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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

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 ). The method of  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  and  the Sturm-Liouville problems with Dirichlet and the general boundary conditions were studied, respectively. Andrew and Paine  found the approximate eigenvalues of regular Sturm-Liouville problem by using the finite element method. Chen and Ho  used the differential transform method to solve the eigenvalue problems. Ghelardoni  named some linear multistep methods as boundary value methods and found the approximate eigenvalues of Sturm-Liouville problem. Ghelardoni and Gheri  used the shooting technique for the calculation of the eigenvalues of Sturm-Liouville problem by considering the Prüfer transformation given in . Kumar , Kumar and Aziz  gave numerical examples to linear or nonlinear boundary value problems by using finite differences method for singular boundary value problems. Kumar and Singh  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 ) the eigenvalues of the operator , where is a complex-valued summable function, consist of the sequence satisfying

In  (see Theorem 1 of ), it is proved that the eigenvalues satisfy the following formula where , , Note that in , 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  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 , using (20) and repeating the proof of equation of  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 .

 1 11,3346 12,0617 13,6112 13,6112 12,0617 11,3346 24,8946 2 41,0082 42,3680 42,5655 42,5655 42,3680 41,0082 54,4909 3 90,3543 92,1439 91,4556 91,4556 92,1439 90,3543 103,912 4 159,389 161,282 161,330 161,330 161,282 159,389 174,822 5 248,089 249,774 249,278 249,278 249,774 248,089 261,654 6 356,418 357,740 357,883 357,883 357,740 356,418 369,832 7 484,332 485,347 486,023 486,023 485,347 484,332 497,594 8 631,775 632,690 632,941 632,941 632,690 631,775 643,821 9 789,683 799,727 800,334 800,334 799,727 789,683 811,972 10 984,984 986,288 986,440 986,440 986,288 984,984 998,400 20 3892,79 3864,330 3894,39 3894,39 3864,330 3892,79 3906,97 30 8599,57 8600,85 8601,04 8601,04 8600,85 8599,57 8612,95 40 14902,3 14903,5 14903,7 14903,7 14903,5 14902,3 14915,3 50 22529,2 22530,6 22530,7 22530,7 22530,6 22529,2 22542,7 60 31151,4 31152,8 31152,8 31152,8 31152,8 31151,4 31165,1 70 40396,8 40398 40398,3 40398,3 40398 40396,8 40410,1 80 49866,8 49867,9 49868,2 49868,2 49867,9 49866,8 49879,9 90 59152,9 59154,3 59154,3 59154,3 59154,3 59152,9 59166,4 100 67854,6 67856,1 67856,1 67856,1 67856,1 67854,6 67868,3

#### 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 .

 1 6,853010948 1,507680075 3,819522987 1,507680075 6,834883138 0,018127810 2 5,436436039 1,649044169 2,135885787 1,649044169 5,433974124 0,002461915 3 6,833127542 1,712792234 3,412323071 1,712792234 6,837907540 0,004779998 4 5,833596917 1,750983257 2,332560213 1,750983257 5,834526726 0,000929810 5 6,821354115 1,777106646 3,269640370 1,777106646 6,823853662 0,002499546 6 6,014222685 1,796412736 2,422163800 1,796412736 6,014989272 0,000766587 7 6,817074514 1,811422205 3,195628783 1,811422205 6,818473194 0,001398680 8 6,122590551 1,823514738 2,476005854 1,823514738 6,123035330 0,000444779 9 6,814993899 1,833516325 3,148677685 1,833516325 6,815710336 0,000716437 10 6,196675434 1,841954226 2,512821873 1,841954226 6,196730326 0,000054891 20 6,378247333 1,885045278 2,601825385 1,885045278 6,371915940 0,006331392

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.

 1 9,8696 16,779308 16,7201 0,059208 6,909708 6,8505 2 39,4784 44,915438 44,9149 0,000538 5,437038 5,4365 3 88,8264 95,665322 95,6569 0,008422 6,838922 6,8305 4 157,9137 163,748052 163,7473 0,000752 5,834352 5,8336 5 246,7401 253,572371 253,5588 0,013571 6,832271 6,8187 6 355,3058 361,320361 361,3199 0,000461 6,014561 6,0141 7 483,6106 490,44101 490,4249 0,01611 6,83041 6,8143 8 631,6547 637,77751 637,7770 0,00051 6,12281 6,1223 9 799,4379 806,267576 806,2502 0,017376 6,829676 6,8123 10 986,9604 993,157379 993,1570 0,000379 6,196979 6,1966 20 3947,8418 3954,223216 3954,2175 0,005216 6,381416 6,3757 30 8882,6440 8889,107347 8889,0782 0,029347 6,463347 6,4342 40 15791,3670 15797,8793 15797,7870 0,0893 6,51236 6,42 50 24674,0110 24680,55663 24680,3312 0,22663 6,54563 6,3202 60 35530,5758 35537,1461 35536,6786 0,4661 6,5703 6,1028 70 48361,0616 48367,65097 48366,7849 0,87097 6,58937 5,7233 80 63165,4682 63172,073 63170,5955 1,473 6,6048 5,1273 90 79943,7956 79950,41327 79948,0466 2,36327 6,61767 4,251 100 98696,0440 98702,67245 98699,0652 3,60245 6,62845 3,0212
 1 9,8696 10,5605748 10,5582 0,0023748 0,6909748 0,6886 2 39,4784 40,0221196 40,0221 1,96 × 10−5 0,5437196 0,5437 3 88,8264 89,5103278 89,5085 0,0018278 0,6839278 0,6821 4 157,9137 158,4971086 158,4971 8,6 × 10−6 0,5834086 0,5834 5 246,7401 247,4233362 247,4214 0,0019362 0,6832362 0,6813 6 355,3058 355,9072187 355,9072 1,87 × 10−5 0,6014187 0,6014 7 483,6106 484,2936551 484,2916 0,0020551 0,6830551 0,681 8 631,6547 632,2669645 632,2668 0,0001645 0,6122645 0,6121 9 799,4379 800,1209185 800,1187 0,0022185 0,6830185 0,6808 10 986,9604 987,580134 987,5798 0,000334 0,619734 0,6194 20 3947,8418 3948,479906 3948,4741 0,005806 0,638106 0,6323 30 8882,6440 8883,2903 8883,2611 0,0291996 0,6463 0,6171 40 15791,3670 15792,01827 15791,9259 0,0923677 0,65127 0,5589 50 24674,0110 24674,66557 24674,4401 0,225465 0,65457 0,4291 60 35530,5758 35531,23287 35530,7654 0,4674694 0,65707 0,1896 70 48361,0616 48361,72051 48360,8544 0,8661055 0,65891 0,2072 80 63165,4682 63166,12865 63164,6511 1,4775505 0,66045 0,8171 90 79943,7956 79944,45741 79942,0907 2,3667109 0,66181 1,7049 100 98696,0440 98696,70685 98693,0996 3,6072546 0,66285 2,9444
 1 9,8696 9,93870144 9,9385 0,00020144 0,06910144 0,0689 2 39,4784 39,53278781 39,5328 1,219 × 10−5 0,05438781 0,0544 3 88,8264 88,89482843 88,8946 0,00022843 0,06842843 0,0682 4 157,9137 157,9720142 157,9720 1,423 × 10−5 0,0583142 0,0583 5 246,7401 246,8084326 246,8082 0,00023264 0,0683326 0,0681 6 355,3058 355,3659045 355,3659 4,46 × 10−6 0,0601045 0,0601 7 483,6106 483,6789196 483,6786 0,0003196 0,0683196 0,068 8 631,6547 631,71591 631,7158 0,00010995 0,06121 0,0611 9 799,4379 799,5062527 799,5058 0,00045269 0,0683527 0,0679 10 986,9604 987,0224095 987,0220 0,00040949 0,0620095 0,0616 20 3947,8418 3947,905575 3947,8998 0,00577499 0,063775 0,058 30 8882,6440 8882,708595 8882,6794 0,02919485 0,064595 0,0354 40 15791,3670 15791,43216 15791,3398 0,09236434 0,06516 0,0272 50 24674,0110 24674,07646 24673,8510 0,22545896 0,06546 0,16 60 35530,5758 35530,64155 35530,1740 0,46754647 0,06575 0,4018 70 48361,0616 48361,12746 48360,2614 0,86605935 0,06586 0,8002 80 63165,4682 63165,53422 63164,0567 1,47751533 0,06602 1,4115 90 79943,7956 79943,86183 79941,4951 2,36672503 0,06623 2,3005 100 98696,0440 98696,1103 98692,5031 3,60719526 0,0663 3,5409

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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