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Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit Ball

Author

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  • Valery Karachik

    (Department of Mathematical Analysis, South Ural State University, 454080 Chelyabinsk, Russia)

Abstract

In the previous author’s works, a representation of the solution of the Dirichlet boundary value problem for the biharmonic equation in terms of Green’s function is found, and then it is shown that this representation for a ball can be written in the form of the well-known Almansi formula with explicitly defined harmonic components. In this paper, this idea is extended to the Dirichlet boundary value problem for the polyharmonic equation, but without invoking the Green’s function. It turned out to find an explicit representation of the harmonic components of the m -harmonic function, which is a solution to the Dirichlet boundary value problem, in terms of m solutions to the Dirichlet boundary value problems for the Laplace equation in the unit ball. Then, using this representation, an explicit formula for the harmonic components of the solution to the Neumann boundary value problem for the polyharmonic equation in the unit ball is obtained. Examples are given that illustrate all stages of constructing solutions to the problems under consideration.

Suggested Citation

  • Valery Karachik, 2021. "Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit Ball," Mathematics, MDPI, vol. 9(16), pages 1-19, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:16:p:1907-:d:611948
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    References listed on IDEAS

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    1. M. Akel & H. Begehr, 2017. "Neumann function for a hyperbolic strip and a class of related plane domains," Mathematische Nachrichten, Wiley Blackwell, vol. 290(4), pages 490-506, March.
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    Cited by:

    1. Valery Karachik & Batirkhan Turmetov & Hongfen Yuan, 2022. "Four Boundary Value Problems for a Nonlocal Biharmonic Equation in the Unit Ball," Mathematics, MDPI, vol. 10(7), pages 1-21, April.
    2. Hongfen Yuan & Valery Karachik, 2023. "Dirichlet and Neumann Boundary Value Problems for Dunkl Polyharmonic Equations," Mathematics, MDPI, vol. 11(9), pages 1-15, May.
    3. Valery Karachik, 2023. "Riquier–Neumann Problem for the Polyharmonic Equation in a Ball," Mathematics, MDPI, vol. 11(4), pages 1-21, February.

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    1. Valery Karachik, 2023. "Riquier–Neumann Problem for the Polyharmonic Equation in a Ball," Mathematics, MDPI, vol. 11(4), pages 1-21, February.

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