APPLICATION OF ADOMAIN DECOMPOSITION METHOD FOR LÜ
SYSTEM
M. Mossa Al-Sawalha *, M.S.M. Noorani
ABSTRACT
Investigating the accuracy of the Adomain decomposition method for solving the Lü system
is of paramount importance and the main aim of this paper. In this study the the fourth-order
Runge-Kutta (RK4) solutions and Comparisons between the decomposition solutions
are explained in details. In particular we look at the accuracy of the ADM as the Lü
system has high Lyapunov exponent.
Keywords: Lü system; Adomain decomposition method; Runge-Kutta method;
Lyapunov exponent.
1- Introduction
The Adomain decomposition method (ADM) (Adomain 1989) is a simple and powerful
method for solving systems of nonlinear differential equations. It has recently been identified
as being efficient for certain nonlinear problems and comparable to the 4th-order RungeKutta (RK4) numerical scheme. No transformation, linearization, perturbation or
discretization is required for the ADM and it yields an analytical solution in the form of a
rapidly convergent infinite power series with easily computable terms.
The accuracy of the ADM method is usually examined with the conventional numerical
method of Runge-Kutta or its variants. Moreover, since the global solution (via ADM or any
numerical method) for nonlinear systems is a bit far fetched to hope for, it is crucial for
researchers to understand the effect of time step on such methods. (Vadasz & Olek 2000)
made numerical comparisons between the ADM and the variable time step Runge-KuttaVerner method for solving the extended Lorenz system. (Guellal et al. 1997) compared the
ADM with RK4 for solving the classical chaotic Lorenz system. (Hashim et al. 2006)
presented a more comprehensive comparative study of ADM and RK4 for the classical
chaotic and non-chaotic Lorenz systems. The accuracy assessment of ADM for the Chen
system was undertaken by (Noorani et al. 2007). (Ghosh et al. 2007) demonstrated the high
accuracy of ADM for strongly nonlinear and chaotic oscillators.
Again, we are interested in the accuracy of the ADM for nonlinear systems of ODEs
capable of exhibiting chaotic behavior. The system which is of interest to us in this
paper is the Lü system:
x = a ( y − x )
y = − xz + cy
z = xy − bz
(1)
which has a chaotic attractor when a =36; b = 3; c = 20. Its dynamics have been analyzed in
detail by Lü. (Lü et al.2002) . where x, y and z are the state variables and a, b, and c are three
positive real constants. system has a typically critical chaotic attractor with Lyapunov
exponents λ1 = 1.5046 , λ2 = 0 and λ3 = −22.5044 .
In this paper, the ADM is again treated as an algorithm for approximating the solutions
of Lu system in a sequence of time intervals. Comparison between the decomposition
solutions and the RK4 solutions shall be made.
2- Adomian decomposition solution
Following (Hashim et al. 2006; Noorani et al. 2007) the explicit analytical solution via ADM
for the Lü system (1) can be written as:
(1.1)
(t − t )
*
∞
x = ∑ am
(t − t )
*
y = ∑ bm
(t − t )
*
z = ∑ cm
m
m!
m =0
∞
,
m!
m =0
∞
m
m
m!
m =0
,
,
where the coefficients are given by the recurrence relations,
a0 = x (t * ), b 0 = y (t * ), c 0 = z (t * ).
am = a (b m −1 − am −1 ),
(3)
m ≥ 1,
ak c m − k −1
+ cb m −1 ,
k = 0 k !( m − k − 1)!
m −1
ak b m − k −1
c m = (m − 1)! ∑
− bc m −1 ,
k = 0 k !( m − k − 1)!
m −1
b m = −(m − 1)! ∑
m ≥ 1,
(4)
m ≥ 1,
As first hinted in (Adomain 1989) and applied in (Vadasz & Olek 2000; Guellal et al. 1997;
Hashim et al. 2006; Noorani et al. 2007), we treat the ADM as an algorithm for
approximating the dynamical response in a sequence of time intervals (i.e. time step),
[0, t ) , [t , t ] ,..., (t
1
1
2
m −1
,T ] , such that the initial condition in ⎡⎣t * , t m +1 ) is taken to be the
condition at t * . For practical computations, a finite number of terms in the series (2) are used
in a time step procedure just outlined.
3-Results and discussions
To demonstrate the ADM accuracy against RK4. The Adomain algorithm is coded in the
computer algebra package Maple. We fix the values of the parameters a = 36, b = 3, and c =
20, we take the initial conditions x(0) = 4, y(0) = 3, z(0) = 2 as considered in (Al-Sawalha &
Noorani 2007). Based on our preliminary calculations, we decided to use 10 terms in the
Adomian decomposition series solutions. The simulations done in this paper are for the time
span t ∈ [0, 7].
First we determine the accuracy of RK4 for the solution of (1) for different sizes of time step
h. From the results presented in Table 1 we see that the maximum differences between the
RK4 (h = 0.001) and RK4 (h = 0.0001) solutions is of the order of magnitude of 10−3 . As
expected, the solutions of the Lü system become less accurate as time progresses. So based
on this observations we choose the RK4 solutions on the time step h= 0.001 as the benchmark
for our comparison purposes.
Having determined the benchmark time-step, we can now investigate the accuracy of the
ADM as compared to RK4. The details of the differences between the 10-term decomposition
solutions (on h= 0.01 and h = 0.001) and the RK4 solutions (on h= 0.001 and h = 0.0001) are
given in Table 2. Further comparisons are shown in Figure1. It can be observed from Figure 1
that increasing the accuracy of the RK4 solutions by decreasing the time step again brought
the Adomian solutions and the RK4 solutions closer to each other. In Fig. 2 we reproduce the
well-known x–y and x–y–z phase portraits of the chaotic Lü system using the 10-term
decomposition solutions on h = 0.001.
Table 1: A determination of the accuracy of RK4 of Lü system
Δ = RK 40.01 − RK 40.001
t
Δx
Δy
Δ = RK 40.001 − RK 40.0001
Δz
Δx
Δy
Δz
1
2.655E-04
1.731E-03
4.395E-03
6.562E-08
1.494E-07
2.575E-07
2
0.01179
0.0132
4.37E-03
8.767E-07
9.591E-07
2.519E-07
3
0.1112
0.09988
0.07486
7.892E-06
7.066E-06
5.341E-06
4
0.23
0.2274
0.09363
1.590E-05
1.549E-05
6.850E-06
5
1.019
1.176
0.7264
7.072E-05
8.347E-05
4.389E-05
6
4.551
5.531
2.723
3.581E-04
4.613E-04
8.414E-05
7
3.693
4.368
6.042E-04
8.379E-04
1.094E-04
6.321
Table 2:Differences between the 10-term decomposition and RK4 solutions of Lü
system
Δ = ADM 0.01 − RK 40.001
t
Δx
1
2
6.563E-08
8.768E-07
Δy
1.494E-07
9.592E-07
3
7.892E-06
7.067E-06
4
1.590E-05
1.549E-05
5
7.073E-05
6
7
Δz
2.576E-07
2.519E-07
5.341E-06
Δ = ADM 0.001 − RK 40.0001
Δx
Δy
6.972E-12
1.464E-11
8.413E-11
9.169E-11
Δz
2.369E-11
2.306E-11
7.525E-10
6.738E-10
5.091E-10
6.851E-06
1.515E-09
1.476E-09
6.532E-10
8.348E-05
4.389E-05
6.728E-09
7.941E-09
4.176E-09
3.581 E-04
4.613 E-04
8.414E-05
3.407E-08
4.389E-08
8.006E-09
6.042 E-04
8.38 E-04
1.094 E-03
5.749E-08
7.973E-08
1.041E-07
Figure 1: A chaotic case with parameters a = 36, b = 3, c = 20: (left) 10-term ADM (h =
0.001) vs RK4 (h = 0.001) and (right) 10-term ADM (h = 0.001) vs RK4 (h = 0.0001) for t ∈
[0, 7].
Figure 2: Phase portraits using 10-term ADM on h = 0.001 for a = 36, b = 3 and c = 20.
5 Concluding remarks
In this work, the ADM was applied to the solutions of the well known Lü system.
Comparisons between the Adomain decomposition solutions and the fourth-order Runge–
Kutta (RK4) numerical solutions were made. We note that the decomposition solutions were
computed via a simple algorithm without any need for perturbation techniques, special
transformations, linearization or discretization. Even though the ADM works well on highly
chaotic systems such as the Lü system, care still has to be taken on the choice of time span,
time step and the number of terms used.
Acknowledgment
This work is financially supported by the Malaysian Ministry of Science, Technology and
Environment Sciencefund grant: 04-01-02-SF0177.
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School of Mathematical Sciences
Faculty of Science and Technology
Universiti Kebangsaan Malaysia
43600 UKM Bangi
Selangor, MALAYSIA
E-mail: sawalha_moh@yahoo.com*.
*
Corresponding author.