Inverse potential problem for a semilinear generalized fractional diffusion equation with spatio-temporal dependent coefficients

Abstract

In this work, we are interested in an inverse potential problem for a semilinear generalized fractional diffusion equation with a time-dependent principal part. The missing time-dependent potential is reconstructed from an additional integral measured data over the domain. Due to the nonlinearity of the equation and arising of a space-time dependent principal part operator in the model, such a nonlinear inverse problem is novel and significant. The well-posedness of the forward problem is firstly investigated by using the well known Rothe’s method. Then the existence and uniqueness of the inverse problem are obtained by employing the Arzelà–Ascoli theorem, a coerciveness of the fractional derivative and Gronwall’s inequality, as well as the regularities of the direct problem. Also, the ill-posedness of the inverse problem is proved by analyzing the properties of the forward operator. Finally a modified non-stationary iterative Tikhonov regularization method is used to find a stable approximate solution for the potential term. Numerical examples in one- and two-dimensional cases are provided to illustrate the efficiency and robustness of the proposed algorithm.