Numerical conformal mapping of doubly connected regions using integral equation via the Kerzman-Stein kernel and Cauchy's integral formula

Abstract

An integral equation method based on the Kerzman-Stein kernelfor conformal mapping of smooth doubly connected regions onto an annulus A = { w : µ < | w | < 1 } is presented. The theoretical development is basedon the boundary integral equation for conformal mapping of doubly connectedregions 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 equationand no numerical experiments are reported. In this paper, we show that usingthe boundary relationship satisfied by a function analytic in a doubly connectedregion, then the previous integral equation can be reduced to a numericallytractable integral equation which however involves the unknown inner radius, µ . For numerical experiments, we discretized the integral equation which leadsto an over determined system of non-linear equations. The system obtained issolved simultaneously using Gauss-Newton method and Lavenberg-Marquardtwith Fletcher’s algorithm for solving the non-linear least squares problems.Numerical implementations on some test regions are also presented.