A boundary integral equation with the generalized Neumann kernel for a mixed boundary value problem in unbounded multiply connected regions

In this paper we propose a new method for solving the mixed boundary value problem for the Laplace equation in unbounded multiply connected regions. All simple closed curves making up the boundary are divided into two sets. The Dirichlet condition is given for one set and the Neumann condition is given for the other set. The mixed problem is reformulated in the form of a Riemann-Hilbert (RH) problem which leads to a uniquely solvable Fredholm integral equation of the second kind. Three numerical examples are presented to show the effectiveness of the proposed method.