Numerical Conformal Mapping of Doubly Connected Regions via the Kerzman-stein Kernel

Abstract: An integral equation method based on the Kerzman-Stein kernel for conformal mapping of smooth doubly connected regions onto an annulus A = {w : μ < |w| < 1} is presented. The theoretical development is based on the boundary integral equation for conformal mapping of doubly connected regions with Kerzman-Stein kernel derived by Razali and one of the authors [8]. However, the integral equation is not in the form of Fredholm integral equation and no numerical experiments are reported. In this paper, we show that using the boundary relationship satisfied by a function analytic in a doubly connected region, then the previous integral equation can be reduced to a numerically tractable integral equation which however involves the unknown inner radius, μ. For numerical experiments, we discretized the integral equation which leads to an over determined system of non-linear equations. The system obtained is solved simultaneously using Gauss-Newton method and Lavenberg-Marquardt with Fletcher’s algorithm for solving the non-linear least squares problems. Numerical implementations on some test regions are also presented.