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1. A PROJECT REPORT ON
NEW METHOD TO SOLVE NONLINEAR EVOLUTIONARY
EQUATION BOTH HIGHER & LOWER ORDERS IN
PLASMAS
A Dissertation to University of Science & Technology, Meghalaya in Partial
Fulfillment of the Requirement for the Degree of Master of Science
SUBMITTED BY:
NAME: SAYEDUL HASSAN
MSM 4th SEMESTER
ROLL NUMBER: 17/MSM/17
EXAMINATION ID: 19240821
ENROLLMENT NUMBER: PG/2017/03295
UNDER THE SUPERVISION OF: Mr. RIDIP SARMA
DEPARTMENT OF MATHEMATICS
SCHOOL OF APPLIED SCIENCES

2. CHAPTER-I
BRIEF DESCRIPTION OF THE





 
G
G
-EXPANSION METHOD & TRAVELLING WAVE SOLUTIONS OF NONLINEAR EVOLUTION EQUATION
INTRODUCTION
The solutions of nonlinear evolution equations by using various different methods is the main goal for many researchers, and
many powerful methods to construct exact solutions of nonlinear evolution equations have been established and developed
such as the inverse scattering transform.
In the present letter we shall propose a new method which is called the





 
G
G
-expansion method to look for travelling wave solutions of nonlinear evolution equations. The main ideas of the proposed
method are that the travelling wave solutions of a nonlinear evolution





 
G
G
 

G
G 
equation can be expressed by a polynomial in , where satisfies a second order
LODE,
  Vt
x
d
dG
G 


 


, , the degree of the polynomial can be determined by considering
the homogeneous balance between the highest order derivatives and nonlinear terms appearing in a given nonlinear
evolution equation, and the coefficients of the polynomial can be obtained by solving a set of algebraic equations resulted
from the process of using the proposed method. It will be seen that more travelling wave solutions of many nonlinear
evolution equation can be obtained by using the 




 
G
G -expansion method

3. 




 
G
G





 
G
G
The rest of the letter is organized as follows. We describe the
-expansion method for finding travelling wave solutions of nonlinear evolution equations, and give the main steps of
the method here. We illustrate the method in detail with the celebrated KdV equation, the features of the
-expansion method are briefly summarized.
CHAPTER-II
DESCRIPTION OF THE





 
G
G
-EXPANSION METHOD





 
G
G
  )
1
.
1
(
0
,......
,
,
,
,
, 

xx
xt
tt
x
t u
u
u
u
u
u
P
In this section we describe the
-expansion method for finding travelling wave solutions of nonlinear evolution equation, say in two independent variables x and t, is given by
 
t
x
u
u ,

 
t
x
u
u ,






 
G
G
Where is an unknown function, P is a polynomial in
and it’s various partial derivatives, in which the highest order derivatives and nonlinear terms are involved. In the following we give the main steps of the
-expansion method

4. Vt
x 


     
2
.
1
,.......
,
, 


 Vt
x
u
t
x
u 

Step 1. Combining the independent variables x and t into one variables , we suppose that
 

u
u 
   
3
.
1
0
,.......
,
,
,
,
, 2











 u
u
V
u
V
u
u
V
u
P
The travelling wave variable (1.2) permits us reducing eq. (1.1) to an ODE for





 
G
G
   
4
.
1
.,
.......... 






 

m
m
G
G
u 

Step 2.Suppose that the solution of ODE (1.3) can be expressed by a polynomial in as follows:
 

G
G 
 
5
.
1
0 





 G
G
G 

Where satisfies the second order LODE in the form

 .,
,.........
m
 0

m
 




 
G
G
& are constants to be determined later, , the unwritten part in (1.4) is also a polynomial in
but the degree of which is generally equal to or less than m-1, the positive integer m can be determined by considering the
homogeneous balance between the highest order derivative and nonlinear terms appearing in ODE (1.3).

5. Step 3. By substituting (1.4) into eq. (1.3) and using second order LODE (1.5), collecting all terms with the same order of 




 
G
G
together, the left-hand side of eq. (1.3) is converted into





 
G
G

 ,
...,
,......... V
m 
another polynomial in . Equating each coefficient of this polynomial to zero, yields a set of algebraic equations for
and
.

 ,
,.......V
m

V
m ,.......,

Step 4. Assuming that the constants and
can be obtained by solving the algebraic equations in step 3, since the general solutions of the second order LODE (1.5) have been well known
for us, then substituting
and the general solutions of eq. (1.5) into (1.4) we have more travelling wave solutions of the nonlinear evolution equation (1.1).
CHAPTER-III
KDV EQUATION
We start with the celebrated kdv equation in the form
)
1
(
0 


 xxx
x
t u
uu
u 
Which arises in many physical problems such as surface water waves and ion-acoustic waves in plasma. The travelling wave variable below
     
2
,
, 


 vt
x
u
t
x
u 

 

u
u 
Permits us converting Equation (1) into an ODE for

6. 0








 u
u
u
u
V 

 
3
0
2
1 2






 u
u
Vu
C 





 
G
G
Integrating it with respect to once yields
Where C is an integration constant that is to be determined later
Suppose that the solution of ODE (3) can be expressed by a polynomial in as follows
   
4
...
.......... 






 

m
m
G
G
u 

 

G
G 
 
5
0 





 G
G
G 

Where
satisfies the second order LODE in the form
By using (4) & (5) it is easily derived that
 
6
..
..........
2
2
2







 

m
m
G
G
u 
..........
1






 




m
m
G
G
m
u 
   
7
.
..........
1
2







 





m
m
G
G
m
m
u 

7. 2
&u
u 

2
2
2 


 m
m
m
Considering the homogenous balance between in equation (3), based on (6) & (7) We required that
, So we can write (4) aS
  



 





 






 

G
G
G
G
u 1
2
2
     
9
2
2
2 2
1
2
2
2
1
3
1
2
4
2
2
2







 






 







 






 
 

 









G
G
G
G
G
G
G
G
u
,
And therefore
By using (8) and (5) it is derived that
         
10
2
2
6
4
3
8
10
2
6 1
2
2
2
1
1
2
2
2
2
1
2
3
2
1
4
2 







 








 








 







 


 




















G
G
G
G
G
G
G
G
u

8. 




 
G
G





 
G
G
By substituting (8)-(10) into equation (3) and collecting all terms with the same power of
together, the left hand side of equation (3) is converted into another polynomial in
Equating each co-efficient of this polynomial to zero, yields a set of simultaneous algebraic equations for




 ,
,
,
,
, 1
2 V
 & C as follows_
 
 
 
 
0
6
2
1
:
4
0
10
2
:
3
2
2
1
:
2
0
2
6
:
1
0
2
2
1
:
0
2
2
2
2
1
1
2
1
2
1
2
2
1
1
2
1
1
1
2
2
2




















































V
V
V
C

9. Solving the algebraic equations above, yields
   
11
12
24
8
2
1
8
,
12
,
12
2
2
2
2
2
2
2
1
2
































C
V

, 

& are arbitrary constant.
   
12
12
12
2







 






 

 




G
G
G
G
u
 t
x 2
8 


 


 
By using (11), expression (8) can be written as
Where,
Equation (12) is the formula of a solution of equation (3), provided that the integration constant C in equation (3) is taken as
that in (11).
Substituting the general solution of equation (5) into (12) we have three types of travelling wave solutions of the kdv equation
(1) as follows_

10. CASE 1: When ,
0
4
2

 

2
4
,
2
4 2
2





 






D















 









 




2
4
2
2
4
1
2
2
. e
C
e
C
F
C
General Solution is-















 









 




2
4
2
2
4
1
2
2
e
C
e
C
G

11. 2
2
4
2
4
2
2
4
2
4
2
4
2
2
4
1
2
4
2
2
4
1
2
2
4
2
2
4
1
2
2
4
2
2
4
1
2
4
2
2
2
4
1
2
2
2
2
2
2
2
2
2
2
2






































































































 










 











 









 









 









 









 









 









 









 









 









 


e
C
e
C
e
C
e
C
G
G
e
C
e
C
e
C
e
C
e
C
e
C
G
2
2
4
2
2
4
2
2
4
1
2
2
4
2
2
4
1
2
2
2
2
2



















































 








 








 








 
e
e
C
e
C
e
e
C
e
C

12. 2
4
2
1
cosh
4
2
1
sinh
4
2
1
cosh
4
2
1
sinh
2
4
2
4
2
1
sinh
4
2
1
cosh
4
2
1
cosh
4
2
1
sinh
4
2
1
sinh
4
2
1
cosh
_
4
2
1
cosh
4
2
1
sinh
2
4
2
2
2
2
2
2
2
2
2
2
1
2
1
2
2
2
2
2
1
2
1
2



































































A
B
B
A
C
C
C
C
C
C
C
C
where, 2
1
2
1 , C
C
B
C
C
A 



    ,
2
2
2
2
2
2
2
1
3
4
2
1
sinh
4
2
1
cosh
4
2
1
cosh
4
2
1
sinh
4
3

a
A
B
Y
Z
Z
Y
A
B




















 












2
1 C
C
Y 
 2
1 C
C
Z 

Where and are the arbitrary constants.

13. ,
,
0
,
0 2
2
Z
Y
Y 


 0


In order to match the solution with the standard form of K-dV soliton solution, we have chosen a particular case with
and , and the solution reads as
  







 
 





2
sec
3 2
2
1 h
A
B
a ,
tanh 1
Y
Z



  
  .
8 2




 


 B
A
a
where
B
V

2
 0


a
Again for the values and Z=0, we find which enables to find the soliton solution as
  







 


B
V
h
A
V
2
sec
3 2
where Vt

 

Or
  








w
m h




 2
sec
A
V
m
3


V
B
w 2


where and
Which is the well-known solitary wave solution of a K-dV equation.

14. Case 2: When ,
0
4
2

 
 the solution of the equation is
   







 












2
4
sin
2
4
cos
2
2
2
1
2
C
C
e
G
From which Gis evaluated as
         







 


















 






















 



2
4
sin
2
4
cos
2
2
4
cos
2
4
sin
2
4 2
2
2
1
2
2
2
2
1
2
2
C
C
e
C
C
e
G
And 




 
G
G is then evaluated as
     
   







 









 




















2
4
sin
2
4
cos
2
4
cos
2
4
sin
2
4
2
2
2
1
2
2
2
1
2
C
C
C
C
G
G

15. 1
C 2
C
Where and are arbitrary constant. Solution has been obtain as
    
a
A
B
C
C
C
C
A
B























 2
2
2
2
2
1
2
2
2
1
2
2
3
4
2
1
sin
4
2
1
cos
4
2
1
cos
4
2
1
sin
4
3

















Because ,
0
4
2

 

the K-dV equation derives the sine-cosine wave. But our focused only to find the solitary wave, i.e, soliton solurtion in isolation,
and discarded the discussions on it.
Case 3: When 0
4
2

 
 , then
 


 2
2
1


 e
C
C
G
G
 

 



2
1
2
2
2
2
C
C
e
e
C
G 





Giving as

16. From which G
G
is then evaluated as
 
 
2
2
2
1
2
2
2
1
2
1
2
2
2






















C
C
C
e
C
C
C
C
e
e
C
G
G
 


 2
3B
A
a 

  1
C 2
C
Where and and are arbitrary constants.
Finally the K-dV equation derives the solution in the following form:
 
  
a
A
B
C
C
C
A
B




 2
2
2
1
2
2
3
3
12




It is a simple solution of the equation, and we are not interested to take up this for the discussions.

17. CONCLUSION





 
G
G
 

G
G 
The -expansion method is firstly proposed, where satisfies a second order
linear ordinary differential equation, by which the travelling wave solutions involving
parameters of the KdV equation, the mKdV equation, the variant boussinesq equations and
the Hirota-Satsuma equations are obtained. When the parameters are taken as special
values the solitary waves are also derived from the travelling waves. The travelling wave
solutions are expressed by the hyperbolic functions, the trigonometric functions and the
rational functions. The proposed method is direct, concise, elementary and effective, and
can be used for many other nonlinear evolution equations.