An inverse Sturm-Liouville problem with a fractional derivative

TL;DR

This paper explores numerically reconstructing potentials in fractional Sturm-Liouville problems using spectral data, demonstrating effective results for various potential types when the fractional order is suitably chosen.

Contribution

It introduces a numerical approach for inverse fractional Sturm-Liouville problems and shows successful potential reconstructions for different potential types.

Findings

Effective potential reconstructions for smooth and discontinuous cases

Reconstruction quality depends on fractional order  away from 2

Eigenvalues and eigenfunctions exhibit distinctive qualitative behaviors

Abstract

In this paper, we numerically investigate an inverse problem of recovering the potential term in a fractional Sturm-Liouville problem from one spectrum. The qualitative behaviors of the eigenvalues and eigenfunctions are discussed, and numerical reconstructions of the potential with a Newton method from finite spectral data are presented. Surprisingly, it allows very satisfactory reconstructions for both smooth and discontinuous potentials, provided that the order $\alpha\in(1,2)$ of fractional derivative is sufficiently away from 2.