# IJPAM: Volume 51, No. 4 (2009)

**NUMERICAL EXPERIMENT ON CONFORMAL MAPPING OF**

DOUBLY CONNECTED REGIONS ONTO

A DISK WITH A SLIT

DOUBLY CONNECTED REGIONS ONTO

A DISK WITH A SLIT

Department of Mathematics

Faculty of Science

University of Technology - Malaysia

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

e-mail: [email protected]

e-mail: [email protected]

**Abstract.**We present a method for computing the conformal mapping function of
doubly connected regions bounded by two closed Jordan curves onto a disk
with a concentric circular slit of radius . Our mapping procedure consists of two parts.
First we solve a system of integral equations on the boundary of the region we wish to map.
The system of integral equations is based on a boundary integral equation involving the
Neumann kernel discovered by the authors satisfied by , , and ,
where is a fixed interior point with predetermined. The boundary values of
are completely determined from the boundary values of through a boundary
relationship. Discretization of the integral equation leads to a system of non-linear
equations. Together with some normalizing conditions, a unique solution to the system
is then computed by means of an optimization method called the Lavenberg-Marquadt algorithm.
Once we have determined the boundary values of , we use the Cauchy integral formula
to compute the interior of the regions. Typical examples for some doubly connected regions
show that numerical results of high accuracy can be obtained for the conformal mapping
problem when the boundaries are sufficiently smooth.

**Received: **February 26, 2009

**AMS Subject Classification: **30C30, 65R20, 65E05, 30C40, 65H10

**Key Words and Phrases: **conformal mapping, integral equations, doubly connected region, Neumann kernel, Lavenberg-Marquardt algorithm, Cauchy's integral formula

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

**ISSN:** 1311-8080

**Year:** 2009

**Volume:** 51

**Issue:** 4