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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.

In this work we present a new numerical technique for solving periodic structure problems. This new approach possesses several advantages. First, it allows for a fast evaluation of the Robin-to-Robin operator for periodic array problems. Secondly, this computational method can also be used for bi-periodic structure problems with local defects. Our strategy is an improvement of the recently developed recursive doubling process by Yuan and Lu. In this paper we consider several problems, such as the exterior elliptic problems with strong coercivity, the time-dependent Schrödinger equation and finally the Helmholtz equation with damping.