Mathematical Problems in Engineering

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Numerical and Soft Computing Methods for Characteristic Value Problems of ODE and ODEs Systems

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Volume 2013 |Article ID 415479 | https://doi.org/10.1155/2013/415479

Guldem Yıldız, Bulent Yılmaz, O. A. Veliev, "Asymptotic and Numerical Methods in Estimating Eigenvalues", Mathematical Problems in Engineering, vol. 2013, Article ID 415479, 8 pages, 2013. https://doi.org/10.1155/2013/415479

Asymptotic and Numerical Methods in Estimating Eigenvalues

Academic Editor: Safa Bozkurt Coskun
Received17 Jan 2013
Accepted08 Mar 2013
Published22 Apr 2013

Abstract

Asymptotic formulas and numerical estimations for eigenvalues of SturmLiouville problems having singular potential functions, with Dirichlet boundary conditions, are obtained. This study gives a comparison between the eigenvalues obtained by the asymptotic and the numerical methods.

1. Introduction

Let be an operator generated in by the expression and by Dirichlet boundary conditions where is a complex-valued summable function.

In this paper, we consider the small and large eigenvalues of the operator when has a finite number of singularities. The large eigenvalues are investigated by the asymptotic method given in [1, 2]. Note that in classical investigations in order to obtain the asymptotic formulas of order it is required that be times differentiable (see [310]). The method of [1] gives the possibility of obtaining the asymptotic formulas of order of eigenvalues and eigenfunctions of when is an arbitrary summable complex-valued function. The small eigenvalues are investigated by numerical and asymptotic methods. Then, we compare the results with the ones obtained by the other methods.

Expression of differential equations in matrix form and the advances in the field of the computers have led to major developments in numerical methods. Regarding the numerical solution of the Sturm-Liouville problems, finite difference method is amongst the popular methods (see [11, 12]). Finite difference method can give effective results for the eigenvalues when it is used in connection with asymptotic correction technique. In [13] and [14] the Sturm-Liouville problems with Dirichlet and the general boundary conditions were studied, respectively. Andrew and Paine [15] found the approximate eigenvalues of regular Sturm-Liouville problem by using the finite element method. Chen and Ho [16] used the differential transform method to solve the eigenvalue problems. Ghelardoni [17] named some linear multistep methods as boundary value methods and found the approximate eigenvalues of Sturm-Liouville problem. Ghelardoni and Gheri [18] used the shooting technique for the calculation of the eigenvalues of Sturm-Liouville problem by considering the Prüfer transformation given in [19]. Kumar [20], Kumar and Aziz [21] gave numerical examples to linear or nonlinear boundary value problems by using finite differences method for singular boundary value problems. Kumar and Singh [22] made a study which collected and classified various calculation techniques for the solution of singular boundary value problems.

2. Asymptotic Formulas for Eigenvalues

It is well known that (see formulas (47a), (47b) in page 65 of [7]) the eigenvalues of the operator , where is a complex-valued summable function, consist of the sequence satisfying

In [1] (see Theorem 1 of [1]), it is proved that the eigenvalues satisfy the following formula where , , Note that in [1], without loss of generality, it was assumed that . Then using (4), the cases and (where , , , , are complex numbers) are investigated in detail.

In this paper, we consider the case where is a positive integer and is a complex number. First using (4) we prove the following.

Theorem 1. The eigenvalue of the operator with potential (7) satisfies the asymptotic formula: where

Proof. At (4) for , let us use the formula where In the last equality, using the transformations and we obtain Let . By (9) we have Arguing as in the proof of of [1] one can readily see that Therefore Thus (8) follows from (4) for . The theorem is proved.

Now assuming that we obtain more precise asymptotic formula by using more subtle estimations.

Theorem 2. If (16) holds, then the eigenvalue of the operator with potential (7) satisfies the asymptotic formula:

Proof. To prove the theorem we use (4) for , (5) and prove that
In (15), instead of and taking and we get From (19) one can readily see that there exists a constant such that for . Therefore, instead of equation of [1], using (20) and repeating the proof of equation of [1] we get the proof of (18). Thus the proof of the theorem follows from (4), (5), and (18). The theorem is proved.

3. Numerical Approximation

Now, we consider the small eigenvalues of the operator by a numerical method.

For the finite difference method [11, 19] take an equally spaced mesh where Writing as , as , and as , we use the centered difference approximation Substituting in (1) we obtain the approximating scheme Incorporating the boundary conditions, we get This can be written in matrix form as where is a tridiagonal matrix and The eigenvalues of (1), (2) are approximated by the eigenvalues of matrix .

In the previous section, the asymptotic formulas for eigenvalues of the operator (1), (2) with the potential (7) are investigated. In this section, we will find the eigenvalues of the operator by using the finite difference method when , for , and . Let us introduce the notation and denote the th eigenvalue of the operator by . The th eigenvalue of the operator , where is denoted by .

In order to be able to apply the Finite Difference method, the nodes should not coincide with the singular points. Let and nodal points be Then .

The approximate eigenvalues of the operators and obtained by the numerical method are denoted and , respectively.

Example 3. In this example we find the eigenvalues of the following boundary value problem for, , and by using Finite Difference method. In Table 1 an example of the computation of the eigenvalues of the operators and is given.
One can see from Table 1 that for the eigenvalues of the operators and are close to each other. This shows that the effect of the potential to the large eigenvalues is small. Moreover the eigenvalues in first, second, and third columns coincide with the eigenvalues in the sixth, fifth, and fourth columns, respectively, since the potential can be reduced to by using the transformation .



111,334612,061713,611213,611212,061711,334624,8946
241,008242,368042,565542,565542,368041,008254,4909
390,354392,143991,455691,455692,143990,3543103,912
4159,389161,282161,330161,330161,282159,389174,822
5248,089249,774249,278249,278249,774248,089261,654
6356,418357,740357,883357,883357,740356,418369,832
7484,332485,347486,023486,023485,347484,332497,594
8631,775632,690632,941632,941632,690631,775643,821
9789,683799,727800,334800,334799,727789,683811,972
10984,984986,288986,440986,440986,288984,984998,400
203892,793864,3303894,393894,393864,3303892,793906,97
308599,578600,858601,048601,048600,858599,578612,95
4014902,314903,514903,714903,714903,514902,314915,3
5022529,222530,622530,722530,722530,622529,222542,7
6031151,431152,831152,831152,831152,831151,431165,1
7040396,84039840398,340398,34039840396,840410,1
8049866,849867,949868,249868,249867,949866,849879,9
9059152,959154,359154,359154,359154,359152,959166,4
10067854,667856,167856,167856,167856,167854,667868,3

4. Comparison of the Asymptotic and Numerical Methods

In this section we compare the estimations obtained by numerical and asymptotic methods of the eigenvalues of the operator , where is defined by (30) and (29). The th eigenvalue of the operator is . The effect of the potential on the th eigenvalue of the operator , that is, the perturbation of the th eigenvalue when is perturbed by is Similarly, the effect of on the th eigenvalue , that is, the perturbation of the th eigenvalue when is perturbed by is The perturbations , evaluated by the numerical and asymptotic methods are denoted by , , , and , respectively.

According to Theorem 2 we define the approximate eigenvalues, denoted by and , of the operators and obtained by the asymptotic method as follows Therefore it is natural to define and by It readily follows from formulas (37) and (30) that It means that for the large eigenvalues the effect of is asymptotically equal to the sum of the effects of the potentials .

The perturbations and evaluated via the finite difference method are given in Table 2. In order to see the effect of the singular points, the number of subintervals is taken as .



16,8530109481,5076800753,8195229871,5076800756,8348831380,018127810
25,4364360391,6490441692,1358857871,6490441695,4339741240,002461915
36,8331275421,7127922343,4123230711,7127922346,8379075400,004779998
45,8335969171,7509832572,3325602131,7509832575,8345267260,000929810
56,8213541151,7771066463,2696403701,7771066466,8238536620,002499546
66,0142226851,7964127362,4221638001,7964127366,0149892720,000766587
76,8170745141,8114222053,1956287831,8114222056,8184731940,001398680
86,1225905511,8235147382,4760058541,8235147386,1230353300,000444779
96,8149938991,8335163253,1486776851,8335163256,8157103360,000716437
106,1966754341,8419542262,5128218731,8419542266,1967303260,000054891
206,3782473331,8850452782,6018253851,8850452786,3719159400,006331392

Table 2 shows that the effect of is approximately within the value range of and , equal to the sum of the effects of the potentials . Thus the perturbation estimations by the numerical methods validate the naturality of (38).

It is well known that if we consider the Sturm-Liouville operator where is a small positive parameter, then the asymptotic methods can be applied more successfully. The th eigenvalue of the operators is denoted by . The approximate eigenvalues obtained by the asymptotic and numerical methods are denoted by and , respectively.

It follows from Theorem 2 and formulas (36), (30) that where

In Tables 3, 4, and 5 the approximate eigenvalues obtained by the asymptotic method and their comparison with and nonperturbated eigenvalues for , , and , respectively, are given.



19,869616,77930816,72010,0592086,9097086,8505
239,478444,91543844,91490,0005385,4370385,4365
388,826495,66532295,65690,0084226,8389226,8305
4157,9137163,748052163,74730,0007525,8343525,8336
5246,7401253,572371253,55880,0135716,8322716,8187
6355,3058361,320361361,31990,0004616,0145616,0141
7483,6106490,44101490,42490,016116,830416,8143
8631,6547637,77751637,77700,000516,122816,1223
9799,4379806,267576806,25020,0173766,8296766,8123
10986,9604993,157379993,15700,0003796,1969796,1966
203947,84183954,2232163954,21750,0052166,3814166,3757
308882,64408889,1073478889,07820,0293476,4633476,4342
4015791,367015797,879315797,78700,08936,512366,42
5024674,011024680,5566324680,33120,226636,545636,3202
6035530,575835537,146135536,67860,46616,57036,1028
7048361,061648367,6509748366,78490,870976,589375,7233
8063165,468263172,07363170,59551,4736,60485,1273
9079943,795679950,4132779948,04662,363276,617674,251
10098696,044098702,6724598699,06523,602456,628453,0212



19,869610,560574810,55820,00237480,69097480,6886
239,478440,022119640,02211,96 × 10−50,54371960,5437
388,826489,510327889,50850,00182780,68392780,6821
4157,9137158,4971086158,49718,6 × 10−60,58340860,5834
5246,7401247,4233362247,42140,00193620,68323620,6813
6355,3058355,9072187355,90721,87 × 10−50,60141870,6014
7483,6106484,2936551484,29160,00205510,68305510,681
8631,6547632,2669645632,26680,00016450,61226450,6121
9799,4379800,1209185800,11870,00221850,68301850,6808
10986,9604987,580134987,57980,0003340,6197340,6194
203947,84183948,4799063948,47410,0058060,6381060,6323
308882,64408883,29038883,26110,02919960,64630,6171
4015791,367015792,0182715791,92590,09236770,651270,5589
5024674,011024674,6655724674,44010,2254650,654570,4291
6035530,575835531,2328735530,76540,46746940,657070,1896
7048361,061648361,7205148360,85440,86610550,658910,2072
8063165,468263166,1286563164,65111,47755050,660450,8171
9079943,795679944,4574179942,09072,36671090,661811,7049
10098696,044098696,7068598693,09963,60725460,662852,9444



19,86969,938701449,93850,000201440,069101440,0689
239,478439,5327878139,53281,219 × 10−50,054387810,0544
388,826488,8948284388,89460,000228430,068428430,0682
4157,9137157,9720142157,97201,423 × 10−50,05831420,0583
5246,7401246,8084326246,80820,000232640,06833260,0681
6355,3058355,3659045355,36594,46 × 10−60,06010450,0601
7483,6106483,6789196483,67860,00031960,06831960,068
8631,6547631,71591631,71580,000109950,061210,0611
9799,4379799,5062527799,50580,000452690,06835270,0679
10986,9604987,0224095987,02200,000409490,06200950,0616
203947,84183947,9055753947,89980,005774990,0637750,058
308882,64408882,7085958882,67940,029194850,0645950,0354
4015791,367015791,4321615791,33980,092364340,065160,0272
5024674,011024674,0764624673,85100,225458960,065460,16
6035530,575835530,6415535530,17400,467546470,065750,4018
7048361,061648361,1274648360,26140,866059350,065860,8002
8063165,468263165,5342263164,05671,477515330,066021,4115
9079943,795679943,8618379941,49512,366725030,066232,3005
10098696,044098696,110398692,50313,607195260,06633,5409

Table 3 shows the eigenvalues of operator obtained by asymptotic method and finite difference method, respectively, for . Here the number of subintervals is taken as .

Table 4 shows the eigenvalues of operator obtained by asymptotic method and finite difference method, respectively, for . Here the number of subintervals is taken as .

Table 5 shows the eigenvalues of operator obtained by asymptotic method and finite difference method, respectively, for . Here the number of subintervals is taken as .

5. Conclusion

It is natural and well known that for small values of the parameter and for large eigenvalues the asymptotic method gives us approximations with smaller errors. The numerical method, in general, gives better results for smaller eigenvalues. The tables show that the results of the asymptotic method also give quiet acceptable results for small eigenvalues, since is small.

Therefore we can easily observe that both of two methods give high-precision results for the calculation of the small eigenvalues. Additionally while the perturbation parameter tends to zero both of the methods are enhanced for smaller eigenvalues, but while this fact is limited to for the numerical approximation, the enhancement continues for the asymptotic method applied to higher eigenvalues. Thus we can conclude that the asymptotic method coupled with a perturbation parameter near to zero provides us a better approximation quality in calculating eigenvalues.

In Tables 35 there are two observations to be considered: the first observation is that for small eigenvalues the perturbated results by numerical and asymptotic methods are close to each other for all . The second observation is that for the large eigenvalues the perturbated results obained by asymptotic methods decrease linearly with respect to small , while the perturbated results obtained by numerical methods are almost the same for all values of . This shows that for small values of the perturbation parameter the asymptotic method is preferable.

Conflict of Interests

The authors of the paper do not have any direct or indirect financial relation with the commercial identities mentioned in the paper.

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Copyright © 2013 Guldem Yıldız et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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