- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 415479, 8 pages
Asymptotic and Numerical Methods in Estimating Eigenvalues
1Department of Mathematics, Nigde University, 51200 Nigde, Turkey
2Department of Math, Faculty of Science and Letters, Marmara University, Göztepe Kampüsü, Kadiköy, 81040 Istanbul, Turkey
3Department of Mathematics, Dogus University, Acibadem, Kadiköy, 81010 Istanbul, Turkey
Received 17 January 2013; Accepted 8 March 2013
Academic Editor: Safa Bozkurt Coskun
Copyright © 2013 Guldem Yildiz 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.
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.
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 [3–10]). 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.
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 .
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 .
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.
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 .
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 3–5 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.
- B. Yilmaz and O. A. Veliev, “Asymptotic formulas for Dirichlet boundary value problems,” Studia Scientiarum Mathematicarum Hungarica, vol. 42, no. 2, pp. 153–171, 2005.
- O. A. Veliev and M. Toppamuk Duman, “The spectral expansion for a nonself-adjoint Hill operator with a locally integrable potential,” Journal of Mathematical Analysis and Applications, vol. 265, no. 1, pp. 76–90, 2002.
- G. D. Birkhoff, “Boundary value and expansion problems of ordinary linear differential equations,” Transactions of the American Mathematical Society, vol. 9, no. 4, pp. 373–395, 1908.
- N. Dunford and J. T. Schwartz, Linear Operators. Part III, Spectral Operators, Wiley-Interscience, New York, NY, USA, 1988.
- W. N. Everitt, J. Gunson, and A. Zettl, “Some comments on Sturm-Liouville eigenvalue problems with interior singularities,” Journal of Applied Mathematics and Physics, vol. 38, no. 6, pp. 813–838, 1987.
- V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel, Switzerland, 1986.
- M. A. Naimark, Linear Differential Operators, George G. Harrap and Company, 4th edition, 1967.
- B. N. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, USA, 1980.
- J. D. Tamarkin, “Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions,” Mathematische Zeitschrift, vol. 27, no. 1, pp. 1–54, 1928.
- E. C. Titchmarsh, Eigenfunction Expansions, vol. I, Oxford University Press, 1962.
- R. L. Burden, Numerical Analysis, Brooks Cole, Pacific Grove, Calif, USA, 7th edition, 2001.
- A. L. Andrew, “Correction of finite difference eigenvalues of periodic Sturm-Liouville problems,” Australian Mathematical Society Journal Series B, vol. 30, no. 4, pp. 460–469, 1989.
- J. W. Paine, F. R. de Hoog, and R. S. Anderssen, “On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems,” Computing, vol. 26, no. 2, pp. 123–139, 1981.
- R. S. Anderssen and F. R. de Hoog, “On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems with general boundary conditions,” BIT, vol. 24, no. 4, pp. 401–412, 1984.
- A. L. Andrew and J. W. Paine, “Correction of finite element estimates for Sturm-Liouville eigenvalues,” Numerische Mathematik, vol. 50, no. 2, pp. 205–215, 1986.
- C.-K. Chen and S.-H. Ho, “Application of differential transformation to eigenvalue problems,” Applied Mathematics and Computation, vol. 79, no. 2-3, pp. 173–188, 1996.
- P. Ghelardoni, “Approximations of Sturm-Liouville eigenvalues using boundary value methods,” Applied Numerical Mathematics, vol. 23, no. 3, pp. 311–325, 1997.
- P. Ghelardoni and G. Gheri, “Improved shooting technique for numerical computations of eigenvalues in Sturm-Liouville problems,” Nonlinear Analysis, vol. 47, pp. 885–896, 2001.
- J. D. Pryce, Numerical Solution of Sturm-Liouville Problems, Clarendon Press, Oxford, UK, 1993.
- M. Kumar, “A new finite difference method for a class of singular two-point boundary value problems,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 551–557, 2003.
- M. Kumar and T. Aziz, “A uniform mesh finite difference method for a class of singular two-point boundary value problems,” Applied Mathematics and Computation, vol. 180, no. 1, pp. 173–177, 2006.
- M. Kumar and N. Singh, “A collection of computational techniques for solving singular boundary-value problems,” Advances in Engineering Software, vol. 40, no. 4, pp. 288–297, 2009.