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PHYSICAL-MATHEMATICAL SCIENCES

On a nonlocal boundary value problem with integral conditions for a fractional telegraph equation

R. A. Pshibikhova

Institute of Applied Mathematics and Automation – branch of the FSBSE "Federal Scientific Center "Kabardin-Balkar Scientific Center of the Russian Academy of Sciences", 360000, KBR, Nalchik, Shortanov street, 89 A

Abstract: In this paper, we study a nonlocal boundary value problem for a generalized telegraph equation with fractional derivatives. Fractional differentiation is specified using the Caputo operator. The equation is considered in a bounded rectangular domain of the plane of two independent variables. Nonlocal boundary conditions are specified in the form of partial integral expressions from the desired solution for each of the variables with given continuous kernels. Using the previously obtained representation for the solution of the Goursat problem for the equation under study in terms of the Wright-type function, the problem under consideration can be reduced to the system of Volterra linear integral equations with respect to the traces of the desired solution on the part of the boundary of the domain. As a result, a theorem on the existence and uniqueness of a solution to the problem under study is proved; its representation is found in terms of solutions to the resulting system of integral equations.

Keywords: non-local problem, Caputo derivative, fractional telegraph equation, integral condition, Wright-type function.

UDC: 17.95

MSC: 35R11

Received: 09.06.2020

DOI: 10.35330/1991-6639-2020-3-95-5-12