Computing eigenvalues of Sturm-Liouville systems of Bessel type (original) (raw)

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.

Spectral Collocation Solutions to Second Order Singular Sturm-Liouville Eigenproblems

ArXiv, 2020

We comparatively use some classical spectral collocation methods as well as highly performing Chebfun algorithms in order to compute the eigenpairs of second order singular Sturm-Liouville problems with separated self-adjoint boundary conditions. For both the limit-circle non oscillatory and oscillatory cases we pay a particular attention. Some "hard" benchmark problems, for which usual numerical methods (f. d., f. e. m., etc.) fail, are analysed. For the very challenging Bessel eigenproblem we will try to find out the source and the meaning of the singularity in the origin. For a double singular eigenproblem due to Dunford and Schwartz we we try to find out the precise meaning of the notion of continuous spectrum. For some singular problems only a tandem approach of the two classes of methods produces credible results.

EIGEN VALUES CALCULATION OF STURM-LIOUVILLE PROBLEM IN BESSEL'S EQUATION

The aim of the study is to calculate eigen values of Sturm-Liouville problem emerged from solving the third boundary problem (Robin Problem), applied on heat flow through hollow cylindrical bodies. The encountered difficulty is manifested in the calculation of the primary eigen values of Bessel's equation. In previous study [3] designed to solve the same problem by using alterative roots property of Bessel's equation, determination of eigen values is time consuming as it is based on trial and error process. Changing the cylinder thickness requires further trial and approaching numerical solutions is even more problematic as guesses can only be made based on visual observations of the graphical representation of Bessel's function roots. This study suggests representing Bessel's equation by Taylor's series. A comparison of results acquired by trial and error approach and Taylor's series shows a high consistency with a high accuracy.