A Boundary integral equation for the Neumann problem in bounded multiply connected region

Abstract

This research determines solutions of the Neumann problem in multiply connected regions by using the method of boundary integral equations. This method is widely used for solving boundary value problems. 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 Neumann problem into the Riemann- Hilbert problem and then derive an integral equation related to the Riemann-Hilbert problem. The derived integral equation is not uniquely solvable. The complete discussion on the solvability of the Neumann problem, the Riemann-Hilbert problem as well as the derived integral equation is presented. As an examination of the present method, some numerical examples for some different test regions are presented. These examples include comparison between the numerical results and the exact solutions.