Numerical conformal mapping of bounded multiply connected regions by an integral equation method

Abstract

Conformal mappings are familiar tools in science and engineering. However exact mapping functions are unknown except for some special regions. In this paper, a boundary integral equation for conformal mapping w = f(z) of multiply connected regions onto an annulus µ1 < |w| < 1 with circular slits µ2,µ3, ..., µM is presented. Our theoretical development is based on the boundary integral equation for conformal mapping of doubly connected region derived by Murid and Razali [12]. The boundary integral equation involved the unknown circular radii. For numerical experiments, the boundary integral equation with some normalizing conditions are discretized which leads to a system of nonlinear equations. This system is solved simultaneously using modi?cation of the Gauss-Newton named Lavenberg-Marquardt with the Fletcher’s algorithm for solving the nonlinear least squares problems. Once the boundary values of the mapping function are calculated, we can use the Cauchy’s integral formula to determine the mapping function in the interior of the region. Numerical implementations on some test regions are also presented