Identification of time-varying source term in time-fractional diffusion equations
Yavar Kian, Éric Soccorsi, Qi Xue, Masahiro Yamamoto
Abstract
This paper is concerned with the inverse problem of determining the time and\nspace dependent source term of diffusion equations with constant-order\ntime-fractional derivative in $(0,2)$. We examine two different cases. In the\nfirst one, the source is the product of two spatial and temporal terms, and we\nprove that both of them can be retrieved by knowledge of one arbitrary internal\nmeasurement of the solution for all times. In the second case, we assume that\nthe first term of the product varies with one fixed space variable, while the\nsecond one is a function of all the remaining space variables and the time\nvariable, and we show that both terms are uniquely determined by two arbitrary\nlateral measurements of the solution over the entire time span. These two\nsource identification results boil down to a weak unique continuation principle\nin the first case and a unique continuation principle for Cauchy data in the\nsecond one, that are preliminarily established. Finally, numerical\nreconstruction of spatial term of source terms in the form of the product of\ntwo spatial and temporal terms, is carried out through an iterative algorithm\nbased on the Tikhonov regularization method.\n