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Integral Equation Approach for Computing Green’s Function on Doubly Connected Regions via the Generalized Neumann Kernel a*
a,b
c
Siti Zulaiha Aspon , Ali Hassan Mohamed Murid , Mohamed M. S. Nasser , Hamisan Rahmat
a
a
Department of Mathematical Sciences, Faculty of Science, UTM, 81310 UTM Johor Bahru, Johor, Malaysia UTM Centre for Industrial and Applied Mathematics (UTM-CIAM), 81310 UTM Johor Bahru, Johor, Malaysia c Department of Mathematics, Faculty of Science, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia b
*Corresponding author:
[email protected] Article history
Abstract
Received :2 February 2014 Received in revised form : 3 August 2014 Accepted :15 October 2014
This research is about computing the Green’s function on doubly connected regions by using the method of boundary integral equation. The method depends on solving a Dirichlet problem. The Dirichlet problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the generalized Neumann kernel. The method for solving this integral equation is by using the Nystrӧm method with trapezoidal rule to discretize it to a linear system. The linear system is then solved by the Gauss elimination method. Mathematica plots of Green’s functions for several test regions are also presented.
Graphical abstract
Keywords Green’s Function; Dirichlet Problem; Integral Equation; Generalized Neumann Kernel Abstrak Kajian ini berkaitan dengan pengiraan fungsi Green pada rantau berkait ganda dua terbatas dengan menggunakan kaedah persamaan kamiran sempadan. Kaedah ini bergantung kepada penyelesaian masalah Dirichlet. Masalah Dirichlet kemudiannya diselesaikan menggunakan persamaan kamiran Fredholm berpenyelesaian unik pada sempadan rantau ini. Inti persamaan kamiran ini adalah inti Neumann teritlak. Kaedah untuk menyelesaikan persamaan kamiran ini ialah dengan menggunakan kaedah Nystrӧm dengan peraturan trapezoid untuk menghasilkan sebuah sistem linear. Sistem linear kemudian diselesaikan dengan kaedah penghapusan Gauss. Plot Mathematica bagi fungsi Green untuk beberapa rantau ujian juga dipersembahkan. Kata kunci: Fungsi Green; Masalah Dirichlet; Persamaan Kamiran; Inti Neumann Teritlak © 2014 Penerbit UTM Press. All rights reserved.
1.0 INTRODUCTION Green’s functions are important since they provide a powerful tool in solving differential equations. They are very useful in several fields such as solid mechanics, applied physics, applied mathematics, mechanical engineering, materials science and quantum field theory.1 Henrici shows three different methods for computing Green’s function for doubly connected regions which leads to three different analytical representations for the Green’s function. The methods are the Fourier series method, infinite product method and Theta series method.2 Crowdy and Marshall have presented an analytical formula for Green’s function for Laplace’s equation in multiply circular domains. The method is constructive and depends on Schottky-Klein prime function associated with multiply connected circular domain.3 Wegmann and Nasser have studied Fredholm integral equation associated with the linear Riemann-Hilbert problems on
multiply connected regions with smooth boundary curves. The kernel of these integral equations is the generalized Neumann kernel. They investigated the existence and uniqueness of solutions of the integral equations by determining the exact number of linear independent solutions and their adjoints.4 Based on Wegmann and Nasser, Nasser et al. have proposed a new boundary integral method for the solution of Laplace’s equation on multiply connected regions using either Dirichlet boundary condition or the Neumann boundary condition. The method is based on two uniquely solvable Fredholm integral equations of the second kind with the generalized Neumann kernel.5 Recently, Alagele has proposed a new method for computing the Green’s function on simply connected region by using the method of boundary integral equation which depends on the solution of a Dirichlet problem.6 Based on paper by Wegmann and Nasser and Nasser et al., the Dirichlet problem is solved using a uniquely solvable Fredholm integral on the equation boundary of the region.4,5 This
71:1 (2014) 49–54 | www.jurnalteknologi.utm.my | eISSN 2180–3722 |
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paper wish to extend Alagele’s work to compute Green’s function for bounded doubly connected regions using integral equation with generalized Neumann kernel.6
where
2.0 AUXILIARY MATERIALS
The interior Dirichlet problem (2.3) is uniquely solvable and can be regarded as a real part of an analytic function F in which is not necessary single-valued.2,5 The function F can be written as
Let Ω be a bounded doubly connected region in the complex plane (Figure 1). The outer boundary 0 has a counter clockwise direction and surrounds the boundary 1 which has clockwise
0 (t ), t J 0 , 1 (t ), t J1.
F ( z ) f ( z ) a1 ln z z1 ,
orientation. So we have 0 1 . where
f
point in
k .5
is a single-valued analytic function in
1
and
a1 is
a1
, z1 is a fixed
real constant uniquely determined by
We assume for bounded
constant
(2.4)
that
Im f ( ) 0.
The
is chosen to ensure that
f '( )d 0. k
1
Figure 1 Bounded doubly connected region
In general the Green’s function for We assume that each boundary
k
has a parameterization
k (t ), t J k , k 0,1, which is a complex periodic function with period 2 , where J k [0, 2 ] is the parametric interval for each k . The parameterization k (t )
G( z, z0 ) u ( z )
d (t ) 0. dt
Therefore the parameterization
k
can be expressed by8
1 ln z z0 , z, z0 , (2.5) 2
where u is the unique solution of the interior Dirichlet problem
2u ( z ) 0, z 1 ln k z0 ,k k . u (k ) 2
also need to be twice continuously differentiable such that
k (t )
(2.6)
(2.1) By computing
F given by (2.4) with
of the whole boundary
0
can be written as
0 :0 (t ), t J 0 [0, 2 ], 1 :1 (t ), t J1 [0, 2 ].
1 ln 0 z0 and 1 ln 0 z1 ,k k , (2.7) 2
the unique solution of the interior Dirichlet problem (2.6) is given (2.2)
in
by
Let u be a real function defined in the region and let z x iy . In our research, for simplicity, we write u(z)
u( z) Re F ( z).
(2.8)
instead of u(x,y). Let H be the space of all real Hölder continuous function with exponent on the boundary Γ. The interior Dirichlet problem is defined as follows:
3.0 INTEGRAL EQUATION FOR THE INTERIOR DIRICHLET PROBLEM
Interior Dirichlet problem:
Let
Let
k H
be a given function. Find the function u harmonic
in , Holder continuous on condition
Ak (t )
for all
be continuously differentiable 2π-periodic functions
t J k , k 0,1. We consider two real functions9
and satisfies the boundary
u(k ) k , k k ,
N l , k ( s, t ) (2.3)
A (s) k (t ) Im l , Ak (t ) k (t ) l ( s) 1
(3.1)
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A (s) k (t ) Re l . Ak (t ) k (t ) l ( s) 1
M l , k ( s, t )
(3.2)
The kernel Nl ,k ( s, t ) is called the generalized Neumann kernel formed with complex-valued function Ak (t ) and When Ak 1, the kernel
Nl ,k
k (t ).
0
0
2
M l ,0 ( s, t )
2 [ p] 0
(t )dt
0
M
l ,1
( s, t ) 1[ p ] (t )dt , s J l , l , p 0,1.
(3.8)
0
4
and
is the classical Neumann kernel
which arise frequently in the integral equations for potential theory and conformal mapping.2 Theorem 3.14 a)
2
l[ p ] ( s) Nl ,0 ( s, t ) 0[ p ] (t )dt Nl ,1 ( s, t ) 1[ p ] (t )dt
hl[ p ]
2 1 2 1 1 [ p] [ p] M l ,k ( s, t ) k (t )dt l Nl ,k ( s, t ) k (t )dt , 2 k 0 0 k 0 0
(3.9)
with
The kernel Nl ,k ( s, t ) is continuous which takes on the
h0 0, [ p]
diagonal the values
h1 h1[ p ] h0[ p ] . [ p]
1 k (t ) Ak (t ) N k ,k (t , t ) Im . 2 k (t ) Ak (t ) 1
(3.3)
It follows from Nasser et al.5 that the unknown constant a1 is the solution of the equation
b)
The kernel
M l ,k ( s, t ) is continuous for l k. When h[0] a1 h[1] 0.
l k , the kernel M k ,k ( s, t ) has the representation
M k , k ( s, t )
1 s t cot M1, k ( s, t ), 2 2
Then (3.4)
h [0] h [0] a1 1 [1] 0[1] . h1 h0
with a continuous kernel M 1,k which takes on the diagonal the values
Hence, the boundary values of the function f is given by5
M1,k (t , t ) where
(3.10)
1 k (t ) Ak (t ) Re , 2 k (t ) Ak (t )
f (k (t )) k (t ) a1 ln | k (t ) z1 | ik (t ),
1
(3.5)
where
k 0,1.
To find the function the function
f ( z ) and
(3.11)
k (t ) k [0] (t ) a1k [1] (t ).
F ( z ) given by (2.4), we need to find the real constant a1 . We define real
By this result we can compute the interior values of over the whole region
f ( z)
by using the Cauchy integral formula
functions
0
k k and where
[1] k
ln k z1
for
f ( z)
k 0,1,
(3.6)
k satisfy (2.7). It follows that5
1 f ( w) dw. 2 i w z
We then compute the function
F ( z)
(3.12) from (2.4) and
u ( z ) from (2.8).
[ p]
f [ p ] (k (t )) k [ p ] (t ) hk (t ) i k [ p ] (t ), for p, k 0,1, are boundary values of analytic function f in k [ p ] is the unique solution of the integral equation [ p]
(3.7) 4.0 NUMERICAL IMPLEMENTATION
where
Denoting the right-hand side of the Equation (3.8) by l ( s) , we get 2
2
0
0
l[ p ] ( s) Nl ,0 ( s, t ) 0[ p ] (t )dt Nl ,1 ( s, t ) 1[ p ] (t )dt l[ p ] ( s). (4.1)
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Since the functions Ak and k are 2 - periodic, the integrals are discretized by the Nystrӧm method with trapezoidal rule.7
P [ Pi j ], Q [Qi j ], R [ Ri j ], S [Si j ]
t j by t j ( j 1)
2 , n
j 1,2,..., n.
(4.2)
Pij
2 2 Nl ,k (0 (ti ),0 (t j )), Qij Nl ,k (0 (ti ),1 (t j )), n n
Rij
2 2 Nl ,k (1 (ti ),1 (t j )), Nl ,k (1 (ti ),0 (t j )), Sij n n
Then, using the Nystrӧm method for (4.1) we obtain the linear system l[ p ] (ti )
2 n
n
N j 1
l ,0
(ti , t j ) 0[ p ] (t j )
2 n
n
N j 1
l ,1
( si , t j ) 1[ p ] (t j ) l [ p ] ( si )
(4.3) where k is an approximation to
2
cot 0
ti t [ p ] 2 k (t j )dt 2 n
2 n
Rx0 ( I S ) x1 y1.
n
M j 1
l ,k
n
M j 1
1, k
(ti , t j ) k [ p ] (t j ),
(4.4)
A (t ) k (t j ) M l ,k (ti , t j ) Re l i , l k, Ak (t j ) k (t j ) l (ti )
We use Wittich method to approximate the integral that contains cotangent function and obtain10
0
xl ,i l (ti ),
solution
then
we
calculate
f (0 (t j )) and f (1 (t j )) by using the following formula: f (0 (t j )) 0 (t j ) a1 ln 0 (t j ) z1 i0 (t j ),
(4.7)
f (1 (t j )) 1 (t j ) a1 ln 1 (t j ) z1 i 1 (t j ),
.
which represents the boundary values of f (z ) on
f ( z)
1 A (t ) 1 k (t j ) ti t j cot , ti t j Re k i 2 Ak (t j ) k (t j ) k (ti ) 2 M 1,k (ti , t j ) 1 k (t j ) Ak (t j ) 1 ti t j . Re , 2 k (t j ) Ak (t j )
cot
unique
We can
compute the interior values of f (z ) over the whole region by using the Cauchy integral formula given in (3.12), i.e.,
1
1 2
(4.6)
To solve the system (4.6) we use the method of Gaussian elimination. Since (4.1) has a unique solution, then for a wide class of quadrature formula the system (4.6) also has a unique solution, as long as n is sufficiently large.7 After we get the
(ti , t j ) k [ p ] (t j )
where l , k , p 0,1, and
2
Hence, the Equation (4.3) can be written as an 2n by 2n system
( I P) x0 Qx1 y0 ,
and
1 2
xl ,i l (ti ), yl ,i l (ti ).
, and
1 A (t ) k (t j ) Im l i , l k , or l k , ti t j Ak (t j ) k (t j ) l (ti ) Nl ,k (ti , t j ) k (t j ) 1 Ak (t j ) 1 l k , ti t j , 2 Im (t ) Im A (t ) , k j k j
l[ p ] (ti )
vectors
xl [ xl ,i ] and yl [ yl ,i ] by
Let n be a given integer and define the n equidistant collocation points
and
ti t k (t )dt K (i, j ) k (t j ) 2 j 1 n
(4.5)
where
0, (i j ) K (i, j ) 2 cot , n n
1 2 i
2
f (0 (t j ))
f (1 (t j ))
2
1
(4.8)
(t ) z (t )dt 2 i (t ) z (t )dt 0
0
0
j
1
j
1
0
j
j
To increase the accuracy of
f (z ) we shall use the following
formula. Based on the fact that
1 1 d 1 , we can write 2 i z
f (z )
as
f ( z)
1 2 i
2
f (0 (t j ))
(t ) z 0
0 (t j )dt
0
j
1 2 i
2
0 (t j )
1 2 i 1
2
f (1 (t j ))
(t ) z (t )dt 1
0 2
1
0
1
(4.9)
j
j
1(t j )
.
(t ) z dt 2 i (t ) z dt 0
0
j
j
Then, using the Nyström method with the trapezoidal rule to discretize the integrals in (4.9), we obtain the approximation
if j-i is even, n
if j-i is odd.
The left-hand side of (4.3) can also be calculated directly by using Mathematica. Define the matrices
f ( z)
f (0 (t j ))
n
f (1 (t j ))
(t ) z (t ) (t ) z (t ) j 1
0
0
j
j
j 1
1
1
j
n
0 (t j )
n
1(t j )
j 1
0
j 1
1
(t ) z (t ) z j
j
(4.10)
j
.
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This has the advantage that the denominator in this formula compensates for the error in the numerator.11 Next, substitute f ( z ) given in (4.10) into Equation F ( z ) in (2.4), and by taking the real part of (2.4) gives (2.8), i.e.
Gn ( z, z0 )
u (z )
.
we can compute the Green’s function
by the following formula (2.5), i.e.
Gn ( z, z0 ) u ( z )
ratio
We describe the error by maximum error norm G( z, z0 ) Gn ( z, z0 ) , where n is the number of nodes
u( z) Re F ( z). Finally, by using
z ei , and the infinite series converges, uniformly for z 1 and 1 at least like a geometric series with
where
and G( z, z0 ) is the numerical approximation of G( z, z0 ) . We choose some test points inside the region. The results are shown in Table 1. Table 1 The error
1 ln z z0 . 2
n
G( z, z0 ) Gn ( z, z0 )
32
64
128
z 5.0 NUMERICAL EXAMPLES Example 1 In Example 1, we consider an annulus as shown (Figure 2). The boundary of this region is parameterized by the function
0 :0 (t ) eit , 1 :1 (t ) peit ,
0 t 2
1.72 10 4 9.77 10 5 1.58 10 5 1.88 10 4 4.8 10 5 4.8 10 5 1.35 10 4 7.78 10 5 1.46 10 5
1.13 10 6 7.86 10 7 4.72 10 7 2.29 10 8 6.3 10 7 6.3 10 7 7.67 10 7 3.69 10 7 7.2110 8
5.8110 11 4.05 10 11 2.54 10 11 1.15 10 11 3.27 10 11 3.27 10 11 3.94 10 11 1.87 10 11 3.03 10 12
The 3D plot of the surface of Gn ( z, z0 ) is shown (Figure 3).
with p 0.5, z0 0.75,
0
0.6 0.7 0.8 0.9 0.74999999999 0.75000000001 0.5+0.5i 0.6+0.6i 0.7+0.7i
1 ln 0 z0 and 1 ln 0 z1 . 2
Figure 3 The 3D plot of Green’s function for Example 1
Example 2 In Example 2, we consider an Epitrochoid as shown (Figure 4). The boundary of this region is parameterized by the function
0 :0 (t ) eit peit ,
Figure 2 The test region Ω for Example 1
1 :1 (t ) qeit , The exact Green’s function of this region is given by2
Log 1 e G( z, z0 ) 1 Log Log Log 1 ei n n n n n cos n n n n 1 n 1
0 t 2
with p 0.3333, q 0.1, z0 0.75, z1 0.01,
i
0
1 ln 0 z0 and 1 ln 0 z1 . 2
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2u ( z ) 0, z 1 ln k z0 ,k k . u (k ) 2
(6.1)
on that region by means of solving an integral equation numerically using Nyström method with the trapezoidal rule. Once we got the solution u ( z ) , the Green’s function of can be computed by using the formula
Gn ( z, z0 ) u ( z )
Figure 4 The test region Ω for Example 2
The 3D plot of the surface of Gn ( z, z0 ) is shown (Figure 5).
1 ln z z0 . 2
(6.2)
The numerical example illustrates that the proposed method can be used to produce approximations of high accuracy. Acknowledgement This work was supported in part by the Malaysian Ministry of Higher Education (MOHE) through the Research Management Centre (RMC), Universiti Teknologi Malaysia (GUPQ.J130000.2526.04H62). References [1] [2] [3] [4] [5] [6]
Figure 5 The 3D plot of Green’s function for Example 2 [7]
6.0 CONCLUSION This study has presented a method for computing the Green’s function on doubly connected regions by using a new approach based on boundary integral equation with generalized Neumann kernel. The idea for computing the Green’s function on is to solve the Dirichlet problem
[8] [9] [10] [11]
Q.H. Qin. 2007. Green’s Function and Boundary Elements of Multifield Materials. Elsevier Science. P. Henrici. 1986. Applied and Computational Complex Analysis. New York: John Wiley. 3: 255–272. D. Crowdy, J. Marshall. 2007. IMA J. Appl. Math. 72: 278–301. R Wegmann, M. M. S Nasser. 2008. J. Comput. Appl. Math. 214: 36– 57. M. M. S. Nasser, A. H. M Murid, M. Ismail, E. M. A Alejaily. 2011. J. Appl. Math. and Comp. 217: 4710–4727. M. M. A. Alagele. 2012. Integral Equation Approach for Computing Green’s Function on Simply Connected Regions. M.Sc Dissertation, Universiti Teknologi Malaysia, Skudai. K. E. Atkinson. 1997. The Numerical Solution of Integral Equations of the Second Kind. Cambridge: Cambridge University Press. L. V. Ahlfors. 1979. Complex Analysis, International Student Edition. Singapore: McGraw-Hill. M. M. S. Nasser. 2009. SIAM J. Sci. Comput. 31: 1695–1715. M. M. S. Nasser. 2009. Comput. Methods Func. Theor. 9: 127–143. J. Helsing, R. Ojala. 2008. J. Comput. Physics. 227: 2899–2921.
