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

This article is cited in 3 scientific papers (total in 3 papers)

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
Received: 25.05.2018
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
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\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
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\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}
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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  mathnet  crossref  crossref  mathscinet  isi  elib
    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  mathnet  crossref
    3. V. V. Karachik, “Reshenie zadachi Dirikhle dlya poligarmonicheskogo uravneniya v share”, Matem. tr., 24:2 (2021), 46–64  mathnet  crossref
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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