Vestnik КRAUNC. Fiz.-Mat. Nauki. 2025. vol. 50. no. 1. P. 39 – 61. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2025-50-1-39-61
Research Article
Full text in Russian
MSC 35M10, 35M12

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Dedicated to the memory of Professor A. K. Urinov

On a nonlocal boundary value problem for a parabolic-hyperbolic equation with two perpendicular lines of type change

I. U. Khaydarov^{1,*}, R. T. Zunnunov^{2,3,4}

^{1}Fergana State University, 150100, Fergana, Murabbiylar street, 19, 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
^3 V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 100174, Tashkent, University street, 9, Uzbekistan
^4International Nordiс University, 100043, Tashkent, Bunyodkor street, 8/2, Uzbekistan

Abstract. The paper addresses a boundary value problem with a shift for a parabolic-hyperbolic equation involving a spectral parameter and characterized by the presence of two mutually perpendicular lines of type change. The domain under consideration is composed of subdomains in which the equation changes type—being parabolic in some regions and hyperbolic in others—making the problem particularly interesting and analytically challenging. Additionally, the equation includes discontinuous coefficients, which necessitates a special formulation of the transmission (gluing) conditions at the interfaces between regions of different types. The study aims to formulate and justify the wellposedness (existence and uniqueness of the solution) of the boundary value problem with a shift in a complex geometrical domain for an arbitrary real value of the spectral parameter \lambda. The authors employ the method of integral transforms and operator theory, including integral and integro-differential operators of the form A^{n,λ}_{mx}, B^{n,λ}_{mx}, and C^{n,λ}_{mx}, whose properties play a key role in the analysis. It is shown that the original problem can be reduced to an equivalent system of Fredholm integral equations of the second kind with continuous kernels. The solvability of this system is established using the Fredholm alternative theorem. Furthermore, it is demonstrated that the solution can be represented explicitly in terms of integral formulas involving Bessel functions and special kernels that capture the behavior of the solution in various parts of the domain. Matching conditions play a crucial role in ensuring the correct joining of solutions across regions where the type of the equation changes and along the line of coefficient discontinuity.

Key words: parabolic-hyperbolic type equation; boundary value problem with a shift; lines of type change; spectral parameter; Fredholm integral equations; nonlocal conditions; Bessel-Clifford functions; well-posedness of the problem; integral and integro-differential operators.

Received: 31.03.2025; Revised: 14.04.2025; Accepted: 17.04.2025; First online: 18.04.2025

For citation. Khaydarov I. U., Zunnunov R. T. On a nonlocal boundary value problem for a parabolic-hyperbolic equation with two perpendicular lines of type change. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 50: 1, 39-61. EDN: RFEXMO. https://doi.org/10.26117/2079-6641-2025-50-1-39-61.

Funding. The second author was supported by a grant from 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: ibrohim0902@gmail.com

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

© Khaydarov I. U., Zunnunov R. T., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

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