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Vol. 24, No. 4. P. 55–61

Adyghe Int. Sci. J. Vol. 24, No. 4. P. 55–61. 

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DOI: https://doi.org/10.47928/1726-9946-2024-24-4-55-61
EDN: ZVUGLU

MATHEMATICS

MSC 35L25 Original Article

On one problem of conjugation

of two hyperbolic equations

of the third ordere

Mazhgikhova Madina Gumarovna
Candidate of Physical and Mathematical Sciences, Junior Researcher Department of Fractional Calculus Institute of Applied Mathematics and Automation KBSC RAS (360000, Kabardino-Balkarian Republic, Nalchik, 89A Shortanov St.), ORCID: https://orcid.org/0000-0001-7612-8850, madina.mazhgihova@yan-dex.ru

Abstract. The problem of conjugation of two equations of hyperbolic type of the third order – the Haller equation in the positive part of the domain with an operator from the wave operator in the negative part – is investigated in the mixed domain. The problem consists in finding a regular solution of the equation under consideration when boundary conditions of the second kind are given on the positive part of the boundary of the region, and Cauchy conditions on one of the characteristics are given in the negative part. In addition, continuous gluing of the sought solution and its derivative on the conjugate line is required. A theorem on the existence of a single regular solution of the problem under study in the mixed domain is proved. To prove the single-valued solvability of the problem, we use the Tricomi method, according to which we obtain the corresponding fundamental relations between the traces of the desired solution and its derivative, transferred from the positive and negative parts of the mixed region to the conjugate line. From the obtained fundamental relations, we come to the second boundary value problem for an inhomogeneous ordinary differential equation of the second order with respect to the trace of the derivative of the desired solution, the solution of which is found and written out in explicit form. Then the solution of the investigated problem is written out in explicit form as the solution of the second boundary value problem for the Haller equation in the positive part of the mixed region and as the solution of the Darbu’s problem for the model equation of hyperbolic type of the third order in the negative part. Sufficient conditions on the given functions, providing regularity of the obtained solutions of the investigated problem in the mixed domain, are found.

Keywords: Hallaire equation, hyperbolic equations, local problem, regular solution.

Funding. This work was carried out within the framework of the state assignment of the Ministry of Education and Science of the Russian Federation under the project: Boundary value and control problems for basic and mixed types of equations and their application to the study of systems with distributed parameters (1021032421196-2).
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. On one problem of conjugation of two hyperbolic equations of the third order. Adyghe Int. Sci. J. 2024. Vol. 24, No. 4. Pp. 55–61.
DOI: https://doi.org/10.47928/1726-9946-2024-24-4-55-61; EDN: ZVUGLU

Submitted 27.11.2024; approved after reviewing 06.12.2024; accepted for publication 13.12.2024.

© Makaova R. Kh., 2024

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