A boundary integral equation for the exterior neumann problem on multiply connected region

Abstract

This research determines solutions of the exterior Neumann problem in multiply connected regions by using the method of boundary integral equations. The method depends on reducing the boundary value problem in question to an integral equation on the boundary of the domain of the problem, and then solves this integral equation. Our approach in this research is to convert the exterior Neumann problem into the exterior Riemann-Hilbert problem. The exterior Riemann-Hilbert problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the Neumann kernel. Once this equation is solved, the auxiliary function and the solution of the exterior Neumann problem can be obtained. As an examination of the proposed method, some numerical examples for some different test regions are presented. These examples include comparison between the numerical results and the exact solutions. Numerical examples reveal that the present method offers an effective solution technique for the exterior Riemann-Hilbert problems when the boundaries are sufficiently smooth.