An integral equation method for solving neumann problems in simply connected regions with smooth boundaries

This paper presents a new method for solving the interior Neumann problem in a simply connected region. The method is based on recent investigations on the interplay of Riemann-Hilbert (RH) problems and Fredholm integral equations with generalized Neumann kernel. In the first step the Neumann problem is reformulated as a uniquely solvable interior RH problem for the derivative of an auxiliary function which is analytic in the region. In the second step, the RH problem is transformed to a uniquely solvable Fredholm integral equation on the boundary of the region. Once this equation is solved, the auxiliary function and the solution of the Neumann problem can be obtained. The efficiency of the method is illustrated by some numerical examples