Sharp asymptotics in a fractional Sturm-Liouville problem

TL;DR

This paper develops explicit asymptotic approximations for solutions of fractional Sturm-Liouville problems, providing sharp estimates for eigenvalues and eigenfunctions, advancing the analytical understanding of these boundary value problems.

Contribution

It introduces a novel approach focusing on explicit asymptotic solutions, complementing existing qualitative and numerical methods in fractional Sturm-Liouville theory.

Findings

Derived asymptotically sharp eigenvalue estimates

Obtained explicit asymptotic eigenfunction approximations

Enhanced analytical understanding of fractional boundary value problems

Abstract

The current research of fractional Sturm-Liouville boundary value problems focuses on the qualitative theory and numerical methods, and much progress has been recently achieved in both directions. The objective of this paper is to explore a different route, namely, construction of explicit asymptotic approximations for the solutions. As a study case, we consider a problem with left and right Riemann-Liouville derivatives, for which our analysis yields asymptotically sharp estimates for the sequence of eigenvalues and eigenfunctions.