ABSTRACT

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.

Keywords:

Adomian Decomposition Method, Duan-Rach-Wazwaz Modified Adomian Decomposition Method, Initial Value Problem, Nonlinear Ordinary Differential Equation, Mathematica Solution, 4-th Order Runge Kutta Method, Padé Approximants

1. 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 approximate 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.

2. Basic Principles of the Adomian Decomposition Method

Consider the general nonlinear ODE in the Adomian’s operator-theoretic form

$Lu+Ru+Nu=g\left(t\right),$ (1)

where g is a given analytic function and u is the unknown solution, and L is the 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 $L=\frac{{\text{d}}^n}{\text{d}{t}^n}(\cdot )$ for

n-th order ODEs, then its inverse $L^{-1}$ follows as the n-fold definite integration operator from $t_0$ to t. Hence, we have $L^{-1}u=u-\psi$ , where $\psi$ is determined using the initial conditions.

Application of $L^{-1}$ to each side of Equation (1) yields

$u\left(t\right)=\gamma \left(t\right)-L^{-1}\left[Ru\left(t\right)+Nu\left(t\right)\right],$ (2)

where $\gamma \left(t\right)=\psi \left(t\right)+L^{-1}g\left(t\right)$ .

The ADM decomposes the solution into a series

$u\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{u}_i\left(t\right),$ (3)

and then decomposes the nonlinear term into a series of Adomian polynomials

$Nu\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{A}_i\left(u_0\left(t\right),\cdots ,u_i\left(t\right)\right),$ (4)

where $A_i\mathrm{,}i\ge 0$ are called the Adomian polynomials and generated by the definitional formula

$A_i=\frac{1}{i!}{\frac{{\partial }^i}{\partial {\lambda }^i}\left[f\left({\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}{u}_k{\lambda }^k\right)\right]|}_{\lambda =0},$ (5)

where $\lambda$ is a grouping parameter of convenience. The formulas of the first four Adomian polynomials for the one-variable simple analytic nonlinearity $Nu\left(t\right)=f\left(u\left(t\right)\right)$ are listed as

$A_0=f\left(u_0\right),$

$A_1=f^{\prime }\left(u_0\right)u_1,$

$A_2=f^{\prime }\left(u_0\right)u_2+f^{″}\left(u_0\right)\frac{u_1^2}{2!},$

$A_3=f^{\prime }\left(u_0\right)u_3+f^{″}\left(u_0\right)u_1{u}_2+f^{\left(3\right)}\left(u_0\right)\frac{u_1^3}{3!},$

$A_4=f^{\prime }\left(u_0\right)u_4+f^{″}\left(u_0\right)\left(\frac{u_2^2}{2!}+u_1{u}_3\right)+f^{\left(3\right)}\left(u_0\right)\frac{u_1^2{u}_2}{3!}+f^{\left(4\right)}\left(u_0\right)\frac{u_1^4}{4!}.$

In the Duan-Rach-Wazwaz modification, by using Duan’s Corollary 3 algorithm [18] the one variable Adomian polynomials are written as

$A_0=f\left(u_0\right),A_i={\displaystyle \underset{k=1}{\overset{i}{\sum }}}{C}_i^k{f}^{\left(k\right)}\left(u_0\right)\text{for}i\ge 1,$ (6)

where the coefficients $C_i^k$ are defined recursively [18] as

$C_i^1=u_i,i\ge 1,C_i^k=\frac{1}{i}{\displaystyle \underset{j=0}{\overset{i-k}{\sum }}}\left(j+1\right)u_{j+1}{C}_{i-1-j}^{k-1},2\le k\le i.$ (7)

The formulae in (7) does not involve the differentiation operator for the coefficients $C_i^k$ [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 ODEs $f\left(u\mathrm{,}{u}^{\prime }\mathrm{,}{u}^{″}\mathrm{,}\cdots \mathrm{,}{u}^{\left(n-1\right)}\right)$ 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 $f\left(u_1\mathrm{,}\cdots \mathrm{,}{u}_m\right)$ , where the $u_k$ , for $1\le k\le m$ are the unknown functions to be determined, the solutions $u_q$ , $q=1,\cdots ,m$ and the nonlinear function $f\left(u_1\mathrm{,}\cdots \mathrm{,}{u}_m\right)$ are decomposed as

$u_q={\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{u}_{q,j},q=1,\cdots ,m\text{and}f\left(u_1,\cdots ,u_m\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{A}_i,$ (8)

where the multivariable Adomian polynomials $A_i$ depend on the $m\left(i+1\right)$ solution components $u_{1,0}{u}_{1,1}\cdots u_{1,i};u_{2,0}{u}_{2,1}\cdots u_{2,i};\cdots ;u_{m,0}{u}_{m,1}\cdots u_{m,i}$ and are defined by [6]

$A_i={\frac{1}{i!}\frac{{\partial }^i}{\partial {\lambda }^i}f\left({\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{u}_{1,j}{\lambda }^j,\cdots ,{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}{u}_{m,j}{\lambda }^j\right)|}_{\lambda =0},$ (9)

where $\lambda$ is a grouping parameter of convenience. The first m-variable Adomian polynomial $A_0$ is $A_0=f\left(u_0\right)$ , where $u_0=\left(u_{1,0}\cdots u_{m,0}\right)$ .

Substitution of the Adomian decomposition series for the solution $u\left(t\right)$ and the series of Adomian polynomials for the nonlinearity Nu, Equations (3) and (4) respectively, into Equation (2) yields

${\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{u}_i\left(t\right)=\gamma \left(t\right)-L^{-1}\left[R{\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{u}_i\left(t\right)+{\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{A}_i\left(t\right)\right].$ (10)

The solution components $u_i\left(t\right)$ may be determined by one of the several advantageous recursion schemes, which differ from another by the choice of the initial solution component $u_0\left(t\right)$ , beginning with the classic Adomian recursion scheme

$u_0\left(t\right)=\gamma \left(t\right),u_{i+1}\left(t\right)=-L^{-1}\left[R{u}_i\left(t\right)+A_i\left(t\right)\right],i\ge 0,$ (11)

where Adomian has chosen $u_0\left(t\right)=\gamma \left(t\right)$ as the initial solution. All the solution components $u_i\left(t\right)\mathrm{,}i\ge 0$ of the solution $u\left(t\right)$ can be determined using Equation (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] .

3. 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 forms.

3.1. IVP of the First-Order Nonlinear ODE

We consider the following first-order nonlinear, nonhomogeneous differential equation subject to a bounded initial condition

$u^{\prime }\left(t\right)+{\alpha }_0\left(t\right)u\left(t\right)+\beta \left(t\right)f\left(u\left(t\right)\right)=g\left(t\right),$ (12)

$u\left(0\right)=C_0.$ (13)

We assume that the nonhomogeneous term g and system coefficients ${\alpha }_0\left(t\right)$ and $\beta \left(t\right)$ are analytic functions.

In Adomian’s operator-theoretic form, the Equation (12) can be written as

$Lu\left(t\right)+Ru\left(t\right)+Nu\left(t\right)=g\left(t\right),$ (14)

where L is the linear operator, R is the linear remainder operator and N is the nonlinear operator such that

$L=\frac{\text{d}}{\text{d}t}(.),Ru\left(t\right)={\alpha }_0\left(t\right)u\left(t\right),Nu\left(t\right)=\beta \left(t\right)f\left(u\left(t\right)\right).$

We note that, accordance by ( [27] , page 105) $f\left(u\right)$ and g are continuously differentiable at any order and they can be expandable in entire series in the neighbourhood of $t=0$ , therefore, sum function $u\left(t\right)$ is real analytic and indicated series is absolutely, uniformly convergent. Therefore, the presentation formula for function $u\left(t\right)$ is meaningful.

From the assumption on the analyticity of the functions ${\alpha }_0\left(t\right)\mathrm{,}\beta \left(t\right)$ and $g\left(t\right)$ we write the respective Taylor expansion series as

${\alpha }_0\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\alpha }_{0,i}{t}^i{\alpha }_{0,i}=\frac{{\alpha }_0^{\left(i\right)}\left(0\right)}{i!},\beta \left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\beta }_i{t}^i,{\beta }_i=\frac{{\beta }^{\left(i\right)}\left(0\right)}{i!},$

$g\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{g}_i{t}^i,g_i=\frac{g^{\left(i\right)}\left(0\right)}{i!}.$

Application of the Adomian decomposition series and Adomian polynomials series result

$u\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{u}_i\left(t\right),Nu\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{A}_i\left(t\right),f\left(u\left(t\right)\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{B}_i\left(t\right),$ (15)

where $f\left(u\right)$ is the simple nonlinearity term and can be any analytic function in u and the corresponding one-variable Adomian polynomials $B_i\left(t\right)$ have the standard formula [6] [16] [17] [18] $B_i\left(t\right)=B_i\left(u_0\left(t\right),\cdots ,u_i\left(t\right)\right)$ .

By calculating the Cauchy products ${\alpha }_0\left(t\right)u\left(t\right)$ and $\beta \left(t\right)f\left(u\left(t\right)\right)$ , respectively we write

$Ru\left(t\right)={\alpha }_0\left(t\right)u\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{m=0}{\overset{i}{\sum }}}{\alpha }_{0,i-m}{t}^{i-m}{u}_i\left(t\right),$ (16)

$Nu\left(t\right)={\beta }_0\left(t\right)f\left(u\left(t\right)\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{m=0}{\overset{i}{\sum }}}{\beta }_{i-m}{t}^{i-m}{B}_i\left(t\right),$ (17)

and hence

$A_i\left(t\right)={\displaystyle \underset{m=0}{\overset{i}{\sum }}}{\beta }_{i-m}{t}^{i-m}{B}_i\left(t\right).$ (18)

We, next solve Equation (14) for $Lu\left(t\right)$ and apply the one-fold definite integral operator $L^{-1}={\displaystyle {\int }_0^t}(.)\text{d}s$ to each side of the resulting equation to get

$u\left(t\right)=C_0+L^{-1}g\left(t\right)-L^{-1}Ru\left(t\right)-L^{-1}Nu\left(t\right),$ (19)

since $L^{-1}Lu\left(t\right)=u\left(t\right)-u\left(0\right)=u\left(t\right)-C_0$ .

Equation (19) is the equivalent nonlinear Volterra integral equation for the solution $u\left(t\right)$ .

Evaluating the integrals $L^{-1}g\left(t\right),L^{-1}Ru\left(t\right)$ and $L^{-1}Nu\left(t\right)$ , we write

$L^{-1}g\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}\frac{g_i{t}^{i+1}}{\left(i+1\right)!},L^{-1}Ru\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{m=0}{\overset{i}{\sum }}}{\alpha }_{0,i-m}{\displaystyle {\int }_0^t}{s}^{i-m}{u}_m\left(s\right)\text{d}s,$ (20)

$L^{-1}Nu\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{m=0}{\overset{i}{\sum }}}{\beta }_{i-m}{\displaystyle {\int }_0^t}{s}^{i-m}{B}_m\left(s\right)\text{d}s.$ (21)

By substituting Equations (15), (20) and (21) into Equation (19) we get the Adomian decomposition series as

$\begin{array}{c}{\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{u}_i\left(s\right)=C_0+{\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}\frac{g_i{t}^{i+1}}{\left(i+1\right)!}-{\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{m=0}{\overset{i}{\sum }}}{\alpha }_{0,i-m}{\displaystyle {\int }_0^t}{s}^{i-m}{u}_m\left(s\right)\text{d}s,\\ -{\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{m=0}{\overset{i}{\sum }}}{\beta }_{i-m}{\displaystyle {\int }_0^t}{s}^{i-m}{B}_m\left(s\right)\text{d}s.\end{array}$

Therefore, the modified recursion scheme is written as

$u_0\left(t\right)=C_0,$

$u_{i+1}\left(t\right)=\frac{g_i{t}^{i+1}}{\left(i+1\right)!}-{\displaystyle \underset{m=0}{\overset{i}{\sum }}}\left[{\alpha }_{0,i-m}u\left(s\right)+{\beta }_{i-m}{B}_m\left(s\right)\right]\text{d}s,$ (22)

for $i\ge 0$ , where the one-variable Adomian polynomials $B_m\left(t\right)$ are

$B_m\left(t\right)=B_m\left(u_0\left(t\right),\cdots ,u_m\left(t\right)\right).$ (23)

As a result, the (m + 1)th-stage solution approximant is given by ${\varphi }_{m+1}\left(t\right)={\displaystyle {\sum }_{i=0}^m}{u}_i\left(t\right)$ , for $m\ge 0$ , in the limit, it yields the exact solution, that is, ${\lim }_{m\to \infty }{\varphi }_{m+1}\left(t\right)={\lim }_{m\to \infty }{\displaystyle {\sum }_{i=0}^m}{u}_i\left(t\right)=u\left(t\right)$ .

By calculating the first several solution components using Equations (22) and (23), we derive the following sequence

$u_0\left(t\right)=C_0=c_0,u_1\left(t\right)=\frac{g_0-{\alpha }_{0,0}{c}_0-{\beta }_0{B}_0}{\left(1\right)}t=c_{1}t,$

$u_2\left(t\right)=\frac{g_1-{\alpha }_{0,1}{c}_0-{\alpha }_{0,0}{c}_1-{\beta }_1{B}_0-{\beta }_0{B}_1}{\left(2\right)}{t}^2=c_2{t}^2,\cdots ,$

By using induction, we find for $i\ge 0$ that

$u_{i+1}\left(t\right)=\frac{g_i-{\displaystyle {\sum }_{m=0}^i\left({\alpha }_{0,i-m}{c}_m+{\beta }_{i-m}{B}_m\right)}}{\left(i+1\right)}{t}^{i+1}=c_{i+1}{t}^{i+1},$ (24)

where the one-variable Adomian polynomials $B_m$ depend solely on the solution coefficients $c_j$ , for $0\le j\le m$ , and are determined as

$B_0=B_0\left(c_0\right),B_1=B_1\left(c_0,c_1\right),B_2=B_2\left(c_0,c_1,c_2\right),\cdots ,B_m=B_m\left(c_0,\cdots ,c_m\right),$ (25)

instead of the solution components $u_j\left(t\right)$ for $0\le j\le m$ .

Therefore, we have derived the desired Taylor expansion series for the solution $u\left(t\right)$ as $u\left(t\right)={\displaystyle {\sum }_{i=0}^{\infty }}{c}_i{t}^i$ . By inspection, from the Equation (25), the solution coefficients $c_i$ are obtained as the nonlinear recurrence relation

$c_0=C_0,c_{i+1}=\frac{g_i-{\displaystyle {\sum }_{m=0}^i\left({\alpha }_{0,i-m}{c}_m+{\beta }_{i-m}{B}_m\right)}}{\left(i+1\right)},i\ge 0,$ (26)

where the one-variable Adomian polynomials $B_m$ 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 $c_{n+1}=c_{n+1}\left(c_0,\cdots ,c_i\right)$ , for $n\ge 0$ .

3.2. 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

$\begin{array}{l}{u}_k^{\left(n_k\right)}\left(t\right)+{\displaystyle \underset{p=1}{\overset{m}{\sum }}}{\displaystyle \underset{r=0}{\overset{n_k-1}{\sum }}}{\alpha }_{k,p,n_q-1-r}\left(t\right)u_p^{n_p-1-r}\left(t\right)\\ +{\beta }_k\left(t\right)f_k\left(u_1\left(t\right),\cdots ,u_1^{\left(n_1-1\right)}\left(t\right);\cdots ,u_m\left(t\right),\cdots ,u_m^{\left(n_m-1\right)}\left(t\right)\right)=g_k\left(t\right),\end{array}$ (27)

$u_k^j\left(0\right)=C_{k,j},1\le k\le m,0\le j\le n_k-1,m\ge 1,n_k\ge 1,$ (28)

where ${\alpha }_{k\mathrm{,}p\mathrm{,}j}\left(t\right)$ and ${\beta }_k\left(t\right)$ for $0\le k\mathrm{,}p\le m$ , $0\le j\le n_p-1$ , $n_k\mathrm{,}{n}_p\ge 1$ are variable system coefficients and $g_k\left(t\right)$ are variable inputs, and $n=\max \left(n_1,n_2,\cdots ,n_m\right),m\ge 1$ . We assume that the system coefficients and the system inputs are analytic functions. We also assume that the problem is subject to appropriate ( ${\displaystyle {\sum }_{p=1}^m}{n}_p$ ) bounded initial conditions, i.e., $n_k$ bounded initial conditions for each n_kth-order nonlinear DE, $1\le k\le m$ .

In Adomian’s operator-theoretic form Equation (27) can written as

$L_k{u}_k\left(t\right)+R_k{u}_1\left(t\right),\cdots ,u_m\left(t\right)+N_k{u}_1\left(t\right),\cdots ,u_m\left(t\right)=g_k\left(t\right),$ (29)

where $L_k$ are the linear operators, $R_k$ are the linear remainder operators, i.e., generally sequential-order differential operators, and $N_k$ are the nonlinear operators such that

$L_k=\frac{{\text{d}}^{n_k}}{\text{d}{t}^{n_k}}(.),R_k{u}_1\left(t\right),\cdots ,u_m\left(t\right)={\displaystyle \underset{p=1}{\overset{m}{\sum }}}{\displaystyle \underset{r=0}{\overset{n_p-1}{\sum }}}{\alpha }_{k,p,n_p-1-r}\left(t\right)u_p^{\left(n_p-1-r\right)}\left(t\right),$

$N_k{u}_1\left(t\right),\cdots ,u_m\left(t\right)={\beta }_k\left(t\right)f_k\left(u_1\left(t\right),\cdots ,u_1^{\left(n_1-1\right)}\left(t\right);\cdots ;u_m\left(t\right),\cdots ,u_m^{\left(n_m-1\right)}\left(t\right)\right).$

For a particular n_kth-order nonlinear DE in the system represented by Equation (27) or Equation (29), we choose the corresponding solution $u_k\left(t\right)$ as the primary solution and the solutions $u_p\left(t\right)$ , for $p\ne k$ , as the secondary solutions with respect to this same n_kth-order DE. We assume that ${\alpha }_{k\mathrm{,}p\mathrm{,}{n}_p-1}\left(t\right)\mathrm{,}\cdots \mathrm{,}{\alpha }_{k\mathrm{,}p\mathrm{,0}}\left(t\right)\mathrm{,}{\beta }_k\left(t\right)$ and $g_k\left(t\right)$ are analytic, and hence have the relating Taylor expansion series

${\alpha }_{k,p,j}\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\alpha }_{k,p,j,i}{t}^i,{\alpha }_{k,p,j,i}=\frac{{\alpha }_{k,p,j}^{\left(i\right)}\left(0\right)}{i!},1\le k,p\le m,0\le j\le n_p-1,$

${\beta }_k\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\beta }_{k,i}{t}^i,{\beta }_{k,i}=\frac{{\beta }_k^{\left(i\right)}\left(0\right)}{i!}$

$g_k\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{g}_{k,i}{t}^i,g_{k,i}=\frac{g_k^{\left(i\right)}\left(0\right)}{i!}.$

The linear differential operators $L_k$ are invertible, and their inverse operators $L_k^{-1}$ are given by the $n_k$ -fold integral $L_k^{-1}={\displaystyle {\int }_0^t}\cdots {\displaystyle {\int }_0^t}\left(\mathrm{.}\right)\text{d}t\cdots \text{d}t$ for the case of a system of m-coupled n_kth-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

$u_k\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{u}_{k,i}\left(t\right),N_k{u}_1\left(t\right),\cdots ,u_m\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{A}_{k,i}\left(t\right),$

$f_k\left(u_1\left(t\right),\cdots ,u_1^{\left(n_1-1\right)}\left(t\right);\cdots ;u_m\left(t\right),\cdots ,u_m^{\left(n_m-1\right)}\left(t\right)\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{B}_{k,i}\left(t\right),$ (30)

where the multi-order differential nonlinearity

$f_k\left(u_1\left(t\right),\cdots ,u_1^{\left(n_1-1\right)}\left(t\right);\cdots ;u_m\left(t\right),\cdots ,u_m^{\left(n_m-1\right)}\left(t\right)\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{B}_{k,i}(t)$

can be any analytic function in $u_1\left(t\right)\mathrm{,}\cdots \mathrm{,}{u}_1^{\left(n_1-1\right)}\left(t\right)\mathrm{<mo>\; ;\; </mo>}\cdots \mathrm{<mo>\; ;\; </mo>}{u}_m\left(t\right)\mathrm{,}\cdots \mathrm{,}{u}_m^{\left(n_m-1\right)}\left(t\right)$ and the relating ${\displaystyle {\sum }_{p=1}^m}{n}_p$ -variable Adomian polynomials $B_{k\mathrm{,}i}\left(t\right)$ have the standard formula [6] [16] [17] [18] .

$\begin{array}{l}{B}_{k,i}\left(t\right)=B_{k,i}(u_{1,0}\left(t\right),\cdots ,u_{1,i}\left(t\right);\cdots ;u_{1,0}^{\left(n_1-1\right)}\left(t\right),\cdots ,u_{1,i}^{\left(n_1-1\right)}\left(t\right);\cdots ;\\ u_{m,0}\left(t\right),\cdots ,u_{m,i}\left(t\right);\cdots ;u_{m,0}^{\left(n_k-1\right)}\left(t\right),\cdots ,u_{m,i}^{\left(n_k-1\right)}(t))\end{array}$

or equivalently,

$\begin{array}{l}{B}_{k,i}\left(t\right)=B_{k,i}(u_{1,0}\left(t\right),\cdots ,u_{1,i}\left(t\right);\cdots ;\frac{{\text{d}}^{n_1-1}}{\text{d}{t}^{n_1-1}}{u}_{1,n_1-1}\left(t\right),\cdots ,\frac{{\text{d}}^{n_1-1}}{\text{d}{t}^{n_1-1}}{u}_{1,i+n_1-1}\left(t\right);\cdots ;\\ u_{m,0}\left(t\right),\cdots ,u_{m,i}\left(t\right);\cdots ;\frac{{\text{d}}^{n_k-1}}{\text{d}{t}^{n_k-1}}{u}_{m,n_k-1}\left(t\right),\cdots ,\frac{{\text{d}}^{n_k-1}}{\text{d}{t}^{n_k-1}}{u}_{m,i+n_k-1}\left(t\right))\end{array}$ (31)

The relating Cauchy products are

$R_k{u}_1\left(t\right),\cdots ,u_m\left(t\right)={\displaystyle \underset{p=1}{\overset{m}{\sum }}}{\displaystyle \underset{r=0}{\overset{n_p-1}{\sum }}}{\alpha }_{k,p,n_p-1-r}\left(t\right)u_p^{\left(n_p-1-r\right)}(t)$

where

${\alpha }_{k,p,j}\left(t\right)u_p^j\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\alpha }_{k,p,j}{t}^i{\displaystyle \underset{s=0}{\overset{\infty }{\sum }}}{u}_{p,s}^{\left(j\right)}\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{s=0}{\overset{i}{\sum }}}{\alpha }_{k,p,j,i-s}{t}^{i-s}\frac{{\text{d}}^j}{\text{d}{t}^j}{u}_{p,s+j}\left(t\right),$

where $j=n_p-1-r$ ,

$\begin{array}{l}{N}_k{u}_1\left(t\right),\cdots ,u_m\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\beta }_{k,i}{t}^i{\displaystyle \underset{p=0}{\overset{\infty }{\sum }}}{B}_{k,s}\left(t\right)\\ ={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{s=0}{\overset{i}{\sum }}}{\beta }_{k,i-s}{t}^{i-s}{B}_{k,s}\left(t\right),\end{array}$

hence it is also determined that

$A_{k,i}\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\beta }_{k,i}{t}^i{B}_{k,s}(t)$

Next we solve Equation (29) for $L_k\left(t\right)$ as

$L_k{u}_k\left(t\right)=g_k\left(t\right)-R_k{u}_1\left(t\right),\cdots ,u_m\left(t\right)-N_k{u}_1\left(t\right),\cdots ,u_m\left(t\right)$ (32)

Applying the n_k-fold integral operator $L_k^{-1}$ to each side of Equation (32), we obtain

$L_k^{-1}{L}_k{u}_k\left(t\right)=L_k^{-1}{g}_k\left(t\right)-L_k^{-1}{R}_k{u}_1\left(t\right),\cdots ,u_m\left(t\right)-L_k^{-1}{N}_k{u}_1\left(t\right),\cdots ,u_m\left(t\right)$ (33)

By integrating left side of Equation (33) and substituting the values specified in Equation (28) we obtain

$L_k^{-1}{L}_k{u}_k\left(t\right)=u_k\left(t\right)-{\displaystyle \underset{r=0}{\overset{n_k-1}{\sum }}}\frac{t^r}{r!}{C}_{k,r}.$ (34)

Substituting this on the left side of Equation (33), we obtain

$u_k\left(t\right)={\displaystyle \underset{r=0}{\overset{n_k-1}{\sum }}}\frac{t^r}{r!}{C}_{k,r}+L_k^{-1}{g}_k\left(t\right)-L_k^{-1}{R}_k{u}_1\left(t\right),\cdots ,u_m\left(t\right)-L_k^{-1}{N}_k{u}_1\left(t\right),\cdots ,u_m\left(t\right).$ (35)

Formula (35) is the equivalent system of m-coupled nonlinear Volterra integral equations.

Evaluating the relating integrals, we get

$L_k^{-1}{g}_k\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}\frac{g_{k,i}{t}^{i+n_k}}{\left(i+1\right)\cdots \left(i+n_k\right)},$ (36)

$L_k^{-1}{R}_k{u}_1\left(t\right),\cdots ,u_m\left(t\right)={\displaystyle \underset{p=1}{\overset{m}{\sum }}}{\displaystyle \underset{r=0}{\overset{n_p-1}{\sum }}}{L}_k^{-1}{\alpha }_{k,p,n_p-1-r}\left(t\right)u_p^{\left(n_p-1-r\right)}\left(t\right),$ (37)

where

$L_k^{-1}{\alpha }_{k,p,j}\left(t\right)u_p^{\left(j\right)}\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{s=0}{\overset{i}{\sum }}}{\alpha }_{k,p,j,i-s}{L}_k^{-1}{t}^{i-s}\frac{{\text{d}}^j}{\text{d}{t}^j}{u}_{p,s+j}\left(t\right),$ (38)

where $j=n_p-1-r$ ,

$L_k^{-1}{N}_k{u}_1\left(t\right),\cdots ,u_m\left(t\right)={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{s=0}{\overset{i}{\sum }}}{\beta }_{k,i-s}{L}_k^{-1}{t}^{i-s}{B}_{k,s}\left(t\right).$ (39)

Substitution of the Equations (30), (36), (37), (38) and (39) into Equation (35) yields the following system of m-coupled modified recursion schemes

$\begin{array}{l}{u}_{k,0}\left(t\right)=C_{k,0},u_{k,1}\left(t\right)=t{C}_{k,1},u_{k,2}\left(t\right)=\frac{t^2}{2!}{C}_{k,2},\cdots ,\\ u_{k,n_k-1}\left(t\right)=\frac{t^{n_k-1}}{\left(n_k-1\right)!}{C}_{k,n_k-1},\cdots \end{array}$

$\begin{array}{c}{u}_{k,i+n_k}\left(t\right)=\frac{t^{i+n_k}}{{\displaystyle {\prod }_{q=1}^{n_k}\left(i+q\right)}}-{\displaystyle \underset{s=0}{\overset{i}{\sum }}}{\beta }_{k,i-s}{L}_k^{-1}{t}^{i-s}{B}_{k,s}\left(t\right)\\ \times {\displaystyle \underset{s=0}{\overset{i}{\sum }}}{\displaystyle \underset{p=1}{\overset{m}{\sum }}}{\displaystyle \underset{r=0}{\overset{i}{\sum }}}{\alpha }_{k,p,n_p-1-r,i-s}{L}_k^{-1}{t}^{i-s}\frac{{\text{d}}^{n_p-1-r}}{\text{d}{t}^{n_p-1-r}}{u}_{p,s+n_p-1-r}(t)\end{array}$

for $1\le k\le m\mathrm{,}{n}_k\mathrm{,}{n}_p\ge \mathrm{1,}i\ge 0$ .

Therefore, the (s + 1)th-stage solution approximants ${\Phi }_{k\mathrm{,}s+1}\left(t\right)$ are given by ${\Phi }_{k,s+1}\left(t\right)={\displaystyle {\sum }_{i=0}^s}{u}_{k,i}\left(t\right)$ for $s\ge 0$ .

From the calculation of the first several solution components, we deduce the following sequence

$u_{k,0}\left(t\right)=C_{k,0}=c_{k,0},u_{k,1}\left(t\right)=t{C}_{k,1}=c_{k,1}t,\cdots ,u_{k,n_k-1}\left(t\right)=\frac{t^{n_k-1}}{\left(n_k-1\right)!}=c_{k,n_k-1}{t}^{n_k-1}$

$\begin{array}{l}{u}_{k,i+n_k}\left(t\right)\\ =\frac{g_{k,i}-{\displaystyle \underset{s=0}{\overset{i}{\sum }}}\left\{{\beta }_{k,i-s}{B}_{k,s}+{\displaystyle {\sum }_{p=1}^m{\displaystyle {\sum }_{r=0}^{n_p-1}\left[{\displaystyle {\prod }_{j=1}^{n_p-1-r}\left(s+j\right)}\right]{\alpha }_{k,p,n_p-1-r,i-s}{c}_{p,s+n_p-1-r}}}\right\}}{{\displaystyle {\prod }_{q=1}^{n_k}\left(i+q\right)}}{t}^{i+n_k}\\ =c_{k,n_k}{t}^{i+n_k},i\ge 0\end{array}$ (40)

where the ( ${\displaystyle {\sum }_{p=1}^m}{n}_p$ )-variable Adomian polynomials $B_{k\mathrm{,}s}$ are now constants and depend merely on the solution coefficients $c_{k\mathrm{,}j}$ for $0\le j\le n_p-1$ . They are determined by induction as

$\begin{array}{l}{B}_{k,s}\left(t\right)=B_{k,s}(c_{1,0},\cdots ,c_{1,s};\cdots ;c_{1,0}^{\left(n_1-1\right)},\cdots ,c_{1,s}^{\left(p_1-1\right)};\cdots ;\\ c_{m\mathrm{,0}}\mathrm{,}\cdots \mathrm{,}{c}_{m\mathrm{,}s}\mathrm{<mo>\; ;\; </mo>}\cdots \mathrm{<mo>\; ;\; </mo>}{c}_{m\mathrm{,0}}^{\left(n_m-1\right)}\mathrm{,}\cdots \mathrm{,}{c}_{m\mathrm{,}s}^{\left(n_m-1\right)})\end{array}$ (41)

or equivalently,

$\begin{array}{l}{B}_{k,s}=B_{k,s}(c_{1,0},\cdots ,c_{1,s};\cdots ;\left\{{\displaystyle {\prod }_{j=1}^{n_1-1}\left(j\right)}\right\}{c}_{1,n_1-1},\cdots ,\left\{{\displaystyle {\prod }_{j=1}^{n_1-1}\left(s+j\right)}\right\}{c}_{1,s+n_1-1};\cdots ;\\ c_{m,0},\cdots ,c_{m,s};\left\{{\displaystyle {\prod }_{j=1}^{n_m-1}\left(j\right)}\right\}{c}_{m,n_m-1},\cdots ,\left\{{\displaystyle {\prod }_{j=1}^{n_m-1}\left(s+j\right)}\right\}{c}_{m,s+n_m-1})\end{array}$ (42)

instead of the solution components $u_{k\mathrm{,}j}\left(t\right)$ and solution derivative components $u_{k\mathrm{,}j}^{\left(i\right)}\left(t\right)$ for $1\le k\le m\mathrm{,0}\le j\le s$ and $0\le q\le n_p-1$ . Thus we have derived the desired Taylor expansion series for each of the m solutions $u_k\left(t\right)$ as $u_k\left(t\right)={\displaystyle {\sum }_{i=0}^{\infty }}{c}_{k,i}{t}^i$ where the solution coefficients $c_{k\mathrm{,}i}$ are given by the system of m-coupled nonlinear recurrence relations, obtained from inspection of Equation (39), as

$c_{k,j}=\frac{c_{k,j}}{j!},1\le k\le m,0\le j\le n_p-1,$

$c_{k,i+n_k}=\frac{g_{k,i}-{\displaystyle \underset{s=0}{\overset{i}{\sum }}}\left\{{\beta }_{k,i-s}{B}_{k,s}+{\displaystyle {\sum }_{p=1}^m{\displaystyle {\sum }_{r=0}^{n_p-1}\left[{\displaystyle {\prod }_{j=1}^{n_p-1-r}\left(s+j\right)}\right]{\alpha }_{k,p,n_p-1-r,i-s}{c}_{p,s+n_p-1-r}}}\right\}}{{\displaystyle {\prod }_{q=1}^{n_k}\left(i+q\right)}},$ (43)

$i\ge \mathrm{0,}m\ge 1$ and $n_k\ge 1$ , where the ( ${\displaystyle {\sum }_{p=1}^m}{n}_p$ )-variable Adomian polynomials $B_{k\mathrm{,}s}$ depend on the solution coefficients $c_{k\mathrm{,}j}$ , for $0\le j\le n_p-1$ , as in Equation (41). Consequently, the rule of recursion for the solution coefficients of the canonical nth-order system of m-coupled n_kth order nonhomogeneous nonlinear IVPs with variable inputs and variable system coefficients are given as

$c_{k,i+n_k}=c_{k,i+n_k}\left(c_{1,0},\cdots ,c_{1,i+n_1-1};\cdots ;c_{p,0},\cdots ,c_{p,i+n_p-1};\cdots ;c_{m,n_m-1},\cdots ,c_{m,s+n_m-1}\right),$

for $i\ge \mathrm{0,1}\le k\mathrm{,}p\le m$ .

4. 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 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] .

$y^{\prime }=\sin t-y^3,y\left(0\right)=\frac{1}{2},$ (44)

over the interval $0\le t\le 40$ . This nonlinear IVP does not have an exact solution but, a detailed qualitative analysis can be found in [31] .

Running PSSOL by taking $n=4$ to output 5th-degree or equivalently 6-term approximation to the solution as

$y_5\left(t\right)=\frac{1}{2}-\frac{t}{8}+\frac{35}{64}{t}^2-\frac{37}{256}{t}^3+\frac{457}{12288}{t}^4-\frac{1823}{16384}{t}^5.$

We note that the order of approximation is $O\left(t^5\right)$ .

Running NSOL for $n=4$ and step size $h=0.05$ to output the numeric solution $y_5$ of order 5 which is depicted with red line as the curve of 5th order approximation $y_5$ 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]

$y^{\prime }=y^2+2y+2,y\left(0\right)=0,$ (45)

on the interval $0\le t\le \text{π}/6$ . It has the exact solution $y\left(t\right)=-1+\tan \left(t+\frac{\text{π}}{4}\right)$ .

Running PSSOL routine for $n=6,7$ and 8 to output the 13-term, 15-term and 17-term approximants of the solution, respectively, as

$\begin{array}{c}{y}_{12}\left(t\right)=2t+2t^2+\frac{8}{3}{t}^3+\frac{10}{3}{t}^4+\frac{64}{15}{t}^5+\frac{244}{45}{t}^6+\frac{2176}{315}{t}^7+\frac{554}{64}{t}^8\\ +\frac{31744}{2835}{t}^9+\frac{202084}{13175}{t}^{10}+\frac{2830336}{155925}{t}^{11}+\frac{2162212}{93555}{t}^{12}\end{array}$