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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.

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 [

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 [

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 [

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.

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

L u + R u + N u = g ( t ) , (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 = d n d t n ( ⋅ ) 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 − ψ , where ψ is determined using the initial conditions.

Application of L − 1 to each side of Equation (1) yields

u ( t ) = γ ( t ) − L − 1 [ R u ( t ) + N u ( t ) ] , (2)

where γ ( t ) = ψ ( t ) + L − 1 g ( t ) .

The ADM decomposes the solution into a series

u ( t ) = ∑ i = 0 ∞ u i ( t ) , (3)

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

N u ( t ) = ∑ i = 0 ∞ A i ( u 0 ( t ) , ⋯ , u i ( t ) ) , (4)

where A i , i ≥ 0 are called the Adomian polynomials and generated by the definitional formula

A i = 1 i ! ∂ i ∂ λ i [ f ( ∑ k = 0 ∞ u k λ k ) ] | λ = 0 , (5)

where λ is a grouping parameter of convenience. The formulas of the first four Adomian polynomials for the one-variable simple analytic nonlinearity N u ( t ) = f ( u ( t ) ) are listed as

A 0 = f ( u 0 ) ,

A 1 = f ′ ( u 0 ) u 1 ,

A 2 = f ′ ( u 0 ) u 2 + f ″ ( u 0 ) u 1 2 2 ! ,

A 3 = f ′ ( u 0 ) u 3 + f ″ ( u 0 ) u 1 u 2 + f ( 3 ) ( u 0 ) u 1 3 3 ! ,

A 4 = f ′ ( u 0 ) u 4 + f ″ ( u 0 ) ( u 2 2 2 ! + u 1 u 3 ) + f ( 3 ) ( u 0 ) u 1 2 u 2 3 ! + f ( 4 ) ( u 0 ) u 1 4 4 ! .

In the Duan-Rach-Wazwaz modification, by using Duan’s Corollary 3 algorithm [

A 0 = f ( u 0 ) , A i = ∑ k = 1 i C i k f ( k ) ( u 0 ) for i ≥ 1 , (6)

where the coefficients C i k are defined recursively [

C i 1 = u i , i ≥ 1 , C i k = 1 i ∑ j = 0 i − k ( j + 1 ) u j + 1 C i − 1 − j k − 1 , 2 ≤ k ≤ i . (7)

The formulae in (7) does not involve the differentiation operator for the coefficients C i k [

The definitional formula of the Adomian polynomials for decomposing multivariable nonlinear functions occurring in either single nonlinear nth-order ODEs f ( u , u ′ , u ″ , ⋯ , u ( n − 1 ) ) or in systems of coupled nonlinear ODEs with multivariable nonlinearities are published by Adomian and Rach in [

u q = ∑ j = 0 ∞ u q , j , q = 1 , ⋯ , m and f ( u 1 , ⋯ , u m ) = ∑ i = 0 ∞ A i , (8)

where the multivariable Adomian polynomials A i depend on the m ( i + 1 ) solution components u 1 , 0 u 1 , 1 ⋯ u 1 , i ; u 2 , 0 u 2 , 1 ⋯ u 2 , i ; ⋯ ; u m , 0 u m , 1 ⋯ u m , i and are defined by [

A i = 1 i ! ∂ i ∂ λ i f ( ∑ j = 0 ∞ u 1 , j λ j , ⋯ , ∑ j = 0 ∞ u m , j λ j ) | λ = 0 , (9)

where λ is a grouping parameter of convenience. The first m-variable Adomian polynomial A 0 is A 0 = f ( u 0 ) , where u 0 = ( u 1 , 0 ⋯ u m , 0 ) .

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

∑ i = 0 ∞ u i ( t ) = γ ( t ) − L − 1 [ R ∑ i = 0 ∞ u i ( t ) + ∑ i = 0 ∞ A i ( t ) ] . (10)

The solution components u i ( t ) 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 ( t ) , beginning with the classic Adomian recursion scheme

u 0 ( t ) = γ ( t ) , u i + 1 ( t ) = − L − 1 [ R u i ( t ) + A i ( t ) ] , i ≥ 0 , (11)

where Adomian has chosen u 0 ( t ) = γ ( t ) as the initial solution. All the solution components u i ( t ) , i ≥ 0 of the solution u ( t ) can be determined using Equation (11) and hence, the solution series follows immediately [

We illustrate the general frameworks of the Duan-Rach-Wazwaz modification of the Adomian decomposition method [

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

u ′ ( t ) + α 0 ( t ) u ( t ) + β ( t ) f ( u ( t ) ) = g ( t ) , (12)

u ( 0 ) = C 0 . (13)

We assume that the nonhomogeneous term g and system coefficients α 0 ( t ) and β ( t ) are analytic functions.

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

L u ( t ) + R u ( t ) + N u ( t ) = g ( t ) , (14)

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

L = d d t ( . ) , R u ( t ) = α 0 ( t ) u ( t ) , N u ( t ) = β ( t ) f ( u ( t ) ) .

We note that, accordance by ( [

From the assumption on the analyticity of the functions α 0 ( t ) , β ( t ) and g ( t ) we write the respective Taylor expansion series as

α 0 ( t ) = ∑ i = 0 ∞ α 0 , i t i α 0 , i = α 0 ( i ) ( 0 ) i ! , β ( t ) = ∑ i = 0 ∞ β i t i , β i = β ( i ) ( 0 ) i ! ,

g ( t ) = ∑ i = 0 ∞ g i t i , g i = g ( i ) ( 0 ) i ! .

Application of the Adomian decomposition series and Adomian polynomials series result

u ( t ) = ∑ i = 0 ∞ u i ( t ) , N u ( t ) = ∑ i = 0 ∞ A i ( t ) , f ( u ( t ) ) = ∑ i = 0 ∞ B i ( t ) , (15)

where f ( u ) is the simple nonlinearity term and can be any analytic function in u and the corresponding one-variable Adomian polynomials B i ( t ) have the standard formula [

By calculating the Cauchy products α 0 ( t ) u ( t ) and β ( t ) f ( u ( t ) ) , respectively we write

R u ( t ) = α 0 ( t ) u ( t ) = ∑ i = 0 ∞ ∑ m = 0 i α 0 , i − m t i − m u i ( t ) , (16)

N u ( t ) = β 0 ( t ) f ( u ( t ) ) = ∑ i = 0 ∞ ∑ m = 0 i β i − m t i − m B i ( t ) , (17)

and hence

A i ( t ) = ∑ m = 0 i β i − m t i − m B i ( t ) . (18)

We, next solve Equation (14) for L u ( t ) and apply the one-fold definite integral operator L − 1 = ∫ 0 t ( . ) d s to each side of the resulting equation to get

u ( t ) = C 0 + L − 1 g ( t ) − L − 1 R u ( t ) − L − 1 N u ( t ) , (19)

since L − 1 L u ( t ) = u ( t ) − u ( 0 ) = u ( t ) − C 0 .

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

Evaluating the integrals L − 1 g ( t ) , L − 1 R u ( t ) and L − 1 N u ( t ) , we write

L − 1 g ( t ) = ∑ i = 0 ∞ g i t i + 1 ( i + 1 ) ! , L − 1 R u ( t ) = ∑ i = 0 ∞ ∑ m = 0 i α 0 , i − m ∫ 0 t s i − m u m ( s ) d s , (20)

L − 1 N u ( t ) = ∑ i = 0 ∞ ∑ m = 0 i β i − m ∫ 0 t s i − m B m ( s ) d s . (21)

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

∑ i = 0 ∞ u i ( s ) = C 0 + ∑ i = 0 ∞ g i t i + 1 ( i + 1 ) ! − ∑ i = 0 ∞ ∑ m = 0 i α 0 , i − m ∫ 0 t s i − m u m ( s ) d s , − ∑ i = 0 ∞ ∑ m = 0 i β i − m ∫ 0 t s i − m B m ( s ) d s .

Therefore, the modified recursion scheme is written as

u 0 ( t ) = C 0 ,

u i + 1 ( t ) = g i t i + 1 ( i + 1 ) ! − ∑ m = 0 i [ α 0 , i − m u ( s ) + β i − m B m ( s ) ] d s , (22)

for i ≥ 0 , where the one-variable Adomian polynomials B m ( t ) are

B m ( t ) = B m ( u 0 ( t ) , ⋯ , u m ( t ) ) . (23)

As a result, the (m + 1)th-stage solution approximant is given by ϕ m + 1 ( t ) = ∑ i = 0 m u i ( t ) , for m ≥ 0 , in the limit, it yields the exact solution, that is, lim m → ∞ ϕ m + 1 ( t ) = lim m → ∞ ∑ i = 0 m u i ( t ) = u ( t ) .

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

u 0 ( t ) = C 0 = c 0 , u 1 ( t ) = g 0 − α 0 , 0 c 0 − β 0 B 0 ( 1 ) t = c 1 t ,

u 2 ( t ) = g 1 − α 0 , 1 c 0 − α 0 , 0 c 1 − β 1 B 0 − β 0 B 1 ( 2 ) t 2 = c 2 t 2 , ⋯ ,

By using induction, we find for i ≥ 0 that

u i + 1 ( t ) = g i − ∑ m = 0 i ( α 0 , i − m c m + β i − m B m ) ( i + 1 ) 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 ≤ j ≤ m , and are determined as

B 0 = B 0 ( c 0 ) , B 1 = B 1 ( c 0 , c 1 ) , B 2 = B 2 ( c 0 , c 1 , c 2 ) , ⋯ , B m = B m ( c 0 , ⋯ , c m ) , (25)

instead of the solution components u j ( t ) for 0 ≤ j ≤ m .

Therefore, we have derived the desired Taylor expansion series for the solution u ( t ) as u ( t ) = ∑ i = 0 ∞ 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 = g i − ∑ m = 0 i ( α 0 , i − m c m + β i − m B m ) ( i + 1 ) , i ≥ 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 ( c 0 , ⋯ , c i ) , for n ≥ 0 .

We consider the following n-th order system of m-coupled n_{k}-th order nonhomogeneous nonlinear IVPs

u k ( n k ) ( t ) + ∑ p = 1 m ∑ r = 0 n k − 1 α k , p , n q − 1 − r ( t ) u p n p − 1 − r ( t ) + β k ( t ) f k ( u 1 ( t ) , ⋯ , u 1 ( n 1 − 1 ) ( t ) ; ⋯ , u m ( t ) , ⋯ , u m ( n m − 1 ) ( t ) ) = g k ( t ) , (27)

u k j ( 0 ) = C k , j , 1 ≤ k ≤ m , 0 ≤ j ≤ n k − 1 , m ≥ 1 , n k ≥ 1 , (28)

where α k , p , j ( t ) and β k ( t ) for 0 ≤ k , p ≤ m , 0 ≤ j ≤ n p − 1 , n k , n p ≥ 1 are variable system coefficients and g k ( t ) are variable inputs, and n = max ( n 1 , n 2 , ⋯ , n m ) , m ≥ 1 . We assume that the system coefficients and the system inputs are analytic functions. We also assume that the problem is subject to appropriate ( ∑ p = 1 m n p ) bounded initial conditions, i.e., n k bounded initial conditions for each n_{k}th-order nonlinear DE, 1 ≤ k ≤ m .

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

L k u k ( t ) + R k u 1 ( t ) , ⋯ , u m ( t ) + N k u 1 ( t ) , ⋯ , u m ( t ) = g k ( t ) , (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 = d n k d t n k ( . ) , R k u 1 ( t ) , ⋯ , u m ( t ) = ∑ p = 1 m ∑ r = 0 n p − 1 α k , p , n p − 1 − r ( t ) u p ( n p − 1 − r ) ( t ) ,

N k u 1 ( t ) , ⋯ , u m ( t ) = β k ( t ) f k ( u 1 ( t ) , ⋯ , u 1 ( n 1 − 1 ) ( t ) ; ⋯ ; u m ( t ) , ⋯ , u m ( n m − 1 ) ( t ) ) .

For a particular n_{k}th-order nonlinear DE in the system represented by Equation (27) or Equation (29), we choose the corresponding solution u k ( t ) as the primary solution and the solutions u p ( t ) , for p ≠ k , as the secondary solutions with respect to this same n_{k}th-order DE. We assume that α k , p , n p − 1 ( t ) , ⋯ , α k , p ,0 ( t ) , β k ( t ) and g k ( t ) are analytic, and hence have the relating Taylor expansion series

α k , p , j ( t ) = ∑ i = 0 ∞ α k , p , j , i t i , α k , p , j , i = α k , p , j ( i ) ( 0 ) i ! , 1 ≤ k , p ≤ m , 0 ≤ j ≤ n p − 1 ,

β k ( t ) = ∑ i = 0 ∞ β k , i t i , β k , i = β k ( i ) ( 0 ) i !

g k ( t ) = ∑ i = 0 ∞ g k , i t i , g k , i = g k ( i ) ( 0 ) 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 = ∫ 0 t ⋯ ∫ 0 t ( . ) d t ⋯ d t 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

u k ( t ) = ∑ i = 0 ∞ u k , i ( t ) , N k u 1 ( t ) , ⋯ , u m ( t ) = ∑ i = 0 ∞ A k , i ( t ) ,

f k ( u 1 ( t ) , ⋯ , u 1 ( n 1 − 1 ) ( t ) ; ⋯ ; u m ( t ) , ⋯ , u m ( n m − 1 ) ( t ) ) = ∑ i = 0 ∞ B k , i ( t ) , (30)

where the multi-order differential nonlinearity

f k ( u 1 ( t ) , ⋯ , u 1 ( n 1 − 1 ) ( t ) ; ⋯ ; u m ( t ) , ⋯ , u m ( n m − 1 ) ( t ) ) = ∑ i = 0 ∞ B k , i (t)

can be any analytic function in u 1 ( t ) , ⋯ , u 1 ( n 1 − 1 ) ( t ) ; ⋯ ; u m ( t ) , ⋯ , u m ( n m − 1 ) ( t ) and the relating ∑ p = 1 m n p -variable Adomian polynomials B k , i ( t ) have the standard formula [

B k , i ( t ) = B k , i ( u 1 , 0 ( t ) , ⋯ , u 1 , i ( t ) ; ⋯ ; u 1 , 0 ( n 1 − 1 ) ( t ) , ⋯ , u 1 , i ( n 1 − 1 ) ( t ) ; ⋯ ; u m , 0 ( t ) , ⋯ , u m , i ( t ) ; ⋯ ; u m , 0 ( n k − 1 ) ( t ) , ⋯ , u m , i ( n k − 1 ) (t))

or equivalently,

B k , i ( t ) = B k , i ( u 1 , 0 ( t ) , ⋯ , u 1 , i ( t ) ; ⋯ ; d n 1 − 1 d t n 1 − 1 u 1 , n 1 − 1 ( t ) , ⋯ , d n 1 − 1 d t n 1 − 1 u 1 , i + n 1 − 1 ( t ) ; ⋯ ; u m , 0 ( t ) , ⋯ , u m , i ( t ) ; ⋯ ; d n k − 1 d t n k − 1 u m , n k − 1 ( t ) , ⋯ , d n k − 1 d t n k − 1 u m , i + n k − 1 ( t ) ) (31)

The relating Cauchy products are

R k u 1 ( t ) , ⋯ , u m ( t ) = ∑ p = 1 m ∑ r = 0 n p − 1 α k , p , n p − 1 − r ( t ) u p ( n p − 1 − r ) (t)

where

α k , p , j ( t ) u p j ( t ) = ∑ i = 0 ∞ α k , p , j t i ∑ s = 0 ∞ u p , s ( j ) ( t ) = ∑ i = 0 ∞ ∑ s = 0 i α k , p , j , i − s t i − s d j d t j u p , s + j ( t ) ,

where j = n p − 1 − r ,

N k u 1 ( t ) , ⋯ , u m ( t ) = ∑ i = 0 ∞ β k , i t i ∑ p = 0 ∞ B k , s ( t ) = ∑ i = 0 ∞ ∑ s = 0 i β k , i − s t i − s B k , s ( t ) ,

hence it is also determined that

A k , i ( t ) = ∑ i = 0 ∞ β k , i t i B k , s (t)

Next we solve Equation (29) for L k ( t ) as

L k u k ( t ) = g k ( t ) − R k u 1 ( t ) , ⋯ , u m ( t ) − N k u 1 ( t ) , ⋯ , u m ( t ) (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 ( t ) = L k − 1 g k ( t ) − L k − 1 R k u 1 ( t ) , ⋯ , u m ( t ) − L k − 1 N k u 1 ( t ) , ⋯ , u m ( t ) (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 ( t ) = u k ( t ) − ∑ r = 0 n k − 1 t r r ! C k , r . (34)

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

u k ( t ) = ∑ r = 0 n k − 1 t r r ! C k , r + L k − 1 g k ( t ) − L k − 1 R k u 1 ( t ) , ⋯ , u m ( t ) − L k − 1 N k u 1 ( t ) , ⋯ , u m ( t ) . (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 ( t ) = ∑ i = 0 ∞ g k , i t i + n k ( i + 1 ) ⋯ ( i + n k ) , (36)

L k − 1 R k u 1 ( t ) , ⋯ , u m ( t ) = ∑ p = 1 m ∑ r = 0 n p − 1 L k − 1 α k , p , n p − 1 − r ( t ) u p ( n p − 1 − r ) ( t ) , (37)

where

L k − 1 α k , p , j ( t ) u p ( j ) ( t ) = ∑ i = 0 ∞ ∑ s = 0 i α k , p , j , i − s L k − 1 t i − s d j d t j u p , s + j ( t ) , (38)

where j = n p − 1 − r ,

L k − 1 N k u 1 ( t ) , ⋯ , u m ( t ) = ∑ i = 0 ∞ ∑ s = 0 i β k , i − s L k − 1 t i − s B k , s ( t ) . (39)

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

u k , 0 ( t ) = C k , 0 , u k , 1 ( t ) = t C k , 1 , u k , 2 ( t ) = t 2 2 ! C k , 2 , ⋯ , u k , n k − 1 ( t ) = t n k − 1 ( n k − 1 ) ! C k , n k − 1 , ⋯

u k , i + n k ( t ) = t i + n k ∏ q = 1 n k ( i + q ) − ∑ s = 0 i β k , i − s L k − 1 t i − s B k , s ( t ) × ∑ s = 0 i ∑ p = 1 m ∑ r = 0 i α k , p , n p − 1 − r , i − s L k − 1 t i − s d n p − 1 − r d t n p − 1 − r u p , s + n p − 1 − r (t)

for 1 ≤ k ≤ m , n k , n p ≥ 1, i ≥ 0 .

Therefore, the (s + 1)th-stage solution approximants Φ k , s + 1 ( t ) are given by Φ k , s + 1 ( t ) = ∑ i = 0 s u k , i ( t ) for s ≥ 0 .

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

u k , 0 ( t ) = C k , 0 = c k , 0 , u k , 1 ( t ) = t C k , 1 = c k , 1 t , ⋯ , u k , n k − 1 ( t ) = t n k − 1 ( n k − 1 ) ! = c k , n k − 1 t n k − 1

u k , i + n k ( t ) = g k , i − ∑ s = 0 i { β k , i − s B k , s + ∑ p = 1 m ∑ r = 0 n p − 1 [ ∏ j = 1 n p − 1 − r ( s + j ) ] α k , p , n p − 1 − r , i − s c p , s + n p − 1 − r } ∏ q = 1 n k ( i + q ) t i + n k = c k , n k t i + n k , i ≥ 0 (40)

where the ( ∑ p = 1 m n p )-variable Adomian polynomials B k , s are now constants and depend merely on the solution coefficients c k , j for 0 ≤ j ≤ n p − 1 . They are determined by induction as

B k , s ( t ) = B k , s ( c 1 , 0 , ⋯ , c 1 , s ; ⋯ ; c 1 , 0 ( n 1 − 1 ) , ⋯ , c 1 , s ( p 1 − 1 ) ; ⋯ ; c m ,0 , ⋯ , c m , s ; ⋯ ; c m ,0 ( n m − 1 ) , ⋯ , c m , s ( n m − 1 ) ) (41)

or equivalently,

B k , s = B k , s ( c 1 , 0 , ⋯ , c 1 , s ; ⋯ ; { ∏ j = 1 n 1 − 1 ( j ) } c 1 , n 1 − 1 , ⋯ , { ∏ j = 1 n 1 − 1 ( s + j ) } c 1 , s + n 1 − 1 ; ⋯ ; c m , 0 , ⋯ , c m , s ; { ∏ j = 1 n m − 1 ( j ) } c m , n m − 1 , ⋯ , { ∏ j = 1 n m − 1 ( s + j ) } c m , s + n m − 1 ) (42)

instead of the solution components u k , j ( t ) and solution derivative components u k , j ( i ) ( t ) for 1 ≤ k ≤ m ,0 ≤ j ≤ s and 0 ≤ q ≤ n p − 1 . Thus we have derived the desired Taylor expansion series for each of the m solutions u k ( t ) as u k ( t ) = ∑ i = 0 ∞ c k , i t i where the solution coefficients c k , i are given by the system of m-coupled nonlinear recurrence relations, obtained from inspection of Equation (39), as

c k , j = c k , j j ! , 1 ≤ k ≤ m , 0 ≤ j ≤ n p − 1 ,

c k , i + n k = g k , i − ∑ s = 0 i { β k , i − s B k , s + ∑ p = 1 m ∑ r = 0 n p − 1 [ ∏ j = 1 n p − 1 − r ( s + j ) ] α k , p , n p − 1 − r , i − s c p , s + n p − 1 − r } ∏ q = 1 n k ( i + q ) , (43)

i ≥ 0, m ≥ 1 and n k ≥ 1 , where the ( ∑ p = 1 m n p )-variable Adomian polynomials B k , s depend on the solution coefficients c k , j , for 0 ≤ j ≤ 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_{k}th order nonhomogeneous nonlinear IVPs with variable inputs and variable system coefficients are given as

c k , i + n k = c k , i + n k ( c 1 , 0 , ⋯ , c 1 , i + n 1 − 1 ; ⋯ ; c p , 0 , ⋯ , c p , i + n p − 1 ; ⋯ ; c m , n m − 1 , ⋯ , c m , s + n m − 1 ) ,

for i ≥ 0,1 ≤ k , p ≤ m .

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 [

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 [

y ′ = sin t − y 3 , y ( 0 ) = 1 2 , (44)

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

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

y 5 ( t ) = 1 2 − t 8 + 35 64 t 2 − 37 256 t 3 + 457 12288 t 4 − 1823 16384 t 5 .

We note that the order of approximation is O ( t 5 ) .

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

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

From

Example 2. Consider the first-order nonhomogeneous nonlinear differential equation with a quadratic nonlinearity [

y ′ = y 2 + 2 y + 2 , y ( 0 ) = 0 , (45)

on the interval 0 ≤ t ≤ π / 6 . It has the exact solution y ( t ) = − 1 + tan ( t + π 4 ) .

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

y 12 ( t ) = 2 t + 2 t 2 + 8 3 t 3 + 10 3 t 4 + 64 15 t 5 + 244 45 t 6 + 2176 315 t 7 + 554 64 t 8 + 31744 2835 t 9 + 202084 13175 t 10 + 2830336 155925 t 11 + 2162212 93555 t 12

y 14 ( t ) = 2 t + 2 t 2 + 8 3 t 3 + 10 3 t 4 + 64 15 t 5 + 244 45 t 6 + 2176 315 t 7 + 554 64 t 8 + 31744 2835 t 9 + 202084 13175 t 10 + 2830336 155925 t 11 + 2162212 93555 t 12 + 178946048 6081075 t 13 + 1594887848 42567525 t 14

y 16 ( t ) = 2 t + 2 t 2 + 8 3 t 3 + 10 3 t 4 + 64 15 t 5 + 244 45 t 6 + 2176 315 t 7 + 554 64 t 8 + 31744 2835 t 9 + 202084 13175 t 10 + 2830336 155925 t 11 + 2162212 93555 t 12 + 178946048 6081075 t 13 + 1594887848 42567525 t 14

Indeed, these are the first 12, 14 and 16 terms of the Taylor series of the

function y ( t ) = − 1 + tan ( t + π 4 ) about the point t = 0 , 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.

Since | − 1 + tan ( t + π 4 ) | ≤ 1 + 3 2 on the interval 0 ≤ t ≤ π 6 , and

| f ( n ) ( 0 ) | = | d n d y n ( y 2 ) | y = 0 | ≤ 2 for any n, accordance by ( [

respective truncation errors are τ = 2 ( 1 + 3 2 ) 12 12 ! = 7.44224 × 10 − 6 ,

τ = 2 ( 1 + 3 2 ) 14 14 ! = 1.42386 × 10 − 7 and τ = 2 ( 1 + 3 2 ) 16 16 ! = 2.06581 × 10 − 8 .

We note that in these computations approximation orders are O ( t 12 ) , O ( t 14 ) and O ( t 16 ) , respectively.

The curves of the computed approximants and the exact solution are plotted in

The MATHEMATICA command P a d e A p p r o x i m a n t [ y 2 n + 1 ( t ) , { t ,0, { n , n } } ] for n = 6 , 7 and 8 output 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

Running NSOL routine for n = 11 , 13 , and 15 and step size h = 0.05 to generate numeric solutions on the interval 0 ≤ t ≤ π / 6 . In

As a comparison RK4 solution is computed and depicted with exact solution in

We denote P n ( t ) = [ n / n ] { y 2 n + 1 ( t ) } and consider the absolute error function | E n ( t ) | = | P n ( t ) − y ( t ) | for P n ( t ) on the interval 0 ≤ t ≤ π / 6 and the maximal error parameter M E n = max 0 ≤ t ≤ π / 6 | E n ( t ) | for n = 6 , 7 , and 8.

In

From

The maximal error parameters M E n for n = 6 , 7 and 8 are given in

Example 3. Consider the 2-dimensional system of nonlinear differential equations with quadratic nonlinearity [

x ′ = x y , y ′ = x 2 , x ( 0 ) = 1 , y ( 0 ) = 0 , (46)

over the interval 0 ≤ t ≤ π 3 . The exact solutions are x ( t ) = sec t , y ( t ) = tan t .

Running PSSOL for n = 10 to generate 11-term approximants x 11 and y 11 of the solutions x ( t ) and y ( t ) , respectively as

x 11 ( t ) = 1 + t 2 2 + 5 t 4 24 + 61 t 6 720 + 277 t 8 8064 + 50521 t 10 3628800 , and y 11 ( t ) = t + t 3 3 + 2 t 5 15 + 17 t 7 315 + 62 t 9 2835 + 1382 t 11 155925 .

We note that the respective orders of the approximation are O ( t 10 ) and

n | 6 | 7 | 8 |
---|---|---|---|

M E n | 1.0747 × 10 − 12 | 1.33227 × 10 − 15 | 8.88178 × 10 − 16 |

O ( t 11 ) .

The MATHEMATICA command P a d e A p p r o x i m a n t [ x 11 ( t ) , { t ,0, { 5,5 } } ] and the MATHEMATICA command P a d e A p p r o x i m a n t [ y 11 ( t ) , { t ,0, { 5,5 } } ] for n = 5 output [5/5] Padé approximants for x 11 ( t ) and y 11 ( t ) as

Pad e x [ 5 / 5 ] = 1 + 11 t 2 252 + 13 t 4 15120 1 − 115 t 2 252 + 313 t 4 15120 , and Pad e y [ 5 / 5 ] = t − t 3 9 + t 5 945 1 − 4 t 2 9 + t 4 63 .

The outputs of the routines PSSOL, NSOL and RK4 for n = 10 , 12 and 14 and correspondingly the outputs of the MATHEMATICA command P a d e A p p r o x i m a n t for n = 5 , 6 and 7 together with the exact solutions x and y are depicted in Figures 4(a)-(c).

In

We list the maximal error parameters M E n for n = 5 , 6 and 7 in

n | 5 | 6 | 7 |
---|---|---|---|

M E n ( x ) | 1.23274 × 10 6 | 4.35763 × 10 11 | 4.35763 × 10 11 |

M E n ( y ) | 3.17247 × 10 7 | 2.25313 × 10 9 | 1.16824 × 10 11 |

From

Example 4. Consider a three-dimensional system of autonomous nonlinear DE with quadratic nonlinearities [

x ′ = y − a x + y z , y ′ = b y − x z , z ′ = c x y − d z − e x 2 , (47)

where x , y , z ∈ R , and the state variables, and a , b , c , d ∈ R + and e ∈ R are constant parameters of the system. This system has many interesting complex behaviors and exhibits chaotic behavior over a wide range of parameters. It can show two coexisting one-wing, a single two-wing, three-and four-wing when its parameters are chosen appropriately.

When we set a = 1.6 , b = 3 , c = 8 , d = 11 and e = 0.5 the system can display two existing one-wing chaotic attractors with different initial conditions as shown in

For the values of the parameters a = 8 , b = 4 , c = 8 , d = 11 and e = 0.5 , the system generates two existing two-wing chaotic attractors as shown in

With the choice of the parameters a = 4.84 , b = 3 , c = 8 , d = 11 and e = 0.5 the system generates a three-wing attractor for the initial values as x ( 0 ) = − 0.5 , y ( 0 ) = − 1 , z ( 0 ) = 0.5 as seen in

interval 0 ≤ t ≤ 40 are displayed.

It can also display a four-wing chaotic attractor with parameters a = 8.8 , b = 3 , c = 8 , d = 11 and e = 0.5 and initial values x ( 0 ) = − 0.5 , y ( 0 ) = − 1 , z ( 0 ) = 0.5 as illustrated in

Another four-wing chaotic attractor exists for a = 9.93 , b = 3 , c = 8 , d = 11 , e = 0.5 and for x ( 0 ) = − 0.5 , y ( 0 ) = − 1 , z ( 0 ) = 0.5 as shown in

Example 5. Consider 4-dimensional system of nonlinear DE with quadratic nonlinearities [

x ′ = − y − z − w , y ′ = x , z ′ = a ( y − y 2 ) − b z , w ′ = c ( z 2 − z 2 ) − d w , (48)

where a, b, c, and d are real parameters.

The system in (48) exhibits hypertoroidal behavior when the parameters are chosen as a = 0.2 , b = d = 0 , c = 0.04 and the initial values are taken as x ( 0 ) = 0 , y ( 0 ) = 0.75 , z ( 0 ) = 0.2 , w ( 0 ) = − 0.75 as shown in

Running NSOL for n = 4 and step-size h = 0.05 outputs the 5th-order numeric solutions for the system on the interval 0 ≤ t ≤ 200 . The 3-dimensional x-y-z (top) and y-z-w (bottom) projections of the modified results

are depicted on the left of

Example 6. Consider the following seven-dimensional third-order hyperchaotic system [

x ′ 1 = − a x 1 + a x 5 − b x 5 x 6 x 7 , x ′ 2 = − c x 2 − d x 6 + x 1 x 6 x 7 , x ′ 3 = − a x 3 + a x 5 − g x 1 x 2 x 7 , x ′ 4 = − a x 4 + e x 1 + x 1 x 2 x 3 ,

x ′ 5 = − a x 5 + e x 7 − x 2 x 3 x 4 , x ′ 6 = − e x 6 + e x 5 + x 3 x 4 x 5 , x ′ 7 = − b x 7 + f x 2 − m x 5 x 6 . (49)

Since it has a chaotic attractor when we set a = 15 , b = 5 , c = 0.5 , d = 25 , e = 10 , f = 4 , g = 0.1 , m = 1.5 and the initial conditions are taken as x 1 ( 0 ) = 0.1 , x 2 ( 0 ) = 0.1 , x 3 ( 0 ) = 0.0 , x 4 ( 0 ) = − 0.2 , x 5 ( 0 ) = − 0.2 , x 6 ( 0 ) = 0.0 , x 7 ( 0 ) = 0.3 we consider the numeric solution of the system (49) for those values over the interval 0 ≤ t ≤ 40 .

Running NSOL by taking n = 4 and step-size h = 0.02 generates 5th-order numeric solutions on the interval for the system. The 2- and 3-dimensional projections of the modified results are plotted on the left of

In this study, we use the Duan-Rach-Wazwaz modified Adomian decomposition method for solving nonlinear IVPs of the first order nonlinear ODEs and two, three, four and seven dimensional systems of nonlinear ODEs. To show the computational accuracy of the technique we consider homogeneous and nonhomogeneous equations with variable and constant systems coefficients. In each example, the solution of the modified technique is compared with that from MATHEMATICA solution and with the exact solution if it is known. In addition, in examples 1, 2 and 3 the numeric solutions are also computed by RK4. We have seen that the modification results closely agree with MATHEMATICA and RK4 solutions and also with exact solutions, if available. However, we have obtained modified solutions over a bigger time step than MATHEMATICA and RK4 solutions. Moreover, different problems have been solved in order to confirm the robustness of the modification over a wide variety of ODEs. Therefore, it may be concluded that the method has the ability of applying all types of nonlinear ODEs provided uniqueness of the solutions.

The authors declare no conflicts of interest regarding the publication of this paper.

Umut, Ö. and Yaşar, S. (2019) Numerical Treatment of Initial Value Problems of Nonlinear Ordinary Differential Equations by Duan-Rach-Wazwaz Modified Adomian Decomposition Method. International Journal of Modern Nonlinear Theory and Application, 8, 17-39. https://doi.org/10.4236/ijmnta.2019.81002