Inverse source problem for 2D nonlinear mixed fractional-order telegraph equations using finite difference method

In this article, an inverse problem for a two-dimensional nonlinear fractional-order telegraph equation (TDNFOTE) involving mixed fractional Caputo derivatives (FCD) is investigated. The model under consideration features a nonlinear diffusion operator formulated in divergence form, coupled with an unknown time-dependent source term. Such mathematical models are essential for describing wave propagation in complex, heterogeneous media, stress relaxation processes in viscoelastic materials, and various anomalous transport phenomena where standard diffusion models prove insufficient. To solve the proposed problem, a finite difference scheme (FDS) based on the L2-1 approximation for the FCD is constructed, while central difference approximations are employed for the spatial discretization. Stability estimates of the proposed scheme are established in the discrete \(L²\) norm, and the convergence of the method is rigorously proven under appropriate mesh constraints. Several numerical experiments are presented to verify the theoretical findings and to illustrate the physical behavior of the solution, including energy evolution, flux density, and nonlinear diffusion effects. The obtained numerical results are also compared with the corresponding exact solution, demonstrating the efficiency and accuracy of the proposed finite difference method for solving the considered inverse fractional telegraph problem.