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JOURNALS // Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika // Archive

Izv. Vyssh. Uchebn. Zaved. Mat., 2024 Number 9, Pages 22–33 (Mi ivm10012)

Inverse problem for a fourth-order differential equation with the fractional Caputo operator

U. D. Durdievab, A. A. Rahmonovba

a Bukhara State University, 11 M.Ikbol str., Bukhara, 200100 Republic of Uzbekistan
b Bukhara Branch of the Institute of Mathematics named after V.I. Romanovskiy, 9 University str., Tashkent, 100174 Republic of Uzbekistan

Abstract: In this paper we consider an initial boundary value problem (direct problem) for a fourth order equation with the fractional Caputo derivative. Two inverse problems of determining the right-hand side of the equation by a given solution of the direct problem at some point are studied. The unknown of the first problem is a one-dimensional function depending on a spatial variable, while in the second problem a function depending on a time variable is found. Using eigenvalues and eigenfunctions, a solution of the direct problem is found in the form of Fourier series. Sufficient conditions are established for the given functions, under which the solution to this problem is classical. Using the results obtained for the direct problem and applying the method of integral equations, we study the inverse problems. Thus the uniqueness and existence theorems of the direct and inverse problems are proved.

Keywords: initial boundary value problem, inverse problem, fractional Caputo derivative, Mittag–Leffler function, eigenfunction, eigenvalue, uniqueness, existence.

UDC: 517.9

Received: 08.09.2023
Revised: 23.04.2024
Accepted: 26.06.2024

DOI: 10.26907/0021-3446-2024-9-22-33


 English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2024, 68:9, 18–28


© Steklov Math. Inst. of RAS, 2025