Inverse source problem with a posteriori boundary measurement for fractional diffusion equations

TL;DR

This paper investigates inverse source problems for time-fractional diffusion equations using boundary measurements, demonstrating unique determination of certain space-time dependent sources through Laplace transform analysis.

Contribution

It introduces a novel approach leveraging the memory effect of fractional diffusion equations to solve inverse problems and establish uniqueness results.

Findings

Successfully determined space-time dependent sources from boundary data.

Proved uniqueness for a broad class of separated variable sources.

Utilized Laplace transform singularity analysis for solution characterization.

Abstract

In this article we study inverse source problems for time-fractional diffusion equations from \textit{a posteriori} boundary measurement. Using the memory effect of these class of equations, we solve these inverse problems for several class of space or time dependent source terms. We prove also the unique determination of a general class of space-time dependent separated variables source terms from such measurement. Our approach is based on the study of singularities of the Laplace transform in time of boundary traces of solutions of time-fractional diffusion equations.