Adyghe Int. Sci. J. Vol. 24, No. 4. P. 34–38.
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DOI: https://doi.org/10.47928/1726-9946-2024-24-4-34-38
EDN: JTOCFS
MATHEMATICS
| MSC 35M10 | Original Article |
On solution uniqueness for Dezin problem for parabolic-hyperbolic
type equation with boundary conditions
of the first kind
Kirzhinov Romazan Anatolevich
Junior Researcher at the Dept. of Mixed–Type Equations Institute of Applied Mathematics and Automation KBSC RAS (360000, Kabardino-Balkarian Republic, Nalchik, 89A Shortanov St.), ORCID https://orcid.org/0000-0001-6645-7175, kirzhinov.r@mail.ru
Abstract. In this paper considered an inhomogeneous second-order parabolic-hyperbolic mixed type equation, represented as one-dimensional heat equation in the parabolic part and the onedimensional wave equation in the hyperbolic part. For the equation, a Dezin problem with boundary conditions of the first kind is investigated, which means to find a solution to the equation that satisfies inner-boundary condition, relating the value of the desired function on the equation type change line to the value of the normal derivative on the hyperbolicity region boundary, and boundary conditions of the first kind. It is established a criterion for the solution uniqueness to the problem. In case when the uniqueness criterion is violated, an example of a nontrivial solution to a homogeneous problem is given, and is obtained a necessary and sufficient condition for the existence of a solution to an inhomogeneous problem.
Keywords: Dezin problem, boundary conditions of the first kind, parabolic-hyperbolic type equation, mixed type equation.
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. Kirzhinov R. A. On solution uniqueness for Dezin problem for parabolic-hyperbolic type equation with boundary conditions of the first kind. Adyghe Int. Sci. J. 2024. Vol. 24, No. 4. Pp. 34–38.
DOI: https://doi.org/10.47928/1726-9946-2024-24-4-34-38; EDN: JTOCFS
Submitted 26.11.2024; approved after reviewing 06.12.2024; accepted for publication 13.12.2024.
© Kirzhinov R. A., 2024
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