vii
TABLE OF CONTENTS
CHAPTER
TITLE
PAGE
THESIS STATUS DECLARATION
SUPERVISOR’S DECLARATION
DECLARATION ON COOPERATION
1
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
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
3
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
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
3.4 Application to Conformal Mapping of Multiply
Connected Regions onto an Annulus with Circular
Slits via the Neumann Kernel
3.5 Application to Conformal Mapping of Multiply
Connected Regions onto a Disk with Circular
Slits via the Neumann Kernel
4
37
42
46
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
x
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
6
116
5.3.1 A System of Integral Equations
116
5.3.2 Numerical Implementation
118
5.3.3 Numerical Results
125
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
143
Sample of Computer Programs
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
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
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
Error norm of the Example 4.5 using Amano’s method
and Symm’s method
77
Error Norm of Interior for the Example 4.5 as given
in Mohamed and Murid (2007a) based on the
Kerzman-Stein kernel
77
4.3
4.10
4.14
4.15
xii
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
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
4.28
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
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
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
Mapping of doubly connected region onto
an annulus
40
Mapping of a multiply connected region Ω of
connectivity M + 1 onto an annulus with circular slits
42
2.2
2.7
3.2
3.3
3.4
4.1
Mapping of a multiply connected region Ω of
connectivity M + 1 onto a disk with circular slits
√
Mapping of the Example 4.1 with a0 = 2 14, a1 = 2,
b0 = 7 and b1 = 1
47
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
for the Example 4.4
73
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
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
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
A rectangular grid in Ω with grid size 0.25 and
its image for the Example 4.8
82
A rectangular grid in Ω with grid size 0.25 and
its image for the Example 4.9
83
A rectangular grid in Ω with grid size 0.05 and
its image with a0 = 2.0 for the Example 4.10
85
A rectangular grid in Ω with grid size 0.05 and
its image for the Example 4.11
97
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
A rectangular grid in Ω with grid and
its image with radius μ = e−2σ for the Example 5.2
111
A rectangular grid in Ω with grid size 0.4 and
its image with radius μ = e−2σ for the Example 5.3
113
A rectangular grid in Ω with grid size 0.25 and
its image with radius μ = e−2σ for the Example 5.4
114
A rectangular grid in with grid size 0.25 and Ω
its image with radius μ = e−2σ for the Example 5.5
115
The Contour for Example 5.6
126
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
5.2
5.3
5.4
5.5
5.6
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...)
√
−1
i Im
-
Imaginary part
Re
-
Real
U
-
Unit disk
-
Epsilon (small number 0 ≤ < 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