Stability analysis and L1-finite difference modeling of inverse problems for fractional Schrödinger equations with variable diffusion

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\leq1 $ 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.