S. J. Shakarovich^{1, 2}

¹Tashkent University of Information Technologies named after Muhammad al-Khwarizmi (Tashkent, Uzbekistan)

²V. I. Romanovskiy Institute of Mathematics AS of the Republic of Uzbekistan (Tashkent, Uzbekistan)

Abstract. The paper studies the inverse problem of the depending on a time variable kernel of the integral term of a multidimensional hyperbolic-type integro-differential equation. First, a direct problem is investigated, assuming that the kernel of the integral term is known. The Fourier method reduces this problem to solving a Volterra-type integral equation of the second kind with respect to the unknown function. A priori estimates for the desired function and for its second-order derivatives are obtained. Next, two inverse problems are studied. The first is determining the memory kernel of a wave process with an integral overdetermination condition. In the second inverse problem, the kernel of the integral term is found from the known solution of the direct problem at some fixed point. In both cases, the inverse problem is reduced to a nonlinear convolution-type Volterra integral equation of the second kind. The method of contracting mappings is used to prove the unique solvability of the posed inverse problems in the space of continuous functions with weighted norms, and an estimate for the conditional stability of the solution is obtained.

Key Words: integro-differential equation, inverse problem, Fourier method, integral kernel, spectral problem, Banach theorem, Gronwall inequality

For citation: S. J. Shakarovich. Problems of Determining the Kernel of an Integro-Differential Equation in a Bounded Domain. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 28:1(2026), 48–66. DOI: https://doi.org/10.15507/2079-6900.28.202601.48-66

Information about the author:

Safarov J. Shakarovich, D. Sci. (Phys. and Math.), Professor, Department of Higher Mathematics, Tashkent University of Information Technologies named after Muhammad al-Khwarizmi (108, Timur St., Tashkent 100084, Uzbekistan), ORCID: http://orcid.org/0000-0001-9249-835X, j.safarov65@mail.ru