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

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 Arzela-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.

Automated summary

This paper investigates an inverse potential problem for a semilinear generalized fractional diffusion equation with a time-dependent principal part. The missing time-dependent potential is reconstructed using additional integral measured data, and a modified non-stationary iterative Tikhonov regularization method is proposed to solve the problem. Numerical experiments demonstrate the effectiveness and robustness of the proposed algorithm.