Boundary Integral Equations with the Generalized Neumann Kernel for the Neumann Problem

This paper presents a new method to solve the interior and the exterior Neumann problems in simply connected regions with smooth boundaries. The method is based on two uniquely solvable Fredholm integral equations of the second kind with the generalized Neumann kernel. Numerical examples reveal that the present method offers an effective numerical method for the Neumann problems when the boundaries are sufficiently smooth.