International Journal of Science & Technology
Volume 2, No 2, 179-188, 2007
A Comparison of Variational Iteration Method with Adomian’s
Decomposition Method in Some Highly Nonlinear Equations
D. D. GANJI, M. NOUROLLAHI and M. ROSTAMIAN
Mazandaran University, Department of Mechanical Engineering, P. O. Box 484, Babol - IRAN.
[email protected]
(Received: 05.07.2007; Accepted: 08.10.2007)
Abstract: In this paper, equations of Generalized Hirota-Satsuma coupled KdV equation, Kawahara equation and
FKdV equations are solved through variational iteration method (VIM) and the results are compared with those of
Adomian’s decomposition method (ADM). In addition, we show that VIM is able to solve a large class of nonlinear
problems effectively, more easily and accurately; and thus it has been widely applicable in engineering and physics.
Keywords: Variational Iteration Method (VIM), Generalized Hirota-Satsuma coupled KdV equation, Kawahara equation, some
FKdV equations.
Bazı Yüksek Mertebeten Lineer Olmayan Denklemlerde Adomian
Decomposition Methodu İle Variational Iterasyon Metodunun Bir
Karşılaştırılması
Özet: Bu makalede, Genelleştirilmiş Hirota-Satsuma coupled KDV denklemi ve Kawahara denklemi ile FKDV
denklemleri Varyasyonel iterasyon metodu ile çözüldü ve sonuçlar Adomian decomposition metodu ve VIM için
karşılaştırıldı. İlave olarak VIM ile, lineer olmayan problemlerin geniş bir sınıfı daha etkili, daha kolay ve doğru bir
şekilde çözülebileceğini gösterdik. Bu nedenle bu yönlemin fizik ve mühendislikte geniş bir uygulaması vardır.
Anahtar Kelimeler: Varyasyonel iterasyon metodu (VIM), Genelleştirilmiş Hirota Satsuma coupled KDV denklemi, Kawahara
denklemi, FKDV denklemleri.
1. Introduction
Nonlinear phenomena play a crucial role in applied
mathematics and physics. The results of solving
nonlinear equations can guide authors to know the
described process deeply. But it is difficult for us
to obtain the exact solution for these problems. In
recent decades, there has been great development
in the numerical analysis [1] and exact solution for
nonlinear partial Equations.
There are many standard methods for solving
nonlinear partial differential equations [24]; for
instance, Backland transformation method [2], Lie
group method [3] and Adomian’s decomposition
method [4], inverse scattering method [5], Hirota’s
bilinear method [6], homogeneous balance method
[7] and He's homotopy perturbation method
(HPM) [9-14, 25].
VIM was first proposed by He [8, 15, 16, 21];
unlike classical techniques, nonlinear equations are
solved easily and more accurately via VIM. This
method has recently been applied to engineering
equations [17]. In this paper we implement VIM in
order to find the solution of the Generalized
Hirota-Satsuma coupled KdV equation, Kawahara
equation [26,27] and FKdV equations.
2. Basic ideas of He's variational iteration
method
To illustrate its basic concepts of variational
iteration method, we consider the following
deferential equation:
D. D. Ganji, M. Nourollahi And M. Rostamian
(2.1)
where L is a linear operator, N a nonlinear operator, and g(x) an heterogeneous term. According to
VIM, we can construct a correction functional as follows:
Lu + Nu = g (x)
(2.2)
u n +1 ( x) = u n ( x) +
x
∫ λ{Lu
0
n (τ ) +
Nu~n (τ ) − g (τ )}dτ
where λ is a general Lagrangian multiplier [15,16], which can be optimally identified via the variational
theory [16], the subscript n indicates the n th order approximation, u~n is considered as a restricted
variation, i.e. δu~n = 0 .
3. Applications
Variational iteration method (VIM) was first
proposed by He [8, 15, 16, 21]. In this method, a
correction functional is constructed by a general
Lagrange multiplier which can be identified
optimally via the variational theory.
This method was successfully applied to
autonomous ordinary differential systems [8], the
relaxation process [20], the nonlinear differential
equations with convolution product nonlinearities
[16], the seepage flow with fractional derivatives
in porous media [15, 23], solitary solution [18, 21]
and the Helmholtz equation [22].
3.1. Generalized Hirota-Satsuma coupled KdV
equation system
We consider a generalized Hirota-Satsuma coupled
KdV equation system as [18]:
(3.1)
1
u xxx − 3uu x + 3(uw) x ,
2
vt = −v xxx + 3uv x ,
ut =
wt = − wxxx + 3uwx .
with the following initial conditions:
1
( β − 2k 2 ) + 2k 2 tanh 2 (kx),
3
4k 2 c 0 ( β + k 2 )
v( x,0) = −
3 c1 2
u ( x,0) =
2
+
(3.2)
2
4k ( β + k )
tanh( kx),
3 c1
w( x,0) = c0 + c1 tanh( kx).
To solve equation system (3.1) using VIM, we consider a correction functional as:
1
2
0 − 3(u n wn ) x } dτ ,
t
u n +1 ( x, t ) = u n ( x, t ) +
∫
λ{u nt − u nxxx + 3u n u nx
t
∫
(3.3)
v n +1 ( x, t ) = v n ( x, t ) + λ{v nt + v nxxx − 3u n v nx } dτ ,
0
t
∫
wn +1 ( x, t ) = wn ( x, t ) + λ{wnt + wnxxx − 3u n wnx } dτ .
0
180
A Comparison of Variational Iteration Method with Adomian’s Decomposition Method in Some Highly Nonlinear Equations
where λ is a general Lagrangian multiplier, and u n u nx , (u n wn ) x , u n v nx and u n wnx point out the restricted
variations; i.e.:
δ (3u n u nx ) = δ (−3u n wn ) x = δ (−3u n v nx ) = δ (−3u n wnx ) = 0
Making the above correction functional stationary, we obtain the following stationary conditions:
1 + λ τ =1 = 0
(3.4.a)
− λ′ = 0
(3.4.b)
The Lagrangian multiplier can therefore be identified as:
(3.5)
λ = -1
Substituting Eq. (3.5) into the correction functional equation system (3.3) results in the following iteration
formula:
1
u nxxx + 3u n u nx
2
0 − 3(u n wn ) x dτ ,
t
u n +1 ( x, t ) = u n ( x, t ) −
∫
u nt −
t
∫
(3.6)
v n +1 ( x, t ) = v n ( x, t ) − v nt + v nxxx − 3u n v nx dτ ,
0
t
∫
wn +1 ( x, t ) = wn ( x, t ) − wnt + wnxxx − 3u n wnx dτ .
0
We start with the initial approximation given the following equation:
1
u 0 ( x, t ) = ( β − 2k 2 ) + 2k 2 tanh 2 (kx),
3
4k 2 c 0 ( β + k 2 ) 4k 2 ( β + k 2 )
tanh( kx),
v0 ( x, t ) = −
+
3 c1
3 c12
(3.7)
w0 ( x, t ) = c0 + c1 tanh( kx).
Using the above iteration formula (3.6) and the initial approximation, we can obtain the following results:
( β − 2k 2 )
2
2
u1 ( x, t ) =
+ 2k tanh (kx)
3
+ t[k 3 tanh(kx)(−8k 2 − 4 β + 12c 0 )
+ k tanh 2 (kx)(8k 4 tanh(kx)
(3.8.a)
+ 4k 2 β tanh(kx) − 12k 2 c 0 tanh( kx)
− 18k 2 c1 tanh 2 (kx) − c1 β + 20k 2 c1 )
+ c1 kβ − 2c1 k 3 ] ,
v1 ( x, t ) =
4k 2
3c1 2
[−c 0 ( β + k 2 ) + c1 tanh( kx)( β + k 2 )
2
(3.8.b)
2
+ t kc1 β (1 − tanh (kx))(k + β )],
w1 ( x, t ) = c0 + c1 tanh(kx) + t c1kβ (1 − tanh 2 (kx)).
(3.8.c)
In order to obtain u 2 , u3 , v 2 , v3 , w2 , w3 ,…, one can use Maple Package and the diagrams. Now we study
the diagrams obtained by VIM and ADM[18].
181
D. D. Ganji, M. Nourollahi And M. Rostamian
Figure 1. The comparison of VIM and ADM for the solution u ( x, t ) for different values of t .
Figure 2. The comparison of VIM and ADM for the solution v( x, t ) for different values of t .
182
A Comparison of Variational Iteration Method with Adomian’s Decomposition Method in Some Highly Nonlinear Equations
Figure 3. The comparison of VIM and ADM for the solution w( x, t ) for different values of t .
183
D. D. Ganji, M. Nourollahi And M. Rostamian
Figure 4. The surfaces on both columns respectively show the solutions,
u ( x, t ), v( x, t ), w( x, t ) , for VIM on the right and ADM on the left.
3.2. Kawahara equation
We consider Kawahara equation as [19]:
u t + uu x + u xxx − u xxxxx = 0
(3.9)
With the following initial conditions:
u ( x ,0 ) =
105
x
sec h 4 (
)
169
2 13
(3.10)
To solve Eq. (3.9) using VIM, we consider the correction functional as:
u n +1 ( x, t ) = u n ( x, t ) +
t
∫ λ{u
0
nt
(3.11)
+ u n u nx + u n xxx − u n xxxxx } dτ
where λ is a general Lagrangian multiplier, and u n u nx indicates the restricted variations; i.e.
δ (u n u nx ) = 0
Making the above correction functional stationary, we obtain the following stationary conditions:
1 + λ τ =1 = 0
(3.12.a)
−λ ′ = 0
(3.12.b)
The Lagrangian multiplier can therefore be identified as:
λ = -1
(3.13)
184
A Comparison of Variational Iteration Method with Adomian’s Decomposition Method in Some Highly Nonlinear Equations
Substituting Eq. (3.13) into the correction functional equation system (3.11) results in the following
iteration formula:
u n+1 ( x, t ) = u n ( x, t )
(3.14)
t
− ∫ u nt + u n u nx + u n xxx − u n xxxxx dτ
0
We start with the initial approximation given the following equation:
u0 ( x, t ) = B sec h 4 ( Ax) ,
A=(
1
13
), B =
105
169
(3.15)
Using the above iteration formula (3.14) and the initial approximation, we can obtain the following
results:
u1 ( x, t ) = B sec h 4 ( Ax) + t (4 B 2 sec h 8 ( Ax ) tanh( Ax)
+ 120 B sec h 4 ( Ax) tanh 3 ( Ax) A 3
− 56 B sec h 4 ( Ax) tanh( Ax ) A 3
4
− 6720 B sec h ( Ax) A
(3.16)
5
− 1504 B sec h 4 ( Ax ) tanh( Ax ) A 5
In order to obtain u 2 , u3 ,…, one can use Maple Package and the diagrams.
Now we study the diagrams obtained by VIM and ADM[19]. The diagrams are as follows:
0.6
0.6
VIM
ADM
0.5
t = 0.2
0.4
u(x,t)
u(x,t)
0.4
0.3
0.3
0.2
0.2
0.1
0.1
-10
VIM
ADM
0.5
t = 0.1
-5
0
5
10
-10
x
-5
0
5
10
x
Figure 5. The comparison of VIM and ADM for the solution u ( x, t ) for different values of t .
185
D. D. Ganji, M. Nourollahi And M. Rostamian
Figure 6. The surfaces respectively show the solution, u ( x, t ) , for VIM and ADM.
3.3. Fifth order KdV equations
3.3.1. Example 1
We consider a fifth order KdV equation as [19]:
ut + uu x − uu xxx + u xxxxx = 0
(3.17)
With the following initial conditions:
u ( x,0) = e x
(3.18)
To solve Eq. (3.17) using VIM, we consider the correctional functional as:
u n +1 ( x, t ) = u n ( x, t )
∫
(3.19)
t
+ λ{u nt + u n u nx − u n u n xxx + u n xxxxx } dτ
0
where λ is a general Lagrangian multiplier, and u n u nx and unun xxx denote the restricted variations; i.e.
δ (u n u nx ) = δ (u n u n xxx ) = 0 .
Making the above correction functional stationary, we obtain the following stationary conditions:
1 + λ τ =1 = 0
(3.20.a)
−λ ′ = 0
(3.20.b)
The Lagrangian multiplier can therefore be identified as:
λ = -1
(3.21)
Substituting Eq. (3.21) into the correction functional Eq. (3.19) results in the following iteration formula:
u n +1 ( x, t ) = u n ( x, t )
∫
(3.22)
t
− u nt + u n u nx − u n u n xxx + u n xxxxx dτ
0
We start with the initial approximation given the following equation:
u0 ( x, t ) = e x
(3.23)
Using the above iteration formula (3.22) and the initial approximation, we can obtain the following
results:
u1 ( x, t ) = e x - te x
(3.24)
In order to obtain u 2 , u3 , …, one can use Maple Package which leads us to:
186
A Comparison of Variational Iteration Method with Adomian’s Decomposition Method in Some Highly Nonlinear Equations
t2 x
e
2
t2
t3
u3 ( x, t ) = e x - te x + e x − e x
2
6
u 2 ( x, t ) = e x - te x +
(3.25)
(3.26)
And the closed form will be as:
u ( x, t ) = e x −t
(3.27)
3.3.2. Example 2
We consider another fifth order KdV equation as [19]:
ut − uu x + u xxx − u xxxxx = cos x + 2t sin x +
t2
sin 2 x
2
(3.28)
With the following initial conditions:
u ( x,0) = 0
(3.29)
To solve Eq. (3.28) using VIM, we consider the correction functional as:
u n +1 ( x, t ) = u n ( x, t )
∫
(3.30)
t
+ λ{u nt − u n u nx + u n xxx − u n xxxxx } dτ
0
where λ is a general Lagrangian multiplier, and u n u nx denotes the restricted variations; i.e. δ (u n u nx ) = 0 .
Making the above correction functional stationary, we obtain the following stationary conditions:
1 + λ τ =1 = 0
(3.31.a)
−λ ′ = 0
(3.31.b)
The Lagrangian multiplier can therefore be identified as:
λ = -1
(3.32)
Substituting Eq. (3.32) into the correction functional Eq. (3.30) results in the following iteration formula:
u n +1 ( x, t ) = u n ( x, t )
∫
(3.33)
t
− u nt + u n u nx − u n u n xxx + u n xxxxx dτ
0
We start with the initial approximation given the following equation:
u0 = t cos x
(3.34)
Using the above iteration formula (3.33) and the initial approximation, we can obtain the following
results:
u n = t cos x , n ≥ 1
(3.35)
4. Conclusions
In this paper, VIM has been successfully applied to
finding the solutions of Generalized HirotaSatsuma coupled KdV equation, Kawahara
equation and FKdV equations. The obtained
solutions are compared with those of ADM. All
the examples show that the results of the present
method are in approximate agreement with those
of ADM. VIM has many merits and advantages
over ADM and can be introduced to overcome the
difficulties arising in calculation of Adomian
polynomials. Besides, VIM does not require small
parameters, thus the limitations of the traditional
perturbation methods can be eliminated, and the
calculations are also simple and straightforward.
The results show that VIM is a powerful
mathematical tool for solving linear and nonlinear
partial differential equations, and therefore, can be
widely applied in engineering problems.
187
D. D. Ganji, M. Nourollahi And M. Rostamian
Reference
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
Burden RL and Faires JD. Numerical analysis.
Boston:PWS Publishing Company,1993.
Rogers C and Shadwich WF.
Bäcklund
Transformations and Their Application. New York:
Academic Press, 1982.
Olver PJ. Applications of Lie Groups to Differential
Equations. Berlin: Springer, 1986.
Adomian G. A review of the decomposition method
in applied mathematics. Math. Anal. Appl, 1988;
135:501–544.
Gardner CS, Green JM, Kruskal MD, Miura RM.
Method for solving the Korteweg-de Vries equation.
Phys Rev Lett 1967; 19: 1095–7.
Hirota R. Exact solution of the Korteweg-de Vries
equation for multiple collisions of solitons. Phys
Rev Lett 1971; 27: 1192–4.
Wang ML. Exact solutions for a compound KdVBurgers equation. Phys Lett A 1996; 213: 279–87.
He JH. Variational iteration method for autonomous
ordinary differential systems. Applied Mathematics
and Computation, 2000; 114: 115-123.
He JH. The homotopy perturbation method for
nonlinear oscillators with discontinuities. Applied
Mathematics and Computation, 2004; 151: 287–
292.
He JH. Comparison of homotopy perturbation
method and homotopy analysis method. Applied
Mathematics and Computation, 2004; 156: 527–
539.
He JH. Asymptotology by homotopy perturbation
method. Applied Mathematics and Computation,
2004; 156(3): 591-596.
He JH. Homotopy perturbation method for
bifurcation of nonlinear problems. International
Journal of Non-linear Science Numerical
Simulation,2005;6(2):207-208.
He JH. Application of homotopy perturbation
method to nonlinear wave equations. Chaos,
Solitons and Fractals, 2005; 26: 695–700.
D. D. Ganji, "The application of He's Homotopy
Perturbation Method to nonlinear equations arising
in heat transfer", Phys. Lett. A, (in press).
He JH. Approximate analytical solution for seepage
flow with fractional derivatives in porous media.
Comput Meth Appl Mech Eng 1998; 167: 57–68.
He JH. Approximate solution of nonlinear
differential equations with convolution product
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
188
nonlinearities. Comput Meth Appl Mech Eng 1998;
167: 69–73.
Marinca V. An approximate solution for onedimensional weakly nonlinear oscillations. Int J
Non-linear Sci Numer Simul 2002; 3:107–10.
D. Kaya, Solitary wave solutions for a generalized
Hirota–Satsuma coupled KdV equation, Appl.
Math. Comput. 2004; 147: 69–78.
D. Kaya, An explicit and numerical solution of some
fifth-order KdV equation by decomposition method,
Appl. Math. Comput. 2003; 144: 353–363.
Draganescu GE and Capalnasan V. Nonlinear
relaxation phenomena in polycrystalline solids.
International Journal of Non-linear Science and
Numerical Simulation, 2003; 4(3):219-225.
He J.H., Wu XH. Construction of solitary solution
and compaction-like solution by variational iteration
method, Chaos, Solitons & Fractals, Volume 29,
Issue 1, July 2006, Pages 108-113.
Shaher Momani and Salah Abuasad. Application of
He's variational iteration method to Helmholtz
equation ,Chaos, Solitons & Fractals, Volume 27,
Issue 5, March 2006, Pages 1119-1123.
Z. M. Odibat, S. Momani, Application of
Variational Iteration Method to Nonlinear
Differential Equations of Fractional Order,
INTERNATIONAL JOURNAL OF NONLINEAR
SCIENCES AND NUMERICAL SIMULATION,
7(1), 2006, 27-34.
N. Bildik, A. Konuralp. The Use of Variational
Iteration Method, Differential Transform Method
and Adomian’s Decomposition Method for Solving
Different Types of Nonlinear Partial Differential
Equations, INTERNATIONAL JOURNAL OF
NONLINEAR SCIENCES AND NUMERICAL
SIMULATION, 7(1), 2006, 65-70.
He JH. Non-Perturbative Methods for Strongly
Nonlinear Problems, Berlin: dissertation.de-Verlag
im Internet GmbH, 2006 6. He JH. Some asymptotic
methods for strongly nonlinear equations,
International Journal of Modern Physics B, 2006
(10) (06) 1141-1199.
N. Polat, D. Kaya and H.I. Tutalar, An Analytic and
Numerical Solution to a Modified Kawahara
Equation and a Convergence Analysis of the
Method, Appl. Math. Comp., 179 (2006), 466-472.
D. Kaya and K. Al-Khaled, A Numerical
Comparison of a Kawahara Equation, Phys. Lett. A,
(in press).