# Vestnik КRAUNC. Fiz.-Mat. nauki. 2023. vol. 44. no. 3. P. 19-29. ISSN 2079-6641

**MATHEMATICS**

**https://doi.org/10.26117/2079-6641-2023-44-3-19-29**

**Research Article**

**Full text in Russian**

**MSC 35L25, 35L80**

**On a Mixed Problem for a Third Order Degenerating Hyperbolic Equation**

**R. Kh. Makaova^\ast**

Institute of Applied Mathematics and Automation KBSC RAS, 89А Shortanova St., Nalchik, 360000, Russia

Abstract. The paper investigates a mixed boundary value problem for a third-order hyperbolic equation with

order degeneration inside the domain In the positive part of the domain, the equation under consideration

coincides with the Hallaire equation, which is a third-order hyperbolic equation, although it is commonly called

an pseudoparabolic equation. In the negative part of the domain, it coincides with the degenerate hyperbolic

equation of the first kind, the special case of the Bizadze-Lyskov equation. For the problem under study, a

theorem on the existence and uniqueness of a regular solution is proved. The uniqueness of the solution is proved by the Tricomi method. Regarding the desired solution, the corresponding fundamental ratios have been found. Using the method of integral equations, the existence of a solution is equivalently reduced to the solvability of the Volterra integral equation of the second kind with respect the derivative of the desired solution. According to the general theory of Volterra integral equations of the second kind, the resulting equation is uniquely solvable in the class of regular functions. The solution to the problem can be stated explicitly as a solution to the mixed problem for the Hallaire equation in the positive part of the domain and as a solution to the Cauchy problem for the degenerate hyperbolic equation of the first kind in the negative part of the domain.

*Key words: degenerate hyperbolic equation, Hallaire equation, fractional integro-differentiation operator.*

Received: 29.09.2023; Revised: 30.10.2023; Accepted: 31.10.2023; First online: 02.11.2023

**For citation.** Makaova R. Kh. On a mixed problem for a third order degenerating hyperbolic equation. Vestnik KRAUNC.

Fiz.-mat. nauki. 2023, 44: 3, 19-29. EDN: WENGSO. https://doi.org/10.26117/2079-6641-2023-44-3-19-29.

**Funding.** The study was carried out without support from foundations.

**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.

**^\astCorrespondence**: E-mail: makaova.ruzanna@mail.ru

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

© Makaova R. Kh., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)

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#### **Information about author**

**Makaova Ruzanna Khasanbievna **– Junior Researcher Department «Equations of Mixed Type», Institute of Applied Mathematics and Automation KBSC RAS, 360000, 89А Shortanova St., Nalchik, Russia ORCID 0000-0003-4095-2332.