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
-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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