An Integral Equation Method for a Boundary Value Problem Arising in Unsteady Water Wave Problems

Abstract

In this paper we consider the 2D Dirichlet boundary value problem for Laplace's equation in a non-locally perturbed half-plane, with data in the space of bounded and continuous functions. We show uniqueness of solution, using standard Phragmen-Lindelof arguments. The main result is to propose a boundary integral equation formulation, to prove equiv- alence with the boundary value problem, and to show that the integral equation is well posed by applying a recent partial generalisation of the Fredholm alternative in Arens et al (J Int. Equ. Appl. 15 (2003) pp1-35). This then leads to an existence proof for the boundary value problem. Keywords. Boundary integral equation method, Water waves, Laplace's equation This paper is concerned with the boundary integral equation method for a problem in potential theory, namely the Dirichlet boundary value problem in a non-locally perturbed half-plane. The main aim of the paper is to discuss the well-posedness of this problem and of a novel second kind boundary integral equation formulation. Our motivation for studying this problem is that it arises in the theory of classical free surface water wave problems, for which boundary integral equation methods are well-established as a computational and theoretical tool (B1, B2, HZ, FD). In particular, accurate numerical schemes, based on boundary inte- gral equation formulations, for the time dependent water wave problem have been proposed and fully analysed in (B1, B2, HZ), these papers providing a full nonlinear stability analysis for the spatial discretisations they propose. A signif- icant component in this analysis is the well-posedness of the boundary integral equation formulation. A key restriction in the analysis in the above papers is the requirement that the free surface be periodic (in 2D) or doubly-periodic (in 3D). This restriction is helpful theoretically and computationally. It enables the boundary integral equation on the free surface to be reduced to one on a (bounded) single peri- odic cell, which can then be discretised with a finite size mesh. Moreover, the