Numerical solution of the problem of finding two unknowns in time-fractional diffusion equations

J. J. Jumayev^ab, Z. R. Bozorov^ab, D. K. Durdiev<sup>ab

a</sup> Institute of Mathematics at the Academy of Sciences of the Republic of Uzbekistan, 46 University str., Tashkent, 100170 Republic of Uzbekistan
^b Bukhara State University, 11 M. Ikbol str., Bukhara, 200118 Republic of Uzbekistan

Abstract: We study an inverse problem for a time-fractional diffusion equation with initial-boundary and overdetermination conditions.This inverse problem aims to determine a time varying coefficient and source in the equation with overdetermination integral conditions. First, we establish the unique existence of the classical solution using the Fourier method, Gronwall inequality for direct problem. Second, by using the fixed point theorem in Banach space, the local existence and uniqueness of this inverse problem are obtained. To verify the theoretical results, a numerical solution to the problem was constructed using the finite difference method. Finally, a numerical example is presented to show the effectiveness of the proposed method.

Keywords: time-fractional diffusion equation, inverse problem, integral equation, Gronwall inequality, finite difference method.

UDC: 517

Received: 30.04.2024
Revised: 30.04.2024
Accepted: 26.06.2024

DOI: 10.26907/0021-3446-2025-7-36-52