An undetermined time-dependent coefficient in a fractional diffusion equation

TL;DR

This paper studies a fractional diffusion equation with a time-dependent coefficient, establishing well-posedness for the direct problem and developing a reconstruction algorithm for the inverse problem, supported by numerical experiments.

Contribution

It introduces a novel operator-based method for recovering the time-dependent coefficient in a fractional diffusion equation, including theoretical analysis and a practical reconstruction algorithm.

Findings

Proved existence and uniqueness for the direct problem.

Developed an operator with fixed points corresponding to the coefficient.

Provided numerical results demonstrating the effectiveness of the reconstruction algorithm.

Abstract

In this work, we consider a FDE (fractional diffusion equation) $${}^C D_t^\alpha u(x,t)-a(t)\mathcal{L} u(x,t)=F(x,t)$$ with a time-dependent diffusion coefficient $a(t)$. For the direct problem, given an $a(t),$ we establish the existence, uniqueness and some regularity properties with a more general domain $\Omega$ and right-hand side $F(x,t)$. For the inverse problem--recovering $a(t),$ we introduce an operator $K$ one of whose fixed points is $a(t)$ and show its monotonicity, uniqueness and existence of its fixed points. With these properties, a reconstruction algorithm for $a(t)$ is created and some numerical results are provided to illustrate the theories.