Adyghe Int. Sci. J. Vol. 25, No. 4. Pp. 14–20.
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DOI: https://doi.org/10.47928/1726-9946-2025-25-4-14-20
EDN: HTXCYF
MATHEMATICS
| MSC 35L25 | Original Article |
The problem of conjugation of two
third-order hyperbolic equations
Makaova Ruzanna Khasanbievna
Junior Researcher Department of Mixed Type Equations Institute of Applied Mathematics and Automation KBSC RAS (360000, Russia, Kabardino-Balkarian Republic, Nalchik, 89A Shortanov St.),
ORCID https://orcid.org/0000-0003-4095-2332, makaova.ruzanna@mail.ru
Abstract. The problem of conjugating two third-order hyperbolic equations is studied in a mixed rectangular domain. In the positive part of the domain, the equation under consideration coincides with the Hallaire equation, and in the negative part, with a thirdorder model equation. The problem consists of finding a regular solution to the equation under consideration when mixed boundary conditions are specified on the positive part of the domain boundary, and Cauchy conditions on one of the characteristics are specified in the negative part. A theorem on the existence of a unique regular solution to the problem under consideration in the mixed domain is proved. To prove the unique solvability of the problem, the Tricomi method is used. According to this method, the corresponding fundamental relations between the traces of the desired solution and its derivative are obtained, transferred from the positive and negative parts of the mixed domain to the conjugation line. From these fundamental relations, by eliminating one of the desired functions, we arrive at a mixed boundary value problem for a second-order inhomogeneous ordinary differential equation with respect to the second desired function—the trace of the derivative of the desired solution. The solution to this problem is found and written out explicitly. Then, the solution to the problem under study is written out explicitly as a solution to a mixed boundary value problem for the Hallaire equation in the positive part of the mixed domain and as a solution to the Darboux problem for a third-order hyperbolic model equation in the negative part. The paper also finds sufficient conditions on the given functions that ensure the regularity of the obtained solutions to the problem under study.
Keywords: model third-order hyperbolic equation, the Hallaire equation, boundary value problem, Tricomi method.
Funding. This work was carried out within the framework of the state assignment of the Ministry of Education and Science of the Russian Federation on the project: Boundary value problems and optimal control problems for local and nonlocal equations of mathematical physics (1024031200120-2).
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.
For citation. Makaova R. Kh. The problem of conjugation of two third-order hyperbolic equations. Adyghe Int. Sci. J. 2025. Vol. 25, No. 4. Pp. 14–20. DOI: https://doi.org/10.47928/1726-9946-2025-25-4-14-20; EDN: HTXCYF
Submitted 29.10.2025; approved after reviewing 26.11.2025; accepted for publication 01.12.2025.
© Makaova R. Kh., 2025
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