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., [2–8]). 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 [12–15]. 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 [2–4, 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).
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.