Vestnik КRAUNC. Fiz.-Mat. nauki. 2024. vol. 48. no. 3. P. 20 – 32. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2024-48-3-20-32
Research Article
Full text in Russian
MSC 35M12

Contents of this issue

Read Russian Version

The First Boundary Value Problem for a Model Equation of Parabolic-Hyperbolic Type of the Third Order

Zh. A. Balkizov^{\ast}

Institute of Applied Mathematics and Automation, Kabardino-Balkarian Scientific Center RAS, 360000, Nalchik, Shortanova st., 89 A, Russia

Abstract. In 1978, the journal Differential Equations published an article by A. M. Nakhushev, which provided a technique for correctly formulating a boundary value problem for a class of second-order parabolic-hyperbolic equations in an arbitrary bounded domain Ω with a smooth or piecewise smooth boundary Σ. The boundary value problem investigated in the above-mentioned work is currently called the first boundary value problem for a second-order mixed parabolic-hyperbolic equation. Within the framework of this work, the first boundary value problem for a third-order model parabolic-hyperbolic equation in a mixed domain is formulated and investigated in the sense in which it was formulated and investigated by A. M. Nakhushev for second-order equations. In one part of the mixed domain, the equation under consideration coincides with a degenerate hyperbolic equation of the first kind of the second order, and in the other part it is an inhomogeneous third-order equation with multiple characteristics of parabolic type. For various values of the parameter λ included in the equation under consideration, theorems of existence and uniqueness of a regular solution of the problem under study are proved. To prove the uniqueness theorem, the method of energy integrals is used in conjunction with the method of A.M. Nakhushev. To prove the existence theorem, the method of integral equations is used. In terms of the Mittag-Leffler function, the solution to the problem is found and written out in explicit form.

Key words: equation of mixed parabolic-hyperbolic type, degenerate hyperbolic equation of the first kind, first boundary value problem for an equation of parabolic-hyperbolic type, integral equations of the second kind, Tricomi problem and method, method of integral equations.

Received: 28.10.2024; Revised: 11.11.2024; Accepted: 13.11.2024; First online: 20.11.2024

For citation. Balkizov Zh. A. The first boundary value problem for a model equation of parabolic-hyperbolic type of the third order. Vestnik KRAUNC. Fiz.-mat. nauki. 2024, 48: 3, 20-32. EDN: RAQLMI. https://doi.org/10.26117/2079-6641-2024-48-3-20-32.

Funding. The work was carried out within the framework of the state assignment of the IPMA KBSC RAS (reg. No. 122041800015-8).

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. Author is solely responsible for providing the final version of the article in print.

^{\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., 2024

© Institute of Cosmophysical Research and Radio Wave Propagation, 2024 (original layout, design, compilation)

References

1. Struchina G. M. Zadacha o sopryazhenii dvukh uravnenij. Inzhenerno-fizicheskij zhurnal. 1961. vol. 4. no. 11. pp. 99-104 (in Russan)
2. Zolina L. A. Boundary value problem for the model equation of the hyperbolic-parabolic type. U.S.S.R. Comput. Math. Math. Phys. 1966. vol. 6. no. 6, pp. 63–78.
3. Sabitov K. B. On the theory of equations of mixed parabolic-hyperbolic type with a spectral parameter. Differ. Equ. 1989. vol. 25. no. 1. pp. 93–100.
4. Nakhushev A. M. On the theory of linear boundary value problems for a second order equation of mixed hyperbolic-parabolic type. Dif. Ur. 1978. vol. 14. no. 1. 66–73 (in Russian)
5. Nakhushev A. M. Zadachi so smeshcheniem dlya uravnenij v chastnykh proizvodnykh [Shift problems for partial differential equations]. Moscow. Nauka. 2006, 287 p. (in Russian)
6. Balkizov Zh. A. The first boundary value problem for a hyperbolic equation degenerating inside a domain. Vladikav. Matemat. Zhurnal. 2016. vol. 18. no. 2. pp. 19-30. DOI: 10.23671/VNC.2016.2.5915 (in Russian)
7. Balkizov Zh. A. Dirichlet boundary value problem for a third order parabolic-hyperbolic equation with degenerating type and order in the hyperbolicity domain. Ufa Math. J. 2017. vol. 9. no. 2. pp. 25–39. DOI:10.13108/2017-9-2-25.
8. Balkizov Zh. A. The first boundary value problem with deviation from the characteristics for a second order parabolic-hyperbolic equation. Bulletin of the Karaganda University. 2018. no. 2(90). pp. 34-42. DOI: 10.31489/2018m2/34-42.
9. Dzhuraev T. D., Sopuev A., Mamazhanov M. Kraevye zadachi dlya uravnenij parabologiperbolicheskogo tipa [Boundary value problems for parabolic-hyperbolic equations]. Tashkent. Fan. 1986. 220 p. (in Russian)
10. Sabitov K. B. Pryamye i obratnye zadachi dlya uravnenij smeshannogo parabologiperbolicheskogo tipa [Direct and inverse problems for equations of mixed parabolichyperbolic type]. Moscow. Nauka. 2016. 272 p. (in Russian)
11. Smirnov M. M. Uravneniya smeshannogo tipa [Mixed type equations]. Moscow. Nauka. 1970. 296 p. (in Russian)
12. Djuraev T. D. Kraevye zadachi dlya uravnenij smeshannogo i smeshanno-sostavnogo tipov
    [Boundary value problems for equations of mixed and mixed-composite types]. Tashkent. Fan. 1979. 238 p. (in Russian)
13. Nakhushev A. M. Uravneniya matematicheskoj biologii [Equations of Mathematical Biology]. Moscow. Visshaya shkola. 1995. 301 p. (in Russian)
14. Smirnov M. M. Vyrozhdayushchiesya giperbolicheskie uravneniya [Degenerate hyperbolic equations]. Minsk. Visheishaya shkola. 1977. 160 p. (in Russian)
15. Nakhushev A.M. Drobnoe ischislenie i ego primenenie [Fractional calculus and its applications]. Moscow. Fizmatlit. 2003. 272 p. (in Russian)
16. Samko S. G., Kilbas A. A., Marichev O. I. Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya [Integrals and derivatives of fractional order and some of their applications]. Minsk. Nauka i tekhnika. 1987. 688 p. (in Russian)
17. Djrbashyan M. M. Integral’nye preobrazovaniya i predstavleniya funkcij v kompleksnoj ploskosti [Integral transforms and representations of functions in the complex plane]. Moscow. Nauka. 1966. 672 p. (in Russian)

---

Information about the author

---