Inverse Source Problems for the Time-Fractional Evolution Equation

TL;DR

This paper studies inverse problems for a time-fractional diffusion-wave equation, focusing on identifying source terms using interior data, and establishes theoretical results for existence, uniqueness, and regularity of solutions.

Contribution

It introduces new methods for solving inverse source problems in fractional PDEs, including proofs of solution properties and approaches for recovering source terms.

Findings

Proved existence and uniqueness of solutions for the direct problem

Developed methods for recovering time-independent sources

Analyzed the identification of time-dependent source coefficients

Abstract

In this paper, we investigate the direct and linear inverse problems of identifying time-dependent and time-independent source terms in a time-fractional diffusion-wave equation, using measured data at an interior point of the time interval. We first establish the existence, uniqueness, and regularity of the solution to the direct problem by employing the spectral method. Then, based on the properties of the direct problem, we study two inverse problems: one involving the recovery of a time-independent source term, and the other concerning the identification of a time-dependent coefficient function in the source term.