Boundary Integral Equation for the Neumann Problem in
Bounded Multiply Connected Region
Ejaily Milad Ahmed Alejaily
A dissertation submitted in partial fulfilment of the
requirements for the award of the degree of
Master of Science (Mathematics)
Faculty of Science
Universiti Teknologi Malaysia
OCTOBER 2009
ACKNOWLEDGEMENTS
Praise be to Allah, Lord of the Worlds, and prayers and peace be upon the
Messenger of Allah. O Allah, to You belongs all praise for your guidance and your
care.
First and foremost, I would like to express my profound thanks to my
supervisor, Assoc. Prof. Dr. Ali Hassan bin Mohamed Murid, who has suggested the
dissertation topic and directed the research. I thank him for his incisive guidance,
supervision and encouragement to improve my work.
I would like also to thank Assist. Prof. Dr. Mohamed M. S. Nasser from Ibb
University, Yemen for his valuable comments and generous support during his one
month visit to IIS, UTM.
Finally, I would like to thank all those who have contributed to this work.
ABSTRACT
This research determines solutions of the Neumann problem in multiply
connected regions by using the method of boundary integral equations. This method
is widely used for solving boundary value problems. The method depends on
reducing the boundary value problem in question to an integral equation on the
boundary of the domain of the problem, and then solves this integral equation. Our
approach in this research is to convert the Neumann problem into the RiemannHilbert problem and then derive an integral equation related to the Riemann-Hilbert
problem. The derived integral equation is not uniquely solvable. The complete
discussion on the solvability of the Neumann problem, the Riemann-Hilbert problem
as well as the derived integral equation is presented. As an examination of the
present method, some numerical examples for some different test regions are
presented. These examples include comparison between the numerical results and the
exact solutions.
ABSTRAK
Penyelidikan ini menyelesaikan masalah Neumann atas satah terkait berganda
untuk menggunakan kaedah persamaan kamiran. Kaedah ini telah digunakan secara
meluas dalam menyelesaikan masalah nilai sempadan. Kaedah ini bergantung kepada
penurunan masalah nilai sempadan ke persamaan kamiran atas sempadan rantau dan
seterusnya menyelesaikan persamaan kamiran tersebut. Pendekatan yang digunakan
dalam penyelidikan ini adalah menukarkan masalah Neumann kepada masalah
Riemann-Hilbert dan seterusnya membina persamaan kamiran yang berkaitan
dengan masalah Riemann-Hilbert. Persamaan kamiran yang diperolehi tidak
mempunyai penyelesaian unik. Perbincangan yang mendalam telah disampaikan
berkaitan kebolehselesaikan masalah Neumann, masalah Riemann-Hilbert, dan
persamaan kamiran yang diperolehi. Untuk mengaji kaedah yang dipersembahkan,
beberapa contoh berangka melibatkan beberapa rantau terpilih disampaikan.
Perbandingan berangka juga diberi antara keputusan berangka dengan penyelesaian
tepat.
TABLE OF CONTAINTS
CHAPTER
TITLE
PAGE
REPORT STATUS DECLARATION
SUPERVISOR’S DECLARATION
1
2
TITLE PAGE
i
DECLARATION
ii
ACKNOWLEDGEMENTS
iii
ABSTRACT
iv
ABSTRAK
v
TABLE OF CONTENTS
vi
LIST OF TABLES
viii
LIST OF FIGURES
ix
LIST OF APPENDICES
x
RESEARCH FRAMEWORK
1
1.1
Introduction
1
1.2
Background of the Problem
2
1.3
Statement of the Problem
4
1.4
Objectives of the Study
4
1.5
Scope of the Study
5
LITERATURE REVIEW
6
2.1
6
Review of Previous Work
3
2.2
Multiply Connected Region
8
2.3
The Neumann Problem
10
2.4
Neumann Kernels
11
2.5
The Riemann-Hilbert Problem
12
2.6
Integral Equation for the RH Problem
15
AN INTEGRAL EQUATION FOR THE NEUMANN
PROBLEM
3.1
Reduction of the Neumann Problem to the RH
Problem
17
Integral Equation Related to the RH Problem
19
Solvability of the RH Problem and the Derived
Integral Equation
19
Solution of the Neumann Problem
23
NUMERICAL IMPLEMENTATIONS
24
4.1
Discretization of the Integral Equation
24
4.2
Numerical Examples
28
3.2
3.3
3.4
4
5
17
CONCLUSION
38
5.1
Summary
38
5.2
Suggestions for Further Research
39
REFERENCES
40
Appendices A-C
43-46
LIST OF TABLES
TABLE NO.
TITLE
′(
)−
′(
)
′(
)−
′(
)
The error ′ ( ) −
for Example 2.
The error ‖ ( ) −
for Example 2.
The error ′ ( ) −
′(
)
4.1
The error
4.2
The error ‖ ( ) −
4.3
4.4
4.5
4.6
4.7
The error
The error ‖ ( ) −
4.9
The error ‖ ( ) −
4.10
The error
4.8
4.11
The error ‖ ( ) −
′(
)−
The error ‖ ( ) −
PAGE
for Example 1
29
( )‖∞ for Example 1
29
for Example 2
31
( )‖∞ for Example 2
31
∞
∞
∞
with ( ) =
( )‖∞ with ( ) =
′(
)
∞
for Example 3
( )‖∞ for Example 3
⁄
⁄
−
−
⁄
⁄
33
33
34
35
( )‖ obtained in [9] for Example 3
35
for Example 4
36
( )‖∞ for Example 4
37
′(
)
∞
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
2.1
Bounded multiply connected region
9
4.1
The test region Ω for Example 1
28
4.2
The test region Ω for Example 2
30
4.3
The high fluctuation of the function
4.4
4.5
The fluctuation of the function
⁄
for Example 2
4.6
The test region Ω for Example 4
The test region Ω for Example 3
for Example 2
with ( ) =
⁄
−
32
32
34
36
LIST OF APPENDICES
APPENDIX
A
B
C
TITLE
Proof of ∫
Re
̇( )
( ) ( )
PAGE
=0
43
Computer program for Example 2
44
Computer program for treatment of the discontinuity of
complex logarithm function
46
CHAPTER 1
RESEARCH FRAMEWORK
1.1
Introduction
Partial differential equations play a vital role in natural sciences and
technology. There are many phenomena in these fields that can be described as
boundary value problems for partial differential equations. However, formulating and
solving such problems is not easy especially when we talk about real modelling of
those phenomena. Furthermore, it is also important to study existence and uniqueness
of the solution of these problems. These issues were and still occupy the minds of
mathematicians and engineers.
The last few decades have witnessed a great progress in computational
mathematics and engineering. This opens the door to more researches in these fields.
Numerical treatment is usually an important and necessary part to deal with
boundary value problems. The nature of some problems imposes on the engineers
and mathematicians some assumptions and limits in order to get solutions for those
problems.
Neumann problem is classified as a boundary value problem associated with
Laplace’s equation and Neumann boundary condition. Different types of Neumann
problems occur naturally in some fields like electrostatics, fluid flow, heat flow and
elasticity.
Boundary integral equation method is one of the common methods for
solving Neumann problem. This method depends on reducing the boundary value
problem in question to an integral equation on the boundary of the domain of the
problem, and then solves this integral equation. This integral equation could be
uniquely solvable or non- uniquely solvable, non-singular or singular. This depends
on the original problem and the way of reduction. This method reduces our task to
solve an integral equation only on the boundary of the region, thus reducing the
dimension of the Neumann problem by one. Numerical treatment is usually needed
to solve the resulting integral equation.
Our approach in this research is to reduce the Neumann problem to the
Riemann-Hilbert problem in multiply connected region, and then derive an integral
equation with the Neumann kernel related to the Riemann-Hilbert problem. This
integral equation is the Fredholm integral equation of the second type.
1.2
Background of the Problem
The partial differential equation which is identified with the name of Pierre
Simon Marquis de Laplace (1749-1827) is one of most important equations in
mathematics which has wide applications to a number of topics relevant to
mathematical physics and engineering [2]. The two-dimensional Laplace equation
has the following form:
∇
( , )=
+
= 0.
(1.1)
There are two important types of boundary value problems for Laplace's
equation: the Dirichlet problem and the Neumann problem.
A function that solves Laplace's equation is called a harmonic function, or
sometimes called potential function in physics and engineering. The problem that we
are interested about is classified as a boundary value problem and it asks for a
harmonic function on a defined region and satisfies a boundary condition related to
the normal derivative of this function on the boundary of that region. Such a problem
is called a Neumann problem (after the German mathematician Carl Gottfried
Neumann (1832-1925)) and sometimes referred to as a Dirichlet problem of the
second kind [12]. Mathematically, Neumann problem is to find a function
defined
in a region Ω with boundary Γ which satisfies
with boundary condition
and solvability condition
Here
∇
=0
= ( )
( )
(1.2)
Γ
= 0.
denotes the directional derivative of
(1.3)
(1.4)
along the outward normal to the
boundary Γ and condition (1.3) is called the Neumann condition. The last condition
(1.4) is known as compatibility condition, which is necessary for the existence of a
solution, where
is the element of arc length on Γ. Even so, the solution of the
Neumann problem is not unique; this is due to the presence of arbitrary constant in
the solution.
Finding an exact analytical solution for the Neumann problem that described
real physics situations is usually impossible. Many approximation and numerical
methods have been developed to solve such type of problems. Boundary integral
method is one of several methods that has been used to solve Neumann problem. Due
to non-uniqueness of the Neumann problem, the boundary integral method leads to
non-uniquely solvable integral equation. However, it is possible to overcome nonuniqueness by imposing additional condition(s) which is (are) consistent with the
nature of the problem.
Through the previous research by Nasser [10], the interior and exterior
Neumann problems are reduced to equivalent Dirichlet problems by using CauchyRiemann equation that are uniquely solvable. Then, boundary integral equations are
derived for the Dirichlet problems.
1.3
Statement of the Problem
Recently Husin [21] and Murid et al. [1] have reduced the Neumann problem
on a simply connected region to the Riemann-Hilbert problem. The Riemann-Hilbert
problem is then formulated as a boundary integral equation which is uniquely
solvable.
The main question here is how can we reduce the Neumann problem in
multiply connected region to the Riemann-Hilbert problem and derive a uniquely
solvable integral equation related to the Riemann-Hilbert problem?
1.4
Objectives of the Study
We can summarize the objectives of this research in the following:

Define and study the solvability of the Neumann problem and
Riemann-Hilbert problem in multiply connected region with smooth
boundary.

Reduce the Neumann problem in multiply connected region to the
corresponding Riemann-Hilbert problem.

Derive a boundary integral equation related to Riemann-Hilbert
problem.

Use appropriate numerical methods to solve this boundary integral
equation and make comparisons with exact solution.
1.5
Scope of the Study
This research endeavours to construct an integral equation method for solving
the Neumann problem in multiply connected region. This includes converting the
Neumann problem into another boundary value problem which is the RiemannHilbert problem, and then solving it using the boundary integral equation method.
Our approach will be based on a complex analysis framework. We will focus on
bounded multiply connected region with smooth boundaries as a domain for the
Neumann problem.
CHAPTER 2
LITERATURE REVIEW
2.1
Review of Previous Work
No doubt that the Laplace’s equation is considered as one of the most
important partial differential equations involved in applied mathematics and
engineering. In the same time of Laplace, Adrien- Marie Legendre (1753-1833)
attempted to solve the equation and he introduced what is now called Legendre
polynomials.
During the nineteenth century, mathematicians spent big efforts to study
several boundary value problems associated with Laplace’s equation. Very little was
known about the general properties of the solutions of Laplace’s equation until 1828
when George Green (1793-1841) and Mikhail Ostrogradsky (1801-1861)
independently investigated properties of class of solution known as harmonic
functions [20].
Since that time up till the present time, studies and researches still continue in
boundary value problems in all fields. Neumann problem is one of boundary value
problems that has caught researchers' attention. Many mathematicians have studied
different type of Neumann problem in several applications in order to get analytic or
numerical solutions. They have used different approaches and applied several
methods like integral transforms methods, finite element methods, asymptotic
methods, etc.
Boundary integral equation methods are usually used to solve boundary value
problems. Different approaches can be used to derive an integral equation related to
the problem under discussion.
No doubt that there are many researchers who worked on finding solution for
Neumann problem and Riemann-Hilbert problem in different defined regions. In the
following we discuss some researches which we think are quite relevant to our study:
In [1, 21], a new method for solving the interior Neumann problem in a
simply connected region was presented. The method is based on recent investigations
on the interplay of Riemann-Hilbert problems and Fredholm integral equations with
generalized Neumann kernel. We will try to adopt the same approach to construct a
method for solving the Neumann problem in multiply connected region.
Nasser [10] has proposed a new method for solving the interior and exterior
Neumann problem in a simply connected region. The method is based on two second
kind Fredholm integral equations with the Neumann kernel. The method consists in
reducing the interior and exterior Neumann problem into equivalent Dirichlet
problem using Cauchy-Riemann equation. Whereas in [9], Nasser and Murid present
a boundary integral method for the solution of both Dirichlet and Neumann problems
in both bounded and unbounded multiply connected regions. The method is based on
a uniquely solvable Fredholm integral equation of the second kind with the
generalized Neumann kernel.
Before that, Mikhlin in [18] reduces the Neumann problem to the modified
problem of Dirichlet which can be formulated in term of integral equation. Henrici in
[15] shows how to reduce the problem of Dirichlet and of Neumann to a set of linear
integral equations of a single real variable. He represents the solution of the
Neumann problem as the potential of a single layer. The resulting integral equation is
the Fredholm integral equation of the second kind with the Neumann kernel.
Krutitskii in [14] suggests a new approach of reduction of the Neumann
problem for the Helmholtz equation (in acoustic scattering) to a uniquely solvable
Fredholm integral equation of the second kind with weakly singular kernel. The basic
idea of this method is to introduce an interior boundary inside the interior domain
(scatterer) bounded by a simple closed surface and to reduce the problem to a
uniquely solvable Fredholm equation on the whole boundary. The solution of the
problem is represented in the form of a single layer potential on the whole boundary.
This research talks about finding a solution for the Neumann problem in
multiply connected region by using boundary integral equations method. Where we
will reduce our problem to the Riemann-Hilbert problem and then derive an integral
equation with the adjoint Neumann kernel.
2.2
Multiply Connected Region
Let Ω be a multiply connected region in the complex plan as shown in Figure
2.1. The boundary of Ω consists of the outer boundary Γ which has positive
orientation and surround the other boundaries Γ , Γ , … , Γ
which have negative
orientations.
So, we have Γ = Γ ∪ Γ ∪ ⋯ ∪ Γ
connected region of connectivity
+ 1,
and we say that Ω is a bounded multiply
≥ 0.
Γ
Γ
Ω
Γ
T⃗
n⃗
T⃗
n⃗
Γ
Figure 2.1: Bounded multiply connected region.
We assume that each boundary Γ has a parameterization
is a complex periodic function with period 2 , where
interval for each
. The parameterization
differentiable such that ̇ ( ) =
( )
We consider the parametric interval
( )=
⎨
⎪
⎩
of the parameterization
( ),
.
.
.
∈
∈
∈
̇( )
,
| ̇ ( )|
while the outward normal direction is defined as
=
( ) of the whole
, where ( ) is defined as
.
The unit tangent vector to the whole boundary Γ is denoted by
=
which
= [0,2 ] is the parametric
≠0.
( ),
( ),
∈
also need to be twice continuously
boundary Γ as a disjoint union of the intervals
⎧
⎪
( ),
− ̇( )
.
| ̇ ( )|
(2.1)
2.3
The Neumann Problem
Consider the multiply connected region Ω with smooth boundary. The
Neumann problem is to find a harmonic function ( , ) defined on Ω such that at
each point of the boundary Γ, the directional derivative of
direction
is equal to a function
In other words, the solution
∇
in the outward normal
defined on this boundary.
must satisfy the following conditions
( ) = 0,
( ( ))
= ( ),
=
+
∈ Ω,
( ) ∈ Γ,
where ( ) is known real continuous function defined on Γ.
The above problem (2.2) and (2.3) may not have a solution for some
(2.2)
(2.3)
. The
necessary condition for a solution to exist can be described via the physical nature of
the problem and written as
where
= | ̇ ( )|
( )
= 0,
is the element of arc length on Γ.
(2.4)
The condition (2.4) is sometimes called the compatibility condition and it can be
interpreted mathematically as the following:
Since
is harmonic in Ω, this implies that
Ω
(∇
)
= 0.
By using the divergence theorem, we get
=0
which is just (2.4).
With the condition (2.4), a solution to (2.2) and (2.3) is guaranteed. However,
the solution is not unique. Indeed, by adding any constant to any particular solution
we still obtain a solution. Therefore, the function
is required to satisfy the
additional condition which can be written as
where
2.4
( ) = 0,
is a fixed point in Ω.
Neumann Kernels
Let
( ) be a nonzero twice continuously differentiable periodic function
with period 2 , and define the following kernels [17]
( , )=
( , )=
The kernel
1
1
Re
Im
( )
̇( )
,
( ) ( )− ( )
( )
̇( )
.
( ) ( )− ( )
(2.5)
(2.6)
is called the generalized Neumann kernel which is continuous with
With ( ) = 1 we have
( , )=
1
Im
̇( )
1 ̈( )
−
.
2 ̇( )
( )
( , )|
( )
= ( , )=
( , )|
( )
= ( , )=
1
1
Re
Im
̇( )
,
( )− ( )
̇( )
.
( )− ( )
(2.7)
(2.8)
The kernel
∗
2.5
,
∗
,
∗
is called the Neumann kernel. The adjoint kernels of the above kernels,
and
∗
are defined as
∗(
, )=
( , )=
∗(
, )=
( , )=
1
1
∗(
, )= ( , )=
∗(
, )= ( , )=
Re
Im
1
1
Re
Im
( )
̇( )
,
( ) ( )− ( )
( )
̇( )
,
( ) ( )− ( )
̇( )
,
( )− ( )
̇( )
.
( )− ( )
The Riemann-Hilbert Problem
The Riemann-Hilbert problem (RH problem) is considered as a boundary
value problem and it is very important since there are many different problems in
physics and mathematics that involve solving RH problem.
RH problem has different forms. The first form was formulated by Bernhard
Riemann (1826-1866). Later, David Hilbert (1862-1943) reduced the Riemann form
to another form and expressed it in term of an integral equation.
The RH problem that we consider is defined as the following:
Let Ω be the multiply connected region which we have mentioned before, and let
( )= ( )+
( ) be a non-zero complex function which is twice continuously
differentiable periodic function with period 2 , and let ( ) and ( ) be real Hölder
continuously periodic functions with period 2 .
=
The RH problem consists of finding a function
continuous in its closure Ω and has boundary values
+
satisfy
( )
( )− ( )
where + denotes the positive (or left) side of Γ.
=
that is analytic in Ω,
+
on Γ which
( ) = ( ),
(2.9)
Equation (2.9) is called the Riemann condition which can also be written as
If we let
( )
Re
( )
Im ( )
= ( ).
( )
(2.10)
= ( )
then the Riemann condition (2.10) is the real part of the equation
( )
( ) =
( )+
( ).
If ( ) = 0, then (2.10) is called the homogeneous Riemann boundary condition and
the problem is called the homogeneous RH problem.
Now, we define the adjoint function
( )=
such that
̇( )
.
( )
(2.11)
Then, by the same manner, we define the adjoint RH problem to be
Re
( )
( )
and the homogeneous adjoint RH problem is
Re
( )
( )
= ( )
= 0.
The solvability of the RH problem depends on the index of the function
which is equivalent to the winding number of
denoted by
and it can be written as [17]
= ind( ) =
with respect to 0. The index of
1
arg ( )|
2
,
is
or
= ind( ) =
1
2
1
=
1
2
̇( )
( )
.
We define the range space of the RH problem denoted by
space of the homogeneous RH problem denoted by
={ ∈
where
={ ∈
:
:
= Re[
=
,
],
and the solution
as follows [18]:
analytic in Ω},
analytic in Ω}.
is the space of real Hölder continuous functions on Γ. The range space
the space of functions
for which the RH problem has a solution. The space
in Ω such that Im (
consists of analytic functions
) = 0.
Similarly for the adjoint problems, we define
=
is
=
∈
∈
:
:
= Re
=
,
,
analytic in Ω ,
analytic in Ω .
Theorem 2.1 [18]
The solvability of the RH problem is connected with the solution space of the
homogeneous adjoint problem by the relations
=(
=(
) ,
) ,
where ⊥ denotes the orthogonal complement space.
Theorem 2.2 [18]
The number of linearly independent solutions of the homogeneous RH problems is
determined by the index of the function
a) If
≤ − , then
in the following way
b) If 1 −
c) If
≤
dim(
≤ 0, then
≥ 1, then
) = −2 + 1 −
−2 + 1 −
2 −1+
dim(
)=0,
≤ dim(
≤ dim(
dim(
,
dim(
) = 0.
)≤− +1,
)≤
+
.
)=2 −1+
.
Theorem 2.3 [18]
The number of linearly independent solution of the homogeneous RH problem and its
adjoint are connected by the formula
dim(
2.6
) − dim(
)=1−
−2 .
Integral Equation for the RH Problem
There is a close relation between the RH problem and the integral equation
with generalized Neumann kernel.
Theorem 2.4 [17]
If
is a solution of the RH problem (2.10) with boundary values
then the imaginary part
where the operators
=
( )+
( ),
(2.12)
=−
,
(2.13)
in (2.12) satisfies the integral equation
and
−
are defined as
(
)( ) =
( , ) ( )
,
(
)( ) =
( , ) ( )
.
The solvability of this integral equation also depends on the index of the
function
and the connectivity of Ω . This is clarified in the following theorem.
Theorem 2.5 [18]
The number of linearly independent solution of the homogeneous integral equations
with operator ±
is given by
dim Null( +
dim Null( −
) = max(0 , 2
− 1) +
max( 0 , 2
) = max(0 , −2
+ 1) +
max( 0 , −2
+ 1),
− 1),
where is the identity operator.
By this theorem and the Fredholm alternative theorem we can determine the
solvability of the integral equation (2.13).
CHAPTER 3
AN INTEGRAL EQUATION FOR THE NEUMANN PROBLEM
3.1
Reduction of the Neumann Problem to the RH Problem
Let ( ) be the solution of the Neumann problem on the multiply connected
region Ω. Hence we can write u as a real part of an analytic function
Ω ∪ Γ, i.e.
where
( )= ( )+
( ),
is the harmonic conjugate of . The derivative of
defined on
with respect to
is given
by
where
=
and
The boundary values of
=
The normal derivative of
.
and
( )=
( )+
( ),
( )= ( )+
( ),
are
( )=
is given by
( )
=
( )+
( )
+
( ).
( )
.
(3.1)
From vector calculus, we can write
=∇ ∙
,
∇ =
+
=∇ ∙
,
∇ =
+
=n
+n ,
If
where n , n
,
.
are the components of the outward normal vector, Then (3.1)
becomes
( )
=
( )
+
( )
=
+
After some rearrangement we get
( )
+
n
( )
Equating real parts on both sides, we get
( )
= ( ) = Re
Which is equivalent to
Re ̇ ( ) −
( )
Equation (3.2) is the RH problem (2.10) with
( ) = ̇ ( ),
( ) =−
+n
=
− ̇( )
| ̇ ( )|
+
+
n
.
( ).
( ),
= | ̇ ( )| ( ).
( ) ,
+n
( ) = | ̇ ( )| ( ).
(3.2)
(3.3)
The above method of reduction has been used in Husin [21] but limited to the case of
interior Neumann problem on a simply connected region.
3.2
Integral Equation Related to the RH Problem
In this section we show how to construct an integral equation related to the
RH problem (3.2) based on Theorem 2.4.
With ( ) = ̇ ( ), equations (2.5) and (2.6) become
( , )|
( )
̇( )
=
( , )|
( )
̇( )
=
where
∗
and
∗
1
1
Re
Im
̇( )
̇( )
−1
=
Re
̇( ) ( ) − ( )
̇( )
̇( )
−1
=
Im
̇( ) ( ) − ( )
are the adjoint kernels of
and
̇( )
=−
( )− ( )
̇( )
=−
( )− ( )
∗(
, ),
∗(
, ),
respectively. Then the integral
equation (2.13) related to the RH problem (3.2) becomes
where the operators
∗
and
( )+(
∗
∗
)( ) = (
are defined as
∗
)( ),
(3.4)
(
∗
)( ) =
∗(
, ) ( )
,
(
∗
)( ) =
∗(
, ) ( )
.
3.3
Solvability of the RH Problem and the Derived Integral Equation
index
Since our RH problem (3.2) is the RH problem (2.10) with ( ) = ̇ ( ), the
Hence
= ind ̇ ( ) for our multiply connected region Ω implies
= 1,
= −1 ,
j = 1,2, … ,
.
= ind ̇ ( ) =
=1−
.
With ( ) = ̇ ( ), the adjoint function (2.11) becomes
( )=
̇( )
= 1.
( )
Thus the solution space of the homogeneous adjoint RH problem is
which means that Im[−
Since
={ ∈
] = 0, since
Re[
:
}
=−
is real. Therefore
] = 0.
is analytic, then it has to be in the form
This implies that
=
. Thus, we can write
, where
is a constant.
= span{1}
i.e
dim(
From Theorem 2.1 we can deduce that
where codim(
=
codim(
) = 1.
) = dim(
) = 1,
) represents the number of conditions for the right-hand side such
that the RH problem (2.10) is solvable.
By applying Theorem 2.3 we get
dim(
)=
.
This implies that the RH problem (3.2) is non-uniquely solvable.
With regard to the integral equation (3.4), Theorem 2.5 implies that
dim Null( +
∗)
= dim Null( −
) =
.
So, we have non-uniquely solvable RH problem (3.2) which gives rise to a nonuniquely solvable integral equation (3.4).
codim(
side
) = 1 means that the RH problem is solvable if and only if the right-hand
satisfies an extra condition. This condition can be derived in the following
way:
Since
= span{1} and
=(
) , then the orthogonality property implies that
( )
=0
i.e
( )| ̇ ( )|
=0
which is same as condition (2.4).
Now, we show how to obtain a unique solution of the integral equation (3.4),
which will give a unique solution of the RH problem (3.2). This means we have to
impose
conditions on the function
to get a unique solution to the integral
equation (3.4).
Let us define the kernel ( , ) for ( , ) ∈
1
,
( , )= 2
0,
, ∈
∈ ,
,
∈
×
,
such that
i = 1,2, … ,
,
i ≠ j, i, j = 0,1,2, … ,
.
This means that ( , ) is equal to 1 when ( ), ( ) belong to same boundary Γ
except the boundary Γ and equal to 0 otherwise. Then we define the operator
(
)( ) =
( , ) ( )
.
We have considered the solution of the Neumann problem
analytic function
form (see[19]):
by
as a real part of an
defined on Ω ∪ Γ. This function can be expressed in the following
∗(
( )=
where
∗(
)+
ln( −
),
) is a single-valued function, analytic in Ω,
is a real constant and
is
an arbitrary point inside Γ . By differentiating and then integrating on the boundary,
we get
( )
Since
∗
∗
=
( )
1
−
+
is a single-valued function, analytic in Ω, then ∫
.
∗
of the Cauchy integral formula we deduce that
1
−
Hence
=
=
Since we have − ̇
=
which implies that
For k = 1,2, … ,
−1
2
=
∫
−
+
,
0,
−1
2
( )
= 0 . In view
j = k,
j ≠ k.
.
, we get
( ) ̇( )
( )
(
−2
( )
=
= 0, since
)( ) =
, hence, the function
−1
2
( )+
( )
is real. This in turn implies
( , ) ( )
satisfies
= 0,
(3.5)
conditions. By adding (3.5) to
our integral equation (3.4), we get
( )+(
which is uniquely solvable.
∗
)( ) + (
)( ) = (
∗
)( ),
(3.6)
3.4
Solution of the Neumann Problem
After solving the integral equation (3.6) for
boundary by means of (2.12) and (3.3) to obtain
( )=
The values of
integral formula as
( ), we then find
− ( ) + | ̇ ( )| ( )
̇( )
on the
(3.7)
inside the region Ω can be evaluated by means of the Cauchy
( )=
=
To calculate the values of
1
2
1
2
( )
−
( ( ))
̇( )
( )−
.
(3.8)
inside the region, we apply the formula of the integral
representation of an analytic function the mth derivative of which is representable by
the Cauchy type integral. The kernel of this formula depends on the integration of the
Cauchy kernel (see [3, p. 303]). In our case, the formula is
( )=
−1
2
′(
) ln 1 −
−
−
Without lost of generality, we can assume that ( ) = 0.
+ ( ) .
Then, the solution of our original Neumann problem, , can be evaluated from
( ) = Re[ ( )].
(3.9)
CHAPTER 4
NUMERICAL IMPLEMENTATIONS
4.1
Discretization of the Integral Equation
Suppose the whole boundary Γ is parameterized by ( ) as defined in (2.1).
The integral equation (3.6) can be written as
( )+
where
and
∗(
, ) ( )
( , ) ( )
+
=
∗(
, ) ( )
,
are defined as
( )=
( )=
⎧
⎪
⎨
⎪
⎩
⎧
⎪
⎨
⎪
⎩
( ),
( ),
( ),
( ),
( ),
( ),
∈
∈
.
.
.
∈
∈
∈
.
.
.
∈
,
.
The above integral equation actually represents a system of integral equations
which can be written as
( )+(
( )+(
∗
∗
)( ) + (
)( ) + (
)( ) = (
)( ) = (
∗
∗
)( ),
)( ),
∈
∈
, ∈
, ∈
.
.
.
The term (
(
∗
∗
)( )|
∈
∈
( )+(
∗
=
∗
)( ) + (
∗
)( ) = (
)( ), for example, is expressed by
+
[
( , ) ( )|
∗
∈
( )]
( , )
=
∈
∈
)( ),
[
∗
( , )
+⋯ +
∈
, ∈ .
( )]
∈
∈
∗
[
+
( )]
( , )
∈
∈
.
We apply the Nyström method with the trapezoidal rule to discretize our
integral equation on an equidistant grid, where each interval
subdivided into n steps of size ℎ = 2 ⁄ . Since i = 0,1, … ,
of
=(
+ 1)
equations in
= [0,2 ] is
, this leads to a system
unknowns. Our choice of the trapezoidal rule was
due to the periodicity of the functions
and , where this method is very accurate
for periodic functions (see[16]). So, the operators
described on an equidistant grid by the trapezoidal rule.
∗
∗
and
tend to be best
Then, we get the following linear system of equations
where
( +
+ )
is the identity matrix of dimension
=
,
while ,
× , derived from the discretization of the operators
and
are
and
∗
,
(4.1)
are matrices of size
and
∗
respectively.
× 1 vectors that approximate the values of functions
and
respectively at the collection points.
To solve the system (4.1) we will use the method of Gaussian elimination.
For
sufficiently large, the uniqueness of our integral equation guarantees the
uniqueness of the system of linear equations.
Since we have the kernel
∗(
∗
is given by
, )=
1
Im
and it is continuous with
∗(
, )=
̇( )
( )− ( )
̈( )
̇( )
1
Im
2
then, we can describe the elements of the matrix
b =
The matrix
where
̇( )
ℎ
⎧ Im
⎪
is described as
is an
ℎ⁄2 .
×
,
( )−
⎨
⎪ ℎ Im
⎩2
⎡
⎢
=⎢
⎢ .
⎣
≠
̈( )
,
̇( )
.
as
,
=
.
.
.
.
.
zero matrix and
is an
⎤
⎥
⎥,
. ⎥
⎦
i, j = 1,2, … , .
=
×
+ 1,
matrix with all elements equal to
The elements of the resulting vector of multiplication of matrix
generated by the singular operator
the following:
Let
=
. The operator
(
∗
)( ) =
Since we have ∫
write
Re
∗(
∗
∗
, by the vector
, which is
can be calculated directly as
is given by
, ) ( )
̇( )
( ) ( )
=
= 0,
1
≠
Re
̇( )
( )− ( )
( )
.
[see Appendix A], then we can
∗
(
1
)( ) =
1
=
where ( ) =
When
(
̇( )
( )
̇( ) ( )
( )− ( )
Re
)( ) =
=
⎨ℎ
⎪ Re
⎩
,
.
1
Re
̇( )
̇( )
( )
Hence, the elements of the vector
ℎ
⎧ Re
⎪
̇( ) ( )
( )− ( )
Re
( )− ( )
( ) ( )
( )− ( )
Re
→ , then by L'Hospital's rule, we get
∗
1
+
̇( )
=
̈( )
− ̇( )
̇( )
( )
Re
are given by
( )
− ̇
( )−
̈( )
( ) − ̇( ) ,
̇( )
For calculating
1
,
≠
,
=
.
i, j = 1,2, … , .
( ) numerically we shall use the following formula
∫
equivalent to (3.8). Based on the fact that
( )=
∫
( ( ))
̇( )
( )−
̇( )
∫ ( )−
= 1, we can write
( ) as
.
Then, by applying the trapezoidal rule we get
( )=
∑
∑
∑
∑
( ) ̇
( )−
̇
( )−
.
(4.2)
This has the advantage that the denominator in this formula compensates for the error
in the numerator (see [5]). The trapezoidal rule will also be used to evaluate the
integration in (3.9). The discontinuity of the logarithmic function in (3.9) will be
treated numerically (see Appendix B).
4.2
Numerical Examples
In this section, we introduce some examples with different test regions to
examine our method. The sample problems are such that the exact solutions are
known. This allows us to compare our numerical results with the exact solutions.
MATLAB 7.7 has been used as the programming language to compute the
numerical results and to generate the graphs for all examples. In Appendix A, we
present the sample computer program for Example 3.
Example 1
In our first example we consider a doubly connected region, Ω as shown in Figure
4.1. The boundaries of this region are parameterized by the functions
Γ:
Γ :
( ) = 3 cos + 5 sin ,
( ) = cos − sin .
4
3
2
1
0
-1
-2
-3
-4
-6
-4
-2
0
2
4
Figure 4.1: The test region Ω for Example 1.
6
We choose the function
( )=
−
which is analytic in Ω . Then the function
( ) = Re
,
( ) =
=
+
cos −
cos
solves the Neumann problem uniquely in this region with the boundary condition
( )=
( )
=
1
Re[−
| ̇ ( )|
and the additional condition ( ) = 0.
We
‖ ( )−
describe
the
( )‖∞ where
error
by
( ) and
′(
( )) ̇ ( )]
infinity-norm
error
‖ ( )−
( ) are the numerical approximations of
and ( ) respectively. We choose four test points with
in Tables 4.1 and 4.2.
Table 4.1: The error ‖ ( ) −
= 16
( )‖∞ for Example 1.
= 32
= 64
9.754977e-04 5.379388e-09 2.448161e-15
−3
5.616242e-04 5.690202e-11 3.006658e-15
−3.1 + 2.1
4.937838e-04 6.609567e-11 2.923383e-15
0.8 − 0. 8
6.163684e-04 7.043463e-08 3.675492e-15
Table 4.2: The error ‖ ( ) −
= 32
( )
= 3. The results are shown
2
= 16
( )‖∞ ,
( )‖∞ for Example 1.
= 64
= 128
2
1.167992e-02 1.484597e-04 8.333704e-08 2.131628e-14
−3
5.088192e-03 1.344374e-08 2.486900e-14 5.684342e-14
−3.1 + 2.1
7.973386e-03 4.840204e-04 1.849295e-05 3.889558e-07
0.8 − 0. 8
4.313050e-04 6.790590e-05 4.156548e-07 5.763212e-11
The results of this example shows that the degree of accuracy of the
computed
( ) is better compared with the computed
use of the improved formula (4.2) to calculate
( ). This is attributed to the
( ), while
( ) is evaluated by
means of the formula (3.9) which is less efficient especially for
close to the
boundaries. The low efficiency of the formula (3.9) can be attributed to the existence
of the logarithmic term ln 1 −
→ ( ) for some
. When
gets closer to the boundary, i.e.
∈ [0,2 ], then the logarithmic term grows faster which cause
some error. However, by decreasing the step size ℎ (increasing the number of nodes),
the effect of this error becomes small, since the number of the points that may cause
this error represent a small percentage of the number of all points distributed on the
whole boundary.
Example 2
Consider the same function
connected region, Ω
as in Example 1 which is analytic in the triply
as shown in Figure 4.2. The parameterizations of the
boundaries of this region are as the following
6
4
2
0
-2
-4
-6
-10
-8
-6
-4
-2
0
2
4
6
Figure 4.2: The test region Ω for Example 2.
8
10
( ) = (6 + 3 cos 2 )
Γ:
,
( ) = 5 + cos − 2 sin ,
Γ:
( ) = −5 + cos − 2 sin .
Γ:
= 0. The results are shown in Tables 4.3 and 4.4.
We choose five test points with
Table 4.3: The error ‖ ( ) −
= 32
( )‖∞ for Example 2.
= 64
= 128
1
1.271292e+00
5.216432e-10
7.495839e-13
7.5
1.049793e+00
1.761464e-09
4.457499e-12
2
1.588851e+00
5.532155e-10
4.266728e-13
−4 + 4
1.132860e+00
2.618570e-10
6.263666e-13
−3 − 3
1.331080e+00
2.841603e-10
4.536686e-13
Table 4.4: The error ‖ ( ) −
= 32
( )‖∞ for Example 2.
= 64
= 128
= 256
1
1.300253e+00 2.502571e-10 6.765699e-13 1.643130e-13
2
4.640931e-02
7.5
1.410005e+01 1.691478e-04 1.175522e-10 5.911716e-12
−4 + 4
6.188060e+00 3.233125e-04 2.757819e-06 1.797701e-07
−3 − 3
3.890448e-09 1.006306e-12 5.464518e-13
4.610711e+00 1.177364e-07 4.186429e-12 1.203482e-12
Comparing with the previous example, we can see that the results of this
example give less accuracy. Observe that there is a big difference in accuracy
between the case with
= 32 and the case with
high fluctuation of the function
= 64. This is actually because of the
̇ ] which needs more points to describe its
= Re[−
behaviour (see Figure 4.3). In Example 1, the results yield better accuracy because the
boundary of the region Ω is simpler than the boundary of Ω in this example. We can
reduce the fluctuation of
if we reduce the rate of growth of the function . This can
be done if we let, for example,
( )=
⁄
−
after this change are listed in Tables 4.5 and 4.6.
⁄
(see Figure 4.4). The results
4
8
x 10
6
( )
4
2
0
-2
-4
-1
0
1
2
3
4
5
Figure 4.3: The high fluctuation of the function
(here we display
6
7
for Example 2
as an instance).
2.5
2
( )
1.5
1
0.5
0
-0.5
-1
-1
0
1
2
3
Figure 4.4: The fluctuation of the function
for Example 2 (here we display
4
5
with ( ) =
6
⁄
as an instance).
7
−
⁄
Table 4.5: The error ‖ ( ) −
= 16
( )‖∞ with ( ) =
= 32
⁄
= 64
−
⁄
for Example 2.
= 128
1
2.584127e-06
6.678885e-08
4.246603e-13 5.684331e-17
7.5
8.483093e-05
1.899593e-07
4.443113e-13 2.784004e-17
2
7.466800e-05
4.352049e-08
1.455999e-13 1.110765e-16
−4 + 4
5.605770e-05
3.502578e-08
1.962364e-12 3.469447e-17
−3 − 3
3.851602e-05
5.035156e-09
1.384630e-12 5.561947e-17
Table 4.6: The error ‖ ( ) −
= 16
( )‖∞ with ( ) =
= 32
⁄
= 64
−
⁄
for Example 2.
= 128
1
1.300536e-05
5.313670e-08
3.176348e-13 2.081668e-16
7.5
5.015306e-04
3.917486e-05
2.116225e-08 1.332268e-14
2
8.782997e-05
6.514248e-07
8.909186e-11 4.510281e-17
−4 + 4
1.050585e-02
4.978452e-03
1.245371e-03 2.989126e-05
−3 − 3
9.464201e-04
5.739429e-05
2.850413e-07 5.952683e-12
Example 3
In this example we consider a triply connected region, Ω as shown in Figure 4.5.
This region has the following parameterizations
Γ:
0
Γ:
1
Γ:
We choose the function
( )=
−
( )=2 ,
( ) = 0.5
,
( ) = 1 + 0.25
which is analytic in Ω , Then the function
( ) = Re
−
,
( ) =
=
−
.
+
+
−
is harmonic in this region and satisfies the Neumann problem with the boundary
condition
( )=
1
Re[−
| ̇ ( )|
′(
( )) ̇ ( )].
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-2
-1
0
1
2
Figure 4.5: The test region Ω for Example 3.
This example has been solved in [9] with the test points
0.6, 1.2, 1.8. We choose the same test points with
Tables 4.7 and 4.8.
Table 4.7: The error ‖ ( ) −
=8
= 16
=
,
=
= and the results are shown in
( )‖∞ for Example 3.
= 32
= 64
0.6
6.373964e-04
4.239329e-05
3.323982e-09
5.978734e-16
7.328139e-04
1.984810e-05
1.147773e-09
1.938921e-15
1.8
1.745826e-04
9.728619e-06
6.539075e-10
3.116545e-15
1.2
⁄
Table 4.8: The error ‖ ( ) −
=8
= 16
( )‖∞ for Example 3.
= 32
= 64
0.6
1.571453e-02 1.557080e-03 4.009499e-05 5.763653e-08
1.8
8.053477e-02 1.538551e-02 1.347985e-03 2.310058e-05
1.2
1.423163e-03 1.582346e-05 2.159699e-09 6.661338e-16
Table 4.9: The error ‖ ( ) −
=8
( )‖∞ obtained in [9] for Example 3.
= 16
= 32
= 64
0.6
1.7e-04
2.6e-06
1.6e-09
5.0e-16
1.4e-03
1.8e-05
5.8e-12
1.1e-16
1.8
5.2e-04
1.1e-05
3.8e-11
2.5e-15
1.2
As we can see from the above tables, the results are comparable for
= 1.2,
while for the points near to the boundaries ( = 0.6, 1.8) there is some differences.
This is due to the same reasons that we have mentioned in Example 1.
Example 4
Let Ω be a multiply connected region of connectivity 5 as shown in Figure 4.6. The
boundaries of Ω are parameterized by the functions
Γ0 :
Γ1 :
Γ:
Γ:
Γ:
0
1
( ) = (9 +
( ) = (3 +
6)
3)
( ) = (3 + 5)
( ) = −6 +
( ) = (3 − 5)
,
−
,
.
,
,
10
8
6
4
2
0
-2
-4
-6
-8
-10
-10
-5
0
5
10
Figure 4.6: The test region Ω for Example 4.
Consider the same function
as in example 3, i.e.
( )=
−
, with
= −3 + 5. We choose nine test points and the results are shown in Tables 4.10
and 4.11.
Table 4.10: The error ‖ ( ) −
= 16
= 32
( )‖ for Example 4.
= 64
= 128
7
6.990375e-02
1.316567e-04
1.739490e-08
7.944109e-15
5+3
4.875295e-02
4.811641e-05
1.659205e-08
5.617334e-15
2+4
5.312716e-02
4.869446e-04
4.592000e-08
3.546046e-14
7
3.592778e-02
1.959344e-04
8.526449e-09
8.881784e-16
−3 − 6
1.447674e-02
6.155298e-05
1.799327e-08
1.776357e-15
−9
4.123396e-02
5.531501e-04
3.820318e-08
3.552714e-15
1−2
5.763292e-02
1.845262e-04
4.100215e-08
2.664535e-15
−2 + 3.7
1.936811e-02
7.406767e-05
4.152010e-08
1.083478e-14
6−6
4.276964e-02
1.119694e-03
4.253846e-08
1.293207e-14
Table 4.11: The error ‖ ( ) −
= 32
= 64
( )‖∞ for Example 4.
= 128
= 512
7
6.628118e-05
1.664843e-07
1.989520e-13
1.563194e-13
5+3
1.011974e-03
7.123924e-06
5.137224e-12
1.065814e-13
2+4
5.268222e-04
1.277896e-07
3.197442e-14
4.263256e-14
7
9.829159e-02
1.623398e-03
8.814859e-07
7.105427e-15
−3 − 6
4.090850e-03
9.002238e-08
4.263256e-14
0.000000e+00
−9
1.381731e-02
7.113128e-05
3.776329e-09
2.131628e-14
1−2
1.629954e-03
2.510883e-04
1.711473e-07
5.329071e-14
−2 + 3.7
7.561762e-02
9.193517e-03
2.055304e-04
4.258816e-13
6−6
2.139823e+00
1.983034e-01
8.013487e-03
4.372076e-10
In this example we find that for more complicated region we need to increase
number of nodes to get good results. In all our examples we subdivide each interval
into the same number
of steps, but as we can see, for instance in region Ω the
boundary Γ needs more points to describe it than the other boundaries Γ , Γ , Γ and
Γ . This unbalance in the discretization could lead to reducing the efficiency of our
method.
CHAPTER 5
CONCLUSION
5.1
Summary
This study has formulated a new boundary integral equation for solving the
Neumann problem in multiply connected regions with smooth boundaries. The idea
of formulation of this integral equation is firstly to reduce the Neumann problem into
the equivalent RH problem from which an integral equation is constructed. The
derived integral equation was in the form
( )+(
∗
)( ) + (
)( ) =
∗
| ̇ ( )| ( )
( ).
(5.1)
We have solved this integral equation numerically using Nyström method
with the trapezoidal rule. Once we got the solution
, the solution of the Neumann
problem was within our reach, where we have used the formula
( ) = Re
−1
2
− ( ) + | ̇ ( )| ( ) ln 1 −
−
( )−
.
(5.2)
The results of the numerical example showed that the efficiency of our
approach was very good. However, the accuracy of values of ( ) for
boundary was not as good as for
( ) for
near to the
far from the boundary. To increase the accuracy of
close to the boundary, one can still use the formula (5.2) for any interior
point , with higher number of nodes.
5.2
Suggestions for Further Research
This dissertation develops a method to solve the Neumann problem within
certain conditions and assumptions. For further research, we suggest the following

We mention in the previous section that we lost some accuracy of values of
( ) as we get closer to the boundaries. This loss, we think, can be reduced if
we study the possibility of improving the formula (5.2) along the same lines
as we have done for evaluating

( ).
In this dissertation we did not calculate the solution of the Neumann problem
on the boundaries
( ). This can be achieved if we calculate
( ) from
( ). For that we suggest to treat the formula (3.7) as a differential
equation and apply, for example, the Runge-Kutta method to solve it.

The domain of the problem in this dissertation was bounded multiply
connected region with smooth boundaries. We propose extending our
approach for unbounded multiply connected region and also for nonsmooth
boundaries.

The general theories of Chapter 3 were developed for solving the interior
Neumann problem on multiply connected region. Other potential applications
need to be explored. Probably some extensions or modifications of the
theories are required to obtain integral equations related to problem like
solving Poisson equation.
REFERENCES
1. A. H. M. Murid, U. T. Husin and H. Rahmat, An Integral Equation Method for
Solving Neumann Problems in Simply Connected Regions with Smooth Boundaries,
Proceeding of ICORAFSS 2009.
2.
A. P. S. Selvadurai, Partial Differential Equations in Mechanics Vo1. 1:
Fundamentals. Laplace's Equation. Diffusion Equation. Wave Equation. SpringerVerlag, Berlin, 2000.
3. F. D. Gakhov, Boundary Value Problem. English translation of Russian edition
1963. Oxford: Pergamon Press, 1966.
4. I. Stakgold, Boundary Value Problems of Mathematical Physics, Vols. II, SIAM,
Philadelphia, 2000.
5. J. Helsing and R. Ojala, On the evaluation of layer potentials close to their
sources, Journal of Computational Physics, Vol. 227 n.5, p.2899-2921, 2008.
6. K. E. Atkinson, The solution of non-unique linear integral equations, Numerische
Mathematik, Vol. 10, p.117-124, 1967.
7. K. E. Atkinson, The Numerical Solution of Integra Equations of the Second Kind,
Cambridge Uni. Press. 1997.
8.
M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and
Applications, Cambridge University Press, 2nd Edition, 2003.
9. M. M. S. Nasser and A. H. M. Murid, A Boundary Integral Equation with the
Generalized Neumann Kernel for Laplace's Equation in Multiply Connected
Regions, submitted for publication.
10. M. M. S. Nasser, Boundary Integral Equations with the Generalized Neumann
Kernel for the Neumann problem, MATEMATIKA, 23, p.83-98, 2007.

11. M. M. S. Nasser, Numerical Solution of the Riemann-Hilbert Problem, Punja
Univ. J. Math. Vol.40, p.9-29, 2008.
12. N. H. Asmar, Applied Complex Analysis with Partial Differential Equations,
Prentice Hall, New. Jersey, 2002.
13. N. I. Muskhelishvili, Singular Integral Equations. English translation of Russian
edition 1953. Leyden: Noordhoff, 1977.
14. P. A. Krutitskii, A new integral equation approach to the Neumann problem in
acoustic scattering, Math. Meth. Appl. Sci. Vol.24, p.1247–1256, 2001.
15. P. Henrici, Applied and Computational Complex Analysis. Vol. 3, John Wiley,
New York, 1986.
16. P. J. Davies and P. Rabinowitz, Methods of Numerical Integration, 2nd Ed.
Academic Press,. New York, 1984.
17. R. Wegmann, A. H. M. Murid and M. M. S. Nasser, The Riemann-Hilbert
problem and the generalized Neumann kernel, J. Comp. Appl. Math. Vol. 182,
p.388-415, 2005.
18. R. Wegmann and M. M. S. Nasser, The Riemann–Hilbert problem and the
generalized Neumann kernel on multiply connected regions, J. Comp. Appl. Math.
Vol. 214, p.36-57, 2008.
19. S. G. Mikhlin, Integral Equations and their Applications to Certain Problems in
Mechanics, Mathematical Physics and Technology, Pergamon Press, New York,
Translated from the Russian by A.H. Armstrong, 1957.
20. T. Myint-U and L. Debnath, Linear partial differential equations for scientists
and engineers, Birkhäuser Boston, 2007.
21. U. T. Husin, Boundary Integral Equation with the Generalized Neumann Kernel
for Solving the Neumann Problem, M.Sc. Dissertation, UTM, 2009.
APPENDIX A
Cauchy integral formula states that for any point z ∈ Ω we have
( )=
For a point
( )
−
1
2
,
analytic.
= ( ) on the boundary Γ we have
( )
−
1
2
=
1
( ),
2
≠ ,
where the integral on the left must be interpreted as a principal value integral.
If
= 1, then
1
2
i.e.
1
1
1
−
1
= ,
2
̇( )
( )− ( )
= 1.
From the above equation and equation (2.7) we can write
1
( ( , )+
( , ))
= 1,
i.e.
( , )
−
( , )
= 1.
Equating imaginary parts on both sides, we get
( , )
=
1
̇( )
( )− ( )
= 0.
APPENDIX B
clc
clear variables
Sigma=0;
m=3;
n=64;
r=m*n;
h=(2*pi)/n;
t=0:h:2*pi-h;
Eta0=(6+3*cos(2*t)).*exp(1i*t);
Eta0p=(-6*sin(2*t)+1i*(6+3*cos(2*t))).*exp(1i*t);
Eta0pp=(-15*cos(2*t)-6-12i*sin(2*t)).*exp(1i*t);
Eta1=5+cos(t)-2i*sin(t);
Eta1p=-sin(t)-2i*cos(t);
Eta1pp=-cos(t)+2i*sin(t);
Eta2=-5+cos(t)-2i*sin(t);
Eta2p=-sin(t)-2i*cos(t);
Eta2pp=-cos(t)+2i*sin(t);
Eta=[Eta0,Eta1,Eta2].';
Etap=[Eta0p,Eta1p,Eta2p].';
Etapp=[Eta0pp,Eta1pp,Eta2pp].';
fe=exp(Eta)-exp(Sigma);
fep=exp(Eta);
fepp=exp(Eta);
Phi=real(-1i.*fep.*Etap);
Phip=real(-1i.*(fep.*Etapp+fepp.*(Etap.^2)));
Psi=imag(-1i.*fep.*Etap);
for I=1:r
for K=1:r
if I==K
Vs(I,K)=(h/(2*pi))*imag(Etapp(K)/Etap(K));
y(I,K)=(h/pi)*real((Etapp(K)/Etap(K))*Phi(K)-Phip(K));
else
Vs(I,K)=(h/pi)*imag(Etap(I)/(Eta(I)-Eta(K)));
y(I,K)=(h/pi)*real((Etap(I)*Phi(K)-Etap(K)*Phi(I))/
(Eta(I)-Eta(K)));
end
end
end
c11=zeros(n,n);
cii=(h/(2*pi)).*ones(n,n);
J=blkdiag(c11,cii,cii);
Y=y*ones(r,1);
Id=eye(r);
Psin=(Id+Vs+J)\Y;
err_Psi=norm(Psi-Psin,inf);
fenp=(Phi+1i.*Psin)./(-1i.*Etap);
err_fenp=norm(fep-fenp,inf);
z=[1; 7.5; 2i; -4+4i; -3-3i];
for I=1:length(z)
fznp(I,1)=(sum(fenp.*Etap./(Eta-z(I))))./(sum(Etap./
(Eta-z(I))));
Lv=1-(z(I)-Sigma)./(Eta-Sigma);
fzn(I,1)=(-h/(2*pi*1i))*sum(fenp.*(log(abs(Lv))+
1i*CArg(Lv)).*Etap);
end
fzp=exp(z);
fz=exp(z)-exp(Sigma);
err_fznp=abs(fznp-fzp);
err_fzn=abs(fzn-fz);
uzn=real(fzn);
uz=real(fz);
err_uzn=abs(uzn-uz);
fprintf('\n
n=%i\n\n
err Psi = %e\n
err fenp = %e\n\n
\n',n,err_Psi, err_fenp)
disp(z);
fprintf('\n
z
err fznp
err fzn
err uzn\n')
for I=1:length(z)
fprintf('\n
%i
%e
%e
%e',I,err_fznp(I),err_fzn(I),err_uzn(I))
end
fprintf('\n\n')
z=
n=1024;
h=(2*pi)/n;
t=0:h:2*pi-h;
Eta0=(6+3*cos(2*t)).*exp(1i*t);
Eta1=5+cos(t)-2i*sin(t);
Eta2=-5+cos(t)-2i*sin(t);
plot(Eta0,'k','LineWidth',1.5)
hold on
plot(Eta1,'k','LineWidth',1.5)
plot(Eta2,'k','LineWidth',1.5)
plot(real(Sigma),imag(Sigma),'xk','MarkerSize',5)
plot(real(z),imag(z),'.k','MarkerSize',10)
plot(real(Eta0(n/8)),imag(Eta0(n/8)),'<k','LineWidth',3,'MarkerFaceC
olor','k','MarkerSize',5)
plot(real(Eta1(n/2)),imag(Eta1(n/2)),'^k','LineWidth',2,'MarkerFaceC
olor','k','MarkerSize',5)
plot(real(Eta2(n/2)),imag(Eta2(n/2)),'^k','LineWidth',2,'MarkerFaceC
olor','k','MarkerSize',5)
axis equal
xlim([-10 10])
ylim([-6 6])
APPENDIX C
function CA=CArg(Z)
L=max(size(Z));
Theta=angle(Z);
for k=2:L
if (abs(Theta(k)-Theta(k-1))>pi)
cc=round(1.0*((Theta(k)-Theta(k-1))/(2*pi)));
Theta(k)=Theta(k)-2*pi*cc;
end
end
CA=Theta;