Int. Journal of Math. Analysis, Vol. 2, 2008, no. 12, 581 - 589
Explicit Approximate Solution of the Coupled KdV
Equations by using the Homotopy Analysis Method
M. M. Rashidi, G. Domairry∗, A. DoostHosseini, S. Dinarvand
Mechanical Engineering Department, Engineering Faculty of Bu- Ali Sina University,
Hamedan, Iran
Mechanical Engineering Department, Engineering Faculty of Mazandaran University,
Babol, P.O. Box 484, Iran
Abstract
In this letter, the homotopy analysis method (HAM), one the most effective
method, is implemented for finding an approximate solution of the Coupled KdV
equations. Also, we try and show how close this method approaches the exact
solution.
Key words: Homotopy analysis method, KdV equations, nonlinear partial differential
equations.
1. Introduction
Modeling of natural phenomena and problems mostly leads to solving nonlinear
equations. Solution of these equations is not easy especially through analytical
approach. Therefore, endless of researchers are directed to decrease error in the
solutions.
In 1992, Liao [1] employed the basic ideas of the homotopy in topology to
propose a general analytical method for nonlinear problems, namely homotopy
analysis method (HAM) [2-7]. Based on homotopy of topology, the validity of the
∗
Corresponding author: G. Domairry
E-mail address: amirganga111@yahoo.com
582
M. M. Rashidi, G. Domairry, A. DoostHosseini, S. Dinarvand
HAM is independent of whether or not there exist small parameters in the considered
equation. The HAM avoids discretization and provides an efficient numerical
solution with high accuracy, minimal calculation, a voidance of physically unrealistic
assumptions.
This method has been successfully applied to solve many types of nonlinear
problems [8-12]. In this paper, the basic idea of HAM is described, and then we apply
it to the coupled KdV equations, which describe the motions of waves in nonlinear
optics, plasma or fluids. After finding a suitable h (auxiliary parameter), the results
are compared with exact results and finally the advantages are discussed.
2. Basic idea of HAM
Let us consider the following differential equation
N [u (τ )] = 0
(1)
Where N is a nonlinear operator, τ denotes in dependent variable, u (τ ) is an
unknown function that is the solution of the equation. We define the function
φ (τ ; p ) = u 0 (τ ),
(2)
p →0
Where, p ∈ [0,1] and u 0 (τ ) is the initial guess which satisfies the initial or boundary
condition and is
lim φ (τ ; p ) = u (τ ).
p →1
(3)
By means of generalizing the traditional homotopy method, Liao [5] constructs the
so- called zero- order deformation equation
(1 − p ) L [φ (τ ; p ) − u 0 (τ )] = p h H (τ ) [φ (τ ; p )],
(4)
where h is the auxiliary parameter which increases the results convergence, H (τ ) ≠ 0
is an auxiliary function and L is an auxiliary linear operator, p increases from 0 to 1,
the solution φ (τ ; p ) changes between the initial guess u 0 (τ ; p ) and solution u (τ ) .
Expanding φ (τ ; p ) in Taylor series with respect to p, we have
∞
φ (τ ; p ) = u 0 (τ ) + ∑ u m (τ ) p m ,
m =1
(5)
where
u m (τ ) =
1 ∂ m φ (τ ; p )
m! ∂ pm
,
p =0
(6)
Explicit approximate solution
583
if the auxiliary linear operator, the initial guess, the auxiliary parameter h, and the
auxiliary function are so properly chosen, the series (5) converges at p = 1 , then we
have
∞
u (τ ) = u 0 (τ ) + ∑ u m (τ ),
(7)
m =1
which must be one of the solutions of the original nonlinear equation, as proved by
Liao [5]. It’s clear that if the auxiliary parameter is h = −1 and auxiliary function is
determined to be H (τ ) = 1, Eq. (4) will be
(1 − p ) L [φ (τ ; p ) − u 0 (τ )] + p N [φ (τ ; p )] = 0.
(8)
This statement is commonly used in HPM procedure. Indeed, in HPM we solve the
nonlinear differential equation by separating every Taylor expansion term.
r
Now we define the vector of u m as follows:
r
r r r
r
u m = {u1 , u 2 , u 3 ,..., u n }
According to the definition Eq. (6), the governing equation and the corresponding
initial condition of u m (τ ) can be deduced from zero-order deformation Eq. (4).
Differentiating Eq.(4) for mtimes with respect to the embedding parameter p and
setting p = 0 and finally dividing by m!, we will have the so-called mth order
deformation equation in the following from
r
L [u m (τ ) − x m u m −1 (τ )] = h H (τ ) R m (u m −1 ),
(9)
where
r
R m (u m −1 ) =
∂
1
m −1
( m − 1)!
N [φ (τ ; p )]
∂p
m −1
,
(10)
p =0
and
⎧0
xm = ⎨
⎩1
m ≤ 1,
m > 1.
(11)
So by applying inverse linear operator to both sides of the linear equation, Eq. (9), we
can easily solve the equation and compute the generation constant by applying the
initial or boundary condition.
3. Applications
Let us first consider the coupled KdV equations [13] in the form
u t = a (u x x x + 6u u x ) + 2bv v x ,
(12-a)
v t = −u x x x − 3uv x ,
(12-b)
with initial conditions [13]
584
M. M. Rashidi, G. Domairry, A. DoostHosseini, S. Dinarvand
1+ a
u ( x , 0) = −
3 + 6a
Me
v ( x , 0) =
(1 + e
2
k + 4k
e
2
kx
(1 + e
kx 2
(13-a)
,
)
kx
(13-b)
,
kx 2
)
1
24a
2
b
Where a ≠ − ( ) , ab < 0 and M = ( −
1/ 2
)
2
k , where k is an arbitrary constant. And we
consider the exact solution of coupled KdV [13]
u (x , t ) = −
v (x , t ) =
1+ a
3 + 6a
M e
(1 + e
k
2
+ 4k
e
2
k ( x +ct )
(1 + e
k ( x +ct ) 2
(14-a)
,
)
k ( x +ct )
k ( x +ct ) 2
(14-b)
,
)
Where M, k, a and c are arbitrary constant. For application of homotopy analysis
method, we choose the initial approximation
u 0 (x , t ) = −
(
v 0 (x , t ) =
1+ a
2
3 + 6a
−24a
k + 4k
1/ 2
2
) k e
b
kx 2
(1 + e )
e
2
kx
(1 + e
kx 2
(15-a)
,
)
kx
(15-b)
,
and the linear operator for equation (12-a)
L [φ1 ( x , t ; p )] =
∂φ1 ( x , t ; p )
∂t
(16-a)
,
and the linear operator for equation (12-b)
L [φ2 ( x , t ; p )] =
∂φ2 ( x , t ; p )
∂t
(16-b)
,
we change equations (12-a), (12-b) to nonlinear form:
⎛ ∂ 3φ1 ( x , t ; p )
∂φ ( x , t ; p )
∂φ ( x , t ; p ) ⎞
N 1[φ1 ( x , t ; p ), ϕ 2 ( x , t ; p )] = 1
−a⎜
+ 6φ1 ( x , t ; p ) 1
⎟
3
∂t
∂x
∂x
⎝
⎠
−2b φ2 ( x , t ; p )
∂ φ2 ( x , t ; p )
(17-a)
∂x
N 2 [φ1 ( x , t ; p ), φ2 ( x , t ; p )] =
∂φ2 ( x , t ; p )
∂x
∂ φ2 ( x , t ; p )
∂φ ( x , t ; p )
+ 3φ1 ( x , t ; p ) 2
3
∂x
∂x
3
+
(17-b)
Assuming H (τ ) = 1, we use the above definition to construct the zero-order
deformation equations
(1 − p ) L [φ1 ( x , t ; p ) − u 0 ( x , t )] = phN 1[φ1 ( x , t ; p ), φ2 ( x , t ; p )],
(18-a)
Explicit approximate solution
585
(1 − p ) L [φ2 ( x , t ; p ) − v 0 ( x , t )] = phN 2 [φ1 ( x , t ; p ), φ2 ( x , t ; p )],
(18-b)
Obviously, when p=0 and p=1,
φ1 ( x , t ; 0) = u 0 ( x , t ),
φ1 ( x , t ;1) = u ( x , t ).
φ2 ( x , t , 0) = v 0 ( x , t ),
φ2 ( x , t ,1) = v ( x , t ).
(19-a)
(19-b)
Differentiating the zero-order deformation equations (18-a), (18-b) mtimes with
respect to p, and finally dividing by m!, we have the mth-order deformation equations
r
r
L [u m ( x , t ) − X m u m −1 ( x , t )] = h R m (u m −1 , v m −1 ),
(20-a)
r
r
L [u m ( x , t ) − X m v m −1 ( x , t )] = h R m′ (u m −1 ,v m −1 ),
(20-b)
where
⎛ ∂ u m −1 ( x , t ) m −1
∂u
(x , t )
∂u
(x , t ) ⎞
r
r
R m (u m −1 ,v m −1 ) = m −1
−a⎜
+ 6 ∑ u n ( x , t ) m −1− n
⎟
3
n =0
∂t
∂x
∂x
⎝
⎠
3
m −1
∂v m −1− n ( x , t )
n =0
∂x
− 2b ∑ v n ( x , t )
(21-a)
,
3
m −1
∂v
( x , t ) ∂ v m −1 ( x , t )
∂v
(x , t )
r
r
+
+ 3 ∑ u n ( x , t ) m −1− n
,
R m′ (u m −1 ,v m −1 ) = m −1
3
0
n
=
∂t
∂x
∂x
(21-b)
and
⎧0
xm = ⎨
⎩1
m ≤ 1,
m > 1.
From (15-a), (15-b) and (21-a), (21-b), we now successively obtain
u 0 (x , t ) = −
(
(1 + a ) k
3 + 6a
−24a
1/ 2
)
2
+
4e
(1 + e
2
k e
5
2
kx 2
(22)
,
)
(23)
kx
) tanh(
kx
2
1 + 2a
u1 ( x , t ) = −
3
2
)
2
(24)
,
a
kx
b
2
( − )3/ 2 b h k 5t sech 2 (
) tanh(
kx
)
2
5
4
kx
2
(25)
,
1 + 2a
a h k t sech (
u 2 (x , t ) =
2
,
a h k t sech (
v 1 (x , t ) =
k
kx
b
kx 2
(1 + e )
v 0 (x , t ) =
kx
3
)(a h k t ( −2 + a cosh( k x )) − 2(1 + 2a )(1 + h ) sinh( k x ))
4(1 + 2a )
2
,
(26)
586
M. M. Rashidi, G. Domairry, A. DoostHosseini, S. Dinarvand
3
a
5
4 kx
3
a − h k t sech (
)(a h k t ( −2 + cosh ( k x )) − 2(1 + 2a )(1 + h ) sinh( k x ))
2
b
2
,
2
4(1 + 2a )
v 2 (x , t ) =
u 3 (x , t ) = −
1
5
24(1 + 2a )
3
5
(a h k t sech (
kx
3
)(18a (1 + 2a ) h (1 + h ) k t cosh(
kx
2
3
− 6a (1 + 2a ) h (1 + h ) k t cosh (
(27)
)
2
3k x
2
2
2
2
6 2
) + 2(6(1 + 2a ) (1 + h ) − 5a h k t
(28)
2
+ (6(1 + 2a ) (1 + h ) + a h k t )cosh( k x ) ) sinh(
2
2
2
2
6 2
kx
))),
2
v 3 (x , t ) =
a 3/ 2
5
5 kx
(( − ) b h k t sech (
)
b
2
4 6 (1 + 2a )
1
3
3
(9a (1 + 2a ) h (1 + h ) k t cosh(
kx
2
2
2
sinh(
kx
3
2
3
kx
)
2
) + a h k t ( −3(1 + 2a )(1 + h )cosh(
a h k t ( −5 + cosh( k x ))sinh(
2
) + 12(1 + 2a ) (1 + h ) cosh (
3k x
)+
(29)
2
kx
)))),
2
M
8
8
k =0
k =0
We used 9 terms in evaluating the approximate solution u approx = ∑ u k ,v approx = ∑ v k .
8
u approx = ∑ u k = u 0 + u 1 + u 2 + ...
k =0
8
v approx = ∑ v k = v 0 + v 1 + v 2 + ...
k =0
(30)
Explicit approximate solution
587
h
Fig.1. h curve of
given by 9th order approximate solution (30) when
u t (1, 0), v t (1, 0)
a = 1, b = −1, k = 1, c = 1
Note that this series contains the auxiliary parameter h, which influence its
convergence region and rate. We should therefore focus on the choice of h by plotting
of h-curve. Fig. 1, shows the h-curve of u t (1, 0) , v t (1, 0) given by approximate solution
(30) when a = 1, b = −1, k = 1, c = 1, in which the solid line represents u t (1, 0) and the
dashed line represents v t (1, 0)
According to h-curve diagram, we select h = −1 and we plot the three-dimensional
plot of approximate solutions obtained by HAM. We can see from the plots that
solutions obtained by HAM have small relative error to the exact solution of the
problem.
0.03
0.02
0.01
0
10
8
6
 200
4
 10
0
x
2
10
t
0.03
0.02
0.01
0
10
8
6
 200
4
 10
0
x
2
10
20 0
(a)
20 0
(b)
t
588
M. M. Rashidi, G. Domairry, A. DoostHosseini, S. Dinarvand
0.04
10
0.02
8
0
 200
6
4
 10
0
x
t
0.04
10
0.02
8
0
 200
6
4
 10
2
10
20 0
(c)
0
x
t
2
10
20 0
(d)
Fig. 2. All graphs are plot with a = 1 , b = −1 , k = 0.2 , c = 0.2 , h = −1 , −20 ≤ x ≤ 20 , 0 ≤ t ≤ 10 .
(a) Approximate solution u ( x , t ) obtained by HAM, (b) Exact solution u ( x , t ) , (c)
Approximate solution v ( x , t ) , (d) Exact solution v ( x , t ) .
4. Conclusion
In this paper, the homotopy analysis method is used to compute the approximate
solutions of the coupled KdV equations. We achieved a very good approximation
with the actual solution of the considered system. And also, this technique is
algorithmic and it is easy to implementation by symbolic computation software, such
as Maple and Mathematica. The numerical results of the above examples display a
fast convergence, with minimal calculations. It shows that the HAM is a very
efficient method. We sincerely hope this method can be applied in a wider range.
References
[1] S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear
problems, Ph.D. Thesis, Shanghai Jiao Tong University,1992.
[2] S.J. Liao, An approximate solution technique not depending on small parameters:
A special example, Int. J. Nonlinear Mech. 30 (1995), 371-380.
Explicit approximate solution
589
[3] S.J. Liao, A kind of approximate solution technique which does not depend upon
small parameters — II. An application in fluid mechanics, Int. J. Nonlinear Mech. 32
(5) (1997), 815-822.
[4] S.J. Liao, An explicit, totally analytic approximate solution for Blasius’ viscous
flow problems, Int. J. Nonlinear Mech. 34 (4) (1999), 759-778.
[5] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method,
Chapman and Hall/CRC Press, Boca Raton (2003).
[6] S.J. Liao, An explicit, totally analytic approximate solution for Blasius’ viscous
flow problems, Appl. Math. Comput. 147 (2004), 499-513.
[7] S.J. Liao, Comparison between the homotopy analysis method and homotopy
perturbation method, Appl. Math. Comput. 169 (2005), 1186-1194.
[8] T. Hayat, M. Khan, M. Ayub, On the explicit analytic solutions of an Oldroyd 6constant fluid, Int. J. Eng. Sci. 42 (2004), 123-135.
[9] S.J. Liao, I. Pop, Explicit analytic solution for similarity boundary layer equations,
Int. J. Heat Mass Transfer, 47 (2004), 75-85.
[10] S. Abbasbandy, The application of homotopy analysis method to nonlinear
equations arising in heat transfer, Phys. Lett. A, 360 (2006), 109-113.
[11] S. Abbasbandy, The application of homotopy analysis method to solve a
generalized Hirota–Satsuma coupled KdV equation, Phys. Lett. A, 361 (2007), 478483.
[12] T. Hayat, M. Sajid, On analytic solution for thin film flow of a fourth grade fluid
down a vertical cylinder, Phys. Lett. A, 361 (2007), 316-322.
[13] Kaya D, Inan IE, Exact and numerical traveling wave solutions for nonlinear
coupled equations using symbolic computation, Appl Math Comput, 151 (2004), 775787.
Received: December 7, 2007