# Vestnik КRAUNC. Fiz.-Mat. Nauki. 2023. vol. 42. no. 1. P. 80-97. ISSN 2079-6641

**MATHEMATICS **

https://doi.org/10.26117/2079-6641-2023-42-1-80-97

Research Article

Full text in Russian

MSC 35A02

**On the Adjoint Problem in a Domain with Deviation Out from the Characteristic for the Mixed Parabolic-Hyperbolic Equation with the Fractional Order Operator**

**B. I. Islomov^*, I. A. Akhmadov^***

National University of Uzbekistan named after Mirzo Ulugbek, Uzbekistan, Tashkent, 100174, st. Universitetskaya, 4.

Abstract. In this article, it was proved the classical, strong solvability and Volterra property of the adjoint problem with departure from the characteristic for an equation of mixed parabolic-hyperbolic type with a fractional order operator in the sense of Gerasimov-Caputo. The aim of the research is to solve the conjugate problem for the equation of a mixed parabolic-hyperbolic type of fractional order. Taking into account the properties of fractional order operators, the adjoint operator is found and the statements of the adjoint problem are applied. To study the formulated problem in the parabolic part of the mixed domain, the first boundary value problem for a parabolic type equation of fractional order in the sense of Gerasimov-Caputo is solved. Using the properties of the Wright function, a functional relation is obtained on the transition line. In the same way, solving the Cauchy problem with the hyperbolic part of the mixed domain, we find a functional relation. Consequently, the problem posed reduces in an equivalent way to a Volterra integral equation of the second kind with a weak singularity. According to the theory of Volterra integral equations of the second kind, the unique solvability of the resulting equation is proved. In addition, using the methods of integro-differentiation operators of fractional order, the theory of special functions, a priori estimates, the theory of integral equations, uniqueness, existence and Volterra theorems for the adjoint problem in a domain with deviation out of the characteristic for a mixed-type equation of fractional order are proved. The results obtained are new and differ from the results of M. A. Sadybekov and A. S. Berdyshev.

*Key words: local boundary conditions, fractional order equation, Wright and Green’s function, strong solvability, deviation out from characteristic.*

Received: 06.12.2022; Revised: 19.03.2023; Accepted: 22.03.2023; First online: 15.04.2023

**For citation.** Islomov B. I., Akhmadov I. A. On the adjoint problem in a domain with deviation out from the characteristic for the mixed parabolic-hyperbolic equation with the fractional order operator. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 42: 1, 80-97. EDN: DCGBAL. https://doi.org/10.26117/2079-6641-2023-42-1-80-97.

**Funding.** Not applicable.

**Competing interests.** The authors declare that 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.

^**Correspondence: E-mail: islomovbozor@yandex.com, ahmadov.ilhom@mail.ru*

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

© Islomov B. I., Akhmadov I. A., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)

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** Information about authors**

**Islomov Bozor Islomovich **– Doctor of Physics and Mathematics, Professor of the Department of “Differential Equations and Mathematical Physics” of the National University of Uzbekistan named after Ulugbek, Uzbekistan, https://orcid.org/0000-0002-3060-3019.

**Akhmadov Ilkhom Ali ugli **– doctoral student (PhD) of the Department

of “Differential Equations and Mathematical Physics” of the National University of Uzbekistan, named after Ulugbek, Uzbekistan, https://orcid.org/0000-0002-5797-7424.