Spectral Theory and Numerical Approximation for Singular Fractional Sturm-Liouville eigen-Problems on Unbounded Domain

TL;DR

This paper introduces singular fractional Sturm-Liouville eigen-problems on unbounded domains, analyzes their spectral properties, and develops fractional polynomial bases for efficient numerical approximation with demonstrated exponential convergence.

Contribution

It presents the first spectral analysis of SFSLP on unbounded domains and constructs generalized Laguerre fractional-polynomials for accurate numerical solutions.

Findings

Eigenvalues are real-valued and eigenfunctions are orthogonal.

The generalized Laguerre fractional-polynomials provide exponential convergence.

Error analysis confirms the effectiveness of fractional spectral methods.

Abstract

In this article, we first introduce a singular fractional Sturm-Liouville eigen-problems (SFSLP) on unbounded domain. The associated fractional differential operators in these problems are both Weyl and Caputo type . The properties of spectral data for fractional operators on unbounded domain has been investigated. Moreover, it has been shown that the eigenvalues of the singular problems are real-valued and the corresponding eigenfunctions are orthogonal. The analytical eigensolutions to SFSLP is obtained and defined as generalized Laguerre fractional-polynomials. The optimal approximation of such generalized Laguerre fractional-polynomials in suitably weighted Sobolev spaces involving fractional derivatives has been derived, which is also available for approximated fractional-polynomials growing fast at infinity. The obtained results demonstrate that the error analysis beneficial of fractional spectral methods for fractional differential equations on unbounded domains. As a numerical example, we employ the new fractional-polynomials bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results.