Vestnik КRAUNC. Fiz.-Mat. Nauki. 2023. vol. 42. no. 1. P. 27-36. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2023-42-1-27-36
Research Article
Full text in Russian
MSC 35M12

On Some Boundary Value Problems with a Shift for a Mixed Type Equation

V. A. Vodahova, F. M. Nahusheva, Z. H. Guchaeva $^*$ , A. H. Kodzokov

Kabardino-Balkarian State University named after H.M. Berbekov, 360004, Nalchick, Chernyshevskogo str., 173, Russia

Abstract. An important stage in the development of the theory of boundary value problems was the proposed
by A.M. Nakhushev in 1969, non-local problems of a new type, which were later called in our country boundary value problems with a shift, and abroad – Nakhushev problems (problems). They are a generalization of the Tricomi problem, and also contain a wide class of well-posed self-adjoint problems. These problems immediately aroused great interest of many authors. In recent years, studies of problems with a shift for equations of mixed type have been carried out especially intensively. But in these works, the boundary conditions, as a rule, contain classical operators, while non-local boundary value problems contain operators of a more complex structure and operators of fractional integro-differentiation. In this paper, we study the unique solvability of problems with mixing for an equation of mixed elliptic-hyperbolic type. Under constraints of unequal type on known functions and different orders of fractional differentiation operators in the boundary condition, uniqueness theorems are proved. The existence of a solution to the problems is proved by reducing the problems to Fredholm equations
of the second kind, the unconditional solvability of which follows from the uniqueness of the solution to the problems.

Key words: problem with shift, Cauchy problem, Dirichlet problem, fractional differentiation operator, fractional integration operator, Fredholm equation, singular integral equation, regularizer.

Received: 03.03.2023; Revised: 27.03.2023; Accepted: 30.03.2023; First online: 15.04.2023

For citation. Vodahova V. A., Nahusheva F. M., Guchaeva Z. H., Kodzokov A. H. On some boundary value problems with a shift for a mixed type equation. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 42: 1, 27-36. EDN: GMQZNY. https://doi.org/10.26117/2079-6641-2023-42-1-27-36.

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: proporwiz@yandex.ru

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

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Information about authors

Vodakhova Valentina Arkadevna – Ph.D. (Phys. & Math. Sci.), Associate Professor, Department of Algebra and Differential Equations, Institute of Physics and Mathematics, Kabardino-Balkarian State University, Nalchik, Russia, ORCID 0009-0001-9990-7467.

Nakhusheva Fatima Muhamedovna – Ph.D. (Phys. & Math. Sci.), Associate Professor, Department of Applied Mathematics and Informatics, Institute of Artificial Intelligence and Digital Technologies, Kabardino-Balkarian State University, Nalchik, Russia, ORCID 0009-0007-5015-965X.

Guchaeva Zera Hamidbievna – Senior Lecturer, Department of Algebra and Differential Equations, Institute of Physics and Mathematics, Kabardino-Balkarian State University, Nalchik, Russia, ORCID 0009-0000-9777-4018.

Kodzokov Azamat Khasanovich – Senior Lecturer, Department of Algebra and Differential Equations, Institute of Physics and Mathematics, Kabardino-Balkarian State University, Nalchik, Russia, ORCID 0009-0007-3431-1228.

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