Inverse problems for the fractional diffusion equation with the Hilfer operator

D. K. Durdiev^ab, H. H. Turdiev^ab, R. R. Rashidov<sup>b

a</sup> V.I. Romanovskiy Institute of Mathematics at the Academy of Sciences of the Republic of Uzbekistan, 9 University str., Tashkent, 100170 Republic of Uzbekistan
^b Bukhara State University, 11 M.Ikbol str., Bukhara, 20018 Republic of Uzbekistan

Abstract: The article discusses inverse problems for the fractional diffusion equation with the Hilfer operator in time. The direct problem is the initial-boundary value problem for this equation with Cauchy-type initial data and Dirichlet boundary conditions. The first inverse problem, which involves determining a time-dependent coefficient, is reduced to an equivalent Volterra-type integral equation. The existence and uniqueness of the solution are proven using the contraction mapping principle. The second inverse problem involves determining a function dependent on the spatial variable on the right-hand side of the equation. This problem is studied using the Fourier method and the properties of the Mittag–Leffler function. The solution is constructed in the form of a series based on eigenfunctions.

Keywords: Riemann–Liouville fractional integral, Riemann–Liouville fractional derivative, direct problem, inverse problem, integral equation, Fourier series, existence, uniqueness, Banach fixed-point theorem.

UDC: 517.958

Received: 31.07.2024
Revised: 31.07.2024
Accepted: 26.09.2024

DOI: 10.26907/0021-3446-2026-1-3-17