Differential Equations and Mathematical Physics

The first boundary value problem in a rectangular domain for a differential equation with the Bessel operator and the Riemann–Liouville partial derivative

F. G. Khushtova

Institute of Applied Mathematics and Automation of Kabardin-Balkar Scientific Centre of RAS, Nal'chik, 360000, Russian Federation

Abstract: The paper is devoted to the first boundary-value problem in a rectangular domain for a differential equation with the singular Bessel operator acting with respect to a spatial variable and the Riemann–Liouville fractional differentiation operator acting with respect to a time variable. An explicit representation of the solution is constructed. The uniqueness of the solution is proved in the class of functions satisfying the Hölder condition with respect to the time variable. When the order of the fractional derivative is equal to unity, and the Bessel operator has no singularity, the studied equation coincides with the heat equation and the obtained results coincide with well-known corresponding classical results.

Keywords: fractional diffusion equation, fractional differentiation operator, Bessel operator, cylindrical function, Mittag–Leffler type function, first boundary-value problem.

UDC: 517.95

MSC: 26A33, 35K20, 35R11

Received: August 18, 2020
Revised: May 18, 2021
Accepted: May 24, 2021
First online: June 30, 2021

DOI: 10.14498/vsgtu1820

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