Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit Ball
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 expli...
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MDPI AG
2021-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/16/1907 |
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| author | Valery Karachik |
| author_facet | Valery Karachik |
| author_sort | Valery Karachik |
| collection | DOAJ |
| description | 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 <i>m</i>-harmonic function, which is a solution to the Dirichlet boundary value problem, in terms of <i>m</i> 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. |
| first_indexed | 2024-03-10T08:37:29Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T08:37:29Z |
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| spelling | doaj.art-cca832bf7f2744968767ad790fc85a6f2023-11-22T08:33:42ZengMDPI AGMathematics2227-73902021-08-01916190710.3390/math9161907Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit BallValery Karachik0Department of Mathematical Analysis, South Ural State University, 454080 Chelyabinsk, RussiaIn 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 <i>m</i>-harmonic function, which is a solution to the Dirichlet boundary value problem, in terms of <i>m</i> 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.https://www.mdpi.com/2227-7390/9/16/1907polyharmonic equationDirichlet problemNeumann problemGreen’s functionAlmansi representation |
| spellingShingle | Valery Karachik Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit Ball Mathematics polyharmonic equation Dirichlet problem Neumann problem Green’s function Almansi representation |
| title | Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit Ball |
| title_full | Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit Ball |
| title_fullStr | Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit Ball |
| title_full_unstemmed | Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit Ball |
| title_short | Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit Ball |
| title_sort | dirichlet and neumann boundary value problems for the polyharmonic equation in the unit ball |
| topic | polyharmonic equation Dirichlet problem Neumann problem Green’s function Almansi representation |
| url | https://www.mdpi.com/2227-7390/9/16/1907 |
| work_keys_str_mv | AT valerykarachik dirichletandneumannboundaryvalueproblemsforthepolyharmonicequationintheunitball |