Determination of source and initial values for acoustic equations with a time-fractional attenuation

TL;DR

This paper addresses the inverse problem of identifying initial conditions or sources in a time-fractionally damped hyperbolic equation, with applications in medical imaging, by establishing stability estimates using Carleman techniques.

Contribution

It introduces a novel stability analysis for inverse problems involving non-local time-fractional derivatives in hyperbolic equations, relevant to medical imaging.

Findings

Proved stability estimates for source and initial state determination

Developed a Carleman estimate tailored for fractional damping equations

Applicable to thermoacoustic and photoacoustic tomography

Abstract

We consider the inverse problem of determining the initial states or the source term of a hyperbolic equation damped by some non-local time-fractional derivative. This framework is relevant to medical imaging such as thermoacoustic or photoacoustic tomography. We prove a stability estimate for each of these two problems, with the aid of a Carleman estimate specifically designed for the governing equation.