Traveling Wave Solutions for Space-Time Fractional Nonlinear Evolution Equations

2. Representation of the method 
This section presents the brief descriptions of the proposed method for finding the exact traveling 
to the nonlinear FPDEs. 
One can consider the nonlinear FPDEs to the following form:
1
,
0
,
0
)
,
,
,
,
,
,
(














u
D
D
u
D
D
u
D
u
D
u
D
u
f
x
x
t
t
x
x
t
,
(1) 
where, 
f
is a function of 
)
,
(
t
x
u
and its partial fractional derivatives, in which higher order 
derivatives and nonlinear terms are involved.
Let us consider the traveling wave variable transformation as 
,
)
1
(
)
1
(
),
(
)
,
(













ct
kx
U
t
x
u
(2) 
where 
k 
and 
c
are constants. 
The Jumarie’s modified Riemann-Liouville [28] derivatives of order 

can be defined by the 
following expression 


















.
1
,
1
;
))
(
(
1
0
;
))
0
(
)
(
(
)
(
)
1
(
1
)
(
)
(
0
n
n
n
z
f
d
f
f
z
dz
d
z
f
D
n
n
z
z









(3) 
Moreover, one can define the modified Riemann-Liouville derivative [29, 30] as
,
0
,
)
1
(
)
1
(
















z
z
D
z
(4) 
Using Eq. (2) and (4), Eq. (1) can be converted as a nonlinear ordinary differential equation 
(ODE) for
)
(

U
U

: 
0
)
,
,
,
,
(







U
U
U
U
F
,
(5) 
where, 
F
is a function of 



,
,
,
,
U
U
U
U



and its derivatives point out the ordinary 
derivatives with respect to 

. 
One can be considered
 
the traveling wave solutions of Eq. (5) as 

---

 
 


0
,
0






N
N
i
i
i
A
e
A
u


(6) 
where the coefficients
)
0
(
N
i
A
i


are constants to be evaluated and 
)
(




satisfies the 
following first order nonlinear ordinary differential equation: 
,
)
(
)
(
)
(
r
e
q
e
p










(7) 
Balancing the higher order derivative with the nonlinear terms of the highest order that appeared 
in Eq. (5), one can get the value of the positive integer
N
. On the other hand, if the degree of 
)
(

U
U

is
n
U
D

)]
(
[

, then the degree of the other expressions can be found by the following 
formulae: 
)
(
]
)
(
[
,
]
)
(
[
k
n
S
nN
d
U
d
u
D
p
N
d
U
d
D
S
K
K
N
N
N















. 
(8) 
Substituting Eq. (6) into Eq. (5) and using (7) rapidly, one can obtain a system of algebraic 
equations for
c
r
q
p
k
N
i
A
i
,
,
,
,
),
0
(


. With the help of symbolic computation, such as Maple, one 
can evaluate the obtaining system and find out the values
c
r
q
p
k
N
i
A
i
,
,
,
,
),
0
(


. It is notable 
that equation (7) has the following general solutions: 
Type 1
: when 
,
1

p
one obtains


,
0
4
,
0
,
2
2
)
(
4
5
.
0
tanh
4
ln
)
(
2
0
2
2


















q
r
q
q
q
q
r
q
r




(9a) 


,
0
4
,
0
,
2
2
)
(
4
5
.
0
tan
4
ln
)
(
2
0
2
2

















q
r
q
q
q
q
r
q
r




(9b) 
,
0
4
,
0
,
0
,
1
))
(
exp(
ln
)
(
2
0















q
r
r
q
r
r



(9c) 
,
0
4
,
0
,
0
,
)
(
)
2
)
(
(
2
ln
)
(
2
0
2
0
















q
r
r
q
r
r





(9d) 
Type 2
: when 
,
0

r
one obtains




0
,
0
,
tan
ln
)
(
0













q
p
pq
q
p



, (9e) 

---





,
0
,
0
,
cot
ln
)
(
0














q
p
pq
q
p



(9f) 




,
0
,
tanh
ln
)
sgn(
)
(
0















pq
pq
q
p
p



(9g) 




0
,
coth
ln
)
sgn(
)
(
0















pq
pq
q
p
p



, (9h) 
Type 3:
when
0

q
and 
,
0

r
one obtains




,
ln
)
(
0






p
(9i) 
where 
0

is the integrating constant. 
Finally, we are obtained the multiple explicit solutions of nonlinear FPDE (1) by combining the 
equations (2), (6) and (9). 
3. Applications of the method 
This section presents four examples to illustrate the applicability of the proposed method to solve 
the space-time FDEs.
3.1 Space-time fractional Burgers equation 
Let us consider the space-time Burgers equation as follows 















A
t
x
x
u
A
x
u
u
t
u
,
0
,
0
,
1
,
0
,
0
2
2
2








(10) 
When 
1




, the fractional equation (10) can be reduced to the well known Burgers equation. 
The space-time FDE (10) have appeared a mathematical model equation not only in fluid flow 
[31] but also it can be applied for describing various types of physical phenomena in the field of 
gas dynamics, heat conduction, elasticity, continuous stochastic processes, etc.
Introducing the following transformation in eq. (10), 
,
)
1
(
)
1
(
),
(
)
,
(













ct
kx
w
t
x
u
(11) 
where 
k
and 
c
are constants, one can rewrite the equation (11) to the following nonlinear ODE as 
follows: 

---

,
0
2
2






w
Ak
w
kw
w
c
(12) 
where primes denote the differentiation with respect to 

. Integrating the eq. (12) once and 
setting the constant of integration to zero for time homogeneity, we get 
,
0
2
2




w
Ak
kw
cw
(13) 
Using the balancing principle between 
w

and 
2
w
in eq. (13) gives 
.
1

N
Therefore the solution 
of (13) can be written as
)
(
1
0
)
(






e
A
A
w
(14) 
where 
1
0
,
A
A
are constants to be determined later and 
)
(


satisfies the auxiliary nonlinear ODE 
(7). 
Substituting eq. (14) into eq. (13) and using (7) frequently, the left-hand side of eq. (13) becomes 
a polynomial in
)
(



e
. Setting the coefficients of this polynomial to zero yields a system of 
algebraic equations as follows:


0
,
0
2
,
0
1
2
2
1
1
2
1
0
1
2
0
1
2
0








p
A
Ak
kA
r
A
Ak
A
kA
cA
kA
q
A
k
A
cA
(15) 
Solving the resulting algebraic equations (15), we have 

















)
4
(
,
,
),
2
4
(
2
2
1
2
0
pq
r
r
Ak
c
k
k
k
A
p
A
pq
r
r
k
A
A
(16) 
where 
k, 
q
p
,
and 
r
are arbitrary constants. 
Combing the solutions of Eq. (7), (11), (14) and (16), one can obtain the following explicit 
solutions to the space-time fractional Burgers equation (10): 
For Type 1: 
,
0
4
,
0
,
))
(
4
5
.
0
tanh(
4
2
2
4
)
(
2
0
2
2
2
1
1



























q
r
q
r
q
r
q
r
q
q
r
r
k
A
u



(17) 
,
0
4
,
0
,
))
(
)
4
(
5
.
0
tan(
)
4
(
2
2
4
)
(
2
0
2
2
2
1
2





























q
r
q
r
q
r
q
r
q
q
r
r
Ak
u



(18) 

---

0
4
,
0
,
1
))
(
exp(
2
4
)
(
2
0
2
1
3























q
r
q
r
r
q
r
r
Ak
u



(19) 
,
0
4
,
0
,
0
,
4
))
(
2
)
(
2
4
)
,
(
2
0
0
2
2
1
4

























q
r
r
q
r
r
q
r
r
Ak
t
x
u




(20) 
where, 
.
)
1
(
)
4
(
)
1
(
2
2





t
q
r
r
Ak
kx








For Type 2:
,
0
,
0
,
)
1
(
2
)
1
(
tanh
1
)
,
(
0
2
1
5






































p
p
t
pq
A
k
kx
pq
pq
Ak
t
x
u






(21) 
,
0
,
0
,
)
1
(
2
)
1
(
coth
1
)
,
(
0
2
1
6






































p
p
t
pq
A
k
kx
pq
pq
Ak
t
x
u






, (22) 
,
0
,
0
,
)
1
(
2
)
1
(
tan
)
,
(
0
2
1
7





































q
p
t
pq
a
k
kx
pq
pq
pq
Ak
t
x
u






(23) 
0
,
)
1
(
2
)
1
(
cot
1
)
,
(
0
2
1
8


































pq
t
pq
a
k
kx
pq
pq
Ak
t
x
u






, 
(24) 
For Type 3:
,
0
,
0
,
)
1
(
)
4
2
(
)
1
(
2
4
)
,
(
1
0
2
2
2
1
9


















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r
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kx
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t
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
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(25)