Open Journal of Applied Sciences, 2012, 2, 193-197
doi:10.4236/ojapps.2012.23029 Published Online September 2012 (http://www.SciRP.org/journal/ojapps)
Approximate Analytical Solutions of Fractional Coupled
mKdV Equation by Homotopy Analysis Method
Orkun Tasbozan, Alaattin Esen, Nuri Murat Yagmurlu
Department of Mathematics, İnonu University, Malatya, Turkey
Email: orkun.tasbozan@inonu.edu.tr
Received June 1, 2012; revised July 3, 2012; accepted July 15, 2012
ABSTRACT
In this paper, the approximate analytical solutions of the fractional coupled mKdV equation are obtained by homotopy
analysis method (HAM). The method includes an auxiliary parameter which provides a convenient way of adjusting
and controlling the convergence region of the series solution. The suitable value of auxiliary parameter is deter-
mined and the obtained results are presented graphically.
Keywords: Homotopy Analysis Method; Approximate Analytical Solution; Fractional Coupled mKdV Equation
1. Introduction
Fractional derivatives provide an excellent tool for the
description of memory and hereditary characteristics of
different materials and processes due to their non-locality
characteristics. This is the main advantage of fractional
derivatives in comparison with integer order model, in
which such effects are in fact neglected [1]. Several defi-
nitions of fractional integration and derivation such as
Riemann-Liouville’s and Caputo’s have been proposed.
The Riemann-Liouville integral operator [1] having order
0
, which is a real number, is defined as
   
1
0
1
d0,
x
Jfxxtt xft

(1)
and as for 0
 
0
J
fx fx
where the real function

,
f
x is said to be in the
space
0x
,C
R
, if there exists a real number p
such that 1


,
p
f
xxfx n
C
where and it
is said to be in the space

(0, ),C
1
fx
if and only if ,
n
hC
Its fractional derivative of order
.Nn0
is gen-
erally used
 
d
,1
d
nn
n
DfxJ fxnn
x


where n is an arbitrary integer. The Riemann-Liouville
integral operator has an important role for the develop-
ment of the theory of both fractional derivatives and in-
tegrals. In spite of this fact, it has certain disadvantages
when it comes to modelling real-world phenomena with
fractional differential equations. This problem has been
solved by M. Caputo first in his article [2] and then in his
book [3]. Caputo definition, which is a modification of
Riemann-Liouville definition, can be given as




 
1
0
1
=d,
Γ
0, 1.
nn
xnn
Dfx JDfx
tt
nf
nn




t
Note that Caputo derivative has the following two im-
portant properties

DJf xfx

and
 



1
0
0!
1.
k
nk
k
x
JDfx fxfk
nn


 
In recent years, many important phenomena in various
scientific and technological areas have been well de-
scribed by fractional differential equations. In general,
since these type equations are not exactly solved, their
numerical solution techniques have become increasingly
important. The HAM, a powerful tool for searching the
approximate solutions which was first proposed by Liao
[4,5], is one of such numerical solution techniques. Un-
like perturbation techniques, the HAM is not limited to
any small physical parameters in the considered equation.
Therefore, the HAM can overcome the foregoing restrict-
tions and limitations of perturbation techniques so it pro-
vides us with a powerful tool to analyze strongly nonlin-
ear problems [6]. The HAM has been proposed and suc-
Copyright © 2012 SciRes. OJAppS
O. TASBOZAN ET AL.
194
cessfully applied to solve several fractional differential
equations modeling problems arising in science and en-
gineering by many authors [6-18] and the references
therein. In this paper, we will apply the HAM to frac-
tional coupled mKdV equation by using Caputo's defini-
tion of fractional differentiation.
2. HAM Solutions of the Time-Fractional
Coupled mKdV Equation
In this section, we implement the HAM to the fractional
coupled mKdV equation defined by




22
22
3330
363
01,0
xx
tx
txxx xxx
x
uuu vvuv
D
Dv vvuuvuuvv
t
 
 
 
0
(2)
with the initial conditions

 
,0tanh 2,
,0tanh 2.
ux x
vx x
(3)
To investigate the HAM solutions of Equation (2) with
the initial conditions given by Equation (3), we can
choose the linear operator

,;,; ,1,2
iti
xtq Dxtqi




having the property
0
i
c
where are constants. From Equation (2),
we can now define nonlinear operators as
1, 2
i
csi
 






1
32
11 1
1
3
2
2
2
12
12
,; ,;,;
1
3,;
2
,;
,;
3
,; ,
,, ,,
;
3,
,
x
tq xtqxtq
xtq x
tx
xtq
xtq x
x
xtq xt
xtq x
q
x
tq
 


 
 






 








21 2
1
2
3
22
3
1
12
22
2
12
,;,,;
,;
,;
,; ,;
3
,;
6,;,;
,;
3,;,;.
xtq xtq
x
tq
xtq x
xtq xtq
x
tx
xtq
xtq xtqx
xtq
xtq xtqx









Therefore, we construct the zero-order deformation
equation as follows


10
11 12
1,;,
,;, ,;,
qxtquxt
qxtqxt


q
(4)


20
221 2
1,;,
,; ,,;.
qxtqvxt
qxtqxt


q
(5)
If we choose 0q
then we get


10
20
,;0 ,,0,
,;0 ,,0
xtuxtux
xtvxtv x


and 1q
, we obtain
 
12
,;1,, ,;1,.
x
tuxt xtvxt


Thus, as the embedding parameter q increases from 0
to 1, the solutions
 
,; 1,2
ixtq i
also change from
the initial values
0,uxt
0,vx and to the solutions
t
,uxt and
,t.i
vx If we expand

,;
x
tq
for 1, 2i
in Taylor series with respect to the embedding parameter
q, we obtain
 

10
1
20
1
,;,, ,
,;,, ,
m
m
m
m
m
m
x
tquxtuxtq
x
tqvxtvxtq


where
 
 
1
0
2
0
,;
1
,,
!
,;
1
,.
!
m
mm
q
m
mm
q
xtq
uxt mq
xtq
vxt mq
If the auxiliary linear operator, the initial guess and the
auxiliary parameter are properly chosen, as pointed
out by Liao [5,8], the above series converges at
1q
,
and one can have
 
0
1
,,m
m
uxtu xtuxt

,,
,
(6)
 
0
1
,,m
m
vxtvxtvxt

(7)
which should be one of the solutions of the original equ-
ation. Let’s define the following vectors
 
 

101
101
,,,, ,,,
,,,, ,,.
mn
mn
uxtuxtuxt
vxtvxt vxt
u
v
By differentiating Equations (4) and (5) m times with
respect to the embedding parameter q, we obtain the mth-
order deformation equations as follows
Copyright © 2012 SciRes. OJAppS
O. TASBOZAN ET AL. 195
 

111,1
,,
mmm mm
uxtuxtR



uv1
,
m
1m
(8)
 

122,1
,, ,
mmm mm
vxtv xtR




uv (9)
subject to the initial conditions
 
 
,0tanh 2,
,0tanh2
m
m
ux x
vx x
where

 
 
 
 
1,1 1
3
11
3
1–1
0
11
00
1
–1
00
,
,,
1
2
,
3,
,
3,,
3,,
mm m
mm
mmn
n
n
mn mn
knk
nk
mn
knk mn
nk
R
uxtuxt
tx
vxt
vxt
xx
uxt
uxtuxtx
uxtv xtvxt
x





















uv
,,

 
 
 
 


2,1 1
3
11
3
1–1
0
1–1
00
1
00
–1
,
,,
,
3,
,
6,,
3,,,
,
,
mm m
mm
mmn
n
n
mn mn
knk
nk
mn
knknk
nk
mn
R
vxtvxt
tx
uxt
vxt
xx
uxt
uxtv xtx
uxtu vxtxvx
vxt
x
t




















uv
t
and
0, 1,
1, 1.
m
m
m
By applying the operator
J
given by Equation (1),
which is the inverse of the operator t
D
, to the both
sides of the mth-order deformation Equations (8) and (9)
for we
obtain
1,m
 
111,11
,, ,
mmm tmmm
uxtu xtJR

uv
,

.

(10)
 
122,11
,, ,
mmm tmmm
vxtvxtJ R

uv (11)
For the purpose of simplicity, setting 12 and
by using Equations (10) and (11) with the initial condi-
tions given by Equation (3), we successively obtain
 
 
  
0
12
,tanh2,
4
,,
Γ1cos 2
uxt x
t
uxt hx

 
 


22
22
2
41
,Γ1cos 2
32tanh 2
,
Γ21cos2
t
uxt hx
tx
hx


and
  
 
 


0
12
22
22
2
,tanh2,
4
,,
Γ1cos 2
41
,Γ1cos 2
32tanh 2
,
Γ21cos2
vxtx
t
vxt hx
t
vxt hx
tx
hx



etc. Therefore, the series solutions expressed by the HAM
given in Equations (6) and (7) can be written in the fol-
lowing forms

 
012
012
,,,,
,,,,
uxtuxt uxt uxt
vxtvxt vxt vxt
,
.


(12)
To demonstrate the efficiency of the method, we com-
pare the HAM solutions of fractional coupled mKdV
equation given by Equation (12) for 0
with its ex-
act solutions [19]

  
,tanh24
,tanh24
uxtx t
vxtx t


,
.
The fact that HAM solution series contains the auxil-
iary parameter providing us with a simply way to
adjust and control the convergence of the solution series
should be noted. To obtain an appropriate range for ,
we consider the so-called -curve to choose a proper
value of which ensures that the solution series is
convergent, as pointed by Liao [5], by finding out the
valid region of corresponding to the line segments
nearly parallel to the horizontal axis. The -curves of
0,0.01u and
0.010v, are given by 3th-order HAM
solution given by Equation (12) for various
parame-
ters in Figure 1. It can be seen from the figure that the
valid range of lies in approximately
1.3 0.7.
Figure 2 shows the numerical solutions of
,uxt
and
,vxt at x = 2 from t = 0 to t = 0.5 for = –0.7, –1
and –1.3 obtained by 3th-order HAM for
1
and
analytical solutions, respectively. Between t = 0 and t =
0.5, it can be seen from the figure that the choice of
–0.7 is an appropriate value.
=
Figure 3 shows the numerical solutions of
,uxt
and
,vxt at x = 2 during for = –0.7
obtained by 3th-order HAM for
00t
0.9
.5
and 0.8
,
Copyright © 2012 SciRes. OJAppS
O. TASBOZAN ET AL.
196
respectively.
Figure 1. The -curves of 3th-order approximate solutions
obtained by the HAM.
Figure 2. The results obtained by the HAM for
= 1
and various by 3th-order approximate solution in com-
parison with the exact solution at x = 2.
Figure 3. The results obtained by the HAM for
= 0.9,
= 0.8 and = – 0.7 by 3th-order approximate solution at x =
2.
3. Conclusion
In this paper, the HAM has been successfully applied to
obtain approximate analytical solution of fractional cou-
pled mKdV equation. It has been also seen that the HAM
solution of the problem converges very rapidly to the
exact one by choosing an appropriate auxiliary parameter
whose valid range is determined using -curves
presented by Liao. In conclusion, this study shows that
the HAM is a powerful and efficient technique in finding
the approximate analytical solution of fractional coupled
mKdV equation and also many other fractional evolution
equations arising in various areas.
 
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