Numerical approach for solving the inverse problem: A two-dimensional time-fractional boundary value problem

This paper presents the inverse problem (IP) for the fractional order two-dimensional parabolic diffusion equation (FOTDPDE) formulated to depend on a initial-boundary value problem (IBVP) with homogeneous Dirichlet boundary conditions (DBC). The model involves a fractional-order Caputo derivative (FOCD) and an inverse time-dependent source term. A Crank-Nicholson finite difference scheme (CN-FDS) is constructed, and stability inequalities and a theorem in the discrete $L^2$ norm are proved to ensure unconditional stability of the proposed scheme. Results calculated by using finite difference methods (FDM) have a temporal convergence rate of $O({\tau }^{2-\alpha })$ and second-order spatial accuracy. Numerical examples are tested to confirm the theoretical stability results and to represent the effectiveness and the accuracy of the method for solving IP for FOTDPDE depending on BVP.