Double-Layer Potentials for a Generalized Bi-Axially Symmetric Helmholtz Equation II

TL;DR

This paper develops the theory of double-layer potentials for a generalized bi-axially symmetric Helmholtz equation, extending previous work by constructing potentials for additional fundamental solutions using hypergeometric functions.

Contribution

It introduces a new theoretical framework for double-layer potentials associated with the second fundamental solution of the equation, expanding the potential theory for this class of elliptic equations.

Findings

Proved limiting theorems for the double-layer potentials.

Derived integral equations related to the denseness of the 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. 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 next 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.