The tri-harmonic Neumann problem

Abstract

In this article investigated the tri-harmonic Neumann function for the unit dics. For harmonics functions the Neumann’s boundary problem is well studied and solved under certain conditions through Neumann’s function, sometimes it is also called Green’s function of the second order. Any case of finding of Green function of the corresponding boundary value problem is very important for this or that area D as it contains extensive information, allowing to write out a large number of analytical solutions in the form of integrated ratios. At the same time the specified procedure makes the main difficulty at the solution Dirichlet and Neumann problems and in an explicit form Green function is known only for a small number of simple areas. The harmonics Green function with itself consistently leads to the subsequent polyharmonic Green function which can be used to solve the subsequent Dirichlet problem for higher order of the Poisson equation. Methods of integrated transformation have received tri-harmonic Neumann function in explicit form for the unit disc of the complex plane with biharmonic Neumann function. With Neumann’s function an integrated idea is given by development for the tri-harmonic operator. Above-mentioned polyharmonic Green function for the unit disc gives rise to the solution some specific polyharmonic objective of Dirichlet problem. In the same way harmonic Neumann function with itself consistently leads to the subsequent polyharmonic Neumann function. Received in the present article result allows to expect interesting prospects in further development of the analytical theory of boundary valua problems in complex analysis for the equations of elliptic type.