This study developed a finite difference method (FDM) for a time-fractional inverse problem associated with Schrödinger partial differential equations. The main objective of the inverse problem is the simultaneous identification of the unknown source function $p(x)$ and the state variable $w(t,x)$. The mathematical model involves the Caputo fractional order derivative (CFOD) of order $0<\alpha \le 1$ and incorporates a spatially variable diffusion coefficient $a(x)$, which significantly increases the complexity of the problem compared with constant-coefficient models. Homogeneous Dirichlet boundary value conditions (DBVCs) were imposed on the spatial domain. For the numerical discretization, the time-CFOD was approximated using a consistent L1-type scheme, while the spatial derivatives were discretized by second-order central finite difference schemes (FDSs). Stability estimates and convergence properties of the proposed numerical scheme are rigorously established using discrete energy techniques. The analysis shows that the method achieves a convergence order of $O({\tau }^{2-\alpha }+h^2)$. To validate the theoretical results, numerical experiments were performed for two benchmark problems with diffusion coefficients $a(x)=x^2+1$ and $a(x)=x^3+1$. The obtained numerical results confirm the effectiveness and robustness of the proposed approach. Graphical comparisons illustrate the behavior of solutions for different fractional orders as time evolves, while error tables demonstrate that the fractional-order solutions provide more accurate approximations to the exact solution than the corresponding integer-order case.

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.