A Hilbert space approach to fractional differential equations

TL;DR

This paper develops a Hilbert space framework for solving fractional differential equations of Riemann-Liouville and Caputo types, establishing existence, uniqueness, and causality of solutions using advanced functional calculus techniques.

Contribution

It introduces a novel approach using exponentially weighted spaces and Fourier transform-based functional calculus to analyze fractional differential equations in Hilbert spaces.

Findings

Proves existence and uniqueness of solutions

Establishes causality of solution operators

Utilizes extrapolation and interpolation spaces

Abstract

We study fractional differential equations of Riemann-Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on $\mathbb{R}$, we define fractional operators by means of a functional calculus using the Fourier transform. Main tools are extrapolation- and interpolation spaces. Main results are the existence and uniqueness of solutions and the causality of solution operators for non-linear fractional differential equations.