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Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2024, Volume 11, Issue 1, Pages 141–151
DOI: https://doi.org/10.21638/spbu01.2024.109
(Mi vspua285)
 

MATHEMATICS

Inverse problem for non-homogeneous integro-differential equation of hyperbolic type

J. Sh. Safarovab

a V.I. Romanovskiy Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, 4B, ul. Universitetskaya, Tashkent, 100174, Uzbekistan
b Tashkent University of Information Technology
Abstract: An inverse problem is considered, which consists in finding a solution and a one-dimensional kernel of the integral term of an inhomogeneous integro-differential equation of hyperbolic type from the conditions that make up the direct problem and some additional condition. First, the direct problem is investigated, while the kernel of the integral term is assumed to be known. By integrating over the characteristics, the given intego-differential equation is reduced to a Volterra integral equation of the second kind and is solved by the method of successive approximations. Further, using additional information about the solution of the direct problem, we obtain an integral equation with respect to the kernel of the integral k(t), of the integral term. Thus, the problem is reduced to solving a system of integral equations of the Volterra type of the second kind. The resulting system is written as an operator equation. To prove the global, unique solvability of this problem, the method of contraction mappings in the space of continuous functions with weighted norms is used. And also the theorem of conditional stability of the solution of the inverse problem is proved, while the method of estimating integrals and Gronoullo's inequality is used.
Keywords: hyperbolic type equation, integro differential equation, the core, the inverse problem, compressed mapping method.
Received: 09.09.2022
Revised: 22.07.2023
Accepted: 31.08.2023
Document Type: Article
UDC: 517.958
MSC: 35C05
Language: Russian
Citation: J. Sh. Safarov, “Inverse problem for non-homogeneous integro-differential equation of hyperbolic type”, Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 11:1 (2024), 141–151
Citation in format AMSBIB
\Bibitem{Saf24}
\by J.~Sh.~Safarov
\paper Inverse problem for non-homogeneous integro-differential equation of hyperbolic type
\jour Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy
\yr 2024
\vol 11
\issue 1
\pages 141--151
\mathnet{http://mi.mathnet.ru/vspua285}
\crossref{https://doi.org/10.21638/spbu01.2024.109}
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