Identification of time-varying source term in time-fractional diffusion equations

TL;DR

This paper investigates the inverse problem of identifying time- and space-dependent source terms in time-fractional diffusion equations, providing theoretical uniqueness results and a numerical reconstruction method.

Contribution

It introduces new uniqueness theorems for source identification in fractional diffusion equations and develops an iterative Tikhonov regularization algorithm for numerical reconstruction.

Findings

Unique determination of source terms from internal and lateral measurements.

Establishment of weak and Cauchy data continuation principles.

Numerical algorithm successfully reconstructs spatial source components.

Abstract

This paper is concerned with the inverse problem of determining the time and space dependent source term of diffusion equations with constant-order time-fractional derivative in $(0,2)$. We examine two different cases. In the first one, the source is the product of two spatial and temporal terms, and we prove that both of them can be retrieved by knowledge of one arbitrary internal measurement of the solution for all times. In the second case, we assume that the first term of the product varies with one fixed space variable, while the second one is a function of all the remaining space variables and the time variable, and we show that both terms are uniquely determined by two arbitrary lateral measurements of the solution over the entire time span. These two source identification results boil down to a weak unique continuation principle in the first case and a unique continuation principle for Cauchy data in the second one, that are preliminarily established. Finally, numerical reconstruction of spatial term of source terms in the form of the product of two spatial and temporal terms, is carried out through an iterative algorithm based on the Tikhonov regularization method.