Vestnik КRAUNC. Fiz.-Mat. nauki. 2024. vol. 48. no. 3. P. 33 – 42. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2024-48-3-33-42
Research Article
Full text in English
MSC 35M10, 35R11

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Bitsadze-Samarskii Type Problem for the Diffusion Equation and Degenerate Hyperbolic Equation

M. Kh. Ruziev¹^{\ast}, R. T. Zunnunov², N. T. Yuldasheva¹, G. B. Rakhimova³

¹V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 100174, Tashkent, University street, 9, Uzbekistan
²Branch of the Russian State University of Oil and Gas (NRU) named after I.M. Gubkin in Tashkent, 100125, Tashkent, Durmon yuli street, 34, Uzbekistan
³Fergana State University, 150100, Fergana, Murabbiylar street, 19, Uzbekistan

Abstract. A boundary value problem of the Bitsadze-Samarskii type is studied in the article for a fractionalorder diffusion equation and a degenerate hyperbolic equation with singular coefficients at lower terms in an unbounded domain. The article considers a mixed domain where the parabolic part of the domain under consideration coincides with the upper half-plane and the hyperbolic part is bounded by two characteristics of the equation under consideration and a segment of the abscissa axis. The uniqueness of the solution to the problem under consideration is proven by the method of energy integrals. The existence of a solution to the problem under consideration is reduced to the concept of solvability of a fractional-order differential equation. An explicit form of the solution to the modified Cauchy problem is given in the hyperbolic part of the mixed domain under consideration. Using this solution, due to the boundary condition of the problem, the main functional relationship between the traces of the unknown function brought to the interval of the degeneracy line of the equation is obtained. Further, using the representation of the solution of the diffusion equation of fractional order, the second main functional relationship between the traces of the sought-for function on the interval of the abscissa axis from the parabolic part of the considered mixed domain is obtained. Through the conjugation condition of the problem under study, an equation with fractional derivatives is obtained from two functional relationships by eliminating one unknown function; its solution is written out in explicit form. In the study of the boundary value problem, generalized fractional integro-differentiation operators with the Gauss hypergeometric function are employed. The properties of the Wright and Mittag-Leffler type functions are extensively utilized in the study.

Key words: boundary value problem, diffusion equation, degenerate hyperbolic equation, Gauss hypergeometric function, Wright function, uniqueness of the solution to the problem, existence of a solution to the problem.

Received: 16.09.2024; Revised: 23.09.2024; Accepted: 27.10.2024; First online: 20.11.2024

For citation. Ruziev M. Kh., Zunnunov R. T., Yuldasheva N. T., Rakhimova G. B. Bitsadze-Samarskii type problem for the diffusion equation and degenerate hyperbolic equation. Vestnik KRAUNC. Fiz.-mat. nauki. 2024, 48: 3, 33-42. EDN: GCWUEC. https://doi.org/10.26117/2079-6641-2024-48-3-33-42.

Funding.The first author is supported by the Grant of the Ministry of Higher Education, Science and Innovation of the Republic of Uzbekistan No. F-FA-2021-424.

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

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

^{\ast}Correspondence: E-mail: mruziev@mail.ru

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

© Ruziev M. Kh., Zunnunov R. T., Yuldasheva N. T., Rakhimova G. B., 2024

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

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