On fundamental solutions for multidimensional Helmholtz equation with three singular coefficients

TL;DR

This paper constructs fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients, expressed via hypergeometric functions, and analyzes their properties for solving boundary value problems.

Contribution

It introduces explicit fundamental solutions for elliptic equations with three singular coefficients using hypergeometric functions, advancing methods for degenerate elliptic equations.

Findings

Fundamental solutions expressed in hypergeometric functions.

Determined the order of singularity.

Analyzed properties for boundary value problem applications.

Abstract

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients, which could be expressed in terms of a confluent hypergeometric function of four variables. In addition, the order of the singularity is determined and the properties of the found fundamental solutions that are necessary for solving boundary value problems for degenerate elliptic equations of second order are found.