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

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.

On Green’s Function of the Dirichlet Problem for the Polyharmonic Equation in the Ball

“…As a continuation of research on the construction of solutions to the Dirichlet problem from [9], we present the following assertion, which follows from Theorem 1.…”

“…, 2m}, the function C|x − ξ| 2m−n is a polynomial of degree less or equal to 2m − 2, which means that it is an m-harmonic function everywhere. Here, C is the corresponding numeric coefficient from (9). Therefore, the function…”

“…[2,3], and the Green's functions in the sector for the biharmonic and 3-harmonic equations are presented in [4,5]. The papers [6][7][8] are devoted to the construction of the Green's function to the Dirichlet problem for the polyharmonic equation in the unit ball, and the paper [9] contains solutions to the Dirichlet and Neumann [10] problems for the homogeneous polyharmonic equation. The papers [11,12] give an explicit form of the Green's functions for the biharmonic and 3-harmonic equations in the unit ball.…”

Integral Representation and Explicit Formula at Rational Arguments for Apostol–Tangent Polynomials

“…Other studies use generating functions to derive Fourier series, integral representation and explicit formula of some special numbers and functions, which is also the main object of this study. It is important to note that the integral representation is necessary in finding explicit formula and asymptotic approximation of a function (see [3,4]).…”

“…These values of z, which we denote by z k , are the singularities of the generating Function (4). We impose that R should be less than the modulus of the nearest singularity, which is…”