The Numerical Solution of First Kind Integral Equation for the Helmholtz Equation on Smooth Open Arcs

Abstract

Consider solving the Dirichlet problem of Helmholtz equation on unbounded region $R^2$\Γ with Γ a smooth open curve in the plane. We use simple-layer potential to construct a solution. This leads to the solution of a logarithmic integral equation of the first kind for the Helmholtz equation. This equation is reformulated using a special change of variable, leading to a new first kind equation with a smooth solution function. This new equation is split into three parts. Then a quadrature method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically. An error analysis in a Sobolev space setting is given. And numerical results show that fast convergence is clearly exhibited.