Solvability of an inverse problem for a time-fractional integro-differential diffusion equation with initial-boundary and integral overdetermination conditions

In the article, we investigate the inverse problem for the integro-differential time-fractional diffusion equation with initial-boundary and integral overdetermination conditions on a rectangular domain. Initially, we give a definition to the classical solution for the original problem. Subsequently, the direct problem is transformed into an equivalent integral equation using the Fourier method. We prove the existence and uniqueness of the solution to this equivalent problem by applying estimates of the Mittag-Leffler function and the successive approximation method. In the second part, we address the inverse problem, which is reduced to an equivalent auxiliary problem. We utilize the contraction mapping principle for proving local existence and uniqueness of the inverse problem solution. Additionally, we provide a practical overview of numerical solutions to the original problem using the finite difference method. The results of a numerical experiment on finer grids and at smaller time steps are demonstrated. The developed algorithm allows one to simultaneously determine the time-dependent kernel in the integral term and the solution to the problem. In conclusion, a test example is given to illustrate the effectiveness of the proposed numerical algorithm.