# IJPAM: Volume 36, No. 2 (2007)

**NUMERICAL CONFORMAL MAPPING OF**

DOUBLY CONNECTED REGIONS VIA

THE KERZMAN-STEIN KERNEL

DOUBLY CONNECTED REGIONS VIA

THE KERZMAN-STEIN KERNEL

Department of Mathematics

Faculty of Science

University of Technology of Malaysia

81310, UTM Skudai, Johor Darul Ta'zim, MALAYSIA

e-mail: ahmm@mel.fs.utm.my

e-mail: akmalmohdy@yahoo.com.my

**Abstract.**An integral equation method based on the Kerzman-Stein kernel for conformal mapping of smooth doubly connected regions onto an annulus
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 [#!murid2!#]. 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, . 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.

**Received: **February 2, 2007

**AMS Subject Classification: **30C30, 30C40, 45G15, 65E05, 65H10

**Key Words and Phrases: **conformal mapping, Doubly connected regions, integral equations, Kerzman-Stein kernel, Gauss-Newton method, Lavenberg-Marquardt algorithm, Fletcher's algorithm

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2007

**Volume:** 36

**Issue:** 2