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S. S. Nourazar et al.

 3( 3 −1) x − 3( 3 −1) x 
e 4
 43 − 24 3
−e 4
( )x



3
3
4
4
3e
9
27 


+
υ ( x, t ) =
t
t2
2
3
3( 3 −1)
3( 3 −1)
2
8
3( 3 −1) 
3( 3 −1) 
 3( 3 −1) x
 3( 3 −1) x
x
x

x
x


e 4
+e 4
e 4

e 4

+e 4
+e 4








−3

3 −1




27 
+ 
16   3(
 e



(

1

)x

3 −1
4

+e



3

(

)x 

3 −1
4

4







   3(

e
  




3

3 3e
The Taylor series expansion for  −
3
2 2e

3 3 − 3 9 3 −12
x+
t
4
4

3 3e

2 2 3
e

3 − 3 9 3 −12
x+
t
4
4

−e
+e



3 3 − 3 9 3 −12
x−
t
4
4



3 3 + 3 9 3 −12
x−
t
4
4

(

)

)x 

3 −1
4

 − 3(
 − 4 + e




2

3 − 3 9 3 −12
x+
t
4
4
3 − 3 9 3 −12
x+
t
4
4

−e
+e

2
)x  

3 −1
4







3 3 − 3 9 3 −12
x−
t
4
4



3 3 + 3 9 3 −12
x−
t
4
4

)

(

)


3
 389 − 225 3 t + .





 is written as:



 3( 3 −1) x − 3( 3 −1) x 
e 4
 43 − 24 3
−e 4


3 3−4
3e 4
9
27 

=

+
t
t2
2
3
3( 3 −1)
3( 3 −1)
2
8




3
3
1
3
3
1
3
3
1
3
3
1
(
)
)
(
(
)
)
(





x
x


x
x
x
x
+e 4
e 4
e 4

e 4

+e 4
+e 4












2

   3( 3 −1) 2
 − 3( 3 −1) x  
x
1
27 

 − 4 + e 4
  389 − 225 3 t 3 + .
e 4
+ 
4 



 
16   3( 3 −1)
3( 3 −1) 
  
x

x


 
4
4


+e
 e

 

 


(

−3

)x

3 −1

(

(3.25)

(

)

(

)

(3.26)

)

Comparing Equation (3.26) with Equation (3.25), thus Equation (3.25) can be reduced to:
3 3 −3

x+

9 3 −12

t



3 3 − 3 9 3 −12
x−
t
4
4

4
3 3e 4
−e

υ ( x, t ) =


3
3
3
9
3
12
3
2 2

x+
t
4
e 4
+e

3 + 3 9 3 −12
x−
t
4
4

3 3 − 3 
3 3
5 − 3 
=
− tanh 
t  
 x +
2 2
2
 4 
 

(3.27)

This is the exact solution of the problem, Equation (3.19). Table 3 shows the trend of rapid convergence of the
results of S1 ( x, t ) = v0 ( x, t ) to S6 ( x, t ) = ∑ i = 0 vi ( x, t ) using the HPM solution toward the exact solution. The
5

maximum relative error of less than 0.038% is achieved in comparison to the exact solution as shown in Table 3.
Deng [12] obtained some travelling solitary wave solutions of Equation (1.1) by applying the first-integral
method as follows:
1

γ γ
 nγ ( ρ  α ) 
  n
(α  ρ ) γ + (α ± ρ )( n + 1)
υ ( x, t ) =
x−
t + x0   
 ± tanh 



2 ( n + 1)
 2 2
  
 4 ( n + 1) 

292

(3.28)