Fractional Sturm-Liouville eigenvalue problems, I

TL;DR

This paper introduces solutions to fractional differential equations of mixed types and analyzes a fractional Sturm-Liouville eigenvalue problem, revealing eigenvalue properties and convergence to classical operators as the fractional order approaches one.

Contribution

It presents the first comprehensive analysis of a Dirichlet type fractional Sturm-Liouville problem with mixed Caputo/Riemann-Liouville operators, including eigenvalue behavior and asymptotic analysis.

Findings

Finite number of real eigenvalues for each fractional order

Infinite non-real eigenvalues exist

Eigenvalues grow unbounded as fractional order approaches 1

Abstract

We introduce and present the general solution of three two-term fractional differential equations of mixed Caputo/Riemann Liouville type. We then solve a Dirichlet type Sturm-Liouville eigenvalue problem for a fractional differential equation derived from a special composition of a Caputo and a Riemann-Liouville operator on a finite interval where the boundary conditions are induced by evaluating Riemann-Liouville integrals at those end-points. For each $1/2<\alpha<1$ it is shown that there is a finite number of real eigenvalues, an infinite number of non-real eigenvalues, that the number of such real eigenvalues grows without bound as $\alpha \to 1^-$, and that the fractional operator converges to an ordinary two term Sturm-Liouville operator as $\alpha \to 1^-$ with Dirichlet boundary conditions. Finally, two-sided estimates as to their location are provided as is their asymptotic behavior as a function of $\alpha$.