Vestnik КRAUNC. Fiz.-Mat. Nauki. 2026. vol. 54. no. 2. P. 26 – 37. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2026-54-2-26-37
Research Article
Full text in English
MSC 35M12

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Dirichlet Problem for a Third-Order Equation of Mixed Hyperbolic-Pseudoparabolic Type in a Rectangular Domain

Zh. A. Balkizov^{\ast}

Institute of Applied Mathematics and Automation, Kabardino-Balkarian Scientific Center of the Russian Academy of Sciences, 89A Shortanova St., Nalchik 360005, Russia

Abstract. This paper formulates and investigates the Dirichlet problem for a third-order equation of mixed hyperbolic-pseudoparabolic type in a rectangular domain D = \{(x, y) : 0 < x < r, −\alpha < y < \beta\}, where one of the matching conditions on the line of type change y = 0 is replaced by a Dirichlet condition on one of the domain boundaries. In the upper part of the mixed domain (relative to the line of type change y = 0), the governing equation is a third-order pseudoparabolic Aller’s equation, while in the remaining part, it becomes a non-homogeneous wave equation. Using Sabitov’s spectral analysis method, we prove the well-posedness of the Dirichlet problem under consideration. We establish sufficient conditions on the given functions that ensure the well-posedness of the problem and the regularity of its solution.

Key words: Aller’s equation, inhomogeneous wave equation, equations of mixed hyperbolic-pseudoparabolic type, Dirichlet problem, regular solution, class of regular functions, spectral analysis method, problem of small denominators.

Received: 08.07.2026; Revised: 25.07.2026; Accepted: 09.09.2026; First online: 14.09.2026

For citation. Balkizov Zh. A. Dirichlet Problem for a Third-Order Equation of Mixed Hyperbolic-Pseudoparabolic Type in a Rectangular Domain. Vestnik KRAUNC. Fiz.-mat. nauki. 2026, 54: 2, 26-37. EDN: ABWTOF. https://doi.org/10.26117/2079-6641-2026-54-2-26-37.

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. The author contributed to this article and is solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by the author.

^{\ast}Correspondence: E-mail: Giraslan@yandex.ru

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

© Balkizov Zh. A., 2026

© Institute of Cosmophysical Research and Radio Wave Propagation, 2026 (original layout, design, compilation)

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