IJPAM: Volume 36, No. 2 (2007)


Ali H.M. Murid$^1$, Nurul Akmal Mohamed$^2$
$^{1,2}$Department of Mathematics
Faculty of Science
University of Technology of Malaysia
81310, UTM Skudai, Johor Darul Ta'zim, MALAYSIA
$^1$e-mail: ahmm@mel.fs.utm.my
$^2$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 $A=\{w:\mu <\vert w\vert<1\}$ 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 [#!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, $\mu$. 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