Spectral Collocation Solutions to Second Order Singular Sturm-Liouville Eigenproblems

TL;DR

This paper compares classical spectral collocation methods and Chebfun algorithms to solve challenging second order singular Sturm-Liouville eigenproblems, analyzing their effectiveness in different singularity and boundary condition scenarios.

Contribution

It provides a comparative analysis of spectral collocation and Chebfun methods for singular Sturm-Liouville problems, including challenging benchmark cases and the interpretation of singularities.

Findings

Spectral methods successfully solve problems where traditional methods fail.

Chebfun algorithms offer high performance and accuracy for singular eigenproblems.

Insights into the nature of singularities and continuous spectrum in specific cases.

Abstract

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.