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3.2 Space-time fractional coupled Burgers equation
Let us consider the space-time fractional coupled Burgers equation as follows
.
0
)
(
2
0
)
(
2
2
2
2
2
x
uv
M
x
v
v
x
v
t
v
x
uv
L
x
u
u
x
u
t
u
(26)
The coupled fractional equations have appeared as model equation in mathematical physics,
which is derived by Esipov [32]. It is very significant that the system is a simple model of
sedimentation or evolution of scaled volume concentrations of two kinds of particles in fluid
suspensions or colloids, under the effect of gravity [33]. The constants
L
and
M
depend on the
system parameters such as the Peclet number, the Stokes velocity of particles due to gravity, and
the Brownian diffusivity.
One can introduce the following transformation
)
1
(
)
1
(
),
(
)
,
(
),
(
)
,
(
ct
x
v
t
x
v
u
t
x
u
(27)
where
c
is a constant, Eq. (26) can be converted to the following form
0
)
(
2
0
)
(
2
uv
M
v
v
v
v
c
uv
L
u
u
u
u
c
(28)
where primes denote the differentiation with respect to
.
According to the balancing principle, the solution of the system of eq. (28) can be expressed by a
polynomial in
)
(
e
as follows:
,
)
(
)
(
)
(
1
0
)
(
1
0
e
B
B
v
e
A
A
u
(29)
where
1
0
1
0
,
,
,
B
B
A
A
are constants to be determined later and
)
(
satisfies the equation (7).
By substituting eq. (29) into eq. (28) and using (7) frequently, one can obtain the following
system of algebraic equations by setting the coefficients of the polynomial in
)
(
e
to zero:
.
0
2
2
2
,
0
3
2
2
2
0
2
2
2
2
,
0
2
,
0
2
2
2
,
0
3
2
2
2
0
2
2
2
2
,
0
2
1
1
2
1
2
1
1
0
0
1
1
1
2
1
0
1
1
1
1
1
0
1
0
1
1
0
1
2
1
1
2
1
1
0
1
0
1
0
1
1
1
1
2
1
2
1
1
0
0
1
1
1
2
1
0
1
1
1
1
1
1
0
0
1
1
0
1
2
1
1
2
1
1
0
1
0
1
0
1
1
p
A
MB
p
B
p
B
A
MpB
A
MpB
pr
B
p
cB
r
B
p
B
B
r
A
MB
q
B
MA
r
B
MA
r
A
MB
r
cB
r
B
B
r
B
pq
B
q
B
q
cB
q
B
MA
q
A
MB
q
B
B
qr
B
p
A
LB
p
A
p
A
A
LpB
A
LpB
pr
A
p
cA
r
A
p
A
A
r
A
LB
q
B
LA
r
B
LA
r
B
LA
r
cA
r
A
A
r
A
pq
A
q
A
q
cA
q
B
LA
q
A
LB
q
A
A
qr
A
(30)
Solving the resulting algebraic equations (30) with aid of symbolic computation, such as Maple,
one obtains
LM
M
p
B
B
B
LM
L
p
A
M
B
L
A
M
B
Mr
r
LMB
c
1
)
1
(
,
,
1
)
1
(
,
)
1
(
)
1
(
,
1
2
2
1
0
0
1
0
0
0
0
(31)
where
0
B
,
q
p
,
and
r
are arbitrary constants.
By combing the equations (9), (27), (29) and (31), the space-time fractional coupled Burger’s
equation (26) has the following traveling wave solutions:
For Type 1:
,
0
4
,
0
,
))
(
4
5
.
0
tanh(
4
2
1
)
1
(
)
,
(
))
(
4
5
.
0
tanh(
4
2
1
)
1
(
)
1
(
)
1
(
)
,
(
2
0
2
2
0
1
0
2
2
0
1
q
r
q
r
q
r
q
r
LM
M
B
t
x
v
r
q
r
q
r
q
LM
L
M
B
L
t
x
u
(32)
,
0
4
,
0
,
))
(
4
5
.
0
tan(
4
2
1
)
1
(
)
,
(
))
(
4
5
.
0
tan(
4
2
1
)
1
(
)
1
(
)
1
(
)
,
(
2
0
2
2
0
2
0
2
2
0
2
q
r
q
r
r
q
r
q
q
LM
M
B
t
x
v
r
r
q
r
q
q
LM
L
M
B
L
t
x
u
(33)
,
0
4
,
0
,
1
))
(
exp(
1
)
1
(
)
,
(
1
))
(
exp(
1
)
1
(
)
1
(
)
1
(
)
,
(
2
0
0
3
0
0
3
q
r
q
r
r
LM
M
B
t
x
v
r
r
LM
L
M
B
L
t
x
u
(34)
,
0
4
,
0
,
0
,
4
))
(
2
)
(
1
)
1
(
)
,
(
4
))
(
2
)
(
1
)
1
(
)
1
(
)
1
(
)
,
(
2
0
0
2
0
4
0
0
2
0
4
q
r
r
q
r
r
LM
M
B
t
x
v
r
r
LM
L
M
B
L
t
x
u
(35)
where,
)
1
(
)
1
(
ct
x
and
.
1
2
2
0
0
M
B
Mr
r
LMB
c
For Type 2:
0
,
0
,
)
)
1
(
)
1
(
(
tan
1
)
1
(
)
,
(
)
)
1
(
)
1
(
(
tan
1
)
1
(
)
1
(
)
1
(
)
,
(
0
0
5
0
0
5
q
p
ct
x
pq
pq
LM
M
B
t
x
v
ct
x
pq
pq
LM
L
M
B
L
t
x
u
(36)
,
0
,
0
,
)
1
(
)
1
(
cot
1
)
1
(
)
,
(
)
1
(
)
1
(
cot
1
)
1
(
)
1
(
)
1
(
)
,
(
0
0
6
0
0
6
q
p
ct
x
pq
pq
LM
M
B
t
x
v
ct
x
pq
pq
LM
L
M
B
L
t
x
u
(37)
,
0
,
0
,
)
)
1
(
)
1
(
(
tanh
1
)
1
(
)
,
(
)
)
1
(
)
1
(
(
tanh
1
)
1
(
)
1
(
)
1
(
)
,
(
0
0
7
0
0
7
q
p
ct
x
pq
pq
LM
M
B
t
x
v
ct
x
pq
LM
L
M
B
L
t
x
u
(38)
,
0
,
0
,
)
1
(
)
1
(
coth
1
)
1
(
)
,
(
)
1
(
)
1
(
coth
1
)
1
(
)
1
(
)
1
(
)
,
(
0
0
8
0
0
8
q
p
ct
x
pq
pq
LM
M
B
t
x
v
ct
x
pq
pq
LM
L
M
B
L
t
x
u
(39)
where
.
1
2
2
0
0
M
B
LMB
c
For Type 3:
,
0
,
0
,
)
1
(
)
1
(
1
)
1
(
)
,
(
)
1
(
)
1
(
1
)
1
(
)
1
(
)
1
(
)
,
(
1
0
0
9
1
0
0
9
ct
x
LM
M
B
t
x
v
ct
x
LM
L
M
B
L
t
x
u
(40)
where,
.
1
2
2
0
0
M
B
LMB
c
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