boundary integral equations approach for...

BOUNDARY INTEGRAL EQUATIONS APPROACH

FOR NUMERICAL CONFORMAL MAPPING

OF MULTIPLY CONNECTED REGIONS

HU LAEY NEE

UNIVERSITI TEKNOLOGI MALAYSIA

BAHAGIAN A – Pengesahan Kerjasama* Adalah disahkan bahawa projek penyelidikan tesis ini telah dilaksanakan melalui kerjasama antara

dengan .

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Tandatangan : ……………………………… Tarikh: ……………..

Nama : ………………………………

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*Jika penyediaan tesis/projek melibatkan kerjasama.

BAHAGIAN B – Untuk Kegunaan Pejabat Sekolah Pengajian Siswazah

Tesis ini telah diperiksa dan diakui oleh: Nama dan Alamat Pemeriksa Luar : Prof. Dr. Bahrom Bin Sanugi Pusat Hubungan Industri dan Khidmat, Canseleri,

Universiti Sains Islam Malaysia, Bandar Baru Nilai, 71800 Nilai, Negeri Sembilan Darul Khusus.

Nama dan Alamat Pemeriksa Dalam I : Prof. Madya Dr. Jamalludin Bin Talib Jabatan Matematik Fakulti Sains,

UTM, Skudai. Pemeriksa Dalam II : Pemeriksa Dalam III : Nama Penyelia I : Prof. Madya Dr. Ali Hassan bin Mohamed Murid

Disahkan oleh Timbalan Pendaftar di Sekolah Pengajian Siswazah:

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Nama : EN KHASSIM B ISMAIL

BOUNDARY INTEGRAL EQUATIONS APPROACH

FOR NUMERICAL CONFORMAL MAPPING

OF MULTIPLY CONNECTED REGIONS

HU LAEY NEE

A thesis submitted in fulfilment of the

requirements for the award of the degree of

Doctor of Philosophy (Mathematics)

Faculty of Science

Universiti Teknologi Malaysia

AUGUST 2009

iii

To my beloved husband Kong King Hwa

and my three children, Wan Kieng, Wan Wee and Jing Yi

who are a great source of joy and inspiration to me

and especially my loving father, Hu Poh Thiew and mother Huang Ngoik Ing.

iv

ACKNOWLEDGEMENT

I wish to express my deepest gratitude to my supervisor, Assoc. Prof.

Dr. Ali Hassan bin Mohamed Murid, who has suggested the research topic and

directed the research. I thank him for his enduring patience. My special thanks

are also due to my co-supervisor, Prof. Dr. Mohd Nor Mohamad, for contributing

ideas, discussing research plans, and encouraging support.

I am high respect to Prof. Rudolf Wegmann of Max-Planck-Institute,

Garching, Germany for helpful discussions on the exact mapping function of

annulus in Section 2.5.1 and for bringing reference von Koppenfels and Stallmann

(1959) to my attention. I wish to thank Prof. Dr. Dai Okano, Prof. Dr. Kaname

Amano and Dr. Mohamed M. S. Nasser for sharing and discussions on multiply

connected regions. I also wish to thank Tn. Hj. Hamisan Rahmat, Dr. Munira

Ismail, Dr. Kamran Fakhar, Dr. Rohanin Ahmad and Assoc. Prof. Dr Ong Chee

Tiong from the Department of Mathematics at Universiti Teknologi Malaysia for

sharing and providing information in the course of this work.

I am indebted to my employer Institut Pendidikan Guru Malaysia (IPGM)

Kampus Sarawak Miri and Ministry of Eduction Malaysia for granting me the

study leave and financial support through the Hadiah Latihan Persekutuan

(HLP). This work was also supported in part by the Malaysian Ministry of Higher

Education (MOHE) through the Research Management Centre (RMC), Universiti

Teknologi Malaysia (FRGS Vote 78089).

Finally, I would like to convey my most sincere gratitude to Dr. Tay Kim

Gaik, Nurul Akmal, my friends and colleagues for their support and friendship

and my family for their love and understanding.

v

ABSTRACT

Several integral equations involving the Kerzman-Stein and the Neumann

kernels for conformal mapping of multiply connected regions onto an annulus with

circular slits and onto a disk with circular slits are presented. The theoretical

development is based on the boundary integral equation for conformal mapping of

doubly connected region. The integral equations are constructed from a boundary

relationship satisfied by a function analytic on a multiply connected region. The

boundary integral equations involve the unknown parameter radii. For numerical

experiments, discretizing each of the integral equations 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. Once the

boundary values of the mapping function are calculated, the Cauchy’s integral

formula has been used to determine the mapping function in the interior of the

region. Typical examples for some test regions show that numerical results of

high accuracy can be obtained for the conformal mapping problem when the

boundaries are sufficiently smooth.

vi

ABSTRAK

Beberapa persamaan kamiran melibatkan inti Kerzman-Stein dan

Neumann untuk pemetaan konformal bagi rantau berkait berganda ke atas

anulus dengan belahan membulat dan ke atas cakera dengan belahan membulat

dipersembahkan. Pembangunan teori adalah berdasarkan persamaan kamiran

sempadan bagi pemetaan konformal rantau berkait ganda dua. Persamaan-

persamaan kamiran adalah dibangunkan dari hubungan sempadan yang ditepati

oleh fungsi yang analisis dalam rantau berkait berganda. Persaman-persamaan

kamiran sempadan ini melibatkan parameter jejari-jejari yang tidak diketahui.

Untuk kajian berangka, setiap persamaan kamiran berkenaan telah didiskretkan

menghasilkan suatu sistem persamaan tak linear. Bersama dengan beberapa

syarat kenormalan, satu penyelesaian unik kepada sistem berkenaan dikira

dengan kaedah pengoptimuman. Setelah nilai sempadan bagi fungsi pemetaan

dikira, formula kamiran Cauchy digunakan untuk menentukan fungsi pemetaan

terhadap rantau pedalaman. Contoh-contoh tipikal untuk beberapa rantau ujikaji

telah menunjukkan keputusan berangka berketepatan tinggi boleh diperoleh

untuk masalah pemetaan konformal dengan sempadan licin.

vii

TABLE OF CONTENTS

CHAPTER TITLE PAGE

THESIS STATUS DECLARATION

SUPERVISOR’S DECLARATION

DECLARATION ON COOPERATION

TITLE PAGE i

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES xi

LIST OF FIGURES xiv

LIST OF SYMBOLS xvii

LIST OF APPENDICES xviii

1 INTRODUCTION 1

1.1 Introduction 1

1.2 Background of The Problem 2

1.3 Problem Statement 5

1.4 Scope of Study 5

1.5 Research Objectives 6

1.6 Thesis Outline 7

viii

2 OVERVIEW OF MAPPING OF MULTIPLY

CONNECTED REGIONS 10

2.1 Introduction 10

2.2 Ideas of Conformal Mapping 10

2.3 The Riemann Conformal Mapping 12

2.4 Conformal Mapping of Multiply Connected Regions 14

2.5 Exact Mapping Function of Doubly Connected

Regions for Some Selected Regions 18

2.5.1 Annulus Onto A Disk With A

Circular Slit 19

2.5.2 Circular Frame 20

2.5.3 Frame of Limacon 21

2.5.4 Elliptic Frame 22

2.5.5 Frame of Cassini’s Oval 22

2.6 Some Numerical Method for Conformal Mapping

of Multiply Connected Regions 23

2.6.1 Wegmann’s Iterative Method 25

2.6.2 Symm’s Integral Equations 26

2.6.3 Charge Simulation Method 26

2.6.4 Mikhlin’s Integral Equation 27

2.6.5 Fredholm Integral Equation 28

2.6.6 Warschawski’s and Gershgorin’s Integral

Equations 28

2.6.7 The Boundary Integral Equation via the

Kerzman-Stein and the Neumann Kernels 29

3 AN INTEGRAL EQUATION RELATED TO

A BOUNDARY RELATIONSHIP 34

3.1 Introduction 34

3.2 The Boundary Integral Equation 35

3.3 Application to Conformal Mapping of Doubly

ix

Connected Regions onto an Annulus via the

Kerzman-Stein Kernel 37

3.4 Application to Conformal Mapping of Multiply

Connected Regions onto an Annulus with Circular

Slits via the Neumann Kernel 42

3.5 Application to Conformal Mapping of Multiply

Connected Regions onto a Disk with Circular

Slits via the Neumann Kernel 46

4 NUMERICAL CONFORMAL MAPPING OF

MULTIPLY CONNECTED REGIONS ONTO

AN ANNULUS WITH CIRCULAR SLITS 50

4.1 Introduction 50

4.2 Conformal Mapping of Doubly Connected Regions

onto an Annulus via the Kerzman-Stein Kernel 51

4.2.1 A System of Integral Equations 51

4.2.2 Numerical Implementation 53

4.2.3 Numerical Results 57


onto an Annulus via the Neumann Kernel 63




4.4 Conformal Mapping of Triply Connected Regions

onto an Annulus with a Circular Slit Via the

Neumann Kernel 86



4.4.3 Interior of Triply Connected Region 95


x

5 NUMERICAL CONFORMAL MAPPING OF

MULTIPLY CONNECTED REGIONS ONTO

A DISK WITH CIRCULAR SLITS 100

5.1 Introduction 100


onto a Disk with a Circular Slit Via the Neumann

Kernel 101



5.2.3 The Interior Mapping 108


5.3 Conformal Mapping of Triply Connected Regions

onto a Disk with Circular Slits Via the Neumann

Kernel 116




6 SUMMARY AND CONCLUSIONS 128

6.1 Summary of the Research 128

6.2 Suggestions for Future Research 131

REFERENCES 133

Appendix A Publications/Presentations 139

Appendix B Sample of Computer Programs 143

xi

LIST OF TABLES

TABLE NO. TITLE PAGE

4.1 Error norm of the the Example 4.1 59

4.2 Error norm of the the Example 4.1 by Murid and

Mohamed (2007) 59

4.3 Error Norm of the the Example 4.1 using Amano’s

method and Symm’s method 59

4.4 The radius comparison of the Example 4.2 60

4.5 The computed approximations of μ and M for the

Example 4.3 62


4.7 Error Norm of the Example 4.4 74

4.8 Error Norm of Interior for the Example 4.4 74

4.9 Error Norm of the Example 4.4 as given in Mohamed

and Murid (2007b) based on the Neumann kernel 74

4.10 Error Norm of the Example 4.4 using Amano’s

method and Symm’s method 75





4.14 Error norm of the Example 4.5 using Amano’s method

and Symm’s method 77

4.15 Error Norm of Interior for the Example 4.5 as given

in Mohamed and Murid (2007a) based on the

Kerzman-Stein kernel 77

xii





4.19 Error Norm of the Example 4.6 using Symm’s method 80







4.25 The computed approximations of μ and M for the

Example 4.10 85

4.26 The radius comparison for the Example 4.10 86

4.27 Radii comparison with Reichel (1986) for

the Example 4.11 97

4.28 Radii comparison with Kokkinos et al. (1990)

for Example 4.11 97

4.29 Radii comparison for Example 4.12 98











5.11 Radii comparison with Reichel (1986) for

the Example 5.6 126

xiii

5.12 Radii comparison with Kokkinos et al. (1990)

for the Example 5.6 126

xiv

LIST OF FIGURES

FIGURE NO. TITLE PAGE

1.1 Canonical regions 4

2.1 The tangents at the point z0 and w0, where f(z) is

an analytic function and f ′(z0) �= 0 12

2.2 The analytic mapping w = f(z) is conformal at point

z0 and w0, where f ′(z0) �= 0 and γ2 − γ1 = β2 − β1 12

2.3 Boundary Correspondence Function θ(t) 14

2.4 An (M + 1) connected region 15

2.5 Mapping of a region of connectivity 4 16

2.6 Mapping of a region of connectivity 4 onto an annulus

with circular slits 16

2.7 Mapping of a region of connectivity 4 onto a disk with

circular slits 17

2.8 The composite g = h ◦ f 18

2.9 The composite function g = h ◦ p 19

3.1 Mapping of a doubly connected region Ω onto

an annulus 38

3.2 Mapping of doubly connected region onto

an annulus 40

3.3 Mapping of a multiply connected region Ω of

connectivity M + 1 onto an annulus with circular slits 42

3.4 Mapping of a multiply connected region Ω of

connectivity M + 1 onto a disk with circular slits 47

4.1 Mapping of the Example 4.1 with a0 = 2√

14, a1 = 2,

b0 = 7 and b1 = 1 58

xv

4.2 Mapping of the Example 4.2 60

4.3 Mapping of the Example 4.3 with a0 = 0.20 61

4.4 A rectangular grid in Ω with grid size 0.25 and

its image with a0 = 2√

14, a1 = 2, b0 = 7, and b1 = 1



its image with a0 = 7, a1 = 5, b0 = 5, and b1 = 1



its image with a0 = 10, a1 = 5, b0 = 3 and b1 = b0/4



its image with c = 0.3 and ρ = 0.1 for the Example 4.7 81


its image for the Example 4.8 82




its image with a0 = 2.0 for the Example 4.10 85





4.13 Mapping of the Example 4.12 based on Ellacott method 99


its image with radius μ = e−2σ for the Example 5.1 110

5.2 A rectangular grid in Ω with grid and






5.5 A rectangular grid in with grid size 0.25 and Ω


5.6 The Contour for Example 5.6 126

xvi

5.7 The image of the mapping for the Example 5.6 127

xvii

LIST OF SYMBOLS

A(z, w) - Kerzman-Stein kernel

H(w, z) - Cauchy kernel

N(z, w) - Neumann kernel

R - Riemann mapping function

C - Set of complex numbers

� - Set of real numbers

e, exp - Exponential(e ≈ 2.718...)

i -√−1

Im - Imaginary part

Re - Real

U - Unit disk

ε - Epsilon (small number 0 ≤ ε < 1)

Γ - Curve (boundary of Ω)

Γ0 - Outer boundary of a doubly connected region

Γ1 - Inner boundary of a doubly connected region

π - Pi (π ≈ 3.142...)

Ω - Connected region

∈ - Component∑- Sum∫- Integration

xviii

LIST OF APPENDICES

APPENDIX NO. TITLE PAGE

A Publications/Presentations 139

B Sample of Computer Programs 143

CHAPTER 1

INTRODUCTION

1.1 Introduction

A conformal mapping, also called a conformal map, a conformal

transformation, angle-preserving transformation, or biholomorphic map is a

transformation w = f(z) that preserves local angle. An analytic function

is conformal at any point where it has nonzero derivatives. Conversely, any

conformal mapping of a complex variable which has continuous partial derivatives

is analytic.

Conformal mappings have been an important tool of science and

engineering since the development of complex analysis. A conformal mapping

uses functions of complex variables to transform a complicated boundary to a

simpler, more manageable configuration. In various applied problems, by means

of conformal maps, problems for certain physical regions are transplanted into

problems on some standardized model regions where they can be solved easily.

By transplanting back we obtain the solutions of the original problems in the

physical regions. This process is used, for example, for solving problems about

fluid flow, electrostatics, heat conduction, mechanics, aerodynamics and image

2

processing. For these and other physical problems that use conformal mapping

techniques, see, for example, the books by Henrici (1974), Churchill and Brown

(1984), Schinzinger and Laura (1991) and Kythe (1998). For theoretical aspects

of conformal mappings, see, e.g., Andersen et al. (1962), Hille (1962), Ahlfors

(1979), Goluzin (1969), Nehari (1975), Henrici (1974), and Wen (1992).

A special class of conformal mappings that map any simply connected

region onto a unit disk is called Riemann map. The Riemann mapping function

is closely connected to the Szego or the Bergman kernels. These kernels can be

computed as a solution of second kind integral equations. Hence to solve the

conformal mapping problem it is sufficient to compute the boundary values of

either the Szego or the Bergman kernel.

An integral equation of the second kind that expressed the Szego kernel

as the solution is first introduced by Kerzman and Trummer (1986) using

operator-theoretic approach. Henrici (1986) gave a markedly different derivation

of the Kerzman-Stein-Trummer integral equation based on a function-theoretic

approach. The discovery of the Kerzman-Stein-Trummer integral equation,

briefly KST integral equation, for computing the Szego kernel later leads to the

formulation of an integral equation for the Bergman kernel as given in Murid

(1997) and Razali et al. (1997). Both integral equations can be used effectively

for numerical conformal mapping of simply connected regions.

1.2 Background of The Problem

The practical limitation of conformal mapping has always been that only

for certain special regions are exact conformal maps known and others have to

be computed numerically.

3

Henrici (1986), Kythe (1998), Murid (1997), Schinzinger and Laura (1991),

Trefethen (1986), Wegmann (2005) and Wen (1992) have surveyed some methods

for numerical approximation of conformal mapping function such as expansion

methods, iterative methods, osculation methods, integral equation method,

Cauchy-Riemann equation methods and charge simulation methods. The integral

equation methods mostly deal with computing the boundary correspondence

function for solving numerical conformal mapping. This correspondence refer

to a particular parametric representation of the boundary (Razali et al., 1997;

Henrici, 1986; Kerzman and Trummer, 1986).

Conformal mapping of multiply connected regions suffer form severe

limitations compared to the simply connected region. There is no exact multiply

equivalent of the Riemann mapping theorem that holds in multiply connected

case. This implies that there is no guarantee that any two multiply connected

regions of the same connectivity are conformally equivalent to each other.

Nehari (1975, p. 335), Bergman (1970) and Cohn (1967) described the

five types of slit region as important canonical regions for conformal mapping of

multiply connected regions, namely

(i) the disk with concentric circular slits (Figure 1.1a),

(ii) an annulus with concentric circular slits (Figure 1.1b),

(iii) the circular slit region (Figure 1.1c),

(iv) the radial slit region (Figure 1.1d), and

(v) the parallel slit region (Figure 1.1e).

The former two are bounded slit regions and the latter three are unbounded

slit regions. It is known that any multiply connected region can be mapped

conformally onto these canonical regions. In general the radii of the circular slits

are unknown and have to be determined in the course of the numerical evaluation.

However, exact mapping functions are not known except for some special regions.

By using a boundary relationship satisfied by a function analytic in a

doubly connected region, Murid and Razali (1999) extended the construction to a

4

0

(a) (b)

(c) (d) (e)

00

Figure 1.1: Canonical regions.

doubly connected region and obtained a boundary integral equation for conformal

mapping of doubly connected regions. Special realizations of this boundary

integral equation are the integral equations for conformal mapping of doubly

connected regions via the Kerzman-Stein and the Neumann kernels. However,

the integral equations are not in the form of Fredholm integral equations and no

numerical experiments are reported in Murid and Razali (1999).

Murid and Mohamed (2007), and Mohamed and Murid (2007b) have

formulated an integral equation method based on the Kerzman-Stein and the

Neumann kernel for conformal mapping of doubly connected regions onto an

annulus. The theoretical development is based on the boundary integral equations

for conformal mapping of doubly connected regions derived by Murid and Razali

(1999). For numerical experiments, the integral equations are discretized which

lead to systems of non-linear equations. The systems obtained are solved

simultaneously using Newton’s iterative method. The numerical implementations

on some test regions are reported only for doubly connected regions onto an

annulus.

5

But Murid and Razali (1999), Murid and Mohamed (2007), and Mohamed

and Murid (2007b) have not yet formulated an integral equation method based on

the Kerzman-Stein and the Neumann kernels for conformal mapping of multiply

connected regions onto the disk with concentric circular slits and an annulus with

concentric circular slits.

1.3 Problem Statement

The research problem is to formulate new integral equations for conformal

mapping of multiply connected regions with smooth boundaries onto the disk

with concentric circular slits and an annulus with concentric circular slits via the

Neumann kernel and the Kerzman-Stein that are suitable for numerical purposes.

1.4 Scope of Study

This research focuses on the integral equation method for the numerical

computation of the conformal mapping of multiply connected regions. The

theoretical development of the integral equation is based on the approach given

by Murid and Razali (1999) for doubly connected regions.

In this study, some new boundary integral equations will be derived for

conformal mapping of multiply connected regions via the Kerzman-Stein and

the Neumann kernels. These integral equations will be applied to multiply

connected regions onto an annulus with concentric circular slits and the disk

with concentric circular slits. For numerical experiments, these integral equations

will be discretized that might leads to a system of equations. Some normalizing

conditions might be needed to help achive unique solutions.

6

The research will also describe a numerical procedure based on Cauchy

integral formula for computing the mapping of interior points. The research will

present numerical examples to highlight the advantages of using the proposed

method.

1.5 Research Objectives

The objectives of this research are:

1. To improve and extend the construction of integral equation related

to a boundary relationship satisfied by a function analytic in a doubly

connected region by Murid and Razali (1999) to multiply connected regions.

2. To derive new boundary integral equation for conformal mapping of

multiply connected regions onto a disk with concentric circular slits via the

Neumann kernel.

3. To derive new boundary integral equations for conformal mapping of

multiply connected regions onto an annulus with circular slits via the

Neumann kernel and the Kerzman-Stein kernel.

4. To use the integral equations to solve numerically the boundary values of

the conformal mapping of multiply connected regions onto an annulus with

concentric circular slits and the disk with concentric circular slits.

5. To use the Cauchy’s integral formula to determine the interior values of

mapping functions.

7

6. To make numerical comparison of the proposed method with exact solution

or with some existing methods.

1.6 Thesis Outline

This thesis consists of six chapters. The introductory Chapter 1 details

some discussion on the introduction, background of the problem, problem

statement, objectives of research, scope of the study and chapter organization.

Chapter 2 gives an overview of methods for conformal mapping in

particular of multiply connected regions as well as the conformal mapping of

multiply connected regions. We discuss some theories of the Riemann mapping

function. We also present some exact conformal mapping of doubly connected

regions for certain special regions like annulus, frame of limacon, elliptic frame,

frame of Cassini’s oval and circular frame. Some numerical methods that have

been proposed in the literature for conformal mapping of multiply connected

regions are also presented in the Section 2.6 of Chapter 2. The boundary integral

equation for conformal mapping of doubly regions derived by Murid and Razali

(1999) is also presented.

In Chapter 3, we construct new boundary integral equation related to a

boundary relationship satisfied by an analytic function on multiply connected

regions. The theoretical development is based on the boundary integral equation

for conformal mapping of doubly connected region derived by Murid and Razali

(1999) who have constructed an integral equation for the mapping of doubly

connected regions onto an annulus involving the Neumann kernel. By using the

boundary relationship satisfied by the mapping function, a related system of

integral equation is constructed, including the unknown parameter radii. We

apply the new boundary integral equation for conformal mapping of multiply

8

connected regions onto a disk with circular slits and onto an annulus with circular

slits via the Neumann and the Kerzman-Stein kernels. Special cases of this

result are the integral equations involving the Kerzman-Stein and the Neumann

kernels related to conformal mapping of doubly connected regions onto an annulus

obtained by Murid and Mohamed (2007), and Mohamed and Murid (2007b).

In Chapter 4, we apply the result of Chapter 3 to derive a new boundary

integral equation related to conformal mapping f(z) of multiply connected region

onto an annulus with circular slits. We discretized the integral equation and

imposed some normalizing conditions different from Murid and Mohamed (2007),

and Mohamed and Murid (2007b) for the case doubly connected region via the

Kerzman-Stein and the Neumann kernels. We also extend the construction of

the boundary integral equation in Chapter 3 to a triply connected regions. The

boundary values of f(z) is completely determined from the boundary values of

f ′(z) through a boundary relationship. Discretization of the integral equation

leads to a system of non-linear equations. Together with some normalizing

conditions, we show how a unique solution to the system can be computed by

means of an optimization method. We report our numerical results and give

comparisons with existing method for some test regions.

In Chapter 5, we apply the result of Chapter 3 to derive a new boundary

integral equation related to conformal mapping f(z) of multiply connected region

onto a disk with circular slits. Discretization of the integral equation leads to a

system of non-linear equations. Together with some normalizing conditions, we

show how a unique solution to the system can be computed by means of an

optimization method. Once the boundary values of the mapping function f are

known, we use the Cauchy’s integral formula to determine the interior values

of the mapping function. Numerical experiments on some test regions are also

reported.

Finally the concluding chapter, Chapter 6, contains a summary of all the

main results and several recommendations. There are two appendices in this

9

thesis. Appendix A presents the list of the papers that have been published,

presented or submitted for publication or presentation during the author’s

candidature. Appendix B presents a sample of the computer program used for a

test region.

boundary integral equations approach for...

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