# Vestnik КRAUNC. Fiz.-Mat. nauki. 2023. vol. 44. no. 3. P. 30-38. ISSN 2079-6641

**https://doi.org/10.26117/2079-6641-2023-44-3-30-38**

**Research Article**

**Full text in Russian**

**MSC 35D99**

**On a Сlass of Non-Local Boundary Value Problems for the Heat Equation**

**F. M. Nakhusheva¹, M. A. Kerefov¹, S. Kh. Gekkieva²^\ast, M. M. Karmokov¹**

¹Kabardino-Balkarian State University named after H.M. Berbekov, 360004, Nalchik, Chernyshevsky st., 173, Russia

²Institute of Applied Mathematics and Automation of Kabardin-Balkar Scientific Center of RAS, 360000, Nalchik, Shortanova st., 89 A, Russia

Abstract. Non-local boundary value problems for parabolic equations, including the equations of thermal

conductivity, have been the object of research for a long time. Interest in such problems is caused by the need

for further development of the theory of boundary value problems with displacement (Nakhushev’s problems), as

well as in connection with their numerous applications. This article is devoted to the study of the question of

the unambiguous solvability of one class of nonlocal boundary value problems for the heat equation. The problem of finding a regular solution of the thermal conductivity equation with a fractional Riemann – Liouville derivative under boundary conditions is considered. The Cauchy problem for an equation equivalent to the original equation is considered, and it is proved that the boundary value problem under consideration is reduced to the first boundary value problem for the heat equation, provided that the Cauchy problem has a unique solution in the class of functions satisfying the conditions of A. N. Tikhonov. In this case, the solution is represented as an integral equation containing the Barrett function in the kernel. Also, by reducing to a system of differential equations with a fractional Riemann-Liouville derivative, the question of the uniqueness and existence of a solution to the problem is solved when the values of the solution at the other end are in the condition. The results obtained in this work will serve as a basis for further research of nonlocal boundary value problems for parabolic differential equations underlying mathematical modeling of processes in systems with fractal structure, as well as the development of the theory of fractional differential equations.

*Key words: class of nonlocal boundary value problems, Tikhonov conditions, regular solution, Cauchy problem, homogeneous problem, fractional differentiation operator, fractional differential equations.*

Received: 28.06.2023; Revised: 27.10.2023; Accepted: 01.11.2023; First online: 02.11.2023

**For citation.** Nakhusheva F. M., Kerefov M. B., Gekkieva S. Kh., Karmokov M. M. On a class of non-local

boundary value problems for the heat equation. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 44: 3, 30-38. EDN:

WFBWCX. https://doi.org/10.26117/2079-6641-2023-44-3-30-38.

**Funding.** The study was carried out without support from foundations

**Competing interests.** There are no conflicts of interest regarding authorship and publication.

**Contribution and Responsibility.** The author participated in the writing of the article and is fully responsible for

submitting the final version of the article to the press.

^\ast**Correspondence**: E-mail: fatima-nakhusheva@mail.ru, kerefov@mail.ru, gekkieva_s@mail.ru, mkarmokov@yandex.ru

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

© Nakhusheva F. M., Kerefov M. B., Gekkieva S. Kh., Karmokov M. M., 2023

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

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#### Information about authors

**Nakhusheva Fatima Mukhamedovna **– Ph. D. (Phys. & Math.), Associate Professor, Kabardino-Balkarian State University named after Kh. M. Berbekov, Department of Applied Mathematics and Computer Science, Nalchik, Russia, fatima-nakhusheva@mail.ru, ORCID 0000-0002-3750-1445.

**Kerefov Marat Aslanbievich** – Ph. D. (Phys. & Math.), Associate Professor, Kabardino-Balkarian State University named after Kh. M. Berbekov, Department of Applied Mathematics and Computer Science, Nalchik, Russia, kerefov@mail.ru, ORCID 0000-0002-7442-5402.

**Gekkieva Sakinat Khasanovna** – Ph. D. (Phys. & Math.), Leading researcher, Institute of Applied Mathematics and Automation of Kabardin-Balkar Scientific Center of RAS, 360000, Nalchik, Shortanova st., 89 A, Russia, gekkieva_s@mail.ru, ORCID 0000-0002-2135-2115.

**Karmokov Mukhamed Matsevich **– Ph. D. (Phys. & Math.), Associate Professor, Kabardino-Balkarian State University named after Kh. M. Berbekov, Department of Applied Mathematics and Computer Science, Nalchik, Russia, mkarmokov@yandex.ru, ORCID 0000-0001-5189-6538.