The Riemann–Hilbert problem and the generalized Neumann kernel on multiply connected regions

This paper presents and studies Fredholm integral equations associated with the linear Riemann–Hilbert problems on multiply connected regions with smooth boundary curves. The kernel of these integral equations is the generalized Neumann kernel. The approach is similar to that for simply connected regions (see R. Wegmann, A.H.M. Murid, M.M.S. Nasser, The Riemann–Hilbert problem and the generalized Neumann kernel, J. Comput. Appl. Math. 182 (2005) 388–415]). There are, however, several characteristic differences, which are mainly due to the fact, that the complement of a multiply connected region has a quite different topological structure. This implies that there is no longer perfect duality between the interior and exterior problems.