Vestnik КRAUNC. Fiz.-Mat. nauki. 2024. vol. 48. no. 3. P. 7 – 19. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2024-48-3-7-19
Research Article
Full text in English
MSC 35A08, 35J05, 35J15, 35J25

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Boundary Value Problems for the Three-Dimensional Helmholtz Equation in the Unbounded Octant, Square and Half Space

Z. O. Arzikulov^{\ast}

Fergana Polytechnic Institute, 150107, Fergana, Ferganskaya str., 86, Uzbekistan

Abstract. At present, the results of the study of boundary value problems for the two-dimensional Helmholtz equation with one and two singular coefficients are known. In the presence of two positive singular coefficients in the two-dimensional Helmholtz equation, explicit solutions of the Dirichlet, Neumann and Dirichlet-Neumann problems in a quarter plane are expressed through a confluent hypergeometric function of two variables. The established properties of the confluent hypergeometric function of two variables allow us to prove the theorem of uniqueness and existence of a solution to the problems posed.In this paper, we study the Dirichlet, Neumann, and Dirichlet-Neumann problems for the three-dimensional Helmholtz equation at zero values of singular coefficients in an octant, a quarter of space, and a half-space. Uniqueness and existence theorems are proved under certain restrictions on the data. The uniqueness of solutions of which is proved using the extremum principle for elliptic equations. Using the known fundamental (singular) solution of the Helmholtz equation, solutions to the problems under study are written out in explicit forms.

Key words: confluent hypergeometric function of three variables; system of partial differential equations; asymptotic formula; three-dimensional Helmholtz equation with three singular coefficients; Dirichlet problem in the first infinite octant.

Received: 27.09.2024; Revised: 26.10.2024; Accepted: 05.11.2024; First online: 20.11.2024

For citation. Arzikulov Z. O. Boundary value problems for the three-dimensional Helmholtz equation in the unbounded octant, square and half space. Vestnik KRAUNC. Fiz.-mat. nauki. 2024, 48: 3, 7-19. EDN: MRZFAU. https://doi.org/10.26117/2079-6641-2024-48-3-7-19.

Funding. The work was carried out without the support of funds.

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

Contribution and Responsibility. Author is solely responsible for providing the final version of the article in print.

^{\ast}Correspondence: E-mail: zafarbekarzikulov1984@gmail.com

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

© Arzikulov Z. O., 2024

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

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