Space-Time-Dependent Source Identification Problem for a Subdiffusion Equation

TL;DR

This paper addresses the inverse problem of identifying a space-time-dependent source in a subdiffusion equation with Caputo derivative, establishing existence and uniqueness of solutions using Fourier and iterative methods, extending prior work to fractional equations.

Contribution

It introduces the first analysis of the inverse source problem for subdiffusion equations with space-time dependence, proving existence and uniqueness of weak solutions.

Findings

Established existence and uniqueness of the inverse problem solution.

Applied Fourier and successive approximation methods for solution computation.

Extended results to fractional-order and parabolic equations.

Abstract

In this paper, we investigate the inverse problem of determining the right-hand side of a subdiffusion equation with a Caputo time derivative, where the right-hand side depends on both time and certain spatial variables. Similar inverse problems have been previously explored for hyperbolic and parabolic equations, with some studies establishing the existence and uniqueness of generalized solutions, while others proved the uniqueness of classical solutions. However, such inverse problems for fractional-order equations have not been addressed prior to this work. Here, we establish the existence and uniqueness of the weak solution to the considered inverse problem. To solve it, we employ the Fourier method with respect to the variable independent of the unknown right-hand side, followed by the method of successive approximations to compute the Fourier coefficients of the solution. Notably, the results obtained are also novel for parabolic equations.