Applied Mathematics and Computation 170 (2005) 285–295

www.elsevier.com/locate/amc

Approximate solution of periodicSturm–Liouville problems

with Chebyshev collocation method

_Ibrahim Celik a,*, Guzin Gokmen b

a Department of Mathematics, Faculty of Arts and Sciences,

Pamukkale University, Denizli 20017, Turkeyb Department of Mathematics, Faculty of Arts and Sciences,

Dokuz Eylul University, Izmir 35000, Turkey

Abstract

This study investigates the eigenvalues of regular Sturm–Liouville problems with

Chebyshev collocation method for the approximate computation of periodic boundary

conditions by a truncated Chebyshev series. Corrected eigenvalues with the Chebyshev

collocation method was obtained by using asymptotic correction technique. The results

were encouraging.

� 2005 Elsevier Inc. All rights reserved.

Keywords: Periodic eigenvalue problem; Collocation method; Chebyshev series

1. Introduction

The general Sturm–Liouville problem can be written as

0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.amc.2004.11.038

* Corresponding author.

E-mail address: i.celik@pamukkale.edu.tr (_I. Celik).

286 _I. Celik, G. Gokmen / Appl. Math. Comput. 170 (2005) 285–295

d

dxpðxÞ du

dx

� �þ ðrðxÞk � q1ðxÞÞu ¼ 0: ð�Þ

This problem arose first in the context of the Separation of Variables method

for partial differential equations. This and related methods continue to give rise

to Sturm–Liouville problems which model phenomena such as the earth�s seis-mic behavior, the propagation of sonar in the water stratified by varying den-

sity, and the stability and velocity of large-scale waves in the atmosphere [9].Sturm–Liouville problem in (*) can be reduced to following form:

u00 þ ðk � qðxÞÞu ¼ 0;which is called the Liouville normal form, or the equivalently the one-dimen-

sional, time-independent form of the all-pervading equation of physics, theSchrodinger equation. If q(x) is a potential function, then an eigenvalue is en-

ergy level and its eigenfunction is the corresponding wavefunction in the quan-

tum physicist or chemist.

This study takes the Liouville normal form and investigates the approximate

solution of the problem. There is no exact solution of Sturm–Liouville prob-

lems or it is difficult to solve. Therefore, various approximate methods need

to be investigated to solve the problem.

Paine et al. [8] presented a method for improving finite difference eigenvaluesestimates for Sturm–Liouville problems of the form as

�y 00 þ qðxÞy ¼ ky; ð1Þ

yð0Þ ¼ yðpÞ ¼ 0: ð2ÞThey showed that when q is a constant, the error in kk has the same asymptoticform as k! 1, the error for general q. To solve (1) and (2), the asymptoticcorrection technique, which is developed by Paine et al. [8] and Andrew and

Paine [3], can be outlined as follows:

Since the exact eigenvalues kk = k2, k = 1, 2, . . . , is known for q(x) 0, theclosed form of the errors can be calculated as

enk ¼ dðnÞk � kk; k ¼ 1; 2; . . . ;

where dðnÞk indicates algebraic eigenvalues obtained by the finite difference

methods.

Also, for q(x)5 0, kðnÞk eigenvalues of the Liouville normal form can be

solved using the finite difference techniques. The corrected eigenvalues can be

obtained as follows:

ekðnÞk ¼ kðnÞ

k � eðnÞk :

The accuracy of the estimates obtained for high eigenvalues may be improved

using the known error for q = 0 to correct the estimates obtained for general q.

_I. Celik, G. Gokmen / Appl. Math. Comput. 170 (2005) 285–295 287

Numerical results indicate that this correction generally improves the accuracy

of the all computed eigenvalues.

Anderssen and Hoog [1] extended the method of Paine et al. [8] to the

general separated boundary conditions

r1y0ð0Þ þ r2yð0Þ ¼ r3y 0ðpÞ þ r4yðpÞ ¼ 0:Three important exceptions not emphasized in [1] are

y 0ð0Þ ¼ y0ðpÞ ¼ 0; ð3Þ

yð0Þ ¼ y0ðpÞ ¼ 0; ð4Þand

y 0ð0Þ ¼ yðpÞ ¼ 0: ð5ÞCorrected eigenvalues are obtained by adding to centered finite difference esti-

mate of kk, (k � 1)2 � 4sin2[(k � 1)h/2]/h2 in case (3) and (k � 1/2)2 �4sin2[(2k � 1)h/4]/h2 in case (4) or (5), where n is number of equal subintervalsand h = p/n.There are also several examples involving non-separated boundary condi-

tions, but not all are suitable candidates for this asymptotic correctiontechnique. For example, with the boundary conditions

yð0Þ ¼ �yðpÞ; y0ð0Þ ¼ y0ðpÞ ð6Þor

yð0Þ ¼ yðpÞ; y0ð0Þ ¼ �y0ðpÞ ð7Þall real numbers are eigenvalues of (1) when q is constant.

The periodic boundary conditions

yð0Þ ¼ yðpÞ; y0ð0Þ ¼ y0ðpÞ ð8Þand the semiperiodic boundary conditions

yð0Þ ¼ �yðpÞ; y0ð0Þ ¼ �y0ðpÞ ð9Þare important non-separated boundary conditions for which the correctiontechnique can be implemented successfully. Periodic boundary conditions arise

in the study of planetary orbits and other periodic physical phenomena. Prob-

lems with these boundary conditions in applications [2,10] usually satisfy

qð0Þ ¼ qðpÞ: ð10ÞVanden Berghe et al. [10] gave a modification of the classical finite difference

scheme and presented numerical examples in which application of the same cor-rection to the modified difference scheme gives more accurate results than [2].

Celik [5,6] investigated the collocation method for the approximate compu-

tation of Sturm–Liouville eigenvalues by a truncated Chebyshev series, and his

results were encouraging.

288 _I. Celik, G. Gokmen / Appl. Math. Comput. 170 (2005) 285–295

This study investigates the solution of the Liouville normal form by correct-

ing to the collocation method for approximate computation of eigenvalues of

periodic Sturm–Liouville problems, where trial solution is taken as the

truncated Chebyshev series.

2. Fundamental relations

The following equation is the Liouville normal form of the general Sturm–

Liouville problem:

y 00ðxÞ þ ðk � qðxÞÞyðxÞ ¼ 0; ð11Þwhere q(x) is defined on the interval of a 6 x 6 b and k is a parameter. Themost general homogeny boundary conditions are

X1j¼0

aijyðjÞðaÞ þ bijyðjÞðbÞ ¼ 0; i ¼ 1; 2; ð12Þ

where aij and bij are real constants.

Approximate trivial solution is expressed in the truncated Chebyshev series

as

yðxÞ ¼X0N

r¼0arT rðxÞ; �1 6 x 6 1;

yðxÞ ¼X0N

r¼0a�r T

�r ðxÞ; 0 6 x 6 1

or

yðxÞ ¼X0N

r¼0aþr T

þr ðxÞ; 0 6 x 6 p;

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

factor 12[7].

Tr(x), T�r ðxÞ and Tþ

r ðxÞ denote the Chebyshev polynomials of the first kind ofdegree r definite by

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

_I. Celik, G. Gokmen / Appl. Math. Comput. 170 (2005) 285–295 289

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

Let us first consider Sturm–Liouville problem (11). 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

r¼0arT rðxÞ; �1 6 x 6 1: ð13Þ

The trial solution (13) can be expressed in the matrix form as

½yðxÞ� ¼ T xA;

where Tx = [T0(x),T1(x), . . . , TN(x)] and A ¼ ½12a0; a1; . . . ; aN ðxÞ�T.

In order to obtain a solution, the Chebyshev collocation method can beused. As a collocation points, it can be taken

xi ¼ cosðN � iÞp

N; i ¼ 1; 2; . . . ;N � 1;

which has called the turning points of Chebyshev polynomials.

Substituting the Chebyshev collocation points into Eq. (11) following

expression can be obtained:

y 00ðxiÞ þ ðk � qðxiÞÞyðxiÞ ¼ 0:The equation could be re-written in the matrix form as

Y 2 þ PY ¼ 0; ð14Þwhere

P ¼

pðx1Þ 0 0

0 pðx2Þ 0

..

. ... ..

.

0 0 pðxN�1Þ

266664

377775; Y ¼

yðx1Þyðx2Þ...

yðxN�1Þ

266664

377775; Y 2 ¼

yð2Þðx1Þyð2Þðx2Þ

..

.

yð2ÞðxN�1Þ

266664

377775:

The second derivative of (13) with respect to x can be written as

yð2ÞðxiÞ ¼X0N

r¼0að2Þr T rðxiÞ; �1 6 x 6 1;

where að2Þr are Chebyshev coefficients and að0Þr ¼ ar and y(0)(x) = y(x). y(2)(xi)

can be expressed in the matrix form as

½yð2ÞðxiÞ� ¼ T xiAð2Þ ð15Þ

or the matrix equation

Y ð2Þ ¼ TAð2Þ; ð16Þ

290 _I. Celik, G. Gokmen / Appl. Math. Comput. 170 (2005) 285–295

where

T ¼

T x1

T x2

..

.

T xN�1

266664

377775¼

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

..

. ... ..

.

T 0ðxN�1Þ T 1ðxN�1Þ T NðxN�1Þ

266664

377775; Að2Þ ¼

12að2Þ0að2Þ1...

að2ÞN

2666664

3777775:

The relation between the Chebyshev coefficient matrix A of y(x) and Cheby-

shev coefficient matrix A(2) of y(2)(x) can be given [11] as

Að2Þ ¼ 22M2A; ð17Þwhere

M ¼

0 120 3

20 5

2 N

2

0 0 2 0 4 0 0

0 0 0 3 0 5 N... ..

. ... ..

. ... ..

. ...

0 0 0 0 0 0 N0 0 0 0 0 0 0

2666666664

3777777775

for odd N ;

M ¼

0 120 3

20 5

2 0

0 0 2 0 4 0 N0 0 0 3 0 5 0

..

. ... ..

. ... ..

. ... ..

.

0 0 0 0 0 0 N0 0 0 0 0 0 0

2666666664

3777777775

for even N :

The matrix representation of Eq. (16) and the matrix equation (14) can be

written respectively as

Y ð2Þ ¼ 22TM2A and 4TM2Aþ PTA ¼ 0:The matrix representation of the boundary conditions (12) can be reduced to

X1j¼0

aijyðjÞð�1Þ þ bijyðjÞð1Þ ¼ 0; i ¼ 1; 2 ð18Þ

on the interval [�1,1]. Using (15) and (17), the following equations,½yjð�1Þ� ¼ 2jT�1MjA; ð19Þ

½yjð1Þ� ¼ 2jT 1MjA ð20Þcan be obtained. Where T�1 = [1,�1,1,�1, . . . ,(�1)N], T1 = [1,1,1,1, . . ., 1].

_I. Celik, G. Gokmen / Appl. Math. Comput. 170 (2005) 285–295 291

Substituting (19) and (20) into Eq. (18)

X1j¼02jðaijT�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 ip

N

� �; i ¼ 1; 2; . . . ;N � 1;

A�ð2Þ ¼ 42M2A�; A� ¼ 1

2a�0; a

�1; . . . ; a

�N

� �

T �0 ¼ ½1;�1; 1;�1; . . . ; ð�1ÞN �; T �

1 ¼ ½1; 1; 1; 1; . . . ; 1�and also when the range is taken as [0,p]

xi ¼p21þ cos ip

N

� �; i ¼ 1; 2; . . . ;N � 1;

Aþð2Þ ¼ 4

p

� �2M2Aþ; Aþ ¼ 1

2aþ0 ; a

þ1 ; . . . ; a

þN

� �;

Tþ0 ¼ ½1;�1; 1;�1; . . . ; ð�1ÞN �; Tþ

p ¼ ½1; 1; 1; 1; . . . ; 1�;the equalities used in the procedure can be obtained.

Using the following representation

W ¼ ½wij� ¼ 4TM2 þ PT ; i ¼ 1; 2; 3; . . . ;N � 1; j ¼ 0; 1; 2; . . . ;N ;

Ui ¼ ½ui0; ui1; . . . ; uiN � ¼X1j¼02jðaijT�1 þ bijT 1ÞMj; i ¼ 1; 2

matrix form can be written as

eW A ¼ 0;where

eW ¼

w10 w11 w1N

w20 w21 w2N

..

. ... ..

.

wN�10 wN�11 wN�1N

u10 u11 u1N

266666666664

377777777775:

u20 u21 u2N

292 _I. Celik, G. Gokmen / Appl. Math. Comput. 170 (2005) 285–295

This set of equations, eW A ¼ 0, has a non-trivial solution only if the determi-nant of the coefficients matrix vanishes. This gives an polynomial of degree

(N � 1) in k and has (N � 1) roots which are the first (N � 1) approximateeigenvalues of the problem.

If q(x) 0, then one can calculate the form errors

eNk ¼ dcollðNÞk � kq¼0

k ; k ¼ 1; 2; . . . ;where kq¼0

k is exact eigenvalues and dcollðNÞk is obtained in the (N + 1 ÆN + 1) sys-

tem of algebraic equations for q(x) 0.For q(x) 6 0, kcollðNÞ

k can be obtained in the (N + 1 ÆN + 1) algebraic equa-tion system. Hence, the corrected eigenvalues of the general Sturm–Liouville

problems can be presented as

ekcollðNÞk ¼ kcollðNÞ

k � eðNÞk :

3. Numerical results

The presented method is used to find the eigenvalues of the form

y00 þ ðk � 10 cos 2xÞy ¼ 0yð0Þ ¼ yðpÞ; y0ð0Þ ¼ y0ðpÞ

ð21Þ

and

y0 þ ðk � x2ðp � xÞÞy ¼ 0yð0Þ ¼ yðpÞ; y0ð0Þ ¼ y0ðpÞ

ð22Þ

Table 1

Approximate solution and errors in computed solution of Eq. (21)

k kk eKð20Þk � kk eKð40Þ

k � kk eKð80Þk � kk ekcollð41Þk kk � ekcollð41Þk

2 2.09946 0.0714 0.0175 0.0044 2.09946 0.00000

3 7.44911 0.0974 0.0235 0.0058 7.44911 0.00000

4 16.64822 0.0745 0.0169 0.0041 16.64822 0.00000

5 17.09658 0.0242 0.0048 0.0011 17.09658 0.00000

6 36.35887 0.0687 0.0134 0.0031 36.35887 0.00000

7 36.36090 0.0675 0.0131 0.0031 36.36090 0.00000

8 64.19884 0.0898 0.0145 0.0033 64.19884 0.00000

10 100.12637 0.1264 0.0158 0.0034 100.12637 0.00000

12 144.08745 0.1965 0.0175 0.0034 144.08745 0.00000

14 196.06412 0.3575 0.0197 0.0036 196.06412 0.00000

16 256.04903 1.1214 0.0225 0.0037 256.04902 0.00001

18 324.03870 3.3904 0.0263 0.0038 324.03697 0.00173

20 400.03133 �5.7942 0.0313 0.0040 400.03022 0.00111

Table 2

Approximate solution and errors in computed solution of Eq. (22)

kk eKð20Þk � kk eCð20Þ

k � kk eKð40Þk � kk eCð40Þ

k � kk eKð80Þk � kk eCð80Þ

k � kk ekcollð41Þk kk � ekcollð41Þk

1 0.0162187 �0.0071436 0.0040215 �0.0017898 0.0010033 �0.0004477 2.0294 0.0000

2 6.5005 0.0144015 �0.0061183 0.0035214 �0.0015784 0.0008756 �0.0003975 6.5005 0.0000

3 7.0151 0.0005517 0.0002637 0.0001520 0.0000792 0.0000389 0.0000206 7.0151 0.0000

4 18.5848 0.0151506 �0.0088496 0.0035783 �0.0023903 0.0008826 �0.0006079 18.5848 0.0000

5 18.6655 0.0024034 0.0023005 0.0006379 0.0006105 0.0001615 0.0001547 18.6655 0.0000

6 38.5816 0.0172910 �0.0077821 0.0038340 �0.0023911 0.0009330 �0.0006215 38.5816 0.0000

7 38.6215 0.0021597 0.0020838 0.0006269 0.0006023 0.0001613 0.0001551 38.6215 0.0000

8 66.5821 0.0200321 �0.0055205 0.0040092 �0.0023175 0.0009563 �0.0006233 66.5821 0.0000

9 66.6054 0.0017155 0.0016938 0.0006047 0.0005879 0.0001593 0.0001550 66.6054 0.0000

10 102.5825 0.0545597 �0.0012836 0.0041890 �0.0021885 0.0009728 �0.0006192 102.5825 0.0000

11 102.5977 0.0009534 0.0009517 0.0005762 0.0005649 0.0001572 0.0001541 102.5977 0.0000

12 146.5829 0.0331513 0.0070930 0.0044024 �0.0020046 0.0009874 �0.0006116 146.5829 0.0000

13 146.5935 �0.0005065 �0.0005683 0.0005407 0.0005333 0.0001557 0.0001528 146.5935 0.0000

14 198.5910 0.0527382 0.0263968 0.0046700 �0.0017563 0.0010023 �0.0006010 198.5831 0.0000

15 198.5910 �0.0040232 �0.0042463 0.0004960 0.0004916 0.0001525 0.0001509 198.5910 0.0000

16 258.5833 0.1155289 0.0887600 0.0050144 �0.0014259 0.0010185 �0.0005876 258.5833 0.0000

17 258.5893 �0.0173409 �0.0181969 0.0004394 0.0004373 0.0001498 0.0001485 258.5893 0.0000

18 326.5834 0.5571154 0.5325781 0.0054656 �0.0009854 0.0010365 �0.0005716 326.5835 0.0001

19 326.5882 �0.2361676 �0.2400345 0.0003670 0.0003665 0.0001467 0.0001456 326.5882 0.0000

20 402.5835 �0.5197643 �0.5245948 0.0060690 �0.0003908 0.0010571 �0.0005525 402.5874 �0.0039

_I.Celik

,G.Gokmen

/Appl.Math.Comput.170(2005)285–295

293

294 _I. Celik, G. Gokmen / Appl. Math. Comput. 170 (2005) 285–295

periodic Sturm–Liouville problems. If we take N = 41, then a polynomial of de-

gree 40 in k is obtained. The approximate corrected eigenvalues ekcollð41Þk and er-

rors ðkk � ekcollð41Þk Þ with the collocation method are given respectively in Tables1 and 2.

4. Conclusions

The collocation method has been used to compute eigenvalues of Eqs. (21)

and (22) for N = 41. The results of the computed approximate eigenvalues havebeen compared with the result of Andrew [2] for Eq. (21). The results of the

computed approximate eigenvalues have been compared with the results of An-

drew [2] and Condon [4] for Eq. (22). The errors of Sturm–Liouville problems

(21) and (22) are given in Tables 1 and 2, respectively. As can be seen in Tables

1 and 2, the results of the corrected collocation method for N = 41 is better

than the results of [2] for N = 40 and [4] for N = 40 and 80. For Eq. (22), in

Table 2, the only the 20th eigenvalue provided by [4] better than the value of

the corrected collocation method with the Chebyshev series.The performance of the corrected collocation method provides better results

than the current methods in terms of values of eigenvalues. All trial solution

involved for the calculation since full matrix is obtained for the collocation

method. 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 trunca-

tion errors because of the computer calculations are more than the truncation

errors caused by the band matrix. It should be noted that the corrected collo-

cation method presented in this paper gives more accurate results than thecurrent methods.

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