On a stable difference scheme for numerically solving a reverse parabolic source identification problem
Ashyralyyev C. Sadybekov M.A.
30 December 2025E.A. Buketov Karaganda University Publish house
Bulletin of the Karaganda University. Mathematics Series
2025#2025Issue 485 - 94 pp.
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.
difference scheme (DS) , inverse problem , partial differential equation (PDE) , reverse parabolic equation , self-adjoint positive definite operator (SAPDO) , source identification problem (SIP) , stability estimate , well-posedness
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Department of Mathematics, Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey
Khoja Akhmet Yassawi International Kazakh-Turkish University, Turkistan, Kazakhstan
National University of Uzbekistan named after Mirzo Ulugbek, Tashkent, Uzbekistan
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Department of Mathematics
Khoja Akhmet Yassawi International Kazakh-Turkish University
National University of Uzbekistan named after Mirzo Ulugbek
Institute of Mathematics and Mathematical Modeling
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