On a stable difference scheme for numerically solving a reverse parabolic source identification problem

Abstract

This article is devoted to the study of source identification problems for reverse parabolic partial differential equations with nonlocal boundary conditions. The principal aim of the work is to construct and analyze stable difference schemes that can be effectively employed for obtaining approximate solutions of such inverse problems. In particular, attention is focused on the Rothe difference scheme, and stability estimates for the corresponding discrete solutions are rigorously derived. These estimates guarantee the reliability and convergence of the proposed numerical method. A stability theorem for the solution of the difference scheme related to the source identification problem is proved. To establish the well-posedness of the underlying differential problem, the operator-theoretic approach is employed, ensuring a solid analytical foundation for the numerical method. Furthermore, the investigation is extended to an abstract setting for difference schemes, which is then applied to the numerical solution of reverse parabolic equations under boundary conditions of the first kind. This unified framework emphasizes both the theoretical justification and the computational effectiveness of the proposed approach. Finally, the efficiency of the developed method is demonstrated through a numerical illustration with a test example.