A matrix method for fractional Sturm-Liouville problems on bounded domain

TL;DR

This paper introduces a matrix-based spectral Galerkin method using orthogonal polynomials to solve fractional Sturm-Liouville problems with Riesz derivatives, analyzing convergence and demonstrating effectiveness through numerical examples.

Contribution

It develops a novel spectral Galerkin approach for fractional Sturm-Liouville problems with Riesz derivatives, including convergence analysis and numerical validation.

Findings

Eigenvalue approximations converge with increasing matrix size

The method is competitive compared to existing approaches

Numerical examples confirm theoretical convergence rates

Abstract

A matrix method for the solution of direct fractional Sturm-Liouville problems on bounded domain is proposed where the fractional derivative is defined in the Riesz sense. The scheme is based on the application of the Galerkin spectral method of orthogonal polynomials. The order of convergence of the eigenvalue approximations with respect to the matrix size is studied. Some numerical examples that confirm the theory and prove the competitiveness of the approach are finally presented.