Abstract and Applied Analysis

Abstract and Applied Analysis / 2014 / Article
Special Issue

Recent Trends in Boundary Value Problems 2014

View this Special Issue

Research Article | Open Access

Volume 2014 |Article ID 695303 | https://doi.org/10.1155/2014/695303

B. Chanane, A. Boucherif, "Computation of the Eigenpairs of Two-Parameter Sturm-Liouville Problems Using the Regularized Sampling Method", Abstract and Applied Analysis, vol. 2014, Article ID 695303, 6 pages, 2014. https://doi.org/10.1155/2014/695303

Computation of the Eigenpairs of Two-Parameter Sturm-Liouville Problems Using the Regularized Sampling Method

Academic Editor: Juan J. Nieto
Received10 Apr 2014
Accepted28 May 2014
Published09 Jun 2014

Abstract

This paper deals with the computation of the eigenvalues of two-parameter Sturm-Liouville (SL) problems using the Regularized Sampling Method, a method which has been effective in computing the eigenvalues of broad classes of SL problems (singular, non-self-adjoint, nonlocal, impulsive, etc.). We have shown, in this work that it can tackle two-parameter SL problems with equal ease. An example was provided to illustrate the effectiveness of the method.

1. Introduction

In an interesting paper published in 1963, Arscott [1] showed that the method of separation of variables used in solving boundary value problems for Laplace’s equation leads to a two-parameter eigenvalue problem for ordinary differential equations with the auxiliary requirement that the solutions satisfy boundary conditions at several points. This has led to an extensive development of multiparameter spectral theory for linear operators (see, e.g., [28]). In [9], the authors give an overview of results on two-parameter eigenvalue problems for second order linear differential equations. Several properties of corresponding eigencurves are given. In [5], the authors have obtained interesting geometric properties of the eigencurves (e.g., the condition of transversal intersections is equivalent to the simplicity of the eigenvalues in the sense of Chow and Hale). All the above works are concerned with the theoretical aspect of the existence of eigenvalues. Also, several authors have dealt with the theoretical numerical analysis of two-parameter eigenvalue problems, [10, 11] and the references therein as well as the works of Plestenjak and his collaborators [1215]. Eigenvalue problems have played a major role in the applied sciences. Consequently, the problem of computing eigenvalues of one-parameter problems has attracted many researchers (see, e.g., [16, 17] and the references therein).

Concerning the computations of eigenvalues of one-parameter Sturm-Liouville problems, the authors in [18] introduced a new method based on Shannon’s sampling theory. It uses the analytic properties of the boundary function. The method has been generalized to a class of singular problems of Bessel type [19]. The books by Atkinson [20], Chow and Hale [21], Faierman [22], McGhee and Picard [23], Sleeman [24], Volkmer [25], and the long awaited monograph by Atkinson and Mingarelli [26] contain several results on eigenvalues of multiparameter Sturm-Liouville problems and the corresponding bifurcation problems. However, no attempt has been made to compute the eigenvalues of two-parameter Sturm-Liouville problems using the approach based on the Regularized Sampling Method introduced recently by the first author in [27] to compute the eigenvalues of general Sturm-Liouville problems and extended to the case of singular [28], non-self-adjoint [29], nonlocal [30], and impulsive SLPs [31]. We will consider, in this paper, the computation of the eigenpairs of two-parameter Sturm-Liouville problems with three-point boundary conditions using the Regularized Sampling Method.

2. The Characteristic Function

Consider the two-parameter Sturm-Liouville problem as follows, where and are positive and in , , and some given constant.

By an eigenvalue of (1) we mean a value of the couple for which problem (1) has a nontrivial solution. Conditions that insure the existence of eigenvalue are given in [24, 7, 32, 33]. In fact, under fairly general conditions, it has been shown (see [21, 34]) that there are smooth curves of eigenvalues (actually eigenpairs). Our objective is to effectively localize the eigencurves in the parameter -plane. We should point out that we have restricted our attention to Dirichlet boundary conditions in order to eliminate technical details that might obscure the ideas.

We will associate with (1) the initial value problem as follows, and deal first with the unperturbed case then with the perturbed case .

2.1. The Unperturbed Case

In this case, (2) reduces to where .

Theorem 1. The solution of (3) is an entire function of for each fixed of order and type and satisfies the following estimate, for each fixed , where for .

Proof. From (3), we get the following integral equation,
Let We will show, by induction on , that It is true for . Assume it is true for . We will show that it is true for . Indeed, from (6), we have Using the fact that the expression attains its maximum at over , we get and using the fact that , we obtain that is, (7) is true for . Hence, it is true for all .
Now, and the series is absolutely and uniformly convergent since where is the modified Bessel function of the first kind order 1 as follows, Using the fact that as , we get Therefore, is an entire function of , as a uniformly convergent series of entire functions, for each fixed , of order and type . This concludes the proof.

We will make use of Liouville-Green’s transformation to bring (2) to the following form, which can be written as an integral equation as follows, where .

Returning to the original variables, we deduce that satisfies the following integral equation, where

We will present next some estimates whose proofs are immediate and left to the reader.

Lemma 2. The function satisfies the following estimate,

Lemma 3. The function defined by where and is a positive integer, is an entire function of for each fixed of order and type . Furthermore, satisfies the following estimate,

Lemma 4. The function satisfies the following estimate,

Combining the above results, we obtain the following theorem.

Theorem 5. The function is an entire function of for each fixed of order and type and satisfies the estimate where depends on but is independent of .

Let denote the Paley-Wiener space as follows, we have the following.

Theorem 6. , as a function of , belongs to the Paley-Wiener space , where for each fixed .

2.2. The Perturbed Case

Let and be two linearly independent solutions of satisfying and ; then, the method of variation of parameters shows that (2) can be written as the integral equation as follows, where .

Here again, it is not hard to show that is an entire function of for each , of order and type . Multiplication by gives a function of in a Paley-Wiener space for each . More specifically, we have the following.

Theorem 7. The function , defined by , belongs to , where as a function of for each , and satisfies the following estimate, where depends on but is independent of .

Proof. Since , , and , we have so that from which we get, after using Gronwall’s lemma [35] on (25) and the estimate for , Furthermore, we have where depends on but is independent of .

To summarize, in both cases, unperturbed and perturbed, the transform of the solution of (2) is in a Paley-Wiener space , where . Thus can be recovered at each from its samples at the lattice points , using the rectangular cardinal series [36, 37].

Theorem 8. Let ; then, with the convergence of the series being uniform and in , and , , .

Let , , , and .

The eigenpairs are therefore , where solve the nonlinear system as follows, where

3. A Numerical Example

We will consider in this section the two-parameter Sturm-Liouville problem with three-point boundary conditions given by

The general solution of the first differential equation can be expressed in terms of Ai and Bi functions [38] and their first derivatives as

Thus the eigenpairs can be obtained from the solutions of the following system,

For numerical purposes, we have truncated the associated series to and took in the function . Thus, the approximate eigenpairs are seen as solutions of the following system,

Table 1 shows the exact eigenpairs together with their approximations using the Regularized Sampling Method (RSM).


(exact) (exact) (RSM) (RSM)


4. Conclusion

In this paper, we have successfully computed the eigenpairs of two-parameter Sturm-Liouville problems using the regularized sampling method, a method which has been very efficient in computing the eigenvalues of broad classes of Sturm-Liouville problems (singular, non-self-adjoint, nonlocal, impulsive, etc.). We have shown in this work that it can tackle two-parameter SL problems with equal ease. An example was provided to illustrate the effectiveness of the method.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful to King Fahd University of Petroleum and Minerals for its usual support.

References

  1. F. M. Arscott, “Paraboloidal co-ordinates and Laplace's equation,” Proceedings of the Royal Society of Edinburgh, vol. 66, pp. 129–139, 1963. View at: Google Scholar | MathSciNet
  2. F. M. Arscott, “Two-parameter eigenvalue problems in differential equations,” Proceedings of the London Mathematical Society, vol. 14, pp. 459–470, 1964. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  3. P. Binding, “Perturbation and bifurcation of nonsingular multiparametric eigenvalues,” Nonlinear Analysis: Theory, Methods & Applications, vol. 8, no. 4, pp. 335–352, 1984. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. P. Binding, “Indefinite Sturm-Liouville theory via examples of two-parameter eigencurves,” in Proceedings of the 1990 Dundee Conference on Differential Equations, R. Jarvis and B. D. Sleeman, Eds., pp. 38–49, Pitman, 1991. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  5. P. Binding and P. J. Browne, “Eigencurves for two-parameter selfadjoint ordinary differential equations of even order,” Journal of Differential Equations, vol. 79, no. 2, pp. 289–303, 1989. View at: Publisher Site | Google Scholar | MathSciNet
  6. P. Binding and H. Volkmer, “Eigencurves for two-parameter Sturm-Liouville equations,” SIAM Review, vol. 38, no. 1, pp. 27–48, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. A. Boucherif, “Nonlinear three-point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 77, no. 2, pp. 577–600, 1980. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. P. J. Browne and B. D. Sleeman, “A note on the characterisation of eigencurves for certain two parameter eigenvalue problems in ordinary differential equations,” Glasgow Mathematical Journal, vol. 35, no. 1, pp. 63–67, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. P. Binding and P. J. Browne, “Spectral properties of two-parameter eigenvalue problems,” Proceedings of the Royal Society of Edinburgh, vol. 89, no. 1-2, pp. 157–173, 1981. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. P. B. Bailey, “The automatic solution of two-parameter Sturm-Liouville eigenvalue problems in ordinary differential equations,” Applied Mathematics and Computation, vol. 8, no. 4, pp. 251–259, 1981. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. E. K. Blum and A. R. Curtis, “A convergent gradient method for matrix eigenvector-eigentuple problems,” Numerische Mathematik, vol. 31, no. 3, pp. 247–263, 1978/79. View at: Publisher Site | Google Scholar | MathSciNet
  12. B. Plestenjak, “A continuation method for a right definite two-parameter eigenvalue problem,” SIAM Journal on Matrix Analysis and Applications, vol. 21, no. 4, pp. 1163–1184, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  13. B. Plestenjak, “A continuation method for a weakly elliptic two-parameter eigenvalue problem,” IMA Journal of Numerical Analysis, vol. 21, no. 1, pp. 199–216, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  14. M. E. Hochstenbach and B. Plestenjak, “A Jacobi-Davidson type method for a right definite two-parameter eigenvalue problem,” SIAM Journal on Matrix Analysis and Applications, vol. 24, no. 2, pp. 392–410, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  15. M. E. Hochstenbach, T. Košir, and B. Plestenjak, “A Jacobi-Davidson type method for the two-parameter eigenvalue problem,” SIAM Journal on Matrix Analysis and Applications, vol. 26, no. 2, pp. 477–497, 2004/05. View at: Publisher Site | Google Scholar | MathSciNet
  16. P. B. Bailey, W. N. Everitt, and A. Zettl, “Computing eigenvalues of singular Sturm-Liouville problems,” Results in Mathematics, vol. 20, no. 1-2, pp. 391–423, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  17. B. Chanane, “High order approximations of the eigenvalues of regular Sturm-Liouville problems,” Journal of Mathematical Analysis and Applications, vol. 226, no. 1, pp. 121–129, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  18. A. Boumenir and B. Chanane, “Eigenvalues of S-L systems using sampling theory,” Applicable Analysis, vol. 62, no. 3-4, pp. 323–334, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  19. A. Boumenir and B. Chanane, “Computing eigenvalues of Sturm-Liouville systems of Bessel type,” Proceedings of the Edinburgh Mathematical Society, vol. 42, no. 2, pp. 257–265, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  20. F. V. Atkinson, Multiparameter Eigenvalue Problems, Academic Press, New York, NY, USA, 1972. View at: MathSciNet
  21. S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, vol. 251, Springer, New York, NY, USA, 1982. View at: MathSciNet
  22. M. Faierman, Two-Parameter Eigenvalue Problems in Ordinary Differential Equations, vol. 205 of Pitman Research Notes in Mathematics Series, Longman, Harlow, UK, 1991. View at: MathSciNet
  23. D. McGhee and R. Picard, Cordes' Two-Parameter Spectral Representation Theory, Pitman Research Notes, Longman, Harlow, UK, 1988.
  24. B. D. Sleeman, Multiparameter Spectral Theory in Hilbert Space, Pitman Research Notes, Longman, Harlow, UK.
  25. H. Volkmer, Multiparameter Eigenvalue Problems and Expansion Theorems, vol. 1356 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1988.
  26. F. V. Atkinson and A. B. Mingarelli, Multiparameter Problems Sturm-Liouville Theory, CRC Press, Boca Raton, Fla, USA, 2011. View at: MathSciNet
  27. B. Chanane, “Computation of the eigenvalues of Sturm-Liouville problems with parameter dependent boundary conditions using the regularized sampling method,” Mathematics of Computation, vol. 74, no. 252, pp. 1793–1801, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  28. B. Chanane, “Computing the eigenvalues of singular Sturm-Liouville problems using the regularized sampling method,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 972–978, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  29. B. Chanane, “Computing the spectrum of non-self-adjoint Sturm-Liouville problems with parameter-dependent boundary conditions,” Journal of Computational and Applied Mathematics, vol. 206, no. 1, pp. 229–237, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  30. B. Chanane, “Computing the eigenvalues of a class of nonlocal Sturm-Liouville problems,” Mathematical and Computer Modelling, vol. 50, no. 1-2, pp. 225–232, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  31. B. Chanane, “Sturm-Liouville problems with impulse effects,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 610–626, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  32. M. Greguš, F. Neuman, and F. M. Arscott, “Three-point boundary value problems in differential equations,” Journal of the London Mathematical Society, vol. 3, pp. 429–436, 1971. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  33. L. Turyn, “Sturm-Liouville problems with several parameters,” Journal of Differential Equations, vol. 38, no. 2, pp. 239–259, 1980. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  34. A. Boucherif and N. Boukli-Hacene, “Two-parameter eigenvalue problems with time-dependent three-point boundary conditions,” Maghreb Mathematical Review, vol. 8, no. 1-2, pp. 57–66, 1999. View at: Google Scholar | MathSciNet
  35. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, NY, USA, 1955. View at: MathSciNet
  36. A. I. Zayed, Advances in Shannon's Sampling Theory, CRC Press, Boca Raton, Fla, USA, 1993. View at: MathSciNet
  37. A. I. Zayed, “A sampling theorem for signals bandlimited to a general domain in several dimensions,” Journal of Mathematical Analysis and Applications, vol. 187, no. 1, pp. 196–211, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  38. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, NY, USA, 1972.

Copyright © 2014 B. Chanane and A. Boucherif. 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.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views811
Downloads498
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.