Communications in Mathematical Sciences

Volume 20 (2022)

Number 1

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

Pages: 53 – 84



Yavar Kian (Aix-Marseille Université, Université de Toulon, Marseille, France)

Éric Soccorsi (Aix-Marseille Université, Université de Toulon, Marseille, France)

Qi Xue (Institut des Sciences de la Terre, CNRS & Université Grenoble-Alpes, Grenoble, France)

Masahiro Yamamoto (Graduate School of Mathematical Sciences, University of Tokyo, Japan; and Peoples’ Friendship University of Russia, Moscow, Russia)


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 a spatial term and a temporal term, and we prove that the term depending on the space variable can be retrieved by observation over the time interval of the solution on an arbitrary sub-boundary. Under some suitable assumptions we can also show the simultaneous recovery of the spatial term and the temporal term. 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 and time variables, and we show that they are uniquely determined by one arbitrary lateral measurement of the solution. These 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 the spatial term in the first case is carried out through an iterative algorithm based on the Tikhonov regularization method.


inverse source problems, diffusion equation, fractional diffusion equation, uniqueness result, numerical reconstruction, Tikhonov regularization method

2010 Mathematics Subject Classification

35R11, 35R30

Received 15 December 2019

Received revised 2 June 2021

Accepted 2 June 2021

Published 10 December 2021