On the Green’s Function for the Third Boundary Value Problem

Karachik, V. V.; Turmetov, B. Kh.

doi:10.3103/s1055134419010036

Cited by 12 publications

(9 citation statements)

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“…This implies identity (15) for the function ( , ) and representation (14) for the solution of problem (5). In the case = 2, this corollary was obtained in work [7], while the case > 2 was resolved in [8].…”

Section: =1mentioning

confidence: 85%

Green function for analogue of Robin problem for polyharmonic equation

Turmetov

2019

Ufimsk. Mat. Zh.

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We propose a method of constructing the Green function for some boundary value problems for a polyharmonic equation in a multi-dimensional unit ball. The considered problem are analogues of the Robin problem for an inhomogeneous polyharmonic equation. For studying the solvability of these problems in the class of smooth in a ball functions, we first provide the properties of integral-differential operators. Then, employing these properties, the considered problems are reduced to an equivalent Dirichlet problem with a special right hand side. Using then known statements on the Dirichlet problem, for the main problems we prove the unique solvability theorems. We also obtain integral representations for solutions of these problems via the solutions of the Dirichlet problem. Employing the explicit form of the Green function, we find an integral representation of the Dirichlet problem with a special right hand side. The obtained integral representation then is used to construct the Green function for analogues of Robin problems. We also provide an approach for constructing the Green function for other main problems. In order to do this, we study the differential properties of the fundamental solution of the polyharmonic operator. The obtained properties of the fundamental solutions are applied for studying the properties of the Green function for the Dirichlet problem. We construct the representations of the Green function for analogues of the Robin problem. While finding the Green functions for these problems, we employ essentially the form of the Green function for the Dirichlet problem for the polyhgarmonic equation. Namely, the Green function of these problems is represented as the sum of the Green function for the Dirichlet problem and some integral term. The obtained results are in agreement with the known results for the Laplace operator.

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Section: =1mentioning

confidence: 85%

Green function for analogue of Robin problem for polyharmonic equation

Turmetov

2019

Ufimsk. Mat. Zh.

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“…In these papers, the explicit form of Green's function is obtained. An explicit representation of Green's function of the third boundary value problem for the Poisson equation is obtained in [9], and in [10], for the three-harmonic equation in a ball, Green's operator acting on polynomial data is presented.…”

Section: Introductionmentioning

confidence: 99%

“…In [9], Green's function of the Neumann problem for the Poisson equation in S for n > 2 is constructed…”

Section: Introductionmentioning

confidence: 99%

Riquier–Neumann Problem for the Polyharmonic Equation in a Ball

Karachik

2023

Mathematics

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“…В дальнейшем изложении нам понадобится функция Грина задачи Неймана для уравнения Пуассона в S . Она была построена в работе[19]…”

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A Solution to the Riquier–neymann Problem for Polyharmonic Equations in a Ball

Karachik

2023

MMPh

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In this paper, an elementary solution for polyharmonic equations is determined and its properties are given. This elementary solution coincides with previously known elementary solutions of biharmonic and triharmonic equations. Using the elementary solution, an integral representation of the solutions of a non-homogeneous polyharmonic equation in a bounded domain with a smooth boundary is found. Based on the integral representation, the solvability of the Riquier–Neumann problem is investigated. First, the concept of the Green's function of the Riquier–Neumann problem is defined, and then the Green's function is proved. Using the integral representation of the solutions of the polyharmonic equation and the Green's function of the Riquier–Neumann problem, the integral representation of the solution of the Riquier–Neumann problem in a unit ball is found. An example of the solution of the Neumann problem for the Poisson equation with the simplest right-hand side is given, which is necessary in what follows. On the basis of the Green's function of the Riquier–Neumann problem, a theorem on the integral representation of the solution of the Riquier–Neumann boundary value problem with boundary data, the integral of which over the unit sphere vanishes, is proved. In conclusion, on the basis of the theorem, an example of calculating the solution of the Riquier–Neumann problem with boundary functions coinciding with the traces of homogeneous harmonic polynomials on a unit sphere is given.

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On the Green’s Function for the Third Boundary Value Problem

Cited by 12 publications

References 12 publications

Green function for analogue of Robin problem for polyharmonic equation

Green function for analogue of Robin problem for polyharmonic equation

Riquier–Neumann Problem for the Polyharmonic Equation in a Ball

A Solution to the Riquier–neymann Problem for Polyharmonic Equations in a Ball

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