¼ ex and qðxÞ ¼ ðxþ 0:1Þ�2. As can be

seen easily from Tables 3 and 4, the results of the collocation method which

uses asymptotic correction techniques is better than the results of the other

methods.

The only the 20th eigenvalue for VD2 method gave better result than the

result of the corrected collocation method. Since full matrix is obtained for the

collocation method, all trial solution involved for the calculation. Because

band matrix is obtained for the other methods, a few terms were used for the

calculations. In the solution, using of full matrix; the truncation errors becauseof the computer calculations are more than the truncation errors caused by the

band matrix. Despite that the corrected collocation method presented is seen to

give more accurate results than the rival methods.

If the errors caused by the computer calculations can be decreased, the

better results can be obtained.

References

[1] A.L. Andrew, Twenty years of asymptotic correction for eigenvalue computation, ANZIAM J.

42 (2000) C96–C116.

[2] A.L. Andrew, J.W. Paine, Correction of Numerov�s eigenvalue estimates, Numer. Math. 47

(1985) 289–300.

[3] G. Vanden Berghe, H. De Mayer, Accurate computation of higher Sturm–Liouville eigenvalues,

Numer. Math. 59 (1991) 243–254.

[4] G. Fix, Asymptotic eigenvalues of Sturm–Liouville systems, Math. Anal. Appl. 19 (1967) 519–

525.

410 _I. C�elik / Appl. Math. Comput. 160 (2005) 401–410

[5] L. Fox, I.B. Paker, Chebyshev Polynomials in Numerical Analysis, Oxford University Press,

London, 1968.

[6] P. Ghelardoni, Approximations of Sturm–Liouville eigenvalues using boundary value methods,

Appl. Numer. Math. 23 (1997) 311–325.

[7] J.W. Paine, F.R. de Hoog, R.S. Andersen, On the correction of finite difference eigenvalue

approximations for Sturm–Liouville problems, Computing 26 (1981) 123–139.