Applied Mathematics, 2011, 2, 118-122
doi:10.4236/am.2011.21013 Published Online January 2011 (http://www.SciRP.org/journal/am)
A Fast Algorithm to Solve the Bitsadze Equation in
the Unit Disk
Daoud Mashat, Manal Alotibi
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
Received October 19, 2010; revised November 18, 2010; accepted November 24, 201 0
Abstract
An algorithm is provided for the fast and accurate computation of the solution of the Bitsadze equation in the
complex plane in the interior of the unit disk. The algorithm is based on the representation of the solution in
terms of a double integral as it shown by Begehr [1,2], some recursive relations in Fourier space, and Fast
Fourier Transforms. The numerical evaluation of integrals at 2
N
points on a polar coordinate grid by
straightforward summation for the double integral would require


2
ON floating point operation per point.
Evaluation of such integrals has been optimized in this paper giving an asymptotic operation count of

In ON per point on the average. In actual implementation, the algorithm has even better computational
complexity, approximately of the order of


1O per point. The algorithm has the added advantage of
working in place, meaning that no additional memory storage is required beyond that of the initial data. This
paper is a result of application of many of the original ideas described in Daripa [3].
Keywords: Singular Integrals, Fast Algorithm, Bitsadze Equation
1. Introduction
The solutions of many elliptic partial differential equa-
tions represent in terms of singular integrals in the com-
plex plane in the interior of the unit disk, as the nonho-
mogeneous Cauchy-Riemann equations, the Beltrami eq-
uation, the Poisson equation, etc. Then solving these eq-
uations requires computing the values of the singular
integrals. There are two main difficulties in the straight-
forward computation of these integrals using quadrature
rules. Firstly, straightforward computation of each of these
integrals requires an operation count of the order


2
ON
per point. This gives a net operation count of


4
ON
for 2 N grid points which is computationally very in-
tensive for large N. Secondly, this method also gives
poor accuracy due to the presence of the singularities in
the integrand. Daripa and Co-workers ([3-9]) presented
fast algorithms to solve the singular integrals that arise in
such solutions. By these algorithms evaluation of singu-
lar integrals has been optimized, giving an asymptotic
operation count of

2ln lnON N for2
N points. More-
over, these algorithms have the added advantages of
working in place, meaning that no additional memory
storage is required beyond that of th e initial data.
In this paper we follow his method to present an algo-
rithm to solve an elliptic partial differential equation
called the Bitsadze equation which define in the unit disk




0;1: 1Bzz

 in complex plane  for any com-
plex valued function w of complex variables ,zz B

by
2
2
2
1;,,
4
wiwfzxiyxy
xy z
 



 
 (1)
Where f is a complex valued function on B. [10]
The Bitsadze equation arises in many areas including
structural mechanics, electrostatics, magneto statics, po-
wer electromagnetic, conductive media, heat transfer and
diffusion ([11-13]). In [1,2], Begehr introduced some
boundary value problems for Bitsadze equation under
some solvability conditions and presented their solutions
as results for applying the Dirichlet and Neumann boun-
dary value problems all those solutions have singular in-
tegrals in their context. For that, we will consider one of
these problems to apply our numerical method for eva-
luating the singular integrals.
Problem (1.1):
zz
wf

in B, 0,w


1
zz zw

on B,


0
z
wc

,
Where


101
;, ,; and fLBCB c 

 .

D. MASHAT ET AL. 119
Its unique solution is given by Equation (1.2) (Below)
Let us rewrite it in the form
  
,wzczuzvzGf z  (1.3)
Where
 
0
1
1,
2
d
uz iz







  
2
1
1
1
1log 1,
2
zd
vz z
iz


 





and
 
22
1
1 .
()
z
Gf zfdd
z








 (1.4)
Our method is basically a recursive routine in Fourier
space that divides the interior of the unit disk into a col-
lection of annular regions and expands the integral in
Fourier series in angular direction with radius dependent
Fourier coefficients. A set of exact recursive relations are
derived which are used to produce the Fourier coeffi-
cients of singular. These recursive relations involve ap-
propriate scaling of one-dimensional integrals in annular
regions. The integrals in (1.4) at all grid points are then
easily obtained from the Fourier coefficients by the FFT.
The rest of the paper is structured as follows. Section 2
presents the solution of the linear integrals u and v. Sec-
tion 3 develops the mathematical foundation of the effi-
cient algorithm to evaluate (1.4) within the unit disk. The
fast algorithms for solving problem (1.1) is discussed in
Section 4. We carry out numerical results with this me-
thod in section 5. Finally, we summarize and conclude in
Section 6.
2. Evaluating the Integrals u and v
1)
 
0
1
1
2
d
uz iz







Let i e


  and i zre

, where ,0r

 then
 
2
0
0
1,
2
i
ii
ii
ied
ure e
iere










Since the integral is on the unit disk then 1


,






20
0
2
0
00
1
,21
1
2
i
i
in
in
n
e
ur d
re
ere d


















Let


0iin n
n
eae





,


2
00
0
1
,2
,
nin inin
n
nn
nin
n
n
urareeed
are
 




 










Then we can write

,in
n
n
ur ue




 (2.1)
Where
if 0,
0 if 0.
n
n
n
ar n
un





2)
  
2
1
1
1
1log 1,
2
zd
vz z
iz


 






 





22
1
0
2 2
1
0
2 2
1
10
11
,log1,
2
1
log1,
2
1
.
2
i
iii
ii
ii
i
n
iin
i
n
ried
vrere e
ire e
rered
re
rr
eed
n
re









 


















Let


1iin
n
n
ebe





,



2 2
10
11 1
1
11
,,
2
.
nin inin
n
i
nn
nnin n
n
br
r
vreee d
n re
brre
n
 





 

 













put n = m+1


2
1
01., mmim
m
m
b
vrrr e
m








Then we can write

,,
im
m
m
vr ve




 (2.3)
Where

2 1 if 0,
1
0 if 0.
mm m
m
brr m
vmm
 






    
222
01
11 1
1
11 1
log 1.
22 ()
zz
dd
wz czzfdd
iziz z
 






 
 

 

 (1.2)

D. MASHAT ET AL. 120
3. Mathematical Foundation of the
Algorithms
In this section, we develop the theory n eed e d to construct
an efficient algorithm for evaluation the singular inte-
grals (1.4). The following theorem is crucial for later
development of the algorithm.
Theorem 3.1. The value of the integral (1.4) with
i
zre

, 0r can be written as
 
in
n
n
Gfzgr e



 (3.1)
Where






22
2
0;10;
2
2
2
0;
; 0
1
; 1
nn
n
BBr
nn
n
Br
rr
fddn
gr rr
fddn






















(3.2)
Proof. If we denote

22
2
0
1
,.
2()
in
z
Qre d
z








 (3.3)
Then it follows from (1.4) and (3.1) that
  
(0;1)
1, .
nB
g
rfQrdd



 (3.4)
The integral in (3.3) can be evaluated by first expand-
ing
22
()
z
z



in power of z

and z

to get



22
22 2
0
2
2
2
1
; ,
; .
n
in
n
n
n
in
n
n
rr
er
z
zrr er





 


















By comparing this result with the Fourier series coef-
ficients of
22
()
z
z



, we get

22
2
2
2
2
0; 0, ,
; 0, ,
,
; 1, ,
0; 1, .
nn
n
nn
n
nr
rr nr
Qr rr
nr
nr


















 


 

(3.5)
Substitution of (3.5) into (3.4) yields the desired result.
Corollary 3.1. Suppose that i
e


  and


f

has
Fourier coefficient


n
f

; then the coefficients


n
g
r
in Equation (3.2) can be written as:



122
2
1
22
2
1
0
–; 0,
=2
; 1.
nn
n
n
r
nrnn
n
n
rr fdn
gr rr
fdn



















(3.6)
Corollary 3.2. It follows directly from Equation (3.6)
that


10
n
g

for 0n,

00
n
g for 1n

.
Corollary 3.3. Let ij rr

. Define


2
1
,
2
1
; 0,
; 1.
j
i
j
i
r
n
n
r
n
ij r
n
n
r
fdn
Cfdn

















(3.7)
and


2
1
,
2
1
; 0,
; 1.
j
i
j
i
r
n
n
r
n
ij r
n
n
r
fdn
Bfdn


















(3.8)
After some algebraic manipulation it follows from
Equations (3 .7) and (3.8) that


.
nini ni
g
rCrBr (3.9)
Where


,
2; 0,
n
nn i
nii ijn j
j
r
CrrCCrn
r





 (3.10)


2
2,
2; 1,
n
nn i
njnij ij
j
r
CrCrrCn
r







 (3.11)


2
2,
2; 0,
n
nn i
niiijn j
j
r
Brr BBrn
r






 (3.12)
and


,
2; 1.
n
nn i
njni jij
j
r
BrBrrB n
r






 (3.13)
Corollary 3.4. Let 01 2 01
M
rrr r

 . It
follows from recursive applications of (3.10)-(3.13) and
from using Corollary 3.2 that



12
,1 ,1
21, 1,
1
2; 0,
2; 1.
Mnnnn
liil ii
il
nl lnn nn
liilii
i
rCrBn
gr
rC rBn















(3.14)

D. MASHAT ET AL. 121
for 1, 2,3,,1lM.
Corollary 3.5. It follows directly from Equations (3.6)
and (3.9) that

110; 0,
nn
CB n (3.15)
 
000; 1.
nn
CB n (3.16)
4. The Fast Algorithm
We construct the fast algorithm based on the theory of
section 3. The unit disk is discredited using
M
N

lat-
tice points with M equidistant points in the radial direc-
tion and N equidistant points in the circular direction.
The following is a formal description of the fast algo-
rithm useful for pr ogramming pur p oses.
Algorithm 4.1. (The fast algorithm for evaluating the
singular integral (1.4))
Input: M, N and





2,1,1, 1, ik N
l
f
rel MkN


Output:





2,1,1,1, ik N
l
Gf relMkN

 .
Step 1. Set 8
K
N, 00r and 1
M
r.
Step 2. Compute the Fourier coefficients


nl
f
r,


0, 1lM and 

,nKK from known values of

2,1,2,, ik N
l
f
re kN

 using the FFT.
Step 3. Compute


,1 1, 1
n
ii
CiM
 and


2,1[0,2] nKK  using Equation (3.7).
Step 4. Compute 

,1 1, 1
n
ii
BiM
 and


2, 1[0,2]nK K  using Equation (3.8).
Step 5. Compute

nl
g
r,


2, 2nKK

 ,


1, 1lM using relations (3.1 4).
set
 


0,0,0, 2
nMnM
CrBrn K
do 0,1, ,2nK
do 1, ,1lM
 
,1 1
1
2,
n
nn l
nll llnl
l
r
Cr rCCr
r






 
2
2,1 1
1
2,
n
nn l
nll llnl
l
r
Brr BBr
r









.
nlnl nl
g
rCrBr
enddo
enddo
set
 


00
0,0,2, 1
nn
CrBrn K

2
110,1
2,
nn
n
Crr C



110,1
2,
nn
n
Br rB






111
.
nnn
g
rCrBr
do 2,,1nK 
do 2, ,1lM
 
2
2
111,
2,
n
nn
l
nlnll ll
l
r
CrCrr C
r







 
111,
2,
n
nn
l
nlnll ll
l
r
BrBr rB
r








.
nlnl nl
g
rCrBr
enddo
enddo
Step 6. Finally, compute



2
22
2;
1,1 1,, .
K
ik Nik N
lnl
nK
Gfzregr e
lM kN


 



Algorithm 4.2 (The fast algorithm for solving the
problem (1.1)):
Input: M, N ,


2
0ik N
e

 ,

2
1ikN
e

 and


2ik N
l
fre

,




1,1 ,1,lM kN .
Output:



2,1, 1,1,. ik N
l
wrel Mk N


Step 1. Set 8
K
N

, 00r and 1
M
r.
Step 2. Compute


2ik N
l
Gf zre

,


1, 1lM,


1,kN using algorithm 4.1.
Step 3. Compute the Fourier coefficients


,
n
an KK from known values of


2
0ik N
e

 ,
1, 2,,kN

 using the FFT.
Step 4. Compute the Fourier coefficients


1, 1
n
bnK K

  from known values of


2
1ikN
e

 ,
1, 2,,kN

 using the FFT.
Step 5. Compute




2
(),1,1,1,
ik N
l
urel MkN


using the relation (2.1).
Step 6. Compute




2
(),1,1,1,
ik N
l
vrel MkN


using the relation (2.3).
Step 7. Compute




2
(),1,1,1,
ik N
l
wrelMk N

 .



222
22
()
ik Nik Nik N
ll l
ik Nik N
ll
wrecre ure
vre Gfre




5. Numerical Results
In this section we solve a boundary value problem for the
inhomogeneous Bitsadze equation in the unit disk by
using the algorithms that presented in section 4.
Example
Consider the problem
2
zz
wz

in B,,wz

2,
zz
zw ,zB


00
z
w

,
The exact solution for this problem is given by


2,wzzz zB


By using algorithm 4.2 we have the max error as illu-
strate in the following Table 1.

D. MASHAT ET AL. 122
Table 1. Maximum relative error.
MAX ERROR
M = 10 M = 50 M = 100 M = 200
N=64 3.371768E-07 2.980596E-07 2.588641E-07 2.245232E-07
N=128 3.371753E-07 2.980553E-07 2.588640E-07 2.245231E-07
N= 256 3.371734E -07 2.980548E-07 2.588639E-07 2.245196E-07
N= 512 3.371718E -07 2.980526E-07 2.588628E-07 2.245153E-07
N= 1024 3.371717E-07 2.980501E-07 2.588627E-07 2.245142E-07
6. Conclusions
We presented a fast algorithm to solve the Bitsadze equ-
ation in the unit disk under special boundary conditions
in the complex plane, by constructed the fast algorithm
to evaluate the singular integral transform (1.4). The
method divides the interior of the unit diskB


:1zz into a collection of annular regions. The in-
tegrals and the function f (z) are expanded in terms of
Fourier series with radius dependent Fourier coefficients.
The good performance of the algorithm is due to the use
of scaling one-dimensional integral in the radial direction
to produce the value of the singular integral over the en-
tire domain. Specifically, scaling factors are used to de-
fine exact recursive relations which evaluate the radius
dependent Fourier coefficients of the singular integral
(1.4). The inverse Fourier transform are applied on each
circle to obtain the value of the singular integrals on all
circles.
7. References
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Analysis I,” Boles’ de la Association’s Mathematical Ve-
nezuelan, Vol. 12, No. 1, 2005, pp. 65-85.
[2] H. Begehr, “Boundary Value Problems in Complex
Analysis II,” Boles’ de la Association’ Mathematical Ve-
nezuelan, Vol. 12, No. 2, 2005, pp. 217-250.
[3] P. Daripa, “A Fast Algorithm to Solve Nonhomogeneous
Cauchy–Riemann Equations in the Complex Plane,”
SIAM Journal on Scientific and Statistical Computing,
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1019117414918
[6] P. Daripa and L. Borges, “A Fast Parallel Algorithm for
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10.1006/jcph.2001.6720
[7] P. Daripa and L. Borges, “A Parallel Version of a Fast
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A:1019143832124
[8] P. Daripa, “A Fast Algorithm to Solve the Beltrami Equ-
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90274-O
[10] A. H. Babayan, “A Boundary Value Problem for Bitsadze
Equation in the Unit Disc,” Journal of Contemporary
Mathematica Analysis, Vol. 42, No. 4, 2007, pp. 177-183.
doi:10.3103/S1068362307040012
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Order Elliptic Equations,” North-Holland Publishing
Company, Amsterdam, 1968.
[12] C. Miranda, “Partial Differential of Elliptic Type,”
Springer, New York, 1970.
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