# Vestnik КRAUNC. Fiz.-Mat. Nauki. 2023. vol. 42. no. 1. P. 37-57. ISSN 2079-6641

MATHEMATICS https://doi.org/10.26117/2079-6641-2023-42-1-37-57
Research Article
Full text in Russian
MSC 26A33; 33E20

Contents of this issue

The Problem for a Mixed Equation with Fractional Power of the Bessel Operator

A. V. Dzarakhokhov ^{*, 1} , E. L. Shishkina ^{*, 2, 3}

¹Gorsky State Agrarian University, Russia, 37 Kirov St., Vladikavkaz 362040.
²Voronezh State University, Russia, 1 Universitetskaya Pl., Voronezh 394018.
³Belgorod State National Research University (BelGU), Russia, 85 Pobedy St., Belgorod 308015.

Abstract. Recently, of particular interest are partial differential equations containing a fractional order differential operator. Similar equations and problems for them find application in the theory of viscous elasticity, electrochemistry, control theory, modeling of epidemics and pandemics, and in various other areas. The present work is devoted to the solution of differential equations containing the Bessel operator of fractional degree. The article discusses the direct and inverse Meyer transforms, modified for the convenience of working with the Bessel operator of a fractional degree. For the considered Meyer transformation, a convolution is obtained. Using the Laplace and Poisson transformations, factorizations of the direct and inverse Meyer transformations are obtained. Using the considered modified Meyer transform, we find a solution to an ordinary differential equation with a Bessel operator of fractional degree. A nonlocal boundary value problem for a mixed parabolic-hyperbolic equation containing a fractional degree Bessel operator is considered. It is proved that, under certain conditions of smoothness of the input functions of the problem and the condition of conjugation on the dividing line of the regions of hyperbolicity and parabolicity, a regular solution of a nonlocal boundary value problem for a mixed parabolic-hyperbolic equation with a Bessel operator of fractional degree exists and is unique.

Key words: the Meyer transform, the Bessel operator of fractional degree, ordinary differential equations of fractional order, partial differential equations of fractional order.

Received: 14.03.2023; Revised: 20.03.2023; Accepted: 22.03.2023; First online: 15.04.2023

For citation. Dzarakhokhov A. V., Shishkina E. L. The problem for a mixed equation with fractional power of the Bessel
operator. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 42: 1, 37-57. EDN: DFSTCW. https://doi.org/10.26117/2079-6641-
2023-42-1-37-57.

Funding. Not applicable.

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing
the final version of the article in print. The final version of the manuscript was approved by all authors.

^*Correspondence: E-mail: azambat79@mail.ru, ilina_dico@mail.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Dzarakhokhov A. V., Shishkina E. L., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)

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