Boundary integral equations with the generalized Neumann kernel for robin problem in simply connected region

Abstract

A mixed boundary value problem with the linear combination of Dirichlet and Neumann conditions is called a Robin problem. In this paper, we consider the Robin problem in a bounded simply connected region G with smooth boundary ωG. It consists of finding a function u harmonic in G and satisfies the Robin boundary condition. This work develops new boundary integral equations for solving the Robin problem. Recently, the interplay of Riemann-Hilbert problems (briefly, RH problems) with conformal mapping, Dirichlet problem and Neumann problem has been studied extensively. The related integral equations involving the generalized Neumann kernel are uniquely solvable. In this paper we show how to reformulate a Robin problem as a Riemann-Hilbert problem. Numerical results are presented to illustrate the solution technique for the Robin problem when the boundaries are sufficiently smooth.