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 Izv. Vyssh. Uchebn. Zaved. Mat., 2019, Number 2, Pages 69–81 (Mi ivm9441)

On unique solvability of boundary-value problems for three-dimentional elliptic equation with three singular coefficients

A. K. Urinov, K. T. Karimov

Fergana State University, 19 Murabbiylar str., Fergana, 150100 Republic of Uzbekistan

Abstract: We consider and investigate a number of boundary-value problems for an elliptic equation with three singular coefficients in a rectangular parallelepiped. By the method of energy integrals, we prove the uniqueness of the solution to the stated problems. To prove the existence of solutions, we use the Fourier spectral method, based on the separation of variables. The solution to the posed problems is constructed as a sum of a double Fourier–Bessel series. In justification of the uniform convergence of the constructed series we use asymptotic estimates of the Bessel functions of the real and imaginary argument. On their basis, we obtain estimates for each term of the series. The obtained estimates made it possible to prove the convergence of the series and its derivatives up to the second order inclusive, and also the existence theorem in the class of regular solutions.

Keywords: Keldysh problem, equations of elliptic type, singular coefficient, spectral method, uniqueness of solution, existence of solution.

DOI: https://doi.org/10.26907/0021-3446-2019-2-69-81

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UDC: 517.956
Revised: 17.01.2018
Accepted: 22.03.2018

Citation: A. K. Urinov, K. T. Karimov, “On unique solvability of boundary-value problems for three-dimentional elliptic equation with three singular coefficients”, Izv. Vyssh. Uchebn. Zaved. Mat., 2019, no. 2, 69–81

Citation in format AMSBIB
\Bibitem{UriKar19} \by A.~K.~Urinov, K.~T.~Karimov \paper On unique solvability of boundary-value problems for three-dimentional elliptic equation with three singular coefficients \jour Izv. Vyssh. Uchebn. Zaved. Mat. \yr 2019 \issue 2 \pages 69--81 \mathnet{http://mi.mathnet.ru/ivm9441} \crossref{https://doi.org/10.26907/0021-3446-2019-2-69-81}