Applied Mathematics and Computation 168 (2005) 125–134

www.elsevier.com/locate/amc

Approximate computation of eigenvalueswith Chebyshev collocation method

_Ibrahim Celik

Faculty of Arts and Sciences, Department of Mathematics, Pamukkale University,

Denizli 20100, Turkey

Abstract

In this study, Chebyshev collocation method is investigated for the approximate

computation of higher Sturm–Liouville eigenvalues by a truncated Chebyshev series.

Using the Chebyshev collocation points, this method transform the Sturm–Liouville

problems and given boundary conditions to matrix equation. By solving the algebraic

equation system, the approximate eigenvalues can be computed. Hence by using asymp-

totic correction technique, corrected eigenvalues can be obtained.

� 2004 Elsevier Inc. All rights reserved.

Keywords: Eigenvalue problem; Collocation method; Chebyshev series

1. Introduction

The concept of an eigenvalue problem is rather important for both in pureand applied mathematics, a physical system, such as a pendulum, a vibrating

and rotating shaft. All these physical systems are connected with eigenpairs

of the system.

0096-3003/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2004.08.024

E-mail address: i.celik@pamukkale.edu.tr

126 I_. Celik / Appl. Math. Comput. 168 (2005) 125–134

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

tial equation

d

dxpðxÞ du

dx

� �þ ðrðxÞk� q1ðxÞÞu ¼ 0:

This equation can be reduced to the canonical Liouville normal form,

u00 þ ðk� qðxÞÞu ¼ 0:

In this study, the Liouville normal form was investigated. If it is difficult to solve

the Sturm–Liouville problems or there are no exact solutions of Sturm–Liou-

ville problems, they can be solved by various approximate methods.

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

out. Fox and Parker [8] used Chebyshev series to solve differential eigenvalue

problems. Pain, de Hoog and Anderson [10] showed that in the case of the sec-ond order centered finite difference method with uniform mesh, the error, when

q(x) is constant (in the Liouville normal form), has the same asymptotic form

(for k! 1) as the error for general q(x). In the study, they gave corrected

finite difference approximation.

Fix [7] investigated the following Sturm–Liouville problem for general q(x).

u00 þ ðk� qðxÞÞu ¼ 0; 0 6 x 6 p

u0ð0Þ � auð0Þ ¼ u0ðpÞ � buðpÞ ¼ 0

or

uð0Þ ¼ uðpÞ ¼ 0

9>>>=>>>;: ð1Þ

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

the function U(x,k), the modifier Prufer phase, which is defined for any given

solution u(x,k) of Eq. (1) by the equation

tanðUÞ ¼ ðk� qðxÞÞ12u0

u

and he found recurrence formulas to obtain eigenvalues of Eq. (1) for general q(x).Andrew and Paine [3] improved the results of Numerov�s method with

asymptotic correction technique. Vanden Berghe and De Meyer [6] have devel-

oped special two step methods producing very accurate results. Ghelardnoi [9]

investigated the approximations of Sturm–Liouville eigenvalues using some

linear multistep methods, called Boundary Value Methods and correction tech-

nique of Andrew–Paine and Paine et al. [10] is extended to these methods.

By using the following Liouville normal form

u00 þ ðk� qðxÞÞu ¼ 0;

uð0Þ ¼ uðpÞ ¼ 0;

the asymptotic correction technique which [3,10] shown can be outlined as

follows.

I_. Celik / Appl. Math. Comput. 168 (2005) 125–134 127

Since the exact eigenvalues kk = k2, k = 1,2, . . . , is known for q(x) � 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 q(x) 6� 0, kðnÞk eigenvalues of more general problem can be solved by

using the finite difference techniques. Hence, the corrected eigenvalues was ob-

tained as follows

~kðnÞk ¼ kðnÞk � eðnÞk :

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

mate computation of eigenvalues, where trial solution is taken as the Chebyshev

polynomials, and the asymptotic correction technique use for collocationmethod.

2. Fundamental relations

Second order differential equations with variable coefficients is of the form

p2ðxÞy00ðxÞ þ p1ðxÞy0ðxÞ þ p0ðxÞyðxÞ ¼ f ðxÞ; ð2Þwhere the p2(x), p1(x), p0(x) and f(x) are defined on the interval of a 6 x 6 b.

The most general homogeny boundary conditions is;X1

j¼0

aijyðjÞðaÞ þ bijyðjÞðbÞ ¼ 0; i ¼ 1; 2; ð3Þ

where aij and bij are real constants.

Approximate trivial solution is expressed in the truncated Chebyshev series

yðxÞ ¼X0N

r¼0

arT rðxÞ; �1 6 x 6 1;

yðxÞ ¼X0N

r¼0

a�r T�r ðxÞ; 0 6 x 6 1

or

yðxÞ ¼X0N

r¼0

aþr Tþr ðxÞ; 0 6 x 6 p;

where the single prime indicates that the first term of the sum is taken withfactor 1

2.

Tr(x), T�r ðxÞ and Tþ

r ðxÞ denote the Chebyshev polynomials of the first kind of

degree r definite by

128 I_. Celik / Appl. Math. Comput. 168 (2005) 125–134

T rðxÞ ¼ cosðr arccos xÞ; �1 6 x 6 1;

T �r ðxÞ ¼ cosðr arccosð2x� 1ÞÞ; 0 6 x 6 1

and

Tþr ðxÞ ¼ cos r arccos

2

px� 1

� �� �; 0 6 x 6 p

and ar, a�r , aþr , r = 0,1, . . . ,N are the undetermined Chebyshev coefficients, [5].

Let us first consider the two order differential equation (2). For any finite

range, a 6 x 6 b, can be transformed to the basic range �1 6 x 6 1. Hence

we assume a trial solution of the form

yðxÞ ¼X0N

r¼0

arT rðxÞ; �1 6 x 6 1: ð4Þ

The trial solution (3) can be expressed in the matrix form as

½yðxÞ� ¼ T xA;

where Tx = [T0(x),T1(x), . . . ,TN(x)] and A ¼ 12a0; a1; . . . ; aN

� �T, [4].

To obtain such a solution, the Chebyshev collocation method can be used. It

can be taken

xi ¼ cosðN � iÞp

N; i ¼ 1; 2; . . . ;N � 1

which has called the turning points of Chebyshev collocation polynomials as

the collocation points.Substituting the Chebyshev collocation points in to Eq. (2) following equa-

tion can be obtained

p2ðxiÞy00ðxiÞ þ p1ðxiÞy0ðxiÞ þ p0ðxiÞyðxiÞ ¼ f ðxiÞor the equation could be written in the matrix form as

P 2Y 2 þ P 1Y 1 þ P 0Y 0 ¼ F ; ð5Þwhere

Pk ¼

pkðx1Þ 0 � � � 0

0 pkðx2Þ � � � 0

..

. ... ..

.

0 0 � � � pkðxN�1Þ

2666437775; F ¼

f ðx1Þf ðx2Þ...

f ðxN�1Þ

2666437775;

Y k ¼

yðkÞðx1ÞyðkÞðx2Þ

..

.

yðkÞðxN�1Þ

2666437775; k ¼ 1; 2:

I_. Celik / Appl. Math. Comput. 168 (2005) 125–134 129

The kth derivative of the function (4) with respect to x can be written as

yðkÞðxiÞ ¼X0N

r¼0

aðkÞr T rðxiÞ; �1 6 x 6 1; k ¼ 0; 1; 2;

where aðkÞr are Chebyshev coefficients and að0Þr ¼ ar and y(0)(x) = y(x). y(k)(xi)can be expressed in the matrix form

yðkÞðxiÞ� �

¼ T xiAðkÞ; k ¼ 0; 1; 2 ð6Þ

or the matrix equation

Y ðkÞ ¼ TAðkÞ; ð7Þwhere

T ¼

T x1

T x2

T xN�1

2666437775¼

T 0ðx1Þ T 1ðx1Þ � � � T Nðx1ÞT 0ðx2Þ T 1ðx2Þ � � � T Nðx2Þ

..

. ... ..

.

T 0ðxN�1Þ T 1ðxN�1Þ � � � T NðxN�1Þ

266664377775; AðkÞ ¼

12aðkÞ0

aðkÞ1

..

.

aðkÞN

2666664

3777775:

It is known from [11] that the relation between the Chebyshev coefficient ma-

trix A of y(x) and Chebyshev coefficient matrix A(k) of y(k)(x) can be given by

AðkÞ ¼ 2kMkA; ð8Þwhere

M ¼

0 12

0 32

0 52

� � � N2

0 0 2 0 4 0 � � � 0

0 0 0 3 0 5 � � � N

..

. ... ..

. ... ..

. ... ..

.

0 0 0 0 0 0 � � � N

0 0 0 0 0 0 � � � 0

266666666664

377777777775for odd N ;

M ¼

0 12

0 32

0 52

� � � 0

0 0 2 0 4 0 � � � 0

0 0 0 3 0 5 � � � N

..

. ... ..

. ... ..

. ... ..

.

0 0 0 0 0 0 � � � N

0 0 0 0 0 0 � � � 0

266666666664

377777777775for even N :

130 I_. Celik / Appl. Math. Comput. 168 (2005) 125–134

Hence the matrix representation of Eq. (7) and the matrix Eq. (5) can be

written respectively as

Y ðkÞ ¼ 2kTMkA andX2

k¼0

2kP kTMkA ¼ F :

Now the matrix representation of the boundary conditions can be found

easily. The boundary condition (3) can be reduced toX1

j¼0

aijyðjÞð�1Þ þ bijyðjÞð1Þ ¼ 0; i ¼ 1; 2; ð9Þ

on the interval [�1,1]. By relation (6) and (8), following equations:

yjð�1Þ½ � ¼ 2jT�1MjA; ð10Þ

yjð1Þ½ � ¼ 2jT 1MjA ð11Þcan be found. Where T�1 = b1,�1,1,�1, . . . , (�1)Nc, T1 = [1,1,1,1, . . . , 1]. Sub-stituting the matrix representations (10) and (11) into Eq. (9),X1

j¼0

2jðaijT�1 þ bijT 1ÞMjA ¼ 0; i ¼ 1; 2;

can be obtained.

When the range is taken as [0,1]

xi ¼1

21þ cos

ipN

� �; i ¼ 1; 2; . . . ;N � 1;

A�ðkÞ ¼ 4kMkA�; A� ¼ 1

2a�0; a

�1; . . . ; a

�N

� �; k ¼ 0; 1; 2;

T �0 ¼ 1;�1; 1;�1; . . . ; ð�1ÞN

;

T �1 ¼ 1; 1; 1; 1; . . . ; 1½ �;

and also when the range is taken as [0,p]

xi ¼p2

1þ cosipN

� �; i ¼ 1; 2; . . . ;N � 1;

AþðkÞ ¼ 4

p

� �k

MkAþ; Aþ ¼ 1

2aþ0 ; a

þ1 ; . . . ; a

þN

� �; k ¼ 0; 1; 2;

Tþ0 ¼ 1;�1; 1;�1; . . . ; ð�1ÞN

;

Tþp ¼ 1; 1; 1; 1; . . . ; 1½ �

equalities are used in the procedure.

I_. Celik / Appl. Math. Comput. 168 (2005) 125–134 131

Using the following representation:

W ¼ ½wij� ¼X2

k¼0

2kP kTMk; i ¼ 1; 2; 3; . . . ;N � 1; j ¼ 0; 1; 2; . . . ;N ;

Ui ¼ ui0; ui1; . . . ; uiN½ � ¼X1

j¼0

2jðaijT�1 þ bijT 1ÞMj; i ¼ 1; 2;

matrix equation can be written aseW A ¼ F ;

where

eW ¼

w10 w11 � � � w1N

w20 w21 � � � w2N

..

. ... ..

.

wN�10 wN�11 � � � wN�1N

u10 u11 � � � u1Nu20 21 � � � u2N

26666666664

37777777775:

Table 1

Approximate solution and errors in computed solution (12)

k kk kk � eKð80Þk kk � ~K

ð40Þk

~kcollð39Þk kk � ~k

collð39Þk

1 0.0000000 1.45E�4 2.32E�3 0.0000000 0.00000000

2 37.7596285 4.42E�4 7.52E�3 37.7593115 3.17E�4

3 37.8059002 9.93E�5 1.61E�3 37.8059007 �5.00E�7

4 37.8525995 4.43E�4 6.88E�3 37.8529195 �3.20E�4

5 70.5475097 1.97E�3 3.22E�2 70.5474910 1.87E�5

6 92.6538177 1.37E�3 2.25E�2 92.6531696 6.48E�4

7 96.2058159 7.28E�4 1.20E�2 96.2060264 �2.11E�4

8 102.254347 2.09E�3 3.45E�2 102.255686 �1.34E�3

9 120.267023

10 136.427351

11 153.729027

12 173.985230

13 196.532946

14 221.128698

15 247.876325

16 276.673983

17 307.415783

18 339.859849

19 375.562223

20 411.784944

132 I_. Celik / Appl. Math. Comput. 168 (2005) 125–134

Note that, if we take p0(x) = k�q(x), p1(x) = 0, p2(x) = 1 and f(x) = 0 then

Eq. (2) is reduce to Sturm–Liouville problems. Hence, the matrix equation

takes the form fW A ¼ 0. This set of equations has a nontrivial solution only

if the determinant of the coefficients matrix vanishes. This gives an equation

of degree (N � 1) in k and has (N � 1) roots which are the first (N � 1) approx-

imate eigenvalues of the problem.If q(x) � 0, then one can calculate the closed form errors

eNk ¼ dcollðNÞk � kq¼0

k ; k ¼ 1; 2; . . . ;

where kq¼0k is exact eigenvalues and dcollðNÞ

k is obtain in the (N + 1 · N + 1) sys-

tem of algebraic equations for q(x) � 0.

For q(x) 6� 0, kcollðNÞk can be obtained in the (N + 1 · N + 1) algebraic equa-

tion system. Hence the corrected eigenvalues of the general Sturm–Liouville

problems can be presented as

~kcollðNÞk ¼ kcollðNÞ

k � eðNÞk :

3. Numerical results

In this part of the paper, the presented method is used to find the eigen-

values of the

Table 2

Approximate solution and errors in computed solution (13)

k kk kk � Kð40Þk kk � eKð40Þ

k~kcollð39Þk kk � ~k

collð39Þk

1 4.89571 2.52E�6 2.52E�6 4.89571 0.00000

2 9.99955 3.04E�5 2.87E�5 9.99955 0.00000

3 15.4685 8.41E�5 4.54E�5 15.4685 0.00000

4 21.0371 3.86E�4 9.35E�5 21.0371 0.00000

5 28.1893 1.80E�3 4.76E�4 28.1893 0.00000

6 37.7907 5.40E�3 9.76E�4 37.7907 0.00000

7 49.6137 1.36E�2 1.53E�3 49.6137 0.00000

8 63.5205 3.07E�2 2.15E�3 63.5205 0.00000

9 79.4646 6.37E�2 2.86E�3 79.4646 0.00000

10 97.4279 1.23E�1 3.65E�3 97.4279 0.00000

11 117.402 2.22E�1 4.54E�3 117.402 0.00000

12 139.384 3.83E�1 5.51E�3 139.384 0.00000

13 163.370 6.33E�1 6.58E�3 163.370 0.00000

14 189.359 1.01E0 7.74E�3 189.359 0.00000

15 217.351 1.55E0 9.01E�3 217.351 0.00000

16 247.344 2.32E0 1.04E�2 247.344 0.00000

17 279.338 3.40E0 1.18E�2 279.338 0.00000

18 313.334 4.85E0 1.34E�2 313.333 1.00E�3

19 349.330 6.81E0 1.51E�2 349.331 �1.00E�3

20 387.326 9.38E0 1.69E�2 387.335 �9.00E�3

I_. Celik / Appl. Math. Comput. 168 (2005) 125–134 133

�y00 þ ð20 cos 2xþ 100 sin22xÞy ¼ ky;

y 0ð0Þ ¼ 0; y0ðpÞ ¼ 0:

(ð12Þ

Coffey–Evans equation and

�y00 þ exy ¼ ky;

y0ð0Þ ¼ 0; y0ðpÞ ¼ 0:

�ð13Þ

Sturm–Liouville equations. For this problems, the approximate corrected

eigenvalues ~kcollð39Þk and errors ðkk � ~k

collð39Þk Þ with the collocation method are

given respectively in Tables 1 and 2 for N = 39.

4. Conclusions

The collocation method has been used to compute eigenvalues of Eq. (12)

and Eq. (13) for N = 39. The results of the computed approximate eigenvalues

have been compared with the result of Andrew [1,2]. The errors of Sturm–

Liouville problems (12) and (13) are given in Tables 1 and 2 respectively. As

can be seen easily from Tables 1 and 2, the results of the corrected collocation

method for n = 39 is better than the results of [2] for N = 40 and n = 80 and is

better than the results Numerov�s method and corrected Numerov method of

[1] for N = 40.The result of the corrected collocation method gave better results than the

other 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 because of the computer calculations

are more than the truncation errors caused by the band matrix. It should be

note that the corrected collocation method presented in this paper gives more

accurate results than the rival methods.

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