Numerical Treatment of Initial Value Problems of Nonlinear Ordinary Differential Equations by Duan-Rach-Wazwaz Modified Adomian Decomposition Method

We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.


Introduction
Phenomena as diverse as the oscillations of a suspension bridge, the spread of a disease, and the motion of the planets are governed by nonlinear differential equations.Most of these nonlinear equations do not have analytical solutions, so approximation and numerical techniques must be used.The Adomian decomposition method (ADM), introduced by Adomian [1] [2] [3] [4] provides immediate and visible symbolic terms of analytic solutions as well as numerical ap-proximate solutions to both linear and nonlinear problems without unphysical restrictive assumptions such as required by linearization, perturbation or discretization [1] [2] [3] [4].It provides the solution in a rapidly convergent series with easily computable components if the equation has a unique solution.The technique uses a decomposition of the nonlinear operator as a series of Adomian functions.Each term of the series is a generalized polynomial, called the Adomian polynomial.The ADM has been successfully applied to a wide class of problems arising in applied sciences and engineering [1]- [14] over three decades.
Adomian decomposition method has led to a number of modifications made by various researchers for different purposes such as to improve the accuracy, or increase the speed of convergence, or expand the application of the original method.Adomian and Rach [11] introduced modified Adomian polynomials which converge slightly faster than the original polynomials and are convenient for computer generation.Wazwaz [12] [13] used Padé approximants to the solution obtained using a modified decomposition method and found that not only does this improve the results, but also that the error decreases with the increase of the degree of the Padé approximants.The later modifications of ADM were proposed by Wazwaz [14], Wazwaz and El-Sayed [15], Duan [16] [17] [18] [19] [20], Duan and Rach [21] [22] [23], Duan, Rach and Wazwaz [24] [25].
In this paper, we consider the applications of the Duan-Rach-Wazwaz modification of ADM to the initial value problems (IVPs) for the systems of nonlinear ordinary differential equations (ODEs).In 2013, Duan, Rach and Wazwaz [25] presented a reliable modification of the ADM which bases on the previous modification schemes [14]- [24], and computes the solutions of variable coefficients higher-order nonlinear initial value problems (IVPs) and solutions of systems of coupled nonlinear IVPs.To implement these algorithms they also designed multistage decomposition and numeric algorithms, and presented MATHEMATICA routines PSSOL and NSOL.
The text is organized as follows.The basic principles of ADM are given in Section 2. For the numerical solutions of the IVPs for the systems of nonlinear differential ODEs, the frameworks of the Duan-Rach-Wazwaz modification are presented in Section 3. In Section 4, numerical treatments of the nonlinear IVPs using the modified technique and MATHEMATICA numerical solution are performed.The solutions of some problems are also computed by using fourth-order Runge Kutta method (RK4) and the comparisons of the results are presented.A brief conclusion is given in Section 5.All computations are carried out in MATHEMATICA.

Basic Principles of the Adomian Decomposition Method
Consider the general nonlinear ODE in the Adomian's operator-theoretic form ( ), Lu Ru Nu g t where g is a given analytic function and u is the unknown solution, and L is the Int.J. Modern Nonlinear Theory and Application linear operator to be inverted, R is the linear remainder operator, and N is an analytic nonlinear operator.We remark that the choice of the linear operator is designed to yield an easily invertible operator with resulting trivial integrations.
This means that the choice is not unique.Generally we choose ( ) n-th order ODEs, then its inverse 1 L − follows as the n-fold definite integration operator from 0 t to t.Hence, we have 1 L u u ψ − = − , where ψ is determined using the initial conditions.
Application of 1 L − to each side of Equation ( 1) yields where ( ) ( ) ( ) The ADM decomposes the solution into a series ( ) ( ) and then decomposes the nonlinear term into a series of Adomian polynomials where , 0 i A i ≥ are called the Adomian polynomials and generated by the defi- nitional formula where λ is a grouping parameter of convenience.The formulas of the first four Adomian polynomials for the one-variable simple analytic nonlinearity In the Duan-Rach-Wazwaz modification, by using Duan's Corollary 3 algorithm [18] the one variable Adomian polynomials are written as ( ) ( ) ( ) where the coefficients k i C are defined recursively [18] as ( ) The formulae in (7) does not involve the differentiation operator for the coefficients k i C [20] [21] [22], but requires only addition and multiplication.So, it is more convenient for computer algebra systems.
The definitional formula of the Adomian polynomials for decomposing multivariable nonlinear functions occurring in either single nonlinear nth-order or in systems of coupled nonlinear ODEs with multivariable nonlinearities are published by Adomian and Rach in [6].By assuming f is an m-ary analytic function ( ) , where the k u , for 1 k m ≤ ≤ are the unknown functions to be determined, the solutions q u , 1, , q m =  and the nonlinear function , 1, , and , , , where the multivariable Adomian polynomials i A depend on the ( ) ; where λ is a grouping parameter of convenience.The first m-variable Ado- mian polynomial 0 A is ( ) , where ( ) Substitution of the Adomian decomposition series for the solution ( ) The solution components where Adomian has chosen ( ) ( ) as the initial solution.All the solution components ( ), 0 i u t i ≥ of the solution ( ) u t can be determined using Equa- tion (11) and hence, the solution series follows immediately [25].We remark that the convergence of the Adomian series has already been proved by several investigators [26] [27].

Duan-Rach-Wazwaz Modification of the Adomian Decomposition Method
We illustrate the general frameworks of the Duan-Rach-Wazwaz modification of the Adomian decomposition method [25] for solving the first-order differential equations and the systems of coupled nonlinear differential equations numerically.Throughout the section we assume the equations are in canonical Int.J. Modern Nonlinear Theory and Application forms.

IVP of the First-Order Nonlinear ODE
We consider the following first-order nonlinear, nonhomogeneous differential equation subject to a bounded initial condition We assume that the nonhomogeneous term g and system coefficients ( ) Application of the Adomian decomposition series and Adomian polynomials series result where ( ) f u is the simple nonlinearity term and can be any analytic function in u and the corresponding one-variable Adomian polynomials We, next solve Equation ( 14) for ( ) Lu t and apply the one-fold definite integral operator ( ) to each side of the resulting equation to get Equation ( 19) is the equivalent nonlinear Volterra integral equation for the solution ( ) ( ) ( ) By substituting Equations ( 15), ( 20) and ( 21) into Equation ( 19) we get the Adomian decomposition series as Therefore, the modified recursion scheme is written as As a result, the (m + 1)th-stage solution approximant is given by ( ) ( ) , for 0 m ≥ , in the limit, it yields the exact solution, that is, . By calculating the first several solution components using Equations ( 22) and ( 23), we derive the following sequence By using induction, we find for where the one-variable Adomian polynomials m B depend solely on the solu- Int.J. Modern Nonlinear Theory and Application tion coefficients j c , for 0 j m ≤ ≤ , and are determined as instead of the solution components ( ) Therefore, we have derived the desired Taylor expansion series for the solution ( ) u t as ( ) . By inspection, from the Equation ( 25), the solution coefficients i c are obtained as the nonlinear recurrence relation ( ) where the one-variable Adomian polynomials m B are the same as shown in Equation (26).So, the rule of recursion for the solution coefficients of the first order canonical nonhomogeneous nonlinear IVP with a variable input and variable system coefficients is obtained as ( )

IVP of the System of Coupled Nonlinear DEs
We consider the following n-th order system of m-coupled n k -th order nonhomogeneous nonlinear IVPs ( ) . We assume that the system coefficients and the system inputs are analytic functions.We also assume that the problem is subject to appropriate ( , , , , , where k L are the linear operators, k R are the linear remainder operators, i.e., generally sequential-order differential operators, and k N are the nonlinear op- erators such that For a particular n k th-order nonlinear DE in the system represented by Equation (27) or Equation ( 29), we choose the corresponding solution The linear differential operators k L are invertible, and their inverse opera- tors 1 k L − are given by the k n -fold integral ( ) for the case of a system of m-coupled n k th-order IVPs, where the initial conditions are all specified at the origin.
Application of the Adomian decomposition series and the series of the Adomian polynomials, yields where the multi-order differential nonlinearity can be any analytic function in  d  , ,  ; ;  , ,  ; ;  d  d   d  d  , ,  ; ;  , ,  d d The relating Cauchy products are Next we solve Equation ( 29) for ( ) , , , , Applying the n k -fold integral operator 1 k L − to each side of Equation ( 32), we obtain , , , , By integrating left side of Equation ( 33) and substituting the values specified in Equation ( 28) we obtain ( ) ( ) Substituting this on the left side of Equation ( 33), we obtain ( Formula ( 35) is the equivalent system of m-coupled nonlinear Volterra integral equations.
Evaluating the relating integrals, we get where where Substitution of the Equations ( 30), (36), (37), (38) and (39) into Equation (35) yields the following system of m-coupled modified recursion schemes Therefore, the (s + 1)th-stage solution approximants ( ) From the calculation of the first several solution components, we deduce the following sequence where the ( ) ) ) , , ; ; , , ; ; , , ; , , instead of the solution components ( ) , k j u t and solution derivative components ≤ ≤ and 0 1 p q n ≤ ≤ − .Thus we have derived the desired Taylor expansion series for each of the m solutions ( ) where the solution coefficients , k i c are given by the system of m-coupled nonlinear recurrence relations, obtained from inspection of Equation (39), as  41).Consequently, the rule of recursion for the solution coefficients of the canonical nth-order system of m-coupled n k th order nonhomogeneous nonlinear IVPs with variable inputs and variable system coefficients are given as , , ; ; , , ; ; , , ,

Examples
In this section, we consider several examples of IVPs for the systems of nonlinear ODEs, which have either quadratic or cubic nonlinearities but, exhibit rather complex behavior.The modified numeric solutions of the problems are obtained by using MATHEMATICA routines PSSOL and NSOL [25].To compare the results, we have calculated the MATHEMATICA numeric solutions for the systems of differential equations by using the command "NDSolve".We also ( ) over the interval 0 40 t ≤ ≤ .This nonlinear IVP does not have an exact solution but, a detailed qualitative analysis can be found in [31].
Running PSSOL by taking 4 n = to output 5th-degree or equivalently 6-term approximation to the solution as ( ) We note that the order of approximation is ( ) Running NSOL for 4 n = and step size 0.05 h = to output the numeric solution 5 y of order 5 which is depicted with red line as the curve of 5th order approximation 5 y and parametric plot on the left in Figure 1.
As the comparison, MATHEMATICA numeric solution and RK4 solution are found and the curves and the parametric plots of the results are sketched with blue and black lines in the middle and on the right, respectively in Figure 1.
From Figure 1, we conclude that this problem has a limit cycle.
Example 2. Consider the first-order nonhomogeneous nonlinear differential equation with a quadratic nonlinearity [32] ( )  .It has the exact solution ( ) Running PSSOL routine for 6, 7 n = and 8 to output the 13-term, 15-term and 17-term approximants of the solution, respectively, as ( ) We note that in these computations approximation orders are , O t O t and ( ) O t , respectively.
The curves of the computed approximants and the exact solution are plotted in Figure 2(a).
The MATHEMATICA command   From Figure 3 we can conclude that the ADM can be combined with the diagonal Padé approximants to estimate the blow-up time [28].Since π 4 t = is the blow-up time for this problem, this can be seen from the figure.
The maximal error parameters n ME for 6, 7 n = and 8 are given in Table 1.
Table 1 shows that the maximal errors for the exact solution decrease approximately at an exponential rate.
Example 3. Consider the 2-dimensional system of nonlinear differential equations with quadratic nonlinearity [32] ( ) ( )    In Figure 5, the absolute errors for the exact solutions ( ) x t and ( ) We list the maximal error parameters n ME for 5, 6 n = and 7 in Table 2.        Example 6.Consider the following seven-dimensional third-order hyperchaotic system [35] with cubic nonlinearity in each equation 43) Int.J. Modern Nonlinear Theory and Application compute numerical solutions using RK4 in examples 1, 2 and 3.Moreover, we use diagonal Padé approximants [28] [29] [30] to improve the modified results and compute errors in the approximations in the examples 2 and 3 since the exact solutions of these problems are known.Example 1.Consider the Abel differential equation of the first kind in canonical form.It is a first order, nonhomogeneous differential equation with a cubic nonlinearity [31].

( 45 )
Int. J. Modern Nonlinear Theory and Application

Figure 1 .
Figure 1.The curves and parametric plots of the 5th-degree MADM approximate solution (red), MATHEMATICA numeric solution (blue) and RK4 solution (black) using the step-size 0.05 h = over the interval 0 40 t ≤ ≤ .

1 ,
the [6/6], [7/7] and [8/8] diagonal Padé-approximants of the 13-term, 15-term and 17-term approximants, generated by the routine PSSOL respectively.The curves of the Padé approximants and the exact solution are plotted in Figure 2(b).Running NSOL routine for 11,13 n = , and 15 and step size 0.05 h = to generate numeric solutions on the interval 0 π 6 t ≤ ≤ .In Figure 2(c), the curves of these NSOL numeric solutions and the exact solution are depicted.As a comparison RK4 solution is computed and depicted with exact solution in Figure 2(d).

Figure 2 .
Figure 2. PSSOL outputs and exact solution in (a), Padé-approximants and exact solution in (b), outputs of the NSOL routine and exact solution in (c) and RK4 results and exact solution in (d) for n = 6, 7 and 8.
with the exact solutions x and y are depicted in Figures 4(a)-(c).
t are sketched in (a) and (b), respectively.

Figure 5 .
Figure 5. (a) The absolute error for the x-component of the solution, ( ) sec x t t = ; (b) The absolute error for the y-component of

Figure 11 .
Figure 11.Hyperchaotic behavior of system (48) two different 3-dimensional x-y-z and y-z-w projections of the system are shown in A and B, respectively.The parameter values are 0.2, 0, 0.04 a b d c = = = = and the initial conditions are ( ) ( ) 0 0, 0 0.75, x y = = the interval for the system.The 2-and 3-dimensional projections of the modified results are plotted on the left of Figure12and Figure13.To compare the results, we also calculated MATHEMATICA numeric solution of the system and plotted on the right of Figure12and Figure13.
, respectively.So, if it is possible to compute all terms of the series we shall see that the Adomian series for this problem is simply that Taylor series.All terms of the series are positive so, absolute convergence is simply the convergence of the series.

Table 1 .
The maximal error parameters ME n for n = 6, 7 and 8.

Table 2 .
The maximal error parameters ME n for n = 5, 6 and 7.From Table2we can conclude that the maximal error parameters for both components of the exact solution decrease approximately at an exponential rate.