American Journal of Computational Mathematics, 2012, 2, 312-315
http://dx.doi.org/10.4236/ajcm.2012.24042 Published Online December 2012 (http://www.SciRP.org/journal/ajcm)
Numerical Solution of Generalized Abel’s Integral
Equation by Variational Iteration Method
R. N. Prajapati1, Rakesh Mohan1, Pankaj Kumar2
1Department of Mathematics, Dehradun Institute of Technology, Dehradun, India
2Kishori Lal Sharma Institue of Engineering & Technology, Chandok, Bijnor, India
Email: narain.prajapati@gmail.com, rkmohan2k4@yahoo.com, Pankaj.rssb@gmail.com
Received June 1, 2012; revised September 13, 2012; accepted October 8, 2012
ABSTRACT
In this paper, a user friendly algorithm based on the variational iteration method (VIM) is proposed to solve singular
integral equations with generalized Abel’s kernel. It is observed that an approximate solutions yn(x) converges to the
exact solution irrespective of the initial choice y0(x). Illustrative numerical examples are given to demonstrate the effi-
ciency and simplicity of the method in solving these types of singular integral equations.
Keywords: Variational Iteration Method; Singular Integral Equation; Abel’s Kernel
1. Introduction
The real world problems in scientific fields such as solid
state physics, plasma physics, fluid mechanics, chemical
kinetics and mathematical biology are nonlinear in gen-
eral when formulated as partial differential equations or
integral equations. In the last two decades, many power-
ful techniques have been proposed to solve the singular
integral equations and differential equation by using VIM
[1-15].
The generalized Abel’s integral equation of the second
kind is given by
 

0
d, 01,
xyt
yxf xtx
xt
 
(1)
where C
is a parameter and 01.

Theorem 1. For each complex ,

  
0
d,
x
yxf xRxtftt
(2)
is the unique solution of Equation (1),
where
 


1
1
1
,
1
n
n
x
Rx xn






(3)
The closed form solution (2) is not very useful in
many cases where it is difficult to evaluate the integral
appearing in (2). So, it is desirable to have numerical
solution for the generalized Abel’s integral Equation (1).
In the present paper, we have proposed an algorithm
based on the variational iteration method to solve the
generalized Abel’s integral Equation (1). It is observed
that the choice of the initial approximation
0
y
x has a
small effect on the efficiency of the method. The appro-
ximate solutions
n
y
x will always converge to the
exact solution.
2. Basic Idea of Variational Iteration
Method
Variational Iteration method was first proposed by He
[2-6] and has been successfully used by many research-
ers to solve various linear and nonlinear models [7-15].
The idea of the method is based on constructing a correc-
tion functional by a general Lagrange multiplier and the
multiplier is chosen in such a way that its correction so-
lution is improved with respect to the initial approxima-
tion or to the trial function.
Now, to illustrate the basic concept of the variational
iteration method, we consider the following general non-
linear system:

,Lyx Nyxfx


(4)
where L is a linear operator, N is a nonlinear operator and
f
x is a known analytic function. The basic character
of the method is to construct a correction functional for
the system, which reads
 
1
0
d,
0
nn
x
nn
yxyx
s
LysN ysfss
n

(5)
C
opyright © 2012 SciRes. AJCM
R. N. PRAJAPATI ET AL. 313
where
is a general Lagrange multiplier, which can be
identified optimally via variational theory, n is the nth
approximate solution, and n is considered as a re-
stricted variation, i.e. namely
y
y
0.
n
y
Successive ap-
proximations,

1,
n
x
0
will be obtained by applying
the obtained Lagrange multiplier and a properly chosen
initial approximation .

y
x
3. Variational Iteration Method of Solution
We consider the following iteration formula for Equation
(1) in the following form
 

1
0
d,0,1,2,3,;
xn
n
yt
yxfx tn
xt
 
(6)
where

n
y
x
is the nth approximate solution of (1) and
0
y
x is an appropriately chosen initial guess. The
value of
is found to be 1.
4. Numerical Examples
The simplicity and accuracy of the proposed method are
illustrated by the following numerical examples by com-
puting the absolute error

,
nn
E
xyxyx where

y
x is the exact solution and

n
y
x is the nth appro-
ximate solution of the problem. The absolute error
has evaluated with examples 1 and 2 for value of
and also the absolute error has calculated with
examples 3 and 4 for different values of
n
Ex
20
n
,
30n
and
respectively.
14n
Example 1. Consider the following generalized Abel’s
integral equation of second kind
 


24
74
34
12
0
Fresnel
16131115 1923π
,,1,, , ,,
214416 16 16 1616
d,
x
yx Cx
x
xF
yt t
xt

 

 

 

(7)
where

2
0
π
Fresnelcosd,
2
xt
Cx t



with exact solution
 
Fresnel
y
xCx solve the above integral equation by
taking 4 different choices of the initial guess
0.
y
x It is
observed that the method always converges to the exact
solution.
Case 1a. Taking the initial guess

0.
y
x as
 
0
24
74
34
Fresnel
16131115 1923π
,,1,, , ,,,
214416 16 16 1616
yx Cx
x
xF

 

 

 

the various approximate solutions

,
n
y
x obtained from
Equation (6), are given as
1
2
52
24
34
Fresnel
3
84137911 13π
,,1,,, ,,
4488 8816
15 π
yx Cx
x
x
F




 


 
 
2
3
13 4
24
34
Fresnel
3
41 317212529π
,,1,, ,,,
17 4416 16161616
4
yx Cx
x
x
F




 


 
  


Case 1b. Now we take a value of the initial
gu
different
ess
0
y
xerfcx and we get
 
34
4x
1
74
2
22
24
74
34
Fresnel 3
32111 15
,1 ,,,
288
21 π
161311 15 1923π
,,1,, , ,,,
214416 16 16 1616
yx Cx
xFx
x
xF


 

 

 

 

 

 
2
32
2
2
52
2
22
2
52
24
34
3
44
Fresnel 3π
3
16 4179
,1 ,,,
15π244
3
84137911 13π
,,1,,, ,,
4488 8816
15 π
x
yx Cx
x
Fx
x
x
F










 
 



 

 





 


 
 
Case 1c. Taking a different value of the initial guess
01
y
xx
and solving we obtain
 
34
416x74
1
24
74
34
Fresnel 321
161311 15 1923π
,,1,, , ,,,
214416 16 16 1616
yx Cxx
x
xF

 

 

 
2
32
2
24
34
3
44
Fresnel 15 π
137911 13π
522,,1,,, ,,,
44 8888 16
x
yx Cx
x
xxF







 
 
 
 
Case 1d. In this case, we are taking the initial guess
0cos
y
xx
lowing approximati
and using Equation (6) we get the fol-
ons of the solution
Copyright © 2012 SciRes. AJCM
R. N. PRAJAPATI ET AL.
314
 

1
34 2
12
Fresnel
4711
1, ,,
x Cx
xx
F





24
74
34
38
84
161311 15 1923π
,,1,, , ,,,
214416 16 16 1616
y
x
xF



 

 

 

 

2
2
32
2
12
2
52
24
34
Fresnel
3
4457
1, , ,
44 4
3π
3
84137911 13π
,,1,,, ,,
4488 8816
15 π
yx Cx
xx
F
x
x
F



 











 
 



 

 

Figures 1-4 show the errors between the exact solution

y
x and the approximate solutions

20
y
x for the
different initials choices of

0
y
x foove four
paper, the variational iteration method has been
a user friendly algorithm
nder the variations of initial guess
r the ab
cases.
5. Conclusion
In this
successfully used to obtain
which is stable u

0
y
x to solve the generalized Abel’s integral equations.
The variational iteration method yields solutions in the
forms of a convergent series with easily calculable terms.
Figure 1. The absolute error E(x) for example 1 (case a).
Figure 3. The absolute error E(x) for example 1 (case c).
Figure 4. The absolute error for E(x) example 1 (case d).
It is shown that the variational iteration method is a
promising tool for such types of singular integral equa-
tions.
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