Approximation of eigenvalues of discontinuous Sturm-Liouville problems with eigenparameter in all boundary conditions (original) (raw)

Eigenvalues of discontinuous Sturm-Liouville problems with symmetric potentials

Computers & mathematics with applications, 1989

Alam'aet-In this paper we consider three examples of discontinuous Sturm-Liouville problems with symmetric potentials. The ¢igcnvalues of the systems were determined using the classical fourth order Runge-Kutta method. These eigenvalues are used to reconstruct the potential function using an algorithm presented in Kobayashi [1, 2]. The results of our numerical experiments are discussed.

A Hermite-Gauss method for the approximation of eigenvalues of regular Sturm-Liouville problems

Journal of Inequalities and Applications

Recently, some authors have used the sinc-Gaussian sampling technique to approximate eigenvalues of boundary value problems rather than the classical sinc technique because the sinc-Gaussian technique has a convergence rate of the exponential order, O(e-(π-hσ)N/2 / √ N), where σ , h are positive numbers and N is the number of terms in sinc-Gaussian technique. As is well known, the other sampling techniques (classical sinc, generalized sinc, Hermite) have a convergence rate of a polynomial order. In this paper, we use the Hermite-Gauss operator, which is established by Asharabi and Prestin (Numer. Funct. Anal. Optim. 36:419-437, 2015), to construct a new sampling technique to approximate eigenvalues of regular Sturm-Liouville problems. This technique will be new and its accuracy is higher than the sinc-Gaussian because Hermite-Gauss has a convergence rate of order O(e-(2π-hσ)N/2 / √ N). Numerical examples are given with comparisons with the best sampling technique up to now, i.e. sinc-Gaussian.

Computational procedure for Sturm-Liouville problems

Journal of Computational Physics, 1983

have been incorporated to develop a procedure for the automatic computation of the eigenvalues and the eigenfunctions of one-dimensional linear Sturm-Liouville boundary value eigenproblems, both singular and nonsingular. The continuous coefftcients of a regular Sturm-Liouville problem have been approximated by a finite number of step functions. In each step the resulting boundary value problem has been integrated exactly and the solutions have then been matched to construct the continuously differentiable solution of the original problem and the corresponding eigencondition.

Analytic vs. numerical solutions to a Sturm-Liouville transmission eigenproblem

Journal of Numerical Analysis and Approximation Theory

An elliptic one-dimensional second order boundary value problem involving discontinuous coefficients, with or without transmission conditions, is considered. For the former case by a direct sum spaces method we show that the eigenvalues are real, geometrically simple and the eigenfunctions are orthogonal. Then the eigenpairs are computed numerically by a local linear finite element method (FEM) and by some global spectral collocation methods. The spectral collocation is based on Chebyshev polynomials (ChC) for problems on bounded intervals respectively on Fourier system (FsC) for periodic problems. The numerical stability in computing eigenvalues is investigated by estimating their (relative) drift with respect to the order of approximation. The accuracy in computing the eigenvectors is addressed by estimating their departure from orthogonality as well as by the asymptotic order of convergence. The discontinuity of coefficients in the problems at hand reduces the exponential order o...

Approximations of Sturm-Liouville Eigenvalues Using Sinc-Galerkin and Differential Transform Methods

2010

In this paper, we present a comparative study of Sinc-Galerkin method and differential transform method to solve Sturm-Liouville eigenvalue problem. As an application, a comparison between the two methods for various celebrated Sturm-Liouville problems are analyzed for their eigenvalues and solutions. The study outlines the significant features of the two methods. The results show that these methods are very efficient, and can be applied to a large class of problems. The comparison of the methods shows that although the numerical results of these methods are the same, differential transform method is much easier, and more efficient than the Sinc-Galerkin method.

Numerical Computation of Spectral Solutions for Sturm-Liouville Eigenvalue Problems

International Journal of Analysis and Applications

This paper focuses on the study of Sturm-Liouville eigenvalue problems. In the classical Chebyshev collocation method, the Sturm-Liouville problem is discretized to a generalized eigenvalue problem where the functions represent interpolants in suitably rescaled Chebyshev points. We are concerned with the computation of high-order eigenvalues of Sturm-Liouville problems using an effective method of discretization based on the Chebfun software algorithms with domain truncation. We solve some numerical Sturm-Liouville eigenvalue problems and demonstrate the efficiency of computations.

The double exponential sinc collocation method for singular Sturm-Liouville problems

Journal of Mathematical Physics, 2016

Sturm-Liouville problems are abundant in the numerical treatment of scientific and engineering problems. In the present contribution, we present an efficient and highly accurate method for computing eigenvalues of singular Sturm-Liouville boundary value problems. The proposed method uses the double exponential formula coupled with sinc collocation method. This method produces a symmetric positive-definite generalized eigenvalue system and has exponential convergence rate. Numerical examples are presented and comparisons with single exponential sinc collocation method clearly illustrate the advantage of using the double exponential formula.