Numerical solutions of source identification problems for telegraph-parabolic equations

This paper presents a numerical study of source identification problems for one-dimensional telegraphparabolic equations subject to Dirichlet and Neumann boundary conditions. In these inverse problems, the unknown source terms are assumed to be space-dependent, which introduces both analytical and computational challenges. The study begins by discretizing the considered problems using the finite difference method – first in space and subsequently in time – resulting in a system of discrete equations. Stability results for the solutions of the resulting finite difference schemes are established to ensure the reliability of the numerical approach. A numerical algorithm is proposed for solving the discrete inverse problems. The algorithm begins by eliminating the unknown source terms, which transforms the original discretized problem into a new nonlocal problem with unknown initial data. To approximate this initial data, an iterative procedure based on fixed-point iterations is constructed. Once the transformed nonlocal problem is solved, the solution of the main finite difference scheme and approximations of the unknown source term are recovered. Numerical results for two test problems are presented to illustrate the proposed method in practice. The findings confirm the accuracy of the approach in solving space-dependent inverse source problems.