Vestnik КRAUNC. Fiz.-Mat. nauki. 2025. vol. 50. no. 1. P. 22 – 39. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2025-50-1-22-38
Research Article
Full text in Russian
MSC 35A02

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On a Boundary Value Problem with Third Kind Boundary Conditions for a Third-Order Mixed-Type Equation with an Elliptic-Hyperbolic Principal Operator

B. I. Islomov, G. K. Kilishbayeva

National University of Uzbekistan named after M. Ulugbek, Tashkent, 100174, st. Universitetskaya 4, Republic of Uzbekistan

Abstract. In this paper, a method for solving a problem with a boundary condition of the third kind for a third-order equation of elliptic-hyperbolic type with a superposition of first- and second-order operators in a rectangular domain is proposed. It is shown that the correctness of the problem statement depends significantly on the ratio of the sides of the rectangle from the hyperbolic part of the mixed domain. An example is given in which the problem with homogeneous conditions has a nontrivial solution. A solution to the problem is constructed as a sum of a series in eigenfunctions of the corresponding one-dimensional spectral problem. A criterion for the uniqueness of the solution is established. When substantiating the uniform convergence of the series, the problem of small denominators arises. In this connection, estimates of small denominators on the distance from zero with the corresponding asymptotics are established. These estimates made it possible to prove the convergence of the series in the class of regular solutions of this equation. Estimates of the stability of the solution from the given boundary functions are proved.

Key words: third order equation, conditions of the second kind, spectral method, small denominators, uniqueness, existence, stability.

Received: 20.01.2025; Revised: 21.03.2025; Accepted: 23.03.2025; First online: 24.03.2025

For citation. Islomov B. I., Kilishbaeva G. K. On a boundary value problem with third kind boundary conditions for a third-order mixed-type equation with an elliptic-hyperbolic principal operator. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 50: 1, 22-39. EDN: XRZCCH. https://doi.org/10.26117/2079-6641-2025-50-1-22-38.

Funding. The study was conducted without the support of foundations.

Competing interests. 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.

^{\ast}Correspondence: E-mail: islomovbozor@yandex.com , kalbaevna85@mail.ru

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

© Islomov B. I., Kilishbaeva G. K., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

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Information about the authors

Islomov Bozor Islomovich – DSc (Phys. & Math.), Professor of the Department of “Differential Equations and Mathematical Physics” of the National University of Uzbekistan named after Mirzo Ulugbek,
Uzbekistan, ORCID 0000-0002-3060-3019.

Kilishbaeva Gulnaz Kalbayevna – doctoral student (PhD) of the Department of “Differential Equations and Mathematical Physics” of the National University of Uzbekistan named after Mirzo Ulugbek,
Uzbekistan, ORCID 0009-0005-9225-2508.

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