The necessary and sufficient condition for the existence and uniqueness of a system of Fredholm integral equations of the first kind

Abstract

In this paper, we consider the existence and uniqueness of a system of Fredholm integral equations of the first kind, which arises from the Dirichlet problem for Laplace equation on 2D multiply-connected domains. We introduce a computable index γ N and prove that the necessary and sufficient condition for the existence and uniqueness of solution of this Fredholm integral equations of the first kind is γ N ≠ 0. The numerical examples demonstrate the validity of our theoretic results.