Approximations of Sturm-Liouville eigenvalues using Boundary Value Methods

doi:10.1016/S0168-9274(96)00073-6

Applied Numerical Mathematics

Volume 23, Issue 3, May 1997, Pages 311-325

https://doi.org/10.1016/S0168-9274(96)00073-6 Get rights and content

Abstract

It is well known that the algebraic approximations to eigenvalues of a Sturm-Liouville problem by the central difference and Numerov's schemes provide only a few estimates restricted to the first element of the eigenvalue sequence. A correction technique, used first by Paine et al. (1981) for the central difference scheme and then by Andrew and Paine (1985) for Numerov's method, improves the results, giving acceptable estimates for a larger number of eigenvalues. In this paper some linear multistep methods, called Boundary Value Methods, are proposed for discretizing a Sturm-Liouville problem and the correction technique of Andrew-Paine and Paine et al. is extended to these new methods.

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Cited by (38)

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In the paper, we determine the eigenvalues and eigenfunctions of the generalized Sturm–Liouville problems, whose potentials may be nonlinear functions of eigen-parameter, by developing new iterative algorithms based on the fictitious time integration method (FTIM) and half-interval method (HIM). We derive two eigen-parameter dependent linear shape functions, from which we can transform the generalized Sturm–Liouville problem to an initial value problem for a new variable, and automatically preserve the prescribed eigen-parameter dependent Sturm–Liouville boundary conditions. A nonlinear equation in terms of the relative norm of two consecutive right-end values of the new variable is derived for iteratively determining the eigenvalue by using the FTIM. The resultant sequence of the iterated eigenvalues are monotonically convergent to the desired eigenvalue, and meanwhile the unknown initial values of the eigenfunction can be determined for computing the eigenfunction by integrating the Sturm–Liouville equation. We propose a high precision point target method by using the HIM to determine the eigenvalue and eigenfunction of the generalized Sturm–Liouville problem, which is transformed to a definite initial value problem with a simpler Dirichlet or Neumann boundary condition on the right-end as a point target equation. Several new theoretical results are proved such that more simple generalized Sturm–Liouville problem can be derived with a single point target on the right-end. Depending on different transformation techniques and the derived target equations, seven types FTIM and four types HIM are developed. Numerical examples confirm that the present FTIM and HIM are effective and accurate.
Solving Sturm–Liouville inverse problems by an orthogonalized enhanced boundary function method and a product formula for symmetric potential
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We develop a fast convergence iterative method after transforming the Sturm–Liouville problem into the weak-form integral equations, and using sinusoidal functions as the testers as well as the bases of eigenfunction. Owing to the orthogonal property of these bases, we can obtain the expansion coefficients in closed-form. A new idea of orthogonalized and enhanced boundary function (OEBF) is introduced to compute eigenvalues. In the Sobolev space, we prove the closeness of the OEBF to the eigenfunction with their distance being reduced when the eigenvalue increases. Moreover, upon expressing the Rayleigh quotient in terms of OEBF, the unknown functions in the Sturm–Liouville operator can be recovered quickly, merely two boundary data of unknown functions and the first eigenvalue are sufficing. The OEBF is a promising method with a few iterations to obtain quite accurate estimations of the unknown potential and weight functions. Thanks to the symmetry property a symmetric matrix eigenvalue problem is derived to recover a symmetric potential. The characteristic equation endowing with a special coefficient matrix is decomposed into a product of two characteristic equations. The Newton iterative method is thus developed by relying on the product formula to reconstruct the unknown symmetric potential function with the aid of a few lower orders eigenvalues.
An efficient method to approximate eigenfunctions and high-index eigenvalues of regular Sturm-Liouville problems
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The eigenvalues are calculated by starting the Homotopy Analysis Method (HAM) with one initial guess [1]. Boundary Value Methods (BVM) [2,9], Sinc-Galerkin, differential transform methods [14] and shooting-type algorithms [12] are other techniques to compute the eigenvalues of SLP. The numerical methods mentioned before deteriorate when a large set of eigenvalues are sought; except the shooting-type method that yields uniform approximation over the whole eigenvalue spectrum.
The computation of the eigenvalues of a Sturm–Liouville problem is a difficult task, when high-index eigenvalues are computed. In most previous methods, it can be seen that the uncertainty of the results increases as the estimated eigenvalues grow larger. This paper is to present some new methods in which, not only the error of calculating the higher eigenvalues does not grow, but it also vanishes as eigenvalues tend to infinity. Moreover, the proposed method gives good estimates of eigenfunctions corresponding to high eigenvalues.
A simple approach for determining the eigenvalues of the fourth-order Sturm-Liouville problem with variable coefficients
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Unfortunately, it is difficult to obtain exact eigenvalues for most cases. Many approximate methods or numerical techniques have been formulated to compute the numerical results of the eigenvalues of Sturm–Liouville problems [3], which include Numerov’s method [4,5], Boundary Value Methods [6], Chebyshev collocation method [7,8], Legendre Galerkin Chebyshev collocation method [9], and others [10]. It should be noted that most of the above work on investigating the second-order Sturm–Liouville problems.
In this paper, a simple and efficient approach is presented to compute the eigenvalues of the fourth-order Sturm–Liouville equations with variable coefficients. By transforming the governing differential equation to a system of algebraic equation, we can get the corresponding polynomial characteristic equations for kinds of boundary conditions based on the polynomial expansion and integral technique. Moreover, the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples for estimating eigenvalues are given. The convergence and effectiveness of the method are confirmed by comparing numerical results with the exact and other existing numerical results.
Two very accurate and efficient methods for computing eigenvalues of Sturm-Liouville problems
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In most cases, it is not possible to obtain the eigenvalues of the problem analytically. However, there are various approximate methods for solving special types of Sturm–Liouville problems as, for example, the weighted residual methods, the Galerkin methods, finite difference [1], finite elements methods [2], differential transformation method [3], Adomian decomposition method [4], Boundary value methods [5], Quatrature methods [6], Haar wavelet [7], the homotopy perturbation method [8] and variational iteration method [9]. DQM was first introduced in the early seventies by Bellman et al. [18] for solving differential equations of engineering mechanics; it is based on the weighted sum of function values as an approximation to the derivatives of that function.
Boundary Value Methods for the reconstruction of Sturm-Liouville potentials
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As further references to methods currently available for the two-spectra problem we mention the ones introduced in [9,15] which make use of the Boundary Value Methods (BVMs) for first or second order ODEs discussed in [26–28].
In this paper we present numerical procedures for solving the two inverse Sturm–Liouville problems known in the literature as the two-spectra and the half inverse problems. The method proposed looks for a continuous approximation of the unknown potential belonging to a suitable function space of finite dimension. In order to compute such an approximation a sequence of direct problems has to be solved. This is done by applying one of the Boundary Value Methods, generalizing the classical Numerov scheme, recently introduced by the authors. Numerical results confirming the effectiveness of the approach proposed are also reported.

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^☆: Work supported by the 40% project of the MURST.

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Approximations of Sturm-Liouville eigenvalues using Boundary Value Methods☆

Abstract

Appl. Numer. Math.

Appl. Numer. Math.

Appl. Numer. Math.

J. Comput. Appl. Math.

J. Comput. Appl. Math.

Comput. Math. Appl.

Stability of some boundary value methods for the solution of initial value problems

BIT

Efficient computation of higher Sturm-Liouville eigenvalues

Correction of finite element eigenvalues for problems with natural or periodic boundary conditions

BIT

Correction of finite difference eigenvalues of periodic Sturm-Liouville problems

J. Austral. Math. Soc. Ser. B