Weak solutions for time-fractional evolution equations in Hilbert spaces

TL;DR

This paper develops a framework for weak solutions to abstract time-fractional evolution equations in Hilbert spaces, establishing existence results and illustrating applications to wave and Petrovsky systems.

Contribution

It introduces a new notion of weak solutions for fractional differential equations and proves their existence, expanding the theoretical understanding of such equations.

Findings

Existence of weak solutions for fractional evolution equations

Application to time-fractional wave equations

Application to time-fractional Petrovsky systems

Abstract

We introduce a notion of weak solution for abstract fractional differential equations, motivated by the definition of Caputo derivative. We prove existence results for weak and strong solutions. We also give two examples as application of our results: time-fractional wave equations and time-fractional Petrovsky systems.