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Vol. 24, No. 4. P. 72–79

Adyghe Int. Sci. J. Vol. 24, No. 4. P. 72–79. 

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DOI: https://doi.org/10.47928/1726-9946-2024-24-4-72-79
EDN: WXTAOJ

MATHEMATICS

MSC 34A08, 34B05 Original Article

Analogue of Bitsadze–Samarskii problem for loaded hyperbolic-parabolic equation with degeneration of order

in the hyperbolicity domain

Kazbek Uzeirovich Khubiev
Candidate of Physical and Mathematical Sciences, Senior Researcher of Department of Mixed Type Equations, Institute of Applied Mathematics and Automation of KBSC RAS (360000, Kabardino-Balkarian Republic, Nalchik, 89A Shortanov St.), ORCID: https://orcid.org/0000-0001-7612-8850, khubiev_math@mail.ru

Abstract. The paper considers a characteristically loaded equation of a mixed hyperbolicparabolic type with degeneration of order in the hyperbolicity part of the domain. In the hyperbolic part of the domain, we have a loaded one-velocity transport equation, known in mathematical biology as the Mac-Kendrick equation, in the parabolic part we have a loaded diffusion equation. In the paper is to study the uniqueness and existence of the solution of the nonlocal inner boundary value problem with Bitsadze–Samarskii type boundary conditions in the parabolic domain and the continuous conjugation conditions; the hyperbolic domain is exempt from the boundary conditions. The problem under investigation is reduced to a nonlocal problem for an ordinary second-order differential equation with respect to the trace of the unknown function in the line of the type changing. The existence and uniqueness theorem for the solution of the problem has been proved; the solution is written out explicitly in the hyperbolic part of the domain. In the parabolic part, the problem under study is reduced to the Volterra integral equation of the second kind, and the solution representation has been found.

Keywords: loaded equation, mixed type equation, hyperbolic-parabolic equation, boundary value problem, non-local problem, Bitsadze–Samarskii problem, inner boundary value problem.

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. Khubiev K. U. Analogue of Bitsadze–Samarskii problem for loaded hyperbolic-parabolic equation with degeneration of order in the hyperbolicity domain. Adyghe Int. Sci. J. 2024. Vol. 24, No. 4. Pp. 72–79.
DOI: https://doi.org/10.47928/1726-9946-2024-24-4-72-79; EDN: WXTAOJ

Submitted 06.12.2024; approved after reviewing 13.12.2024; accepted for publication 16.12.2024.

© Khubiev K. U., 2024

REFERENCES

1. Nakhushev A. M. Nagruzhennye uravneniya i ikh primeneniya [Loaded equations and their applications]. Moscow: Nauka, 2012. 232 p.
2. Nakhushev A. M. Uravneniia matematicheskoi biologii [Equations of mathematical biology]. Moscow: Vysshaia Shkola, 1995. 301 p.
3. Jenaliyev M. T., Ramazanov M. I. Nagruzhennye uravneniya kak vozmushcheniya differencial’nykh uravnenii [Loaded equation — how perturbed differential equationsy]. Almaty: Gylym, 2010. 334 p.
4. Attaev A. Kh. The Cauchy Problem for the MC Kendrick–Von Foerster Loaded Equation. International Journal of Pure and Applied Mathematics. 2017. Vol. 113. No. 4. Pp. 47–52.
5. Berezgova R. Z. On a nonlocal boundary-value problem for the Mckendrick von Foerster loaded equation with Caputo operator. Vestnik KRAUNC. Fiz.-Mat. Nauki. 2017. Vol. 19. No. 3. Pp. 5–9.
6. Napso A. F. The Bicadze–Samarskii problem for an equation of parabolic type. Differ. Uravn. 1977. Vol. 13, No. 4. Pp. 761–762.
7. Napso A. F. A nonlocal problem for an equation of mixed parabolic-hyperbolic type. Differ. Uravn. 1978. Vol. 14, No. 1. Pp. 185–186.
8. Khubiev K. U. Inner boundary value problem for the loaded equation of mixed type. Izvestiya Vuzov. Severo-Kavkazskii Region. Estestvennye nauki. 2008. No. 6 (148). Pp. 23–25.
9. Khubiev K. U. Boundary-value problem for a loaded equation of hyperbolic-parabolic type with degeneracy of order in the domain of hyperbolicity. Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. 2018. Vol. 149. Pp. 113–117.
10. Vogahova V. A. A boundary value problem with A. M. Nakhushev’s nonlocal condition for a pseudoparabolic equation of moisture transfer. Differ. Uravn. 1982. Vol. 18, No. 2. Pp. 280–285.
11. KhubievK. U. The Bitsadze–Samarskii problem for some characteristically loaded hyperbolicparabolic
equation. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki. 2019. Vol. 23, No. 4. Pp. 789–796.
12. Khubiev K. U. The Bitsadze–Samarskii problem for a loaded hyperbolic-parabolic equation with degeneracy of order in the hyperbolicity Domain. Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz. 2021. Vol. 198. Pp. 123–132.
13. Kozhanov A. I. Solvability of Boundary Value Problems with the Nonlocal Bitsadze–Samarskii Condition for Linear Hyperbolic Equations. Doklady Mathematics. 2010. Vol. 81, No. 3. Pp. 467–470.
14. Mirsaburov M., Khairullaev I. N., Bobomurodov U. E. A generalization of Bitsadze–Samarskii problem for mixed type equation. Russian Mathematics (Izvestiya VUZ. Matematika). 2016. Vol. 60, No. 10. Pp. 29–32.

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