Spatial boundary problem with the Dirichlet-Neumann condition for a singular elliptic equation

TL;DR

This paper derives an explicit solution for a boundary value problem involving a singular elliptic equation with Dirichlet-Neumann conditions in a quarter ball, utilizing Green's function and hypergeometric functions.

Contribution

It introduces a method to explicitly solve a singular elliptic boundary problem using Green's function and hypergeometric functions, which is novel for this type of problem.

Findings

Explicit Green's function involving Appell hypergeometric functions derived

Solution expressed in terms of hypergeometric functions and their properties

Method applicable to similar singular elliptic boundary problems

Abstract

The present work devoted to the finding explicit solution of a boundary problem with the Dirichlet-Neumann condition for elliptic equation with singular coefficients in a quarter of ball. For this aim the method of Green's function have been used. Since, found Green's function contains a hypergeometric function of Appell, we had to deal with decomposition formulas, formulas of differentiation and some adjacent relations for this hypergeometric function in order to get explicit solution of the formulated problem.