An inverse problem for 1D fractional integro-differential wave equation with fractional time derivative

A. A. Rahmonov<sup>ab

a</sup> Bukhara State University, 11 M.Ikbol str., Bukhara, 200100, Republic of Uzbekistan
^b Institute of Mathematics, Uzbekistan Academy of Science, 9 University St, Olmazor District, Tashkent 100174, Republic of Uzbekistan

Abstract: This paper is devoted to obtaining a unique solution to an inverse problem for a one-dimensional time-fractional integro-differential equation. First, we consider the direct problem, and the unique existence of the weak solution is established, after that, the smoothness conditions for the solution are obtained. Secondly, we study the inverse problem of determining the unknown coefficient and kernel, and the well-posedness of this inverse problem is proved. The local existence and global uniqueness results are based on the Fourier method, fractional calculus, properties of the Mittag-Leffler function, and Banach fixed point theorem in a suitable Sobolev space.

Keywords and phrases: fractional integro-differential wave equation, Gerasimov-Caputo fractional derivative, Fourier method, Mittag-Leffler function, Bessel inequality.

MSC: 34A55, 34B05, 58C40

Received: 05.01.2024

Language: English

DOI: 10.32523/2077-9879-2025-16-2-74-97