Solving time-fractional diffusion equations with singular source term

TL;DR

This paper investigates the mathematical properties of time-fractional diffusion equations with singular, time-dependent source terms, establishing existence, uniqueness, and solution representations under complex fractional derivative conditions.

Contribution

It proves the existence and uniqueness of weak solutions for complex time-fractional diffusion equations with singular sources, extending previous results to multi-term, distributed, and space-dependent derivatives.

Findings

Unique weak solutions exist under specified conditions.

Solutions admit a Duhamel representation.

Applicable to multi-term, distributed, and space-dependent derivatives.

Abstract

This article deals with time-fractional diffusion equations with time-dependent singular source term. Whenever the order of the time-fractional derivative is either multi-term, distributed or space-dependent, we prove that the system admits a unique weak solution enjoying a Duhamel representation, provided that the time-dependence of the source term is a distribution.