Trace operators for Riemann--Liouville fractional equations

TL;DR

This paper investigates the Riemann-Liouville fractional derivative in the context of fractional wave equations, establishing well-posedness, regularity, and trace results, and exploring duality with Caputo derivatives using Mittag-Leffler functions.

Contribution

It introduces trace operators for Riemann-Liouville fractional equations and analyzes their properties, including duality with Caputo derivatives, advancing the mathematical understanding of fractional PDEs.

Findings

Established well-posedness and regularity results.

Derived trace results in interpolation spaces.

Explored duality with Caputo derivatives.

Abstract

We begin with a brief overview of the most commonly used fractional derivatives, namely the Caputo and Riemann-Liouville derivatives. We then focus on the study of the fractional time wave equation with the Riemann-Liouville derivative, addressing key questions such as well-posedness, regularity, and a trace result in appropriate interpolation spaces. Additionally, we explore the duality relationship with the Caputo fractional time derivative. The analysis is based on expanding the solution in terms of Mittag-Leffler functions.