An inverse coefficient problem for the fractional telegraph equation with the corresponding fractional derivative in time

D. K. Durdiev^ab, T. R. Suyarov^ab, Kh. Kh. Turdiev<sup>ab

a</sup> V.I. Romanovskiy Institute of Mathematics Uzbekistan Academy of Sciences, 9 University str., Tashkent, 100174 Republic of Uzbekistan
^b Bukhara State University, 11 M. Ikbal str., Bukhara, 200117 Republic of Uzbekistan

Abstract: This work investigates an initial-boundary value and an inverse coefficient problem of determining a time dependent coefficient in the fractional wave equation with the conformable fractional derivative and an integral. In the beginning, the initial boundary value problem (direct problem) is considered. By the Fourier method this problem is reduced to equivalent integral equations. Then, using the technique of estimating these functions and the generalized Gronwall inequality, we get apriori estimate for the solution via the unknown coefficient which will be used to study the inverse problem. The inverse problem is reduced to an equivalent integral equation of Volterra type. To show the existence and uniqueness of the solution to this equation, the Banach principle is applied. The local existence and uniqueness results are obtained.

Keywords: inverse problem, conformable fractional derivative, integral equation, Fourier series, Banach fixed point theorem.

UDC: 517.923: 517.958

Received: 30.01.2024
Revised: 30.01.2024
Accepted: 20.03.2024

DOI: 10.26907/0021-3446-2025-2-39-52

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2025, 69:2, 32–44