Numerical experiment on conformal mapping of doubly connected regions onto a disk with a slit

Abstract

We present a method for computing the conformal mapping function of doubly connected regions bounded by two closed Jordan curves onto a disk with a concentric circular slit of radius . Our mapping procedure consists of two parts. First we solve a system of integral equations on the boundary of the region we wish to map. The system of integral equations is based on a boundary integral equation involving the Neumann kernel discovered by the authors satisfied by , , and , where is a fixed interior point with predetermined. The boundary values of are completely determined from the boundary values of through a boundary relationship. Discretization of the integral equation leads to a system of non-linear equations. Together with some normalizing conditions, a unique solution to the system is then computed by means of an optimization method called the Lavenberg-Marquadt algorithm. Once we have determined the boundary values of , we use the Cauchy integral formula to compute the interior of the regions. Typical examples for some doubly connected regions show that numerical results of high accuracy can be obtained for the conformal mapping problem when the boundaries are sufficiently smooth