This paper deals with an inverse problem on simultaneously determining a time-dependent potential term and a time source function from two-point measured data in a multi-term time-fractional diffusion equation. First we study the existence, uniqueness and some regularities of the solution for the direct problem by using the fixed point theorem. Then a nice conditional stability estimate of inversion coefficients problem is obtained based on the regularity of the solution to the direct problem and a fine property of the Caputo fractional derivative. In addition, the ill-posedness of the inverse problem is illustrated and we transfer the inverse problem into a variational problem. Moreover, the existence and convergence of the minimizer for the variational problem are given. Finally, we use a modified Levenberg–Marquardt method to reconstruct numerically the approximate functions of two unknown time-dependent coefficients effectively. Numerical experiments for three examples in one- and two-dimensional cases are provided to show the validity and robustness of the proposed method.

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 Arzel`a-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.

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