Application of Optimal Homotopy Asymptotic Method for the

Applied Mathematical Sciences, Vol. 8, 2014, no. 18, 875 - 884
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2014.312706
Application of Optimal Homotopy Asymptotic
Method for the Approximate Solution
of Kawahara Equation
Bothayna S. Kashkari
Department of Mathematics, Sciences Faculty for Girls
King Abdulaziz University, Jeddah, Saudi Arabia
Copyright © 2014 Bothayna S. Kashkari. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, nonlinear Kawahara equation is solved by Optimal Homotopy
asymptotic method (OHAM). Comparisons are made between the exact solution,
Homotopy Perturbation method (HPM), Variational homotopy perturbation
method (VHPM) and variational iteration method (VIM). The results reveal that
the proposed method is very useful for problems with large domain, very
effective and easy to use.
Keywords: Kawahara equatio; Optimal Homotopy analysis method; Homotopy
perturbation method; Variational homotopy perturbation method; variational
iteration method.
1. Introduction
Nonlinear phenomena play a crucial role in applied mathematics and physics.
The Kawahara equation was first proposed by Kawahara in 1972, as a model
equation describing solitary wave propagation in media [7]. The Kawahara
equation occurs in the theory of magneto-acoustic waves in plasma and in the
theory of shallow water waves with surface tension. The existence and uniqueness
876
Bothayna S. Kashkari
of solutions are obtained by Shuangping and Shuangbin [9]. In the literature, this
equation is also referred as a fifth–order KdV equation [3].
In recent decades, some works have been done in order to find the numerical
solution of this equation, for example: Adomain decomposithon method;
Variational iteration [1]; Dual-Perov-Galerkin method [2]; Homotopy analysis
method [6]. Later, in 2008, Marinca et al. [11-13] introduced a new analytical
method known as OHAM to obtaining an approximate solution of nonlinear
problems. This method is straightforward and reliable.
In this work, OHAM has been used to solve Kawahara equation given by:
u t  uu x  u 3x   u 5 x  0
(1.1)
with the initial condition:
u (x ,0)  g (x )
(1.2)
where  ,  and  are nonzero positive arbitrary constants.
Recently, the hyperbolic function solutions of Kawahara eq. (1.1) was derived by
using G  - expansion method [8] and is given by:
G
u (x , t )  
3 12(  2   ) 105  2
13

sech 4 (
169

169 
26

36
(x 
t ))

169
(1.3)
In order to communicate the reliability of OHAM, we deal with different
examples in the subsequent section. Finally, numerical comparison between
OHAM and other existing methods shows the efficiency of OHAM.
2. Analysis of the method
Let us consider the partial differential equation of the following form:
L (u (x ,t ))  f (x ,t )  N (u (x ,t ))  0,
B (u ,
u
)  0,
t
x 
(2.1)
x 
(2.2)
where L is the simpler part of the partial differential equation which is easier to
solve, N is a nonlinear operator, B is a boundary operator, u (x , t ) is an
unknown function, x and t denote spatial and temporal independent variables,
respectively,  is the boundary of the domain  and f (x ,t ) is a known analytic
function.
Now, we construct an optimal homotopy  (x ,t ; p ) : [0,1]   . which satisfies
Application of optimal homotopy asymptotic method
877
H ( (x , t ; p ), p )  (1  p ){L ( (x , t ; p ))  f (x , t )}
(2.3)
H ( p ){L ( (x , t ; p ))  f (x , t )  N ( (x , t ; p ))}  0
 (x , t ; p )
(2.4)
B ( (x , t ; p ),
)0
x
where, p [0,1] is an embedding parameter, H ( p ) is a nonzero auxiliary
function for p  0 and H (0)  0 . Clearly, we have:
(2.5)
p  0  H ( (x ,t ;0)),0)  L ( (x ,t ;0))  f (x ,t )  0
(2.6)
p  1  H ( (x ,t ;1)),1)  H (1){L ( (x ,t ;1))  f (x ,t )  N( (x ,t ;1)}  0
When p  0 and p  1 , it holds that  (x , t ; 0)  u 0 (x , t ) and  (x ,t ;1)  u (x ,t )
respectively. Thus, as p varies from 0 to 1, the solution  (x ,t ; p ) goes from
u 0 (x , t ) to u (x ,t ) , where u 0 (x , t ) is obtain from eq. (2.3) and eq. (2.4) with
p  0 given
u
(2.7)
L ( (x , t ;0))  f (x , t )  0 ,
B (u 0 , 0 )  0
t
The auxiliary function H ( p ) is chosen in the form
H ( p )  pC1  p 2C 2  p 3C 3  ,
(2.8)
where C 1 ,C 2 ,C 3 , are constants to be determined later. To get an approximate
solution,  (x , t ; p ,C i ) is expanded in a Taylors series about p as

 ( x , t ; p ,C i )  u 0 ( x , t )  u k ( x , t , C i ) p k ,
i  1, 2,3,
(2.9)
k 1
Substituting eq.(2.9) into eq.(2.3) and equating the coefficients of like powers of
p , the first and second – order problems are given as
u
(2.10)
L (u1 (x , t ))  C 1N 0 (u 0 (x , t )) ,
B (u1 , 1 )  0
t
and
L (u 2 (x , t ))  L (u 1 (x , t ))  C 2 N 0 (u 0 (x , t ))  C 1[L (u 1 (x , t ))  N 1 (u 0 (x , t ),u 1 (x , t ))] ,
u 2
)0
t
the general governing equations for u k ( x , t ) are given by
L (u k (x , t ))  L (u k 1 (x , t ))  C k N 0 (u 0 (x , t )) 
B (u 2 ,
k 1
C [L (u
i 1
i
k i
(x , t ))  N k i (u 0 (x , t ),u 1 (x , t ),
(2.11)
,u k i (x , t ))] ,
u k
(2.12)
)0 ,
k  2,3,
t
where N k i (u 0 (x , t ), u1 ( x , t ), , u k i ( x , t )) is the coefficient of p k i in the
expansion of N ( (x ,t ; p ) about the parameter p and
B (u k ,
878
Bothayna S. Kashkari
N ( (x , t ; p ,C i )  N 0 (u 0 (x , t ))   N k (u 0 ,u 1 ,u 2 ,
k 1
,u k ) p k , i  1, 2,3
(2.13)
The convergence of the series (2.9) depends upon the auxiliary constants
C 1 ,C 2 ,C 3 , if it is convergent at p  1 , one has
u (x , t ;C i )  u 0 (x , t )  u k (x , t ,C i ) ,
i  1, 2,3,
(2.14)
k 1
Substituting eq.(2.14) into eq.(2.1) there results the following residual
(2.15)
R (x , t ;C i )  L (u (x , t ;C i ))  f (x , t )  N (u (x , t ;C i ))
If R ( x , t ;C i ) , then u (x , t ;C i ) will be the exact solution. Generally such a case
will not arise for nonlinear problem.
For the determinations of auxiliary constants, C 1 ,C 2 ,C 3 ,
there are many
methods like Galerkin’s Method, Ritz Method, Least Squares Method and
Collocation Method to find the optimal values of C 1 ,C 2 ,C 3 , One can apply the
Method of Least Squares as under
t
J (C i )    R 2 (x , t ,C i ) dx dt ,
i  1, 2,3,
(2.16)
0
and
J
J
J
(2.17)

 
0
C 1 C 2
C n
The constants C 1 ,C 2 ,C 3 , can also be determined as under
R (x 1 , t;C i )  R (x 2 , t;C i )   R (x n , t;C i )  0 ,
i  1, 2, , n (2.18)
at any time t ,where x i   .
It is clear that for the low order of n, the nonlinear algebraic system can be solved
with some ease but if n is large it becomes more difficult to solve.
3. Solution of Kawahara equation by OHAM
Example 1:
Consider Kawahara equation (1.1) with       1 and with initial condition [10]
ut  uu x  u 3x  u 5x  0
(3.1)
72 105

sech 4 (kx )
169 169
The exact solution of this equation is
72 105
u (x , t ) 

sech 4 (k (x  ct ))
169 169
1
36
where k 
and c 
.
169
2 13
u 0 (x , 0)  
(3.2)
(3.3)
Application of optimal homotopy asymptotic method
Following the OHAM formulation presented in Section 2 we start with
 (x , t ; p )
L ( (x , t ; p )) 
t
 (x , t ; p ) 3 (x , t ; p ) 5 (x , t ; p )
N ( (x , t ; p ))   (x , t ; p )


x
x 3
x 5
f (x , t )  0
with the boundary condition
72 105
 (x , 0; p )  

sech 4 (kx )
169 169
Zeroth Order Problem
72 105
x
u o (x , t )
u 0 (x ,0)  

sech 4 (
)
0,
t
169 169
2 13
its solution is
3 
x 
u 0 (x , t ) 
24  35sech 4 (
)

169 
2 13 
First-order problem
x
x
7560C 1 sech 4 (
) tanh(
)
u1 (x , t )
2
13
2
13

u1 (x , 0)  0
,
t
28561 13
its solution is
x
x
7560C 1 t sech 4 (
) tanh(
)
2
13
2
13
u1 (x , t ) 
28561 13
879
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
Second order problem
u 2 (x , t )

t
3780sech 6 (
x 
x
x 
)  108C 12 t  72C 12 t cosh(
)  169 13 C 1  C 12  C 2  sinh(
)
2 13 
13
13 
,
62748517
u 2 (x , 0)  0
its solution is
(3.12)
u 2 (x , t ) 
3780 t sech 6 (

x
x
x 
)  54C 12 t  36C 12 t cosh(
)  169 13 C 1  C 12  C 2  sinh(
)
2 13 
13
13 
62748517
(3.13)
Substituting eq. (3.9), eq.(3.11) and eq.(3.13) in (2.14), we get the third-order
approximate solution of (3.1) and (3.2).
u ( x , t ;C 1 , C 2 )  u 0 ( x , t )  u 1 ( x , t )  u 2 ( x , t )
(3.14)
880
Bothayna S. Kashkari
For the calculations of the constants C 1 and C 2 using (3.14) in (2.15) and applying
the procedure mentioned in (2.18), we get
C 1  0.9502755734372723 , C 2  0.0027377082434278804
The approximate solutions and the corresponding errors are shown in Table 1.
The results produced by the OHAM are in a very good agreement with the best of
the results of the HPM, This is clear in Table 2. Thus, the OHAM is seen to be
very efficient in solving Kawarara equation.
Table 1. Absolute error corresponding to example 1 at time
and
.
2.0
4.0
6.0
8.0
10.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.
0
Table 2. Result of the proposed method compared with results in [11] at time
and
.
OHAM
HPM
1.0
4.945485119
2.0
1.889124802
3.0
2.183730037
4.0
9.099936950
5.0
1.313920964
Example 2:
Now, we consider the Equation (3.1) with initial condition [4] , [5]
72 420  sech  kx 
u 0 (x , 0)  

169 169 1  sech kx 2
  
2


2
The exact solution of this equation is
2
72 420 sech (k  x  ct )
u (x , t )  

169 169 1  sech 2 (k  x  ct ) 2
where k 
1
36
and c 
.
169
2 13
(3.15)
(3.16)
Application of optimal homotopy asymptotic method
881
again we apply the OHAM formulation with the boundary condition
72 420 sech 2 ( kx )
 (x , 0; p )  

169 169 1  sech 2 (kx ) 2
(3.17)
Zeroth Order Problem
u o (x , t )
0,
t
x
)
72 420
2
13
u 0 (x , 0)  

2
169 169 
x 
2
)
1  sech (
2 13 

sech 2 (
its solution is
x
2x 

12 13  34cosh(
)  3cosh(
)
13
13 

u 0 (x , t ) 
2
x 

169  3  cosh(
)
13 

First-order problem
(3.18)
(3.19)
u (x , t )
1

t
x
2x
3x
4x
5x
6x 

315C  3216 sinh (
)  6979 sinh (
)  5366 sinh (
)  362 sinh (
)  10 sinh (
)  5 sinh (
)
1
13
13
13
13
13
13 


x 

28561 13  3  cosh(
)
13 

u1 (x , 0)  0
its solution is
7
,
(3.20)
u 1 (x , t ) 
x
2x
3x
4x
5x
6x 

315C t  3216 sinh(
)  6979 sinh(
)  5366 sinh(
)  362 sinh(
)  10 sinh(
)  5 sinh(
)
1
13
13
13
13
13
13 


7
x 

28561 13  3  cosh(
)
13 

(3.21)
Substituting eq. (3.19) and eq.(3.21) in (2.14), we get the second-order
approximate solution of (3.1) and (3.15).
u ( x , t ;C 1 )  u 0 ( x , t )  u 1 ( x , t )
(3.22)
For the calculations of the constant C 1 using (3.22) in (2.15) and applying the
procedure mentioned in (2.18), we get
C 1  1.0970949215398926
The approximate solutions and the corresponding errors are shown in Table 3.
The results produced by the OHAM are in a very good agreement with the best of
the results of the methods listed in Table 4.Thus, the OHAM is seen to be very
efficient in solving Kawarara equation.
882
Bothayna S. Kashkari
Table 3. Absolute error corresponding to example 1 at time
and
.
2.0
4.0
6.0
8.0
10.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.
0
Table 4. Result of the proposed method compared with results in [12] and [13] at time
and
OHAM
.
VIM
VHPM
0.1
2.4345
6.0868
0.2
4.9774
9.4059
0.3
7.6287
1.3000
0.4
1.0388
1.6000
0.5
1.3256
1.9000
0.1
7.1835
3.5000
0.2
1.4475
4.5000
0.3
2.1875
5.4000
0.4
2.9383
6.4000
0.5
3.6999
7.4000
0.1
1.1898
8.5000
0.2
2.3904
1.0100
0.3
3.6017
1.1700
0.4
4.8239
1.3400
0.5
6.0568
1.5000
Application of optimal homotopy asymptotic method
883
4. Conclusion
In this paper, the Optimal Homotopy Asymptotic Method has been proposed to
obtain numerical solution of Kawahara equation. It is investigated that the
technique is explicit, reliable and easy to use. This method is very useful for
problems with large domain. This method provides us a convenient way to control
the convergence and we can easily adjust the desired convergence regions.
Absolute error in Table 1 and showed that the results are very consistent with the
increasing time. The proposed method has been compared with the other
published methods listed in Table 2 and 4 seen to be very accurate. This method
has a great potential to attract researchers, scientists and engineer of every field.
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Received: December 15, 2013

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