Vestnik КRAUNC. Fiz.-Mat. Nauki. 2026. vol. 54. no. 1. P. 93 – 103. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2026-54-1-93-103
Research Article
Full text in English
MSC 35M12

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Tricomi Problem for a Mixed-Type Equation of the Second Kind in a Domain the Elliptic Part of Which is the First Quadrant of the Plane

R. T. Zunnunov^{1,2,\ast}, Sh. A. Bektosheva^3, R.I. Parovik^{1,4}

^1Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, 100174, Tashkent, Universitetskaya str., 9, Uzbekistan
^2Tashkent Branch of the Gubkin Russian State University of Oil and Gas (NRU), 100125, Tashkent, Mirzo-Ulugbek district, Durmon yuli str., 34, Uzbekistan
^3 Kokand Pedagogical Institute, 150700, Kokand, Movarounnahr str., 77, Uzbekistan
^4 Institute of Cosmophysical Research and Radio Wave Propagation, Far Eastern Branch of the Russian Academy of Sciences, 684034, Paratunka village, Mirnaya str., 7, Russia

Abstract. In the theory of mixed-type equations, most studies have been carried out for bounded domains with smooth boundaries and for equations of the first kind. In the present paper, for a mixed-type equation of the second kind u_{xx} + \text{sign}y|y|^mu_{yy} = 0, 0 < m < 1, a Tricomi problem is studied in an unbounded domain whose elliptic part is the first quadrant of the plane. The uniqueness of the solution is proved using the extremum principle. The existence of the solution is established by the Green’s function method in the elliptic part and by the integral equation method in the hyperbolic part. In constructing the Green’s function, properties of the modified Bessel functions of the second kind and the Gaussian hypergeometric function are employed. A Fredholm integral equation of the second kind is derived for the trace of the solution on the degeneracy line; its solvability follows from the proven uniqueness. Numerical calculations are performed to visualize the solution, and the results are presented as three-dimensional surfaces and contour plots. A mathematical and physical interpretation of the solution is given for various values of the parameter m.

Key words: Tricomi problem, mixed-type equation of the second kind, extremum principle, Green’s function method, integral equation method, first quadrant of the plane

Received: 17.03.2026; Revised: 26.03.2026; Accepted: 28.03.2026; First online: 29.03.2026

For citation. Zunnunov R. T., Bektosheva Sh. A., Parovik R. I. Tricomi problem for a mixed-type equation of the second kind in a domain the elliptic part of which is the first quadrant of the plane. Vestnik KRAUNC. Fiz.-mat. nauki. 2026, 54: 1, 93-103. EDN: SSEJBO. https://doi.org/10.26117/2079-6641-2026-54-1-93-103.

Funding. The work was carried out within the framework of the Agreement between the V. I. Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan and the Federal State Budgetary Scientific Institution “Institute of Cosmophysical Research and Radio Wave Propagation of the Far Eastern Branch of the Russian Academy of Sciences”on cooperation in the field of mathematical research (No. 117 of April 28, 2022).

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: zunnunov@mail.ru

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

© Zunnunov R. T., Bektosheva Sh. A., Parovik R. I., 2026

© Institute of Cosmophysical Research and Radio Wave Propagation, 2026 (original layout, design, compilation)

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Information about the authors

Zunnunov Rakhimjon Temirbekovich – D. Sci. (Phys. & Math.), Leading Researcher, V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan, ORCID 0000-0001-9392-5464.

Bektosheva Shohsanam Akhrorjon kizi – PhD student, Department of Mathematics, Kokand State University, Kokand, Uzbekistan, ORCID 0009-0009-3378-9662.

Parovik Roman Ivanovich – D. Sci. (Phys. & Math.), Assistant Professor, Leading Researcher, Lab. of Modeling Physical Processes, Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, S. Paratunka, Russia, ORCID 0000-0002-1576-1860.

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