Approximation of Eigenvalues of Sturm-Liouville Problems by Using Hermite Interpolation

Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. In this paper, we use the derivative sampling theorem “Hermite interpolations” to compute approximate values of the eigenvalues of Sturm-Liouville problems with eigenvalue parameter in one or two boundary conditions. We use recently derived estimates for the truncation and amplitude errors to compute error bounds. Also, using computable error bounds, we obtain eigenvalue enclosures. Also numerical examples, which are given at the end of the paper, give comparisons with the classical sinc method and explain that the Hermite interpolations method gives remarkably better results.

Computing Eigenvalues of Discontinuous Sturm-Liouville Problems with Eigenparameter in All Boundary Conditions Using Hermite Approximation

The eigenvalues of discontinuous Sturm-Liouville problems which contain an eigenparameter appearing linearly in two boundary conditions and an internal point of discontinuity are computed using the derivative sampling theorem and Hermite interpolations methods. We use recently derived estimates for the truncation and amplitude errors to investigate the error analysis of the proposed methods for computing the eigenvalues of discontinuous Sturm-Liouville problems. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented. Moreover, it is shown that the proposed methods are significantly more accurate than those based on the classical sinc method.