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Transcript of Approximate calculation of eigenvalues with the method of weighted residuals–collocation method
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: [email protected], [email protected] (_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.