Third Double-layer Potential for generalized bi-axially symmetric Helmholtz equation

TL;DR

This paper develops the theory of double-layer potentials for the third fundamental solution of the generalized bi-axially symmetric Helmholtz equation, expanding potential theory beyond the first known solution.

Contribution

It constructs the potential theory for the third fundamental solution, utilizing properties of Appell's hypergeometric functions, and establishes related limiting theorems and integral equations.

Findings

Proved limiting theorems for the third fundamental solution.

Derived integral equations related to the double-layer potentials.

Extended potential theory to a new fundamental solution.

Abstract

The double-layer potential plays an important role in solving boundary value problems for elliptic equations, and in the study of which for a certain equation, the properties of the fundamental solutions of the given equation are used. All the fundamental solutions of the generalized bi-axially symmetric Helmholtz equation were known, and only for the first one was constructed the theory of potential. Here, in this paper, we aim at constructing theory of double-layer potentials corresponding to the third fundamental solution. By using some properties of one of Appell's hypergeometric functions in two variables, we prove limiting theorems and derive integral equations concerning a denseness of double-layer potentials.