AN INTEGRAL EQUATION METHOD FOR SOLVING EXTERIOR
NEUMANN PROBLEMS ON SMOOTH REGIONS
AZLINA BINTI JUMADI
UNIVERSITI TEKNOLOGI MALAYSIA
AN INTEGRAL EQUATION METHOD FOR SOLVING EXTERIOR NEUMANN
PROBLEMS ON SMOOTH REGIONS
AZLINA BINTI JUMADI
A dissertation submitted in partial fulfillment of the
requirements for the award of the degree of
Master of Science (Mathematics)
Faculty of Science
Universiti Teknologi Malaysia
NOVEMBER 2009
To Mak, Abah, Jeli, Arul, Azhar, Ainul and abang.
ACKNOWLEDGEMENT
First of all, I thank ALLAH (SWT), the Lord Almighty, for giving me the
health, strength and ability to complete this report.
I would like to express my sincere gratitude to my supervisor Dr. Hjh.
Munira binti Ismail, who has directed the research and spend her times throughout
this period, also for her guidance and advice.
My special thanks are also due to Assoc. Prof. Dr Ali Hassan Mohamed
Murid for suggesting the topic and contributing ideas on this research. I would also
like to thank Assist. Prof. Dr. Mohamed M. S. Nasser, Department of Mathematics,
Faculty of Science, Ibb University, Yemen for lending his time and providing a lot
of ideas in Chapter 4 during his one month visit.
My deepest gratitude further goes to my family for being with me in any
situation, their encouragement, endless love and trust. Finally with my best feelings I
would like to thank all my close friends who helped me during this research.
ABSTRACT
This work develops a boundary integral equation method for numerical
solution of the exterior Neumann problem. An integral equation for solving the
exterior Neumann problem in a simply connected region is derived in this
dissertation based on the exterior Riemann-Hilbert problem. In the first step the
exterior Neumann problem is reduced to an exterior Riemann-Hilbert problem for
the derivative of an auxiliary function which is analytic in the region. Then, the
exterior Riemann-Hilbert problem is transformed to a uniquely solvable Fredholm
integral equation on the boundary of the region. Once this equation is solved, the
auxiliary function and the solution of the exterior Neumann problem can be
obtained. The efficiency of the method is illustrated by some numerical examples.
ABSTRAK
Kajian ini bertujuan untuk membina suatu kaedah persamaan kamiran
sempadan untuk penyelesaian berangka bagi masalah Neumann luaran. Satu
persamaan kamiran untuk menyelesaikan masalah Neumann luaran dalam rantau
terkait mudah dibentuk dalam disertasi ini berdasarkan masalah Riemann-Hilbert
luaran. Dalam langkah pertama, masalah Neumann luaran akan diturunkan kepada
masalah Riemann-Hilbert luaran dalam rantau tersebut. Kemudian, masalah
Riemann-Hilbert luaran akan dijelmakan kepada satu persamaan kamiran Fredholm
yang mempunyai penyelesaian unik dalam rantau tersebut. Apabila persamaan ini
diselesaikan jawapan untuk masalah Neumann luaran boleh dicari. Keberkesanan
kaedah ini diilustrasikan dengan beberapa contoh berangka.
TABLE OF CONTENTS
CHAPTER
TITLE
PAGE
REPORT STATUS DECLARATION
SUPERVISOR’S DECLARATION
1
TITLE PAGE
i
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
x
LIST OF FIGURES
xii
LIST OF APPENDICES
xiii
RESEARCH FRAMEWORK
1
1.1
General Introduction
1
1.2
Background of the Problem
3
1.3
Statement of the Problem
3
1.4
Objectives of the Study
4
1.5
Scope of the Study
5
1.6
Significance of the Study
5
1.7
Dissertation Organization
5
2
HISTORICAL DEVELOPMENT OF BOUNDARY
7
INTEGRAL EQUATION METHOD
2.1
Introduction
7
2.2
Historical Background
7
2.3
Boundary Integral Equation on Solving Neumann
9
Problem
2.4
Boundary Integral Equation on Solving Riemann-Hilbert
11
Problem
2.5
3
Conclusion
NEUMANN PROBLEM AND RIEMANN-HILBERT
12
13
PROBLEM
3.1
Introduction
13
3.2
The Neumann Problem
13
3.2.1
The Exterior Neumann Problem
14
3.2.2
Definition of Normal Derivative
16
3.3
The Riemann-Hilbert Problem
17
3.3.1
The Exterior Riemann-Hilbert Problem
18
3.3.2
The Solvability of Riemann-Hilbert Problems
19
3.3.3
Integral Operators
21
3.3.4
Integral Equation for the Exterior Riemann-
23
Hilbert Problem
3.4
4
Conclusion
MODIFICATION OF THE EXTERIOR NEUMANN
25
26
PROBLEM
4.1
Introduction
26
4.2
Reduction of Exterior Neumann Problem to the Exterior
27
Riemann-Hilbert Problem
4.3
The Solvability of the Riemann-Hilbert Problem
29
4.4
Integral Equation for the Exterior Riemann-Hilbert
29
Problem
4.5
Modified Integral Equation for the Exterior Riemann-
30
Hilbert Problem
5
4.6
Modifying the Singular Integral Operator
34
4.7
Computing f ( z ) from f ( z )
36
4.8
Conclusion
37
NUMERICAL IMPLEMENTATIONS OF THE
38
BOUNDARY INTEGRAL EQUATION
6
5.1
Introduction
38
5.2
Numerical Implementation
38
5.3
Examples
40
5.4
Numerical Computation and Results
43
5.5
Discussion of the Results
48
5.6
Conclusion
49
CONCLUSIONS AND SUGGESTIONS
50
6.1
Conclusion
50
6.2
Suggestions for Future Research
50
REFERENCES
52
Appendices A–C
54
LIST OF TABLES
TABLE NO.
5.1
TITLE
The Error f ( z )  f n ( z )

for the Exterior Neumann Problem
PAGE
43
on the Boundaries 1 , 2 and 3
5.2
The Error f ( z )  f n( z ) for the Exterior Neumann Problem at
44
the Test Points Exterior to the Boundary 1
5.3
The Error f ( z )  f n( z ) for the Exterior Neumann Problem at
44
the Test Points Exterior to the Boundary 2
5.4
The Error f ( z )  f n( z ) for the Exterior Neumann Problem at
44
the Test Points Exterior to the Boundary 3
5.5
Comparison of the Error f ( z )  f n ( z ) with Nasser [3] for the
45
Exterior Neumann Problem at the Test Points Exterior to the
Boundary 1
5.6
Comparison of the Error f ( z )  f n ( z ) with Nasser [3] for the
46
Exterior Neumann Problem at the Test Points Exterior to the
Boundary 2
5.7
Comparison of the Error f ( z )  f n ( z ) with Nasser [3] for the
Exterior Neumann Problem at the Test Points Exterior to the
Boundary 3
47
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
3.1
The Exterior Neumann Problem
15
5.1
The Curve 1 and the Exterior Test Points
41
5.2
The Curve 2 and the Exterior Test Points
42
5.3
The Curve 3 and the Exterior Test Points
42
LIST OF APPENDICES
APPENDIX
A
TITLE
Computer Program for Solving Neumann Problem on the
PAGE
54
Region Exterior to 1
B
Computer Program for Solving Neumann Problem on the
57
Region Exterior to 2
C
Computer Program for Solving Neumann Problem on the
Region Exterior to 3
60
CHAPTER 1
RESEARCH FRAMEWORK
1.1
Introduction
The problem of finding a function which is harmonic in a specified domain
and which satisfies prescribed conditions on the boundary of that domain is abound
in applied mathematics. If the values of the function are prescribed along the
boundary, the problem is known as a boundary value problem of the first kind, or a
Dirichlet problem. If the values of the normal derivative of the function are
prescribed on the boundary, the boundary value problem is one of the second kind,
or a Neumann problem (in honor of German mathematician Carl Gottfried
Neumann). Modifications and combinations of those types of boundary condition
also arise.
The Neumann problem is a class of fundamental boundary value problems
for analytic functions, a living subject with a fascinating history and interesting
applications. It is a boundary value problem for determining a harmonic function,
u ( x , y ) interior or exterior to a region with prescribed values of its normal
derivative,
u
on the boundary. Some examples are heat problems in an insulated
n
plate, electrostatic potential in a cylinder and potential of flow around airfoil.
In its simplest form, the Neumann problem consists in finding a function
u ( x , y ) satisfying the following conditions:
i.
u is continuous and differentiable in  and  .
ii.
u is harmonic in  .
iii.
 u ( z )  0 for all z in  .
iv.
If

denotes differentiation in the direction of the exterior normal,
n
then
u
  (t ),
n  (t )
 (t )   .
(1.1)
which is known as a Neumann condition (see [1]).
The Neumann problem is often solved by conformal mapping for arbitrary
simply connected region [2]. The basic technique is to transform a given boundary
value problem in the xy plane into a simpler one in the uv plane where they can be
solved easily. By transforming back to the original region, the desired answer is
obtained.
An important fact about conformal mapping which accounts for much of its
applications is that the Laplace’s equation is invariant under conformal mapping.
Although conformal mappings have been an important tool of science and
engineering since the development of complex analysis, the practical use of the
conformal maps has always been limited by the fact that exact conformal mappings
are only known for special regions.
Other than conformal mapping, there are many techniques from various
mathematics fields for solving the Neumann problem. Since 1900, lots of
mathematicians started to investigate the types of Neumann problems and using
different methods and approaches in order to formulate its solution such as finite
difference method, finite element method, iterative method, collocation method and
boundary integral equation method.
In this report we will only cover the boundary integral equation methods
which can be used to solve the boundary value problem in its original region. The
reformulation of the boundary value problem as an equivalent integral equation over
the boundary reduces the dimensionality of the problem which makes the method an
efficient tool for complicated engineering problems.
1.2
Background of the Problem
The boundary integral equation method is a classical method for solving the
Neumann problem. The classical boundary integral equations for the Neumann
problems are derived by representing the solutions of Neumann problems as the
potential of a single layer; however, the integral equation for the interior Neumann
problem is not uniquely solvable. Furthermore, extra calculations are required for
determining the boundary values of the solutions of the Neumann problems from the
solutions of the integral equation.
In this project, boundary integral equations will be derived for the exterior
Neumann problem in simply connected regions   . The derived integral equations
are Fredholm integral equations of the second kind with continuous kernels provided
that the boundaries are sufficiently smooth.
1.3
Statement of the Problem
The research on boundary integral equations with the generalized Neumann
kernel is still continuing. Through the previous research by Ummu Tasnim in [3], the
interior Neumann problem is reduced to equivalent Riemann-Hilbert problem by
using Cauchy-Riemann equations. Then, the boundary integral equation is derived
for the Riemann-Hilbert problem based on an earlier work by Nasser [4]. In [4], the
interior and exterior Neumann problems are reduced to equivalent Dirichlet
problems by using Cauchy-Riemann equations. Then, the boundary integral
equations are derived for the Dirichlet problem. This research continues the study on
Neumann problems based on [3, 4, 5, and 6]. The aim of the study is to derive an
integral equation for the exterior Neumann problem by reducing it to the exterior
Riemann-Hilbert problem using Cauchy-Riemann equations. Furthermore, the
analysis on the solvability for this integral equation will be determined as well.
1.4
Objectives of the Study
This study embarks on the following objectives:
i. To reduce the exterior Neumann problem to the exterior Riemann-Hilbert
problem.
ii. To study the boundary integral equation for the exterior Riemann-Hilbert
problem.
iii. To derive a Fredholm integral equation of the second kind for the
Neumann problem based on the exterior Riemann-Hilbert problem in Ω−.
iv. To determine the solvability of the formulated integral equation.
v. To provide a numerical technique for the boundary integral equation
using software MATLAB.
1.5
Scope of the Study
This research will focus on the development of a numerical method for the
exterior Neumann problem in a simply connected region Ω−. Firstly, the exterior
Neumann problem will be reduced to the exterior Riemann-Hilbert problem. Then,
the boundary integral equation for the Neumann problem will be derived based on
the exterior Riemann-Hilbert problem. Next, the solvability of the derived integral
equation will be discussed. Then, a numerical method will be employed to solve the
problem numerically through several examples.
1.6
Significance of the Study
The purpose of the study is to develop a new method for solving the exterior
Neumann problem. The method is based on recent investigations on Ummu
Tasnim’s approach [3] for the interior Neumann problem and on the interplay of
Riemann-Hilbert problems and Fredholm integral equations with generalized
Neumann kernel [4, 6]. This approach will enrich the numerical procedure of solving
exterior Neumann problem and enhance the numerical effectiveness of solving it.
1.7
Dissertation Organization
Chapter 1 contains the general introduction and background regarding of one
type of Laplace’s problem called the Neumann problem. In this chapter, we define
our statement of the problem, objectives of the study, scope of the study and also
significance of the study.
In Chapter 2 we give the historical background of the Neumann problem and
the Riemann-Hilbert and a brief review of the boundary integral equation method for
solving the Neumann problem. We also discuss some important facts that will be
required later and also some auxiliary material related to Neumann problem and
Riemann-Hilbert problem in Chapter 3.
In Chapter 4 we provide the theoretical formulation for the overall research.
First, the reduction of the exterior Neumann problem into the exterior RiemannHilbert problem is explained. Then, the solvability of the derived Riemann-Hilbert
problem is reviewed. Then the derived integral equation is used to solve numerically
the exterior Neumann problem based on the derived Riemann-Hilbert problems.
In Chapter 5, we present the numerical implementations of the derived
integral equation. A Nyström method will be used based on the trapezoidal rule.
After that, some numerical examples on some test regions are reviewed. We then
conclude this chapter with displaying some comparison result with the exact solution
and some discussions on the result obtained.
Finally in Chapter 6, we conclude our project by summarizing every chapter
and then stating some suggestions for the future research.
CHAPTER 2
HISTORICAL DEVELOPMENT OF BOUNDARY INTEGRAL EQUATION
METHOD
2.1
Introduction
This chapter gives the historical background and development of the
boundary integral equation method. It is divided into five sections. Section 2.2
includes the historical background of boundary integral equation method. The
literature regarding the boundary integral equation method on solving the Neumann
problem and the Riemann-Hilbert problem is given respectively in Section 2.3 and
2.4. Finally, Section 2.5 gives the conclusion of this chapter.
2.2
Historical Background
The theory and application of integral equations is an important subject
within applied mathematics. Integral equations are used as mathematical models for
many and varied physical situations, and integral equations also occur as
reformulations of other mathematical problems. It is sufficient, for example, to
notice works on the static theory of elasticity and on the problem of immersed bodies
in hydrodynamics. It is well known also how important a role is played by the
method of integral equations in the theory of oscillations, in problems on stability of
struts and in many other problems.
Integral equations were studied in the nineteenth century as one means of
investigating boundary value problems for Laplace’s equation and other elliptic
partial differential equations. It is an inexpensive, flexible technique to solve the
elliptic boundary value problems and has the advantage of reducing the
dimensionality of the problem, making it an efficient tool for complicated
engineering problems.
One of the first results, if not the first one, which can be related to integral
equations were Fourier inversion formulae obtained in 1811 [7]. Another integral
equation was obtained by Abel, the integral equation which one or the other concrete
problem of physics and mechanics reduces.
An important moment in the development of integral equations was the work
of Volterra (1896), in which he investigated an equation of the form
x
 x     K x , s  s ds  f ( x ) ,
(2.1)
a
where   x  is an unknown function, K  x, s  and f (s ) given functions, and  a
numerical parameter, and he proved that if K  x, s  and f ( s ) are continuous in some
interval [a , b], then the equation (2.1) has in this interval one and only solution which
can be constructed by the method of successive approximations.
It is rather difficult to investigate the integral equation of the form
b
 x     K  x, s  s ds  f ( x ),
a
a  x  b,
(2.2)
which differs from Volterra equation only in that the variable upper limit of the
integral x has been replaced by the constant limit b . An equation of the form (2.2)
is now called a Fredholm equation. For f  0, we have  and f given, and we
seek  ; this is the non-homogenous problem. If its right-hand side is identically
equal to zero, then (2.2) is called homogenous. During the period of 1960-1990,
there has been much work on developing and analyzing numerical methods for
solving linear Fredholm integral equations of the second kind, with the integral
operator being compact on a suitable space of functions.
Various boundary integral equation formulations have long been used as a
means of solving Laplace’s equation numerically although this approach has been
less popular than the use of finite difference methods and finite element methods.
From 1960 to the present day, there has been a significant increase in the popularity
of using boundary integral equations to solve Laplace’s equation and many other
elliptic equations, including the biharmonic equation, the Helmholtz equation, the
equations of linear elasticity and the equations for Stokes’ fluid flow.
Next, we will review the historical background and development of the
boundary integral equation method on solving the Neumann problem.
2.3
Boundary Integral Equation on Solving Neumann Problem
The classical boundary integral equations for the Neumann problems are the
two second kind Fredholm integral equations with the Neumann kernel. The
Neumann kernel usually appears in the integral equations related to the Dirichlet
problem, the Neumann problem and conformal mappings. However, the integral
equation for the interior Neumann problem is not uniquely solvable due to the lack
of unique solvability for the Neumann problem itself.
Recently, Nasser [4] has developed two uniquely solvable second kind
Fredholm integral equations with the generalized Neumann kernel that can be used
to solve the interior and exterior Neumann problems in simply connected regions
with smooth boundaries. The method is based on two uniquely solvable Fredholm
integral equations of the second kind with the generalized Neumann kernel. He
reduces the interior and the exterior Neumann problem into equivalent Dirichlet
problems using the Cauchy-Riemann equation. Then, the Neumann problem is
solved using the integral equation for the Dirichlet problem. The derived integral
equations are uniquely solvable which can provide boundary values of the solution
of the Neumann problem without any extra calculations.
Later, Ummu Tasnim [3] presents a new method using boundary integral
equation for solving the interior Neumann problem in a simply connected region Ω.
In the first step the Neumann problem is reduced to an interior Riemann-Hilbert
problem for the derivative f ' of an auxiliary function f which is analytic in Ω.
Then the Riemann-Hilbert problem is transformed to a Fredholm integral equation
on the boundary of Ω which always has a unique solution.
Actually, the approach used by Ummu Tasnim in [3] has been used before in
[5]. The interior Neumann problem is reduced to an interior Riemann-Hilbert
problem. However, the Riemann-Hilbert problem is solved in [5] using a nonuniquely solvable integral equation. So, the results in [3] have significant advantages
over the result given in [5].
Later, we will review the historical background and development of the
boundary integral equation method on solving the Riemann-Hilbert problem using
the proposed method since later the Neumann problem will be reduced to the
Riemann-Hilbert problem.
2.4
Boundary Integral Equation on Solving Riemann-Hilbert Problem
Riemann-Hilbert problems are one of the most important classes of boundary
value problems for analytic functions. The Dirichlet problem for Laplace’s equation,
which is one of the classical elliptic boundary value problems, is a special case of the
Riemann-Hilbert problem. When the Dirichlet problem is solved by the double layer
potential representation, a boundary integral equation with a continuous kernel
results. The kernel is known as the Neumann kernel.
There are not many methods for solving the Riemann-Hilbert problem and
most of the available methods are limited to only the Riemann-Hilbert problem in
circular regions. With the aid of conformal mapping, one can solve the RiemannHilbert problems in arbitrary simply connected regions.
The Riemann-Hilbert problem can also be solved using boundary integral
equation. A boundary integral method for the Riemann-Hilbert problem in simply
connected regions has been proposed by Sherman where he derived a Fredholm
integral equation of the second kind with continuous kernel for the interior RiemannHilbert problem. Sherman’s method does not require the availability of conformal
mappings. However, it is limited only to the Riemann-Hilbert problem with zero
index (see [8]).
Nasser in [8] has derived boundary integral equations for the RiemannHilbert problem in arbitrary simply connected regions as well as for a certain class of
Riemann-Hilbert problem in multiply connected regions. The derived integral
equations are Fredholm integral equations of the second kind with continuous
kernels provided that the boundaries are sufficiently smooth.
Wegmann et al. [6] have also proposed a method for solving RiemannHilbert problems on simply connected regions with smooth boundaries. The method
is basically same as [8] but with further improvement in theoretical and numerical
results.
Recently, Nasser [9] has proposed the numerical treatment of the integral
equations with generalized Neumann kernel and the applications of the integral
equations to solve the Riemann-Hilbert problem.
2.5
Conclusion
We have reviewed the literature regarding the boundary integral equation
method on solving the Neumann problem and the Riemann-Hilbert problem in this
chapter and also explained some historical background and the development of the
boundary integral equation method.
CHAPTER 3
NEUMANN PROBLEM AND RIEMANN-HILBERT PROBLEM
3.1
Introduction
This chapter gives some auxiliary materials related to Neumann problem and
Riemann-Hilbert problem. It is divided into four sections. Section 3.2 and 3.3 gives
some important facts on Neumann problem and Riemann-Hilbert problem
respectively. Finally, Section 3.4 gives the conclusion of this chapter.
3.2
The Neumann Problem
Let  be a bounded simply connected Jordan region bounded by  and let
the exterior of  be denoted by   with 0   and ∞ belongs to   . The boundary
 :   is assumed to have a positively oriented parametrization
(t ) where (t ) is
a 2  periodic twice continuously differentiable function with  (t ) 
parameter t need not be the arc length parameter.
d
 0. The
dt
For a fixed
with 0 <
< 0, the Hölder space
consists of all 2 -
periodic real functions which are uniformly Hölder continuous with exponent
. It
becomes a Banach space when provided with the usual Hölder norm. A Hölder
continuous function ℎ on Γ can be interpreted via ℎ( ) ≔ ℎ( ( )) as a Hölder
continuous function ℎ of the parameter
and vise versa. Let  be a real-valued
function defined on   .
The Neumann problem is sometimes referred to as the Dirichlet problem of
second kind. The exterior Neumann problem is defined in the following section.
3.2.1 The Exterior Neumann Problem
Definition 3.1 ([4]).
Let n be the exterior normal to  and let   H  be a given function such that
2
    |  ( ) | d  0.
(3.1)
0
Find the function u harmonic in   , Hölder continuous on  and satisfies on the
boundary condition (see Figure 3.1)
u
  (t ),
n  (t )
 (t )  .
(3.2)
The function u is also required to satisfy the additional condition
u ( z )  O (| z |) 1 as z   .
(3.3)
Lemma 3.1 ([10, p. 313]). The exterior Neumann problem (3.2) is uniquely
solvable.
The exterior Neumann problem has a unique solution u in   . The solution
u can be considered as a real part of a single-valued analytic function f . But this is
not necessary. The function u can also be written as the real part of
m
f ( z )  F ( z )   a j log( z  z j )
(3.4)
j 1
where F is a single-valued analytic function in   and a1 , a 2 ,..., a m are real
constants which are chosen such that F is a single-valued analytic function in   ,
i.e.,
 F ( )d
 0,
j  1,2,..., m.
(3.5)
j
For   , the constants a1 , a 2 ,..., a m satisfy
m
a
j 1
m
j
1
1
f ( )d  
f ( )d  0.


2i  j
j 1 2i j
 
In view of (3.3), we can assume without loss the generality that f ()  0.
 :(t )
n
0

u
  (t )
n


Figure 3.1 : The Exterior Neumann Problem
(3.6)
3.2.2 Definition of Normal Derivative
Suppose that  is a simply connected region bounded by a simple path  .

We recall from vector calculus that if r (t ) is a parametrization of  , then the unit

tangent T is defined as

T

r (t )
.

r (t )
(3.7)
Since a complex number can also be regarded as a vector, then the complex unit

tangent vector T is defined by
  (t )
T
,
 (t )
(3.8)

where (t ) is a complex parametrization of  . We call T (t ) the tangent directional
derivative to  at (t ) . In vector calculus [11], there are two unit vectors that are


perpendicular to the unit tangent vector T (t ) . If T (t )  0 , then we define the unit

normal vector n to the curve at t to be the vector that is perpendicular to T (t ) and



has the same direction as T (t ) . By rotating the tangent T clockwise by , we
2
obtain
n 
e
i

2
 (t )
 i (t )

.
 (t )
 (t )
(3.9)
The directional derivative of u( x, y) in the direction of the outward unit
normal vector to the path at point (t ) is denoted by
u
. In calculus of several
n
variables [12], the directional derivative
u
may also be expressed in the following
n
way using gradient:
u
 u  n,
n
(3.10)


where u ( x, y )  u x i  u y j . If (t )  x(t )  iy(t ) is the parametrization of  , then
(t )  x(t )  iy(t ). Therefore, (3.9) becomes
n 


 i x(t )  iy(t ) 
y (t )   x(t ) 

i 
j  n xi  n y j.
 (t )
 (t )
 (t )
(3.11)
Substituting (3.11) to (3.10), we obtain
u x y (t )  u y x(t )
u

.
n
 (t )
3.3
(3.12)
The Riemann-Hilbert Problem
Suppose that  and   are the interior and exterior of  , respectively, such
that the origin of the coordinate system belongs to  and  belongs to   . If a
function g(z) is defined in a region containing  , then the limiting values of the
function g(z) when the point z tends to the point    from the interior or the
exterior of  will be denoted by g  ( z ) and g  (z ) , respectively. Let A(t ) be a
complex continuously differentiable 2   periodic function with A  0 . With
C ,   H  , let the function (z) be defined by
 z  
1
2i


C ( )  i ( ) d
,
A( )
 z
z  .
(3.13)
Then (z) is analytic in  as well as in   and the boundary values   from
inside and   from outside belong to H  and can be calculated by Plemelj’s
formula [13]
     
1 C    i   1 C    i   d

,
2
A 
2i 
A 
z
  .
(3.14)
The integral in (3.14) is a Cauchy principal value integral. The boundary values
satisfy the jump relation
A   A   C  i .
(3.15)
3.3.1 The Exterior Riemann-Hilbert Problem
Definition 3.2. ([13])
For C  H  , the exterior Riemann-Hilbert problem is defined as follows:
Given functions A and C , it is required to find a function g analytic in  and
continuous on the closure Ω with g()  0 such that the boundary values g  satisfy
Re[ A(t ) g  ( (t ))]  C (t ),
(t )   .
(3.16)
The boundary condition (3.16) can be written in the equivalent form
g  (t )  
A(t ) 
2C (t )
g ( (t )) 
,
A(t )
A(t )
(t )  .
(3.17)
The homogenous boundary condition of the exterior Riemann-Hilbert problem is
given by
Re[ A(t ) g  ( (t ))]  0,
(t )  .
(3.18)
3.3.2 The Solvability of Riemann-Hilbert Problems
The solvability of the Riemann-Hilbert problems is determined by the index,
 of the function A . The index of the function A is defined as the winding number
of A with respect to zero.
1
2
≔ ind( ) ≔
arg( A) 0 ,
2
(3.19)
i.e. the change of the argument of A over one period divided by 2 . The following
lemma from [14] shows the relation between the solvability of the Riemann-Hillbert
problems and the index
which is also regarded as the index of the Riemann-Hilbert
problems.
Lemma 3.2.
In the case   0 , the homogenous exterior Riemann-Hilbert problem (3.18) in the
unbounded simply connected region   has 2  1 linearly independent solutions
and the non-homogenous problem (3.16) is solvable and its solution depends linearly
on 2  1 arbitrary real constant. In the case   0, the homogenous problem has
only trivial solution and the non-homogenous problem is solvable only if  2  1
conditions are satisfied. If the latter conditions are satisfied, the non-homogenous
problem has a unique solution.
The index can also be expressed by the integral
≔ ind ( ) =
1
2
 d arg A(t )


1
2i
 d arg A(t ),
(3.20)

where the integral is understood in the sense of the Stieltjes [14]. If the function
A (t ) is continuously differentiable on  , then
≔ ind ( ) =
1
2i
 d ln A(t ) 

1
2i


A (t )
dt.
A(t )
(3.21)
Since  is closed and A (t ) is non-vanishing continuous function on  , the index
≔ ind ( ) is an integer.
We define the range spaces of the Riemann-Hilbert problem, i.e., the spaces
of function C for which the Riemann-Hilbert problem are solvable,

: C  H
R  : C  H  : C (t )  Re[ A(t ) g  ( (t ))], g analytic in ,.
R

(3.22)

: C (t )  Re[ A(t ) g  ( (t ))], g analytic in   , g    0 .
(3.23)
We define also the solution spaces of the homogeneous Riemann-Hilbert
problem:

: C  H
S  : C  H  : C (t )  A(t ) g   t , g analytic in ,.
S

(3.24)

: C (t )  A(t ) g   t , g analytic in   , g    0 .
(3.25)
Then the number of arbitrary real constants in the solutions of the
homogenous Riemann-Hilbert problems, i.e. dim(
±
) and the number of conditions
on the function C so that the non-homogenous Riemann-Hilbert problems are
solvable, i.e. codim(
theorem from [6].
±
) are given in term of the index  as in the following
Theorem 3.1.
The codimensions of the spaces R  and the dimensions of the spaces S  are given
by the formulas
codim(
) = max0,2  1,
(3.26)
codim(
) = max0, 2  1,
(3.27)
dim( S  )  max( 0, 2  1).
(3.28)
dim( S  )  max( 0,2  1).
(3.29)
3.3.3 Integral Operators
Let A (t ) be a continuously differentiable 2 -periodic function with A  0.
We define two real functions N and M by
N ( , t ) :

1  A( )
 (t )
,
Im
  A(t )  (t )   ( ) 
(3.30)
M ( , t ) :

1  A( )
 (t )
.
Re
  A(t )  (t )   ( ) 
(3.31)
The kernel N ( , t ) is called the generalized Neumann kernel form with A and 
(see [6]). We also define the kernels U and V (when A  1 for the kernels N and M )
as
V ( , t ) :

1 
 (t )
,
Im
   (t )   ( ) 
(3.32)
U ( , t ) :

1 
 (t )
.
Re
   (t )   ( ) 
(3.33)
The kernel V is the Neumann kernel which arises frequently in the integral
equations for potential theory and conformal mapping (see [5, p. 286]).
We then define the kernels U * and V * (the adjoint kernels of U and V
respectively) as
V * ( , t ) :

1   ( )
,
Im
   ( )   (t ) 
(3.34)
U * ( , t ) :

1   ( )
.
Re
   ( )   (t ) 
(3.35)
Lemma 3.3 ([6]).
a) The kernel N ( , t ) is continuous with
 1 (t )
1
A (t ) 
.
Im


A(t ) 
 2  (t )
N (t , t ) :
(3.36)
b) The kernel M ( , t ) has the representation
M ( , t ) :
1
 t
cot
 M 1 ( , t )
2
2
(3.37)
with a continuous kernel M1 which takes on the diagonal the values
M 1 (t , t ) :
 1 (t )
1
A (t ) 
.
Re


A(t ) 
 2  (t )
(3.38)
We define the Fredholm integral operators with kernel N , V , and V * as
2
N :
 N ( , t ) (t )dt ,
0
(3.39)
2
V :
 V ( , t ) (t )dt ,
(3.40)
0
2
*
V  :
V
*
( , t ) (t )dt ,
(3.41)
0
and the singular operators with kernels M , U , and U * as
2
M  :
 M ( , t ) (t )dt ,
(3.42)
0
2
U :
 U ( , t ) (t )dt ,
(3.43)
0
2
U *  :
U
*
( , t ) (t ) dt.
(3.44)
0
The eigenproblem of the generalized Neumann kernel has been studied in
[6]. The next theorem gives , the eigenvalue of N.
Theorem 3.2 ([6]).
(a)   1 is an eigenvalue of N with multiplicity max (0, 2  1).
(b)   1 is an eigenvalue of N with multiplicity max (0, 2  1).
It follows from Theorem 3.2 that N can have only one of the eigenvalues   1 or
  1 but not both at the same time.
3.3.4 Integral Equation for the Exterior Riemann-Hilbert Problem
There is a close connection between exterior Riemann-Hilbert problem and
integral equations with the generalized Neumann kernel. For a given function
C ,   H  , let the function   z  be defined by (3.13). Based on the application of
Plemelj’s formula, a boundary integral equation with generalized Neumann kernel
has been derived for exterior Riemann-Hilbert problem by Wegmann et al. [6] as in
the following theorem.
Theorem 3.3.
If g z    z  is a solution of the exterior problem (3.16) with boundary values
A(t ) g   t   C (t )  i (t ),
(3.45)
then the imaginary part  in (3.45) satisfies the integral equation
  N  M C .
(3.46)
We now discuss the relevance of the solution  of the integral equation
(3.46) for the exterior Riemann-Hilbert problem.
Theorem 3.4 ([6]).
Let C  H  be a given function,  be a solution of (3.46) and  be defined by
(3.13).
(a) If   0 then g   in   is a solution of the exterior Riemann-Hilbert
problem (3.16). The boundary values of are given by (3.45).
(b) If   0 then g   in   satisfies the Riemann-Hilbert condition
Re[A(t ) g  ( (t ))]  C (t )  C0 (t ),
where C0  S  .
(3.47)
Now we discuss the solvability of the integral equations in full generality with
general right-hand side, .
Theorem 3.5 ([6]).
(a) If   0 , then the integral equation
  N  
(3.48)
has a solution if and only if   R  .
(b) If   0 , then the integral equation (3.48) has a unique solution for any
 H.
(c) Equation (3.46) is solvable for each C  H  .
3.4
Conclusion
We have discussed extensively some important facts and concepts of
Neumann problem as well as the Riemann-Hilbert problem that are needed to
conduct this study in this chapter.
CHAPTER 4
MODIFICATION OF THE EXTERIOR NEUMANN PROBLEM
4.1
Introduction
This chapter provides the theoretical formulation of the overall research. It is
divided into eight sections. Section 4.2 explains on reduction of the exterior
Neumann problem to the exterior Riemann-Hilbert problem. The solvability of the
Riemann-Hilbert problem is reviewed in Section 4.3. The derived integral equation
and the study of the solvability of the integral equation are then explained in Section
4.4. Some modification on the derived integral equation and the singular integral
operator are explained respectively in Section 4.5 and 4.6. Section 4.7 gives the
formula to obtain ( ). Finally, Section 4.8 gives the conclusion of this chapter.
4.2
Reduction of Exterior Neumann Problem to the Exterior Riemann
Hilbert Problem
Suppose that u is the solution of the exterior Neumann problem and v is a
harmonic conjugate of u in   . The function f ( z )  u ( x , y )  iv ( x , y ) is analytic


in     where  :  (t )  x (t )  iy (t )  x (t )i  y (t ) j if and only if the partial
derivatives of u and v are continuous and satisfy the Cauchy-Riemann equations
ux  v y
u y  v x .
and
(4.1)
The derivative f ( z ) is given either of
f ( z )  u x ( x , y )  iv x ( x, y )
or
f ( z)  v y ( x, y)  iu y ( x, y) .
(4.2)
The directional derivative of f in the direction of the outer unit normal
vector to the path  is given by
f ( )
v 
 u

i  .
n  ( t )  n
n   ( t )
(4.3)
Applying (3.10) and (3.11) to (4.3), we obtain
f ( )
 u  n  iv  n   ( t )
n  ( t )








 ux i  u y j  n xi  n y j  i vx i  v y j  n x i  n y j


 
 u x  iv x n x  iv y  iu y n y   (t )
Applying (4.2) to (4.4), we get


 (t )
(4.4)
f ( )
 f ( )n x  in y  .
 (t )
n  (t )
(4.5)
Therefore, we can write (3.2) as the real part of (4.5),
 f ( ) 
Re 
   (t )
 n  ( t ) 
Re[ f ( )(n x  in y )  ( t ) ]   (t )
(4.6)
(4.7)
Substituting (3.11) and (3.9) into (4.7), we obtain
  i (t )

Re nf ( )  ( t )  Re 
f ( )    (t ) .
  (t )


(4.8)
Re i (t ) f ( )   (t )  (t ) .
(4.9)

Thus, we obtain
Letting g  ( (t ))  if ( (t )) and  (t )   (t )  (t ) , we obtain
Re[ (t ) g  ( (t ))]   (t )
(4.10)
which is the exterior Riemann-Hilbert problem as defined in Section 3.2.1.
Comparison of (4.10) with (3.16) yields A (t )   (t ) and C (t )   (t ).
4.3
The Solvability of the Riemann-Hilbert Problem
From Section 3.2.2, it is well known that the solvability of the RiemannHilbert problem (4.10) depends upon the index
= ind( ). Applying (3.21) with
A ( t )   (t ) , we have
= ind ( ) =
1
2i
 d ln A(t ) 

1
2i


A (t )
1
dt 
A(t )
2i
(t )
 (t )dt.

Letting    (t ) and d   (t ) dt , and by the Cauchy-Integral formula, we obtain
  1.
From Theorem 3.1, we obtain
codim(
) = max(0, −1) = 0
(4.11)
dim( S  )  max( 0,1)  1.
Equation (4.11) means that there is no condition for
(4.12)
in (4.10), so that the non-
homogenous exterior Riemann-Hilbert problem is solvable. Equation (4.12) implies
that, there is a real constant in the solution of the homogenous exterior RiemannHilbert problem so that the solution of the exterior Riemann-Hilbert problem is not
unique.
4.4
Integral Equation for the Exterior Riemann-Hilbert Problem
By Theorem 3.3, if g   t  is the solution of the exterior Riemann-Hilbert
problem (4.10) with boundary values
 (t ) g   t    (t )  i (t )
(4.13)
then the imaginary part  in (4.13) satisfies the integral equation
(4.14)
  N  M 
i.e.
2
     N ( , t ) (t ) dt 
0
2
 M ( , t ) (t )dt
(4.15)
0
The next theorem represents the solvability of integral equation (4.14) and (4.15).
Theorem 4.1 ([13]).
If   0 , then the integral equation (4.14) and (4.15) is uniquely solvable. If   0 ,
then the integral equation (4.14) and (4.15) is non-uniquely solvable.
Therefore from the above theorem, we concluded that the integral equation
(4.14) and (4.15) is non-uniquely solvable. To overcome the non-uniqueness, we
need to impose additional constraint which can be embedded into integral equation
that will yield a uniquely solvable integral equation.
4.5
Modified Integral Equation for the Exterior Riemann-Hilbert Problem
Recall from Section 3.2.3 that with A (t )   (t ) , the generalized Neumann
kernel N becomes
1
    
,
 Im



t









N ( , t )  
 1
 t  
,
Im

 2
  t  
 V * ( , t ).
t  
(4.16)
t 
(4.17)
While the kernel M becomes
M ( , t )  
1     

Re
         
 U * ( , t ).
(4.18)
(4.19)
So, our non-uniquely solvable integral equation (4.14) and (4.15) respectively
become
  V *    U * ,
2
   
V
(4.20)
2
*
( , t ) (t ) dt    U * ( , t ) (t ) dt.
0
(4.21)
0
Recall from Section 4.1, that the function f is analytic on    . Then at
each point z in that domain when z 0  0, f ( z ) can be represented by the following
Laurent series expansion such that f   0 :
( )=
+
+
+⋯
(4.22)
Differentiate once respect to z and multiply with  iz ,
−
( )=−
Since g  ( (t ))  if ( (t )) ,
−
−
− ⋯ = ( ).
(4.23)
g  z  
F ( z)
.
z
(4.24)
By means of Cauchy-Formula and the fact from [14, p.24],
F ( )
d  F ()  0

1
2i

1
2i
 g  d


 0
(4.25)
(4.26)

 g  d

 0.
(4.27)

From (4.13), g   t  
2

0
2
 (t )  i (t )
, so (4.27) becomes
 (t )
 (t )  i (t )
 (t ) dt  0
 (t )
(4.28)
2
  (t )dt  i   (t )dt  0.
0
(4.29)
0
Note that, the exterior Neumann problem need to satisfy (3.1). From (4.9),
 (t )   (t )  (t ) . Therefore (3.1) becomes
2
  (t )dt  0 .
(4.30)
0
We consider (4.30) as the additional condition for the exterior Riemann-Hilbert
problem which the right-hand side of the Riemann-Hilbert problem (4.10) need to
satisfy. Thus from (4.30), (4.29) becomes
2
  (t )dt  0 .
0
(4.31)
Let the kernel   , t  be defined as
  , t  
1
2
(4.32)
and let the operator J be defined by
2
J       , t  t dt ,
 .
(4.33)
0
We can write (4.31) as (4.33) where,
J ( ) 
1
2
2
  t dt  0.
(4.34)
0
Adding (4.34) to (4.20) yields the new integral equation
  V *   J   U *
(4.35)
i.e.
2
     V * ( , t ) (t ) dt 
0
1
2
2
2
*
  (t )dt    U ( , t ) (t )dt
0
(4.36)
0
According to [15, p.137-141], this new integral formula is uniquely solvable but with
different right-hand side.
The proof of the solvability of the integral equation can be followed from the paper
written by Atkinson [16, p.119].
4.5
Modifying the Singular Integral Operator
Consider the right-hand side of (4.36):
2
  U * ( , t ) (t ) dt  
0
1

2

  

 Re      t   (t )dt .
(4.37)
0
From Lemma 3.3, we can write the kernel of the right-hand side of (4.36) as
 U * ( , t )  M ( , t ) 
1
 t
cot
 M 1 ( , t ),
2
2
(4.38)
where (4.38) is singular since it is unbounded when   t. So in this section, we will
make some arrangement and modification to remove the difficulty.
Using the fact that
2
 U ( , t )dt 
0
1

2

 t 

(4.39)
 Re  t      dt  0,
0
(4.37) becomes,
2
1
  U ( , t ) (t ) dt  

0
*

where for t   ,
1

2

0
2

0
1
    
 t dt    
Re
      t  


  t      




 t      


dt 
Re    t 
  t       







2

 t 

 Re  t       dt
0
2
 B( , t )dt,
0
(4.40)

  t     




 t     
1    t    t    
1



Re
. (4.41)
B ( , t )  Re    t 
  t       
 
 t     









Letting  
  

and  
, so that

 2
B ( , t ) 
lim
→
  (t )   ( ) 
1 

Re    t 

  t      

( )
( )
( )
( )
=
̇( )
̇( )
and
.
Thus, the values of B ( , t ) on the diagonal are

  t  t   t  t  



2
1 
 t 




B(t , t )  Re  t  t 


 
 t 







1


    
 t    t  Re    .
  t   

(4.42)
The function B ( , t ) was actually defined in [6, p.409] but with A(t )   t .
Then, we define the integral operator with kernel B as
2
B :
 B( , t ) (t )dt .
(4.43)
0
So, the uniquely solvable integral equation (4.35) and (4.36) respectively becomes
  V *   J  B ,
(4.44)
2
   
*
 V ( , t ) (t )dt 
0
4.6
1
2
2
2
 (t )dt 

0
 B( , t )dt.
(4.45)
0
Computing f ( z ) from f ( z )
From (4.10) and (4.13), the boundary values of the function f  are given by
 i (t ) f  t    (t )  i (t ) .
(4.46)
By obtaining  , in view of (4.46), the Cauchy integral formula implies that the
function f ( z ) can be calculated for z    by
f z  
1
2i
f  
1
   z d  2i
2

0
 (t )  i (t )  (t )dt
.
 i (t )  (t )  z
(4.47)
i.e., the function f z  is represented by a Cauchy type integral. Since the order of
the derivative is 1 and since ind( ) = 1, it follows from [14, p.303] that the function

f ( z ) can be expressed as an anti-derivative function of f z  for z   . So, we
have
1
f (z) 
2i
2

0
 (t )  i (t )
 (t ) 

log 1 
 (t ) dt .
 i (t )
z 

(4.48)
This formula can also be obtained by interpolating both sides of (4.47) with respect
to z and apply the fundamental theorem. Hence, the unique solution of the Neumann
problem is given for bounded  by
u ( z )  Re( f ( z ))
(4.49)
or
u( z ) 
4.8
1
2


 Re t   i t  log1 

 t  
 dt.
z 
(4.50)
Conclusion
Theoretical aspects for the research are discussed in this chapter. The
reductions of the exterior Neumann problem into the exterior Riemann-Hilbert
problem are derived as a non-uniquely solvable as well as the derived boundary
integral equation. Additional constraint to the solution of the problem is required so
that the problem is uniquely solvable. Then the related integral equation is modified
to obtain uniquely solvable integral equation.
CHAPTER 5
NUMERICAL IMPLEMENTATIONS OF THE BOUNDARY INTEGRAL
EQUATION
5.1
Introduction
This chapter gives the numerical implementation of the integral equation and
numerical results in some test regions (see Section 5.3). It is divided into six
sections. Section 5.2 gives the numerical implementation of the integral equation.
All the results obtained are shown in Section 5.4 and the discussion of the results is
given in Section 5.5. Finally, Section 5.6 gives the conclusion of this chapter.
5.2
Numerical Implementation
Since the function A(t )   t  and  t  are 2  periodic, the integrals in the
integral equation (4.45) can be best discretized on an equidistant grid by the
trapezoidal rule, i.e., the integral operators are discretized by the Nyström method
with the trapezoidal rule as the quadrature rule [9].
Let n be a given integer and define the n equidistant collocation points t k
by
t k  k  1
2
,
n
(5.1)
k  1,2,..., n.
Then, using the Nyström method with the trapezoidal rule to discretized the integral
equation (4.45), we obtain the linear systems
 n t j  
2
n
n
V * t j , t k  n t k  
k 1
2
n
n
 J t j , t k  n t k  
k 1
2
n
n
 Bt
j
, t k , (5.2)
k 1
with j  1,2,..., n and  n is an approximation to  .
×
Let be the
identity matrix. Also let
and
be the
×
matrix and
× 1 vector respectively whose elements are all unity. Defining the matrix
, =
,
=
=
=
=
2 *
V t k , t j 
n
],
=[
] by
1


t k 
,
 Im






t



t

j 
 k
2 


n 
1  t j  

,
 2   t j  
2
2  1 
J t k , t j  
 
n
n  2 
2
Bt k , t j 
n
=[
and vector
1


2 


n 
1

 
=
tk  t j
(5.3)
tk  t j
(5.4)
,
  t k  t j    t j  t k  
, t k  t j
Re







t


t
j
k


(5.5)

 t   
  t    t  Re j  ,
j
j
  t   

 j 

tk  t j
=  n t k  , and
(5.6)
=
(5.7)
.
Hence, the application of Nyström method to the uniquely solvable integral
equations (4.45) leads to the following
( −
by
linear system
+ ) = .
(5.8)
By solving the linear systems (5.8), we obtain  n t k  for k  1, 2,..., n. Then the
approximate solution  n t  can be calculated for all t  0,2  using the Nyström
interpolating formula, i.e., the approximation  n t  of the integral equation (4.43) is
given by
 n t  
2
n
n
 B(t, t
j 1
j
)
2
n
n
V t, t 
*
j
j 1
n
(t j ) 
2
n
n
 1
  2 
j 1
n

(t j )  .

(5.9)
By obtaining  , (4.47) and (4.48) implies that the function f ( z ) and f ( z ) can be
calculated for z    .
5.3
Examples
For our examples, we use three boundary curves: an ellipse, the ovals of
Cassini, and an “amoeba”. These examples were also used in [4]. In this paper, we
used these boundaries to show that our method can also work efficiently as the
method proposed in [4] for such regions.
For the ellipse (see Figure 5.1), the boundary has parametrization
1 :  t   cos t  i5 sin t , 0  t  2 .
(5.10)
For the ovals of Cassini (see Figure 5.2), the boundary parametrization is
2 :  t   R (t )e it , 0  t  2 ,
(5.11)
where R (t )  2 .5  2 cos 2t . For the “amoeba” (see Figure 5.3), the boundary
parametrization is
3 :  t   R(t )e it , 0  t  2 ,
with R (t )  e cos t cos 2 2t  e sin t sin 2 2t.
Figure 5.1: The Curve 1 and the Exterior Test Points.
(5.12)
Figure 5.2: The Curve 2 and the Exterior Test Points.
Figure 5.3: The Curve 3 and the Exterior Test Points.
5.4
Numerical Computation and Results
In this dissertation, the latest version of MATLAB available, i.e., MATLAB
7.6.0 (R2008a) is used. The formulated boundary integral equation is discretized to
get equation (5.8). The linear system is then solved using the MATLAB “\” operator
that makes use of the Gauss elimination method.
The maximum error norm
f ( z )  f n ( z )

between the exact values of
f ( z ) and the approximate value of f n (z ) on the boundary is presented in Table 5.1
and the absolute error f ( z )  f n( z ) at four test points z outside  for the exterior
Neumann problem is listed in Tables 5.2 – 5.4. The absolute error f ( z )  f n ( z ) at
four test points z outside  for the exterior Neumann problem is listed in Tables
5.5 – 5.7. The results obtained are also compared to the results obtained in [4]. In
Tables 5.5 – 5.7, the symbol
refers to our results, while the symbol
refers to
the results given in Nasser [4]. The numerical results are presented for various values
of n where n is the number of node points given in (5.1).
Table 5.1: The Error f ( z )  f n ( z )

for the Exterior Neumann Problem on the
Boundaries 1 , 2 and 3 .
n
1
2
3
16
6.64(-02)
7.29(-01)
4.59(-01)
32
3.71(-03)
1.28(-01)
2.27(-01)
64
6.34(-06)
3.35(-03)
3.55(-02)
128
1.50(-11)
2.00(-06)
3.71(-04)
256
2.66(-15)
6.46(-13)
1.54(-08)
512
2.94(-15)
5.87(-14)
6.14(-14)
Table 5.2: The Error f ( z )  f n ( z ) for the Exterior Neumann Problem at the Test
Points Exterior to the Boundary 1.
n
z  4
z  2
z2
z4
16
1.68(-02)
5.58(-02)
5.58(-02)
1.68(-02)
32
7.99(-04)
2.78(-03)
2.78(-03)
7.99(-04)
64
1.24(-06)
4.61(-06)
4.61(-06)
1.24(-06)
128
2.88(-12)
1.21(-11)
1.21(-11)
2.88(-12)
256
9.99(-17)
1.90(-16)
2.21(-16)
5.83(-17)
Table 5.3: The Error f ( z )  f n ( z ) for the Exterior Neumann Problem at the Test
Points Exterior to the Boundary 2 .
n
z  2i
z  i
zi
z  2i
16
1.64(-03)
1.01(-01)
1.01(-01)
1.64(-03)
32
2.56(-04)
3.33(-02)
3.33(-02)
2.56(-04)
64
7.49(-06)
1.72(-04)
1.72(-04)
7.49(-06)
128
2.41(-09)
6.73(-07)
6.73(-07)
2.41(-09)
256
3.40(-16)
2.26(-13)
2.25(-13)
2.24(-16)
Table 5.4: The Error f ( z )  f n ( z ) for the Exterior Neumann Problem at the Test
Points Exterior to the Boundary 3 .
n
z  1
z  1  i
z  1 i
z  2i
32
1.89(-02)
8.39(-03)
3.21(-02)
1.67(-02)
64
5.41(-04)
1.84(-04)
1.30(-03)
1.18(-03)
128
6.94(-07)
3.65(-07)
2.99(-06)
7.72(-06)
256
1.39(-11)
1.67(-11)
3.80(-11)
2.80(-10)
512
2.23(-15)
2.65(-16)
2.00(-16)
5.00(-17)
Table 5.5: Comparison of the Error f ( z )  f n ( z ) with Nasser [4] for the Exterior
Neumann Problem at the Test Points Exterior to the Boundary 1 .
n
z  4
z  2
z2
z4
16
8.45(-02)
1.42(-01)
1.42(-01)
8.45(-02)
1.02(-02)
7.51(-02)
7.51(-02)
1.02(-02)
3.94(-03)
6.70(-03)
6.70(-03)
3.94(-03)
5.10(-04)
6.23(-03)
6.23(-03)
5.10(-04)
6.10(-06)
1.04(-05)
1.04(-05)
6.10(-06)
7.85(-07)
1.94(-05)
1.94(-05)
7.85(-07)
1.41(-11)
2.40(-11)
2.40(-11)
1.41(-11)
1.82(-12)
1.00(-10)
1.00(-10)
1.82(-12)
2.52(-16)
5.63(-16)
6.18(-16)
2.80(-16)
2.78(-16)
7.77(-16)
1.11(-16)
4.44(-16)
32
64
128
256
Our Numerical Result
Based on Nasser’s Result in [4]
Table 5.6: Comparison of the Error f ( z )  f n ( z ) with Nasser [4] for the Exterior
Neumann Problem at the Test Points Exterior to the Boundary 2 .
N
z  2i
z  i
zi
z  2i
16
4.25(-03)
7.88(-02)
7.88(-02)
4.25(-03)
2.51(-04)
2.56(-02)
2.56(-02)
2.51(-04)
8.30(-04)
3.20(-03)
3.20(-03)
8.30(-04)
3.57(-06)
6.25(-04)
6.25(-04)
3.57(-06)
1.28(-05)
1.85(-05)
1.85(-05)
1.28(-05)
6.88(-12)
2.86(-07)
2.86(-07)
6.86(-12)
4.09(-09)
4.11(-10)
4.11(-10)
4.10(-09)
1.62(-14)
4.36(-13)
4.37(-13)
1.81(-14)
8.88(-16)
3.55(-15)
3.78(-15)
10.00(-16)
2.10(-14)
2.37(-14)
2.37(-14)
2.18(-14)
32
64
128
256
Our Numerical Result
Based on Nasser’s Result in [4]
Table 5.7: Comparison of the Error f ( z )  f n ( z ) with Nasser [4] for the Exterior
Neumann Problem at the Test Points Exterior to the Boundary 3 .
n
z  1
z  1  i
z  1 i
z  2i
32
6.68(-03)
7.64(-03)
7.75(-03)
5.14(-03)
1.79(-02)
5.62(-03)
6.29(-02)
2.60(-02)
1.67(-04)
1.33(-04)
1.64(-04)
1.35(-04)
5.88(-04)
2.13(-04)
1.09(-03)
9.83(-04)
4.32(-07)
3.28(-07)
5.80(-07)
2.27(-07)
1.55(-07)
7.76(-07)
4.21(-06)
2.31(-06)
1.52(-11)
1.35(-11)
1.87(-11)
4.23(-12)
8.72(-11)
6.52(-11)
1.29(-10)
1.91(-10)
6.92(-16)
6.75(-16)
2.29(-16)
2.36(-16)
2.22(-16)
1.22(-15)
2.22(-16)
3.89(-16)
64
128
256
512
Our Numerical Result
Based on Nasser’s Result in [4]
5.6
Discussion of the Results
In order to give an impression how the method works for different regions,
we have presented the results of several test calculations. The region used in this
example started with the smooth region (nearly circle) to a non-convex and
complicated region.
Tables 5.1 to 5.7 show that with increasing n, the convergence is extremely
fast where convergence is measured as n. We see that the trapezoidal technique
used is reasonably satisfactory for the calculated integral equation. This situation is
expected. Although the quadrature rule used is low-order rules, for an ill-behaved
function; the case of a periodic function integrated over a complete period will
converge at about the same rate with the high-order rules (see [17, p.57]).
The maximum error norm f ( z )  f n( z )  for different value of n is shown
in Table 5.1. For the elliptical boundary, accurate results can be obtained using small
values of n but when the region changes to a more complicated shape, the number of
node points n required to get an accurate results increases. This is normal, because
when the region changes to a more complex shape, the tangent vector near the
vertices also changes rapidly. This requires the use of a large number of node points
if the Nyström method with the equidistant trapezoidal rule is used.
We also compared the error between f ( z )  f n ( z ) and f ( z )  f n ( z ) at
every test points. The degree of accuracy between the compute f n is less than the
computed f n when the test point is near the boundary.
In Tables 5.5, 5.6 and 5.7, the comparison between our proposed methods
with the method proposed in [4] yields comparable accuracy which shows that our
method also work efficiently for such regions.
The convergence of the method is excellent except for points away the
boundary and on the boundary. The degree of accuracy will decline when the test
point is near or on the boundary. This situation is expected since the function f ( z )
in (4.47) and the function f ( z ) in (4.48) has singularity in their equations [18].
As can be seen, the agreement between the numerical and the exact solutions
is excellent. The numerical examples show clearly that the developed method gives
results of high accuracy for solutions of the Neumann problem in the three test
regions with smooth boundaries.
5.7
Conclusion
We presented the numerical implementations of the derived boundary
integral equation in this chapter. The integral in the integral equation were solved by
the Nyström method with the trapezoidal rule as the quadrature rule. The exterior
Neumann problem were then solved numerically in three test regions using the
proposed method. The numerical examples illustrated that the proposed method
yields approximation of high accuracy.
CHAPTER 6
CONCLUSIONS AND SUGGESTIONS
6.1
Conclusion
In this dissertation, we discussed mainly on the numerical solution of the
exterior Neumann problem using boundary integral equation method. Unlike the
classical methods for solving the Neumann problem which require the availability of
a conformal mapping from the problem domain to a ‘simpler’ one, the present
method can be used to solve the Neumann problem in its original domain.
Furthermore, the presented method has the advantage that it needs less numerical
operations and is easier to program.
6.2
Suggestion for Future Research
Here we list out several possibilities for future research. In this dissertation,
the scope of the study had been limited to the simply connected regions with smooth
boundaries. The extension of the presented method can be done in multiply
connected regions with or without corners.
Further work on improving the numerical techniques used for computing the
function f ( z ) for points near the boundary and the function f ( z ) for points near
and on the boundary needs to be carried out. One can use a suitable subtraction
technique explained in [17, p.268]. Note that these subtraction techniques weaken
but do not entirely remove the singularity. It is possible to continue the subtraction
process further in a systematic way, to smooth out the equation to any degree
desired.
The numerical experiments in this dissertation were based on solving the
Fredholm integral equations of the second kind. The integral equation was
discretized by the Nyström method with the trapezoidal rule. The matrices equation
is then solved using the MATLAB “\” operator that makes use of Gauss elimination.
Further work on improving the numerical techniques used in this dissertation needs
to be carried out.
With the above summary, conclusions and future recommendations, we
conclude this dissertation.
REFERENCES
[1]
Asmar, N. H. Applied Complex Analysis with Partial Differential Equations,
Prentice Hall Inc., New Jersey, 2002
[2]
Churchill, R. V. and Brown, J. W. Complex Variables and Applications, 4th
ed., McGraw-Hill Book Company, Inc., New York, 1984
[3]
Ummu Tasnim Husin. Boundary Integral Equation with the Generalized
Neumann Kernel for the Interior Neumann Problem, M.Sc Dissertation,
Universiti Teknologi Malaysia, 2009
[4]
Nasser, M. M. S. Boundary Integral Equations with the Generalized
Neumann Kernel for the Neumann Problem, MATEMATIKA, 23(2007), 8398.
[5]
Henrici, P. Applied and Computational Complex Analysis, Vol. 3, John
Wiley, New York, 1986
[6]
Wegmann, R., Murid, A. H. M. and Nasser, M. M. S. The Riemann-Hilbert
Problem and the Generalized Neumann Kernel, Journal of Computational
and Applied Mathematics, 182(2005), 388-415.
[7]
Mikhlin, S. G. Linear Integral Equations, English Translation of Russian
edition 1948, Hindustan Publishing Corporation, India, 1960
[8]
Nasser, M. M. S. Boundary Integral Equations Approach for the Riemann
Problem, Ph.D. Thesis, Universiti Teknologi Malaysia, 2005
[9]
Nasser, M. M. S. Numerical Solution of the Riemann-Hilbert Problem,
Journal of Mathematics, 40(2008), 9-29.
[10]
Atkinson, K. E. The Numerical Solution of Integral Equations of the Second
Kind, Cambridge University Press, 1997
[11]
Begehr, H. G. W. Complex Analytic Methods for Partial Differential
Equations, World Scientific, 1994
[12]
Ellis, R. and Gullick, D. Calculus with Analytic Geometry, Harcourt Brace
Jovanorich, Inc., 1978
[13]
Zamzamir, Z., Murid, A. H. M. and Ismail, M. Particular Solution for NonUniquely Solvable Exterior Riemann-Hilbert Problem on a Simply
Connected region with Corners, Proceedings of the 5th Asian Mathematical
conference, Malaysia, 2009
[14]
Gakhov, F. D. Boundary Value Problems, English Translation of Russian
Edition 1963, Pergamon Press, Oxford, 1966
[15]
Mikhlin, S. G. Integral Equations, English Translation of Russian edition
1948, Pergamon Press, Armstrong, 1957
[16]
Atkinson, K. E. The Solution of Non-Unique Linear Integral Equations,
Numerische Mathematik, 10(1967), 117-124.
[17]
Delves, L. M. and Mohamed, J. L. Computational Methods for Tntegral
Equations, Cambridge University Press, New York, 1985
[18]
Muskhelishvili, N. I. Singular Integral Equations, Noordhoff, Groningen,
1953
APPENDIX A
COMPUTER PROGRAM FOR SOLVING NEUMANN PROBLEM ON THE
REGION EXTERIOR TO
clear
%%
n = 256
h = 2*pi/n;
s = (0:h:2*pi-h).';
%%
et
= cos(s)+5.*i.*sin(s);
etp = -sin(s)+5.*i.*cos(s);
etpp = -cos(s)-5.*i.*sin(s);
%%
fex
= 1./et;
fpex = -1./(et.^2);
fppex = 2./(et.^3);
mue = imag(-i.*etp.*fpex);
psi = real(-i.*etp.*fpex);
psip = real(-i.*etpp.*fpex-i.*(etp.^2).*fppex);
%%
for k=1:n
for j=1:n
if (k==j)
V(k,j)=h*(1/(2*pi))*imag(etpp(j)/etp(j));
else
V(k,j)=h*(1/pi)*imag(etp(k)/(et(k)-et(j)));
end
end
end
%%
J = (h/(2*pi)).*ones(n,n);
I = eye(n);
%%
for k=1:n
for j=1:n
if (k==j)
B(k,j)=h*(1/pi)*(psip(j)-psi(j)*real(etpp(j)/etp(j)));
else
B(k,j)=h*(1/pi)*real((etp(k)*psi(j)-etp(j)*psi(k))/(et(j)-et(k)));
end
end
end
y = B*ones(n,1);
%%
A
= I-(V-J);
mun = A\y;
err_norm_mu = norm(mue-mun,inf);
%%
fpn = (psi+i.*mun)./(-i.*etp);
%%
zo = [-4 ; -2 ; 2 ; 4];
for k=1:length(zo)
fpnzw(k,1) = -(h/(2*pi*i))*sum(fpn.*etp./(et-zo(k)));
end
%%
for k=1:length(zo)
fnzw(k,1) = (h/(2*pi*i))*sum(fpn.*etp.*Log(1-et./zo(k)));
un(k,1) = (h/(2*pi))*sum(imag((-mun+i.*psi).*Log(1-et./zo(k))));
end
%%
err_norm_fp = norm(fpex-fpn,inf)
[zo abs(fpnzw-(-1./zo.^2))]
[zo abs(fnzw-1./zo)]
%%
% Before proceeding, the following function files must be saved.
% filename : Log.m
%%
%function w = Log (z,alp)
%%%
%if (nargin == 1),
% w = log(abs(z))+i.*Arg(z);
%elseif (nargin == 2),
% w = log(abs(z))+i.*Arg(z,alp);
%end
%%
%filename : Arg.m
%%
%function v = Arg (z,alp)
%%%if (nargin == 1),
% alp = -pi;
%end
%%%
%v = atan2(imag(z),real(z));
%%%
%n = length(v);
%for k=1:n
% while (v(k)<=alp || v(k)>2*pi+alp)
%
v(k) = v(k)+sign(alp-v(k))*2*pi;
% end
%end
%%%
%y = v;
%%%
APPENDIX B
COMPUTER PROGRAM FOR SOLVING NEUMANN PROBLEM ON THE
REGION EXTERIOR TO
clear
%%
n = 256
h = 2*pi/n;
s = (0:h:2*pi-h).';
%%
et
= cos(s)+5.*i.*sin(s);
etp = -sin(s)+5.*i.*cos(s);
etpp = -cos(s)-5.*i.*sin(s);
%%
t
= exp(i.*s);
R
= 2.5+2.*cos(2.*s);
Rp = -4.*sin(2.*s);
Rpp = -8.*cos(2.*s);
et
= R.*t;
etp = (Rp+i.*R).*t;
etpp = (Rpp-R+2*i.*Rp).*t;
%%
for k=1:n
for j=1:n
if (k==j)
V(k,j)=h*(1/(2*pi))*imag(etpp(j)/etp(j));
else
V(k,j)=h*(1/pi)*imag(etp(k)/(et(k)-et(j)));
end
end
end
%%
J = (h/(2*pi)).*ones(n,n);
I = eye(n);
%%
for k=1:n
for j=1:n
if (k==j)
B(k,j)=h*(1/pi)*(psip(j)-psi(j)*real(etpp(j)/etp(j)));
else
B(k,j)=h*(1/pi)*real((etp(k)*psi(j)-etp(j)*psi(k))/(et(j)-et(k)));
end
end
end
y = B*ones(n,1);
%%
A
= I-(V-J);
mun = A\y;
err_norm_mu = norm(mue-mun,inf);
%%
fpn = (psi+i.*mun)./(-i.*etp);
%%
zo = [-2i ; -i ; i ; 2i];
for k=1:length(zo)
fpnzw(k,1) = -(h/(2*pi*i))*sum(fpn.*etp./(et-zo(k)));
end
%%
for k=1:length(zo)
fnzw(k,1) = (h/(2*pi*i))*sum(fpn.*etp.*Log(1-et./zo(k)));
un(k,1) = (h/(2*pi))*sum(imag((-mun+i.*psi).*Log(1-et./zo(k))));
end
%%
err_norm_fp = norm(fpex-fpn,inf)
[zo abs(fpnzw-(-1./zo.^2))]
[zo abs(fnzw-1./zo)]
%%
% Before proceeding, the following function files must be saved.
% filename : Log.m
%%
%function w = Log (z,alp)
%%%
%if (nargin == 1),
% w = log(abs(z))+i.*Arg(z);
%elseif (nargin == 2),
% w = log(abs(z))+i.*Arg(z,alp);
%end
%%
%filename : Arg.m
%%
%function v = Arg (z,alp)
%%%if (nargin == 1),
% alp = -pi;
%end
%%%
%v = atan2(imag(z),real(z));
%%%
%n = length(v);
%for k=1:n
% while (v(k)<=alp || v(k)>2*pi+alp)
%
v(k) = v(k)+sign(alp-v(k))*2*pi;
% end
%end
%%%
%y = v;
%%%
APPENDIX C
COMPUTER PROGRAM FOR SOLVING NEUMANN PROBLEM ON THE
REGION EXTERIOR TO
clear
%%
n = 256
h = 2*pi/n;
s = (0:h:2*pi-h).';
%%
et
= cos(s)+5.*i.*sin(s);
etp = -sin(s)+5.*i.*cos(s);
etpp = -cos(s)-5.*i.*sin(s);
%%
t
= exp(i.*s);
s1
s2
s4
c1
c2
c4
es
ec
=
=
=
=
=
=
=
=
sin(s);
sin(2.*s);
sin(4.*s);
cos(s);
cos(2.*s);
cos(4.*s);
exp(sin(s));
exp(cos(s));
%R = exp(cos(s)).*((cos(2.*s)).^2)+exp(sin(s)).*((sin(2.*s)).^2);
R
= ec.*c2.^2+es.*s2.^2;
Rp
= -s1.*c2.^2.*ec+c1.*s2.^2.*es+2.*s4.*(es-ec);
Rpp = (-c1.*c2.^2+2.*s1.*s4+s1.^2.*c2.^2).*ec+(s1.*s2.^2+2.*c1.*s4+c1.^2.*s2.^2).*es+8.*c4.*(es-ec)+2.*s4.*(es.*c1+ec.*s1);
et
= R.*t;
etp = (Rp+i.*R).*t;
etpp = (Rpp-R+2*i.*Rp).*t;
%%
for k=1:n
for j=1:n
if (k==j)
V(k,j)=h*(1/(2*pi))*imag(etpp(j)/etp(j));
else
V(k,j)=h*(1/pi)*imag(etp(k)/(et(k)-et(j)));
end
end
end
%%
J = (h/(2*pi)).*ones(n,n);
I = eye(n);
%%
for k=1:n
for j=1:n
if (k==j)
B(k,j)=h*(1/pi)*(psip(j)-psi(j)*real(etpp(j)/etp(j)));
else
B(k,j)=h*(1/pi)*real((etp(k)*psi(j)-etp(j)*psi(k))/(et(j)-et(k)));
end
end
end
y = B*ones(n,1);
%%
A
= I-(V-J);
mun = A\y;
err_norm_mu = norm(mue-mun,inf);
%%
fpn = (psi+i.*mun)./(-i.*etp);
%%
zo = [-1 ; -1-i ; 1-i ; 2-i];
for k=1:length(zo)
fpnzw(k,1) = -(h/(2*pi*i))*sum(fpn.*etp./(et-zo(k)));
end
%%
for k=1:length(zo)
fnzw(k,1) = (h/(2*pi*i))*sum(fpn.*etp.*Log(1-et./zo(k)));
un(k,1) = (h/(2*pi))*sum(imag((-mun+i.*psi).*Log(1-et./zo(k))));
end
%%
err_norm_fp = norm(fpex-fpn,inf)
[zo abs(fpnzw-(-1./zo.^2))]
[zo abs(fnzw-1./zo)]
%%
% Before proceeding, the following function files must be saved.
% filename : Log.m
%%
%function w = Log (z,alp)
%%%
%if (nargin == 1),
% w = log(abs(z))+i.*Arg(z);
%elseif (nargin == 2),
% w = log(abs(z))+i.*Arg(z,alp);
%end
%%
%filename : Arg.m
%%
%function v = Arg (z,alp)
%%%if (nargin == 1),
% alp = -pi;
%end
%%%
%v = atan2(imag(z),real(z));
%%%
%n = length(v);
%for k=1:n
% while (v(k)<=alp || v(k)>2*pi+alp)
%
v(k) = v(k)+sign(alp-v(k))*2*pi;
% end
%end
%%%
%y = v;
%%%