American Journal of Computational Mathematics, 2011, 1, 235-239
doi:10.4236/ajcm.2011.14027 Published Online December 2011 (http://www.SciRP.org/journal/ajcm)
Analytical Solution of Two Extended Model Equations for
Shallow Water Waves by He’ s Va riational Iteration
Method
Mehdi Safari1, Majid Safari2
1Department of Mechanical Engineering, Islamic Azad University, Aligoodarz Branch, Aligoodarz, Iran
2Department of Management, Islamic Azad University, Aligoodarz Branch, Aligoodarz, Iran
Received July 1, 2011; revised September 26, 2011; accepted October 8, 201 1
Abstract
In this paper, we consider two extended model equations for shallow water waves. We use He’s variational
iteration method (VIM) to solve them. It is proved that this method is a very good tool for shallow water
wave equations and the obtained solutions are shown graphically.
Keywords: He’s Variational Iteration Method, Shallow Water Wave Equation
1. Introduction
Clarkson et al. [1] investigated the generalized short wa-
ter wave (GSWW) equation
d
x
txxtt xtx
uuuuu uxu   
0, (1)
where

and

are non-zero constants.
Ablowitz et al. [2] studied the specific case 4


and 2

 where Equation (1) is reduced to
42 d
x
txxtt xtx
uuuuu uxu 
0,
0,
(2)
This equation was introduced as a model equation
which reduces to the KdV equation in the long small
amplitude limit [2,3]. However, Hirota et al. [3] exam-
ined the model equation for shallow water waves
33 d
x
txxt txt x
uuuuuuxu 
 (3)
obtained by substituting 3

  in (1).
Equation (2) can be transformed to the bilinear fo rms

2
1
3
0,
xt txxtsx
DD DDDDDD
ff

 



3
(4)
where s is an auxiliary variable, and f satisfies the bilin-
ear equation

30,
xs x
DDD ff (5)
However, Equation (3) can be transformed to the bi-
linear form


20,
xt txx
DDDDD ff

 (6)
where the solution of the equation is

 
,2ln ,
x
x
uxt f (7)
where f(x, t) is given by the perturbation ex p ansion
 
1
,1 ,
n
n
n
,
f
xtf xt 



 (8)
where ε is a bookkeeping non-small parameter, and


,
n
f
xt , n = 1, 2, ··· are unknown fun ctions that will be
determined by substituting the last equation into the bi-
linear form and solving the resulting equations by equat-
ing different powers of

to zero.
The customary definition of the Hirota’s bilinear op-
erators are given by


.,
,|, .
nm
nm
tx
DD aba xt
tt xx
bxt xxt t
 
 
 
 

 
 
 

(9)
Some of the properties of the D-operators are as fol-
lows

23
22
24 2
2
22
6
23
42
22
dd,3d ,
,3,
ln,1515 ,
ttx
ttxt t
xx
x
tx x
xx
xt
DffDDf f
uxxu uxux
ff
Df fDf f
uuu
ff
DDf fDf f
f
uuuu
ff







 

(10)

M. SAFARI ET AL. 236
where


 
,2ln,

,
x
x
uxtf xt (11)
Also extended model of Equation (2) is obtained by
the operator 4
x
D to the bilinear forms (4) and (5)

23
1
3
0,
xt txxxtsx
DD DDDDDDD
ff

 



3
(12)
where s is an auxiliary variable, and
f
satisfies the
bilinear equation

3.0
xs x
DDD ff,
0,
(13)
Using the properties of the D operators given above,
and differentiating with respect to x we obtain the ex-
tended model for Equation (2) given by
4
2d 6
txxt t
x
xtx xxxx
uu uu
uuxuu uu


 (14)
In a like manner, we extend Equation (3) by adding
the operator 4
x
D to the bilinear forms (6) to obtain

23
.0
xt txxx
DDDDDD ff 

,
0,
(15)
Using the properties of the D operators given above
we obtain the extended model for Equation (3) given by
3
3d 6
txxt t
x
xt xxxxx
uu uu
uuxuu uu


 (16)
In this paper, we use the He’s variational iteration
method (VIM) to obtain the solution of two considered
equations above for shallow water waves. The varia-
tional iteration method (VIM) [4-10] established in (1999)
by He is thoroughly used by many researchers to handle
linear and nonlinear models. The reliability of the
method and the reduction in the size of computational
domain gave this method a wider applicability. The
method has been proved by many authors [11-15] to be
reliable and efficient for a wide variety of scientific ap-
plications, linear and nonlinear as well. The method
gives rapidly convergent successive approximations of
the exact solution if such a solution exists. For concrete
problems, a few numbers of approximations can be used
for numerical purposes with high degree of accuracy.
The VIM does not require specific transformations or
nonlinear terms as required by some existing techniques.
2. Basic Idea of He’s Variational Iteration
Method
To clarify the basic ideas of VIM, we consider the fol-
lowing differential equation:

LuNugt

 (17)
where is a linear operator, a nonlinear operator
and LN


g
t an inhomogeneous term.
According to VIM, we can write down a correction
functional as follows:
  

10d
tnn
nn
Lu Nu
utut g






 


 (18)
where

is a general Lagrangian multiplier which can
be identified optimally via the variational theory. The
subscript indicates the th approximation and
is considered as a restricted variation
nnn u

0
n
u


.
3. VIM Implement for First Extended Model
of Shallow Water Wave Equation
Now let us consider the application of VIM for first ex-
tended model of shallow water wave equation with the
initial condition of:


211
1sech 21
,0 22
c
cx
c
ux c








(19)
Its correction variational functional in x and t can be
expressed, respectively, as follows:
 
  
  
3
12
0
0
3
3
,,
(,) (,)
,,,
4, 2d
,, ,
6, d
tnn
nn
x
nnn
n
nn n
n
ux ux
uxtuxtx
uxux u
ux x
ux uxux
ux
xx
x






 




 







 
 





 (20)
where

is general Lagrangian multiplier.
After some calculations, we obtain the following sta-
tionary conditions:


0



 (21a)


1
t 

0

 (21b)
Thus we have:


1,t   (22)
As a result, we obtain the following iteration formula:
  
  
  
3
12
0
0
3
3
,,
,,
,,,
4, 2d
,, ,
6, d
tnn
nn
x
nnn
n
nn n
n
ux ux
uxtuxt x
uxux u
ux x
ux uxux
ux
xx x






 












 
 

 


 (23)

M. SAFARI ET AL.237
We start with the initial approximation of


,0ux
given by Equation (19). Using the above iteration for-
mula (23), we can directly obtain the other components
as follows:


2
0
11
1sech21
,22
c
cx
c
uxt c








(24)



12
3
1
,11
cosh 1
21
1111
1 coshcosh
2212
11 1
2sinh ,
21 1
uxt
cxc
c
cc
cc xx
cc
cc
xtc
cc




















 






1
1




(25)
  
  
 
3
11
21 2
0
111
10
3
111
1
3
,,
,,
,,),
4, 2d
,, ,
6, d
t
x
ux ux
uxt uxtx
uxux u
ux x
ux uxux
ux
xx x






 



 







 
 

 

 (26)
In Figure 1 we can see the 3-D result of first extended
model of shallow water wave equation by VIM.
4. VIM Implement for Second Extended
Model of Shallow Water Wave Equation
At last we consider the application of VIM for second
extended model of shallow water wave equation with the
initial condition given by Equation (19).
Its correction variational functional in x and t can be
expressed, respectively, as follows:
  
  
  
3
12
0
0
3
3
,,
,,
,,,
3, 3d
,, ,
6, d
tnn
nn
x
nnn
n
nn n
n
ux ux
uxtuxt x
uxux u
ux x
ux uxux
ux
xxx








 




 







 
 




 (27)
where

is general Lagrangian multiplier.
After some calculations, we obtain the following sta-
tionary conditions:

0 
 (28)

1
t 



Figure 1. For the first extended model of shallow water
wave equation with the first initial condition (24) of Equa-
tion (14), when c = 2.
Thus we have:


1,t 

 (30)
As a result, we obtain the following iteration formula:




   
  
 
1
3
2
0
0
3
3
,,
,,
3,
,, ,
3d
,,
6, d
nn
tnn n
n
x
nn n
nn
n
uxtuxt
ux uxux
ux
x
ux uux
xx
ux ux
ux x
x
,




 





 




 




 

 

 (31)
We start with the initial approximation of


,0ux
given by Equation (19). Using the above iteration for-
mula (31), we can directly obtain the other components
as follows:


2
0
11
1sech21
,22
c
cx
c
uxt c








(32)


12
3
1
,11
cosh 1
21
1111
(1)cosh cosh
2212
11 1
2sinh ,
21 1
uxt
cxc
c
cc
cc xx
cc
cc
xtc
cc




















 






1
1
(33)
0 (29)

M. SAFARI ET AL. 238
Figure 2. For the second extended model of shallow water
wave equation with the first initial condition (32) of Equa-
tion (16), when c = 2.
 
  
  
 
21
3
11 1
1
2
0
11 1
0
311
1
3
,,
,,
3,
,, ,
3d
,,
6, d
t
x
uxtuxt
ux uxux
ux
x
ux uux
xx
ux ux
ux xx
,




 










 

 

 

 (34)
In Figure 2 we can see the 3-D result of second ex-
tended model of shallow water wave equation by VIM.
5. Acknowledgements
In this paper, He’s variational iteration method has been
successfully applied to find the solution of two extended
model equations for shallow water. The obtained results
were showed graphically it is proved that He’s varia-
tional iteration method is a powerful method for solving
these eq uatio ns . I n our work ; w e use d th e Map l e P ack ag e
to calculate the functions obtained from the He’s varia-
tional iteration method.
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M. SAFARI ET AL. 239
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