Inverse problem for a hyperbolic integro-differential equation in a bounded domain

J. Sh. Safarov^a, D. K. Durdiev^ab, A. A. Rakhmonov<sup>a

a</sup> V. I. Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, Tashkent, 100174, Uzbekistan
^b Tashkent University of Information Technologies, Tashkent, 100084, Uzbekistan

Abstract: In this paper, we consider the inverse problem of determining the kernel of an integral term in an integro-differential equation. The problem of determining the memory kernel in the wave process is reduced to a nonlinear Volterra integral equation of the first kind of convolution type, then over determination condition it brings to the Volterra integral equation of the second kind. The method of contraction maps proves the unique solvability of the problem in the space of continuous functions with weight norms, and an estimate of the conditional stability of the solution is obtained.

Key words: integro-differential equation, inverse problem, kernel, spectral problem, fixed point theorem, Gronwall inequality.

UDC: 517.958

Received: 18.01.2023
Revised: 08.04.2024
Accepted: 17.05.2024

DOI: 10.25205/1560-750X-2024-27-1-139-162

 English version: Siberian Advances in Mathematics, 2024, 34:2, 154–166