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

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.

Unique recovery of the initial state of distributed order time fractional diffusion equation

“…When the highest order time derivative of the equation under consideration is one and the lower order time derivative is less than one, we refer to the paper from Huang, Li and Yamamoto [10] in which the stability of the inverse source problem was established by using the Carleman estimates. However, their methods heavily relies on the first order time-derivative so that cannot work for our case.…”