Applied Mathematics, 2014, 5, 1-6
Published Online January 2014 (http://www.scirp.org/journal/am)
http://dx.doi.org/10.4236/am.2014.51001
OPEN ACCESS AM
Exact Solution to Nonlinear Differential Equations of
Fractional Order via (Gʹ/G)-Expansion Method
Muhammad Younis, Asim Zafar
Centre for Undergraduate Studies, University of the Punjab, Lahore, Pakistan
Email: younis.pu@gmail.com, asimzafar@hotmail.com
Received August 16, 2013; revised September 16, 2013; accepted September 23, 2013
Copyright © 2014 Muhammad Younis, Asim Zafar. This is an open access article distributed under the Creative Commons Attribu-
tion License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited. In accordance of the Creative Commons Attribution License all Copyrights © 2014 are reserved for SCIRP and the owner of
the intellectual property Muhammad Younis, Asim Zafar. All Copyright © 2014 are guarded by law and by SCIRP as a guardian.
ABSTRACT
In this article, a new application to find the exact solutions of nonlinear partial time-space fractional differential
Equation has been discussed. Firstly, the fractional complex transformation has been implemented to convert
nonlinear partial fractional differential Equations into nonlinear ordinary differential Equations. Afterwards,
the (Gʹ/G)-expansion method has been implemented, to celebrate the exact solutions of these Equations, in the
sense of modified Riemann-Liouville derivative. As application, the exact solutions of time-space fractional Bur-
gers’ Equation have been discussed.
KEYWORDS
Exact Solutions; Complex Transformation; (Gʹ/G)-Expansion Method; Nonlinear BURGERS’ Equation;
Fractional Calculus Theory
1. Introduction
Nonlinear partial differential equations have shown a variety of applications in almost every field, such as in
electromagnetic, acoustics, electrochemistry, cosmology, biological and material science [1-4]. Fractional dif-
ferential equations can be considered as the generalization form of the differential equations, as they are in-
volved with the derivatives of any real or complex order (for details see [3]).
In literature a lot of work has been done on nonlinear partial differential equations [5-8] rather than nonlinear
partial differential equations (NPDEs) of fractional order. Recently, some new techniques have been introduced,
by different authors, to find traveling wave solutions for NPDEs of fractional order [9-11].
In this article, the
GG
-expansion method [5] has been applied in the sense of modified Riemann-
Liouville derivative to find the exact solutions of space-time fractional nonlinear Burgers’ Equation, which has
the following form [12]:
2
20, 0, 0, 1,
uuu
ut
txx


 


 (1)
The rest of the letter is organized as follows, in Section 2 the basic definitions and properties of the fractional
calculus are considered regarding to modified Riemann-Liouville derivative. In Section 3, the
GG
-expan-
sion method has been proposed to find the exact solutions for NPDEs of fractional order with the help of frac-
tional complex transformation. As an application, the new exact solutions of nonlinear Burgers’ Equation have
been discussed in Section 4. In last Section 5, the conclusion has been drawn.
2. Preliminaries and Basic Definitions
In this section, the method has been applied in the sense of the Jumarie’s modified Riemann-Liouville derivative
M. YOUNIS, A. ZAFAR
OPEN ACCESS AM
2
[13,14] of order
. For this, some basic definitions and properties of the fractional calculus theory are being
considered (for details see [3]). Thus, the fractional integral and derivatives can be defined following [13,14] as:
Definition 2.1 A real function

,>0fss , is said to be in the space ,CR
, if there exists a real number
>
p
such that
1
p
f
ssfs, where
10,fsC
, and it is said to be in the space m
C
if ,
m
f
C
mN.
Definition 2.2 The Jumarie’s modified Riemann-Liouville derivative, of order
, can be defined by the
following expression:
   




0
1d 0d, 01,
1d
,1,1.
s
sn
n
sff
s
Dfs
fsn nn
 
 

 
Moreover, some properties for the modified Riemann-Liouville derivative have also been given as follows:


 

 
 

1,
1
,
.
rr
s
sss
sgss
r
Ds s
r
Dfsgs fsDgsgsDts
DfgsfgsDgsDfgsgt








 
 
3. The (Gʹ/G)-Expansion Method for Nonlinear Fractional Partial Differential Equations
In this section, the
GG
-expansion method [15,16] has been discussed to obtain the solutions of nonlinear
fractional partial differential equations.
For this, we consider the following NPDE of fractional order:

,,,,,,,,,0,
for 0,,1,
tsx tttssssx
PuDuDuDu DDuDDuDDuDDu
 



(2)
where u is an unknown function and P is a polynomial of u and its partial fractional derivatives along
with the involvement of higher order derivatives and nonlinear terms.
To find the exact solutions, the
GG
-expansion method can be performed using the following steps.
Step 1: First, we convert the NFPDE into nonlinear ordinary differential Equations using fractional complex
transformation introduced by Li et al. [17].
The travelling wave variable
 

,,, 111
KtLx My
utxy u

 
  

(3)
where ,KL
and
M
are non-zero arbitrary constants, permits us to reduce Equation (2) to an ODE of
uu
in the following form
,, ,,0.Puuu u

(4)
If the possibility occurs, then Equation (4) can be integrated term by term one or more times.
Step 2: Suppose that the solution of Equation (4) can be expressed as a polynomial of
GG
in the form:

, 0,
i
m
im
im
G
uG
 




(5)
where i
s are constants and
G
satisfies the following second order linear ordinary differential equation.
0,GGG

 

(6)
with
and
as constants.
Step 3: The homogeneous balance can be used, to determine the positive integer m, between the highest or-
M. YOUNIS, A. ZAFAR
OPEN ACCESS AM
3
der derivatives and the nonlinear terms appearing in (4).
Moreover, the degree of
u
can be defined as
Du m

, which gives rise to the degree of the other
expressions as follows:

dd
,.
dd
s
qq
p
uu
D
mqDu mqsqm
qq

 

 

 
 

Therefore, the value of m can be obtained for the Equation (5).
Step 4: After the substitution of (5) into (4) and using Equation (6), we collect all the terms with the same or-
der of
GG
together. Equate each coefficient of the obtained polynomial to zero, yields the set of algebraic
equations for ,, ,,KLM
and

0, 1, 2,,
iim
.
Step 5: After solving the system of algebraic equations, and using the Equation (6), the variety of exact so-
lutions can be constructed.
4. Application of Burgers’ Equation
In this section, the improved
GG
-expansion method have been used to construct the exact solutions for
nonlinear space-time fractional Burgers’ Equation (1).
2
20,0, 0,1.
uuu
au bt
txx

 


 (7)
It can be observed that the fractional complex transform
 

,, 11
Kx Lt
uxt u

 
 

(8)
where K and L are constants, permits to reduce the Equation (7) into an ODE. After integrating once, we
have the following form:
22
0,
2
a
CLUKUbKU
 
(9)
where C is a constant of integration. Now by considering the homogeneous balance between the highest order
derivatives and nonlinear term presented in the above Equation, we have the following form

1
101 1
, 0,
GG
uGG
 

 
 
 
 (10)
where 101
,,,K

and L are arbitrary constants. To determine these constants substitute Equation (10)
into (9), and collecting all the terms with the same power of
GG
together, equating each coefficient equal
to zero, yields a set of algebraic Equations.


22
11
2
1101
22
00111
2
110 1
22
11
10,
2
0,
1210,
2
0,
10.
2
aKbK
LaKbK
CL aKbK
LaKbK
aKbK

 

 


 






After solving these algebraic equations with the help of software Maple, yields the following results.
Case 1 For the values:
22
101
2
, , and 0.
aC bLbK L
KabLa aK
 

M. YOUNIS, A. ZAFAR
OPEN ACCESS AM
4
Straightforward simplification of Equation (10) yields, the following equation

2.
bK GL
uaG aK



 (11)
From Equations (6) and (11), we have the following travelling wave solutions.
If 24>0

, then we have the following hyperbolic solution

22
2
22
cosh 4sinh 4
22
4.
sinh 4cosh 4
22
AB
bK L
uaaK
AB

 

 


 




 

 




If 24<0

, then we have the following trigonometric solution

22
2
22
sin4cos 4
22
4
cos 4sin4
22
AB
bK L
uaaK
AB

 
 
 


 




 

 




and if 240

, then we have the following solution

2,
bK BL
uaABaK




where

2
.
11
aCbLxLt
abL
 


Case 2 For the values:

22 22
01011
1
1
2
01
2
, ,
1 and 0.
bK bK
Ca
LbK
a
 


 

 
Straightforward simplification of Equation (10) yields, the following equation


1
2
12
,
bKG
uLbK
aaG


 
 (12)
From Equations (6) and (12), we have the following travelling wave solutions.
If 24>0

, then we have the following hyperbolic solution


1
22
1
22
2
22
cosh4 sinh4
122
4.
sinh4cosh 4
22
AB
bK
uLbK
aaAB

 
 



 




 
 




If 24<0

, then we have the following trigonometric solution


1
22
1
22
2
22
sin4cos4
122
4
cos4sin4
22
AB
bK
uLbK
aa AB

 
 
 







 
 




M. YOUNIS, A. ZAFAR
OPEN ACCESS AM
5
and if 240

, then we have the following solution


1
2
12 ,
bK B
uLbK
aaAB


 

where

.
11
Kx Lt



 
where A and B are arbitrary constants.
5. Conclusion
The
GG
-expansion method has been extended to solve the nonlinear partial differential equation of frac-
tional order, in the sense of modified Riemann-Liouville derivative. First, the fractional complex transformation
has been used to convert the fractional equations into ordinary differential equation. Then
GG
-expansion
method has been used to find exact solutions. As an application, the new exact solutions for the space-time frac-
tional Burgers’ Equations have been found. It can be concluded that this method is very simple, reliable and
proposes a variety of exact solutions to NPDEs of fractional order.
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