Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 39. no. 2. pp. 7–19. ISSN 2079-6641

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MSC 35M12

Research Article

Boundary value problem for a mixed-type equation with a higher order elliptic operator

R. R. Ashurov¹, M. B. Murzambetova²

¹Institute of Mathematics named after V. I. Romanovskiy, Academy of Sciences of the Republic of Uzbekistan, 100174, Tashkent, University str., 9, Uzbekistan
²Nukus state pedagogical institute named after Ajiniyaz, 230100, Nukus, P. Seytov str.,104, Uzbekistan


In this paper, we consider a boundary value problem for a mixed-type equation with a positive,
formally self-adjoint, high order elliptic operator. The results of the work were obtained using
the Fourier method. Theorems on the existence and uniqueness of the classical solution of the
problem are proved. In this case, the positivity of elliptic operator turned out to be essential.
At the end of the paper, a mixed-type equation with a non-negative elliptic operator is considered,
and it is shown that the solution of the corresponding problem is not unique.

Key words: boundary value problem, method Fourier, elliptic operator.

DOI: 10.26117/2079-6641-2022-39-2-7-19

Original article submitted: 06.06.2022

Revision submitted: 07.07.2022

For citation. Ashurov R. R., Murzambetova M. B. Boundary value problem for a mixed-type equation with a higher order elliptic operator. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 39: 2, 7-19. DOI: 10.26117/2079-6641-2022-39-2-7-19

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.

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

©Ashurov R. R., Murzambetova M. B., 2022


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Ashurov Ravshan Radjabovich – D. Sci. (Phys. & Math.), Professor, Head of Laboratory of Differential equations and their applications, Institute of Mathematics, Academy of sciences of Uzbekistan, Tashkent, Uzbekistan, ORCID 0000-0001-5130-466X.

Murzambetova Mexriban Begdullaevna – Teacher of the Faculty of Physics and Mathematics of the Nukus State pedagogical Institute named after Ajiniyaz, Nukus, Uzbekistan, ORCID 0000-0001-6704-0785.