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 Zh. Vychisl. Mat. Mat. Fiz., 2019, Volume 59, Number 1, Pages 71–86 (Mi zvmmf10818)

The Green function of the Dirichlet problem for the biharmonic equation in a ball

V. V. Karachik

South Ural State University, Chelyabinsk, 454080 Russia

Abstract: An elementary solution of the biharmonic equation is defined. By using the properties of the Gegenbauer polynomials, series expansions of this elementary solution and an associated function with respect to a complete system of homogeneous harmonic polynomials orthogonal on a unit sphere are obtained. Then the Green function of the Dirichlet problem for the biharmonic equation in a unit ball is constructed in the case when the space dimension n is larger than 2. For $n>4$, a series expansion of the Green function with respect to a complete system of homogeneous harmonic polynomials orthogonal on a unit sphere is obtained. This expansion is used to calculate the integral, over a unit ball, of a homogeneous harmonic polynomial multiplied by a positive power of the norm of the independent variable with a kernel being the Green function. The Green function is found in the case $n = 2$.

Key words: Green function, biharmonic equation, Dirichlet problem.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 02.A03.21.0011 This study was supported by the Government of the Russian Federation, resolution no. 211 of March 16, 2013, and agreement no. 02.A03.21.0011.

DOI: https://doi.org/10.1134/S0044466919010113

English version:
Computational Mathematics and Mathematical Physics, 2019, 59:1, 66–81

Bibliographic databases:

UDC: 517.575
Revised: 23.07.2018

Citation: V. V. Karachik, “The Green function of the Dirichlet problem for the biharmonic equation in a ball”, Zh. Vychisl. Mat. Mat. Fiz., 59:1 (2019), 71–86; Comput. Math. Math. Phys., 59:1 (2019), 66–81

Citation in format AMSBIB
\Bibitem{Kar19} \by V.~V.~Karachik \paper The Green function of the Dirichlet problem for the biharmonic equation in a ball \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2019 \vol 59 \issue 1 \pages 71--86 \mathnet{http://mi.mathnet.ru/zvmmf10818} \crossref{https://doi.org/10.1134/S0044466919010113} \elib{https://elibrary.ru/item.asp?id=36954033} \transl \jour Comput. Math. Math. Phys. \yr 2019 \vol 59 \issue 1 \pages 66--81 \crossref{https://doi.org/10.1134/S0965542519010111} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000468086500006} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85065782983} 

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• http://mi.mathnet.ru/eng/zvmmf/v59/i1/p71

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. V. Karachik, “The Green Function of the Dirichlet Problem for the Triharmonic Equation in the Ball”, Math. Notes, 107:1 (2020), 105–120
2. V. V. Karachik, “Predstavlenie resheniya zadachi Dirikhle dlya bigarmonicheskogo uravneniya v share cherez funktsiyu Grina”, Chelyab. fiz.-matem. zhurn., 5:4(1) (2020), 391–399
3. V. V. Karachik, “Reshenie zadachi Dirikhle dlya poligarmonicheskogo uravneniya v share”, Matem. tr., 24:2 (2021), 46–64