BOUNDARY INTEGRAL EQUATIONS APPROACH FOR...
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BOUNDARY INTEGRAL EQUATIONS APPROACH
FOR NUMERICAL CONFORMAL MAPPING
OF MULTIPLY CONNECTED REGIONS
HU LAEY NEE
UNIVERSITI TEKNOLOGI MALAYSIA
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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
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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.
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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.
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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.
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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
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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
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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
4.3 Conformal Mapping of Doubly Connected Regions
onto an Annulus via the Neumann Kernel 63
4.3.1 A System of Integral Equations 63
4.3.2 Numerical Implementation 65
4.3.3 Numerical Results 71
4.4 Conformal Mapping of Triply Connected Regions
onto an Annulus with a Circular Slit Via the
Neumann Kernel 86
4.4.1 A System of Integral Equations 86
4.4.2 Numerical Implementation 88
4.4.3 Interior of Triply Connected Region 95
4.4.4 Numerical Results 96
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5 NUMERICAL CONFORMAL MAPPING OF
MULTIPLY CONNECTED REGIONS ONTO
A DISK WITH CIRCULAR SLITS 100
5.1 Introduction 100
5.2 Conformal Mapping of Doubly Connected Regions
onto a Disk with a Circular Slit Via the Neumann
Kernel 101
5.2.1 A System of Integral Equations 101
5.2.2 Numerical Implementation 103
5.2.3 The Interior Mapping 108
5.2.4 Numerical Results 109
5.3 Conformal Mapping of Triply Connected Regions
onto a Disk with Circular Slits Via the Neumann
Kernel 116
5.3.1 A System of Integral Equations 116
5.3.2 Numerical Implementation 118
5.3.3 Numerical Results 125
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
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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
4.6 The radius comparison of 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
4.11 Error Norm of the Example 4.5 76
4.12 Error Norm of Interior for the Example 4.5 76
4.13 Error Norm of the Example 4.5 as given in Mohamed
and Murid (2007b) based on the Neumann kernel 77
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
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4.16 Error Norm of the Example 4.6 79
4.17 Error Norm of Interior for the Example 4.6 79
4.18 Error Norm of the Example 4.6 as given in Mohamed
and Murid (2007b) based on the Neumann kernel 79
4.19 Error Norm of the Example 4.6 using Symm’s method 80
4.20 Error Norm of the Example 4.7 81
4.21 Error Norm of Interior for the Example 4.7 81
4.22 Error Norm of the Example 4.7 as given in Mohamed
and Murid (2007b) based on the Neumann kernel 82
4.23 The radius comparison of the Example 4.8 83
4.24 The radius comparison of the Example 4.9 84
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
5.1 Error Norm of the Example 5.1 110
5.2 Error Norm of Interior for the Example 5.1 110
5.3 Error Norm of the Example 5.2 112
5.4 Error Norm of Interior for the Example 5.2 112
5.5 Error Norm of the Example 5.3 113
5.6 Error Norm of Interior for the Example 5.3 113
5.7 Error Norm of the Example 5.4 114
5.8 Error Norm of Interior for the Example 5.4 114
5.9 Error Norm of the Example 5.5 116
5.10 Error Norm of Interior for the Example 5.5 116
5.11 Radii comparison with Reichel (1986) for
the Example 5.6 126
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5.12 Radii comparison with Kokkinos et al. (1990)
for the Example 5.6 126
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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
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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
for the Example 4.4 73
4.5 A rectangular grid in Ω with grid size 0.25 and
its image with a0 = 7, a1 = 5, b0 = 5, and b1 = 1
for the Example 4.5 76
4.6 A rectangular grid in Ω with grid size 0.4 and
its image with a0 = 10, a1 = 5, b0 = 3 and b1 = b0/4
for the Example 4.6 78
4.7 A rectangular grid in Ω with grid size 0.05 and
its image with c = 0.3 and ρ = 0.1 for the Example 4.7 81
4.8 A rectangular grid in Ω with grid size 0.25 and
its image for the Example 4.8 82
4.9 A rectangular grid in Ω with grid size 0.25 and
its image for the Example 4.9 83
4.10 A rectangular grid in Ω with grid size 0.05 and
its image with a0 = 2.0 for the Example 4.10 85
4.11 A rectangular grid in Ω with grid size 0.05 and
its image for the Example 4.11 97
4.12 A rectangular grid in Ω with grid size 0.05 and
its image for the Example 4.12 98
4.13 Mapping of the Example 4.12 based on Ellacott method 99
5.1 A rectangular grid in Ω with grid size 0.05 and
its image with radius μ = e−2σ for the Example 5.1 110
5.2 A rectangular grid in Ω with grid and
its image with radius μ = e−2σ for the Example 5.2 111
5.3 A rectangular grid in Ω with grid size 0.4 and
its image with radius μ = e−2σ for the Example 5.3 113
5.4 A rectangular grid in Ω with grid size 0.25 and
its image with radius μ = e−2σ for the Example 5.4 114
5.5 A rectangular grid in with grid size 0.25 and Ω
its image with radius μ = e−2σ for the Example 5.5 115
5.6 The Contour for Example 5.6 126
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5.7 The image of the mapping for the Example 5.6 127
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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
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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
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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.
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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
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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.
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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.
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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.
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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
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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
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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.