International Journal of Mathematics and Computational Science

Vol. 2, No. 2, 2016, pp. 43-54

http://www.aiscience.org/journal/ijmcs

ISSN: 2381-7011 (Print); ISSN: 2381-702X (Online)

* Corresponding author

E-mail address: qyh2004@zjnu.edu.cn (Y. H. Qian)

Construction of Analytic Solution for Coupled Cubic Nonlinear Systems Using Homotopy Analysis Method

J. M. Guo1, Y. H. Qian1, *, S. P. Chen2

1College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, Zhejiang, P. R. China

2College of Mathematics, Xiamen University of Technology, Xiamen, P. R. China

Abstract

In this paper, the existence solution of initial value problem for coupled cubic nonlinear systems is proved at first. Then by using

the homotopy analysis method (HAM), an analytical approximation of those systems can be obtained. It is full of freedom to

choose a set of base functions when using the HAM, and as the set of base functions is chosen differently, the analytical

approximation solutions which will have some different effect. Therefore, it is an interesting and meaningful task to get a more

efficient analytical approximation by a better set of base functions. Furtherly, by combining the HAM with padé approximation,

the result can be obtained on broader region of convergence. To illustrate the accuracy of the present method, the solutions

obtained in this paper are compared with those of Runge-Kutta method, which shows the HAM is effective and feasible.

Keywords

Initial Value Problem, Coupled Cubic Nonlinear Systems, the Homotopy Analysis Method, Homotopy Padé Approximation

Received: April 1, 2016 / Accepted: April 18, 2016 / Published online: May 12, 2016

@ 2016 The Authors. Published by American Institute of Science. This Open Access article is under the CC BY license.

http://creativecommons.org/licenses/by/4.0/

1. Introduction

Coupled nonlinear systems are one of the most important

systems in nonlinear dynamical problems, which are pervaded

in different disciplines such as science, engineering,

mechanical, physical, structural applications and even in

aeronautical technology. However, it is hardly to seek the

analytical exact solutions in normal circumstances. As a result,

the analytical approximation has received lots of attention in

these problem few decades. In reference [1], Fang and Guo

systematically studied the existence of time periodic solutions

for a damped generalized coupled nonlinear wave equations.

And in reference [2], Liu and his associates explored general

solution for the coupled equations of transversely isotropic

magnetoelectroelastic solids. At the same time, Al-Saif and

Zhu applied mixed differential quadrature method to solve the

coupled two-dimensional incompressible Navier-stokes

equation and heat equation in reference [3]. Then, Lazhar

Bougoffa and Smail Bougouffa successfully applied Adomian

decomposition method (ADM) to solve some coupled systems

in reference [4]. But inconveniently, ADM has its own

limitation and it is not suitable for strongly nonlinear

problems.

The homotopy analysis method (HAM) systematically

proposed in reference [5] by Professor Liao, can be without

the above insufficient of ADM. For more than one decade, a

number of scholars have adopted the HAM to a variety of

nonlinear problems in engineering and physical science. Liao

and his associates [6-8] furnished the analytical formulas for

various nonlinear dynamical system. Xu [9] derived the

explicit solutions of the free convection flow over a vertical

flat plate embedded in a porous medium. Allan and Syam [10]

solved the nonhomogeneous Blasius problem. Abbasbandy

[11] generalized the HAM to the problem of nonlinear heat

44 J. M. Guo et al.: Construction of Analytic Solution for Coupled Cubic Nonlinear Systems

Using Homotopy Analysis Method

transfer equations. Hayat [12-14] deduced the solutions of

grade fluid problems. Song and Zhang [15] unraveled the

problem of fractional KdV-Burgers-Kuramoto differential

equations. Moreover, Inc [16] considered the Laplace

equation having Dirichlet and Neumann boundary conditions

by using the HAM. Li and his associates [17] investigated the

in-phase and out-of-phase periodic solutions of coupled van

der Pol oscillators. Zhang and his associates [18] derived the

highly convergent solutions for both two- and three-degree-of

freedom van der Pol oscillators via the HAM. Cobiaga and

Reartes [19] employed the method to search for periodic orbits

in delay differential equations. Wang and Lu [20] investigated

the application of HAM to nonlinear short-crested waves in a

fluid of finite depth.

Currently, some optimal HAM approaches are developed,

which can get faster convergent homotopy series solution [21,

22]. Employing the homotopy padé approximation,

Pirbodaghil and Hoseini [23] obtained the accurate solutions

of a nonlinear free vibration of conservative 2DOF systems.

López and Ceniceros [24] studied type I Fourier-padé

approximation for vector-valued analytic functions formed by

Nikishin systems and gave the exact rate of convergence of the

corresponding approximants. Wang and Au [25] combined

padé approximation and the generalized padé approximation

of the matrix exponential function in precise integration and

developed a new generalized family of precise time step

integration methods. In order to solve the problems of some

conventional methods, Le [26] presented two complex padé

approximations of wide-angle beam propagator. The resulted

approaches allow more accurate approximations to the

Helmholtz equation than the real padé approximation. Li [27]

presented a generalized Padé approximation method, which

can be used to solve homoclinic and heteroclinic orbits of

strongly nonlinear autonomous oscillators. Zhao and his

associates [28] used the homotopy perturbation method (HPM)

and Padé approximation method to study analytical solutions

of velocity and temperature in laminar boundary layer over a

flat plate with wall injection flow. H.Vazquez-Leal and his

associates [29] presented a comparison of HPM, NDHPM,

Picard and Picard-Padé methods for solving

Michaelis-Menten equation.

In this paper, attention is taken on the coupled cubic nonlinear

systems, which can be expressed as

3 2 2 3

1 1 1 2 1 2 3 1 2 4 2

3 2 2 3

2 1 1 2 1 2 3 1 2 4 2

,x a x a x x a x x a x

x b x b x x b x x b x

= + + + = + + +

ɺɺ

ɺɺ (1)

with the initial conditions

1 1 2 2 1 3 2(0) , (0) , (0) , (0) 0.x c x c x c x= = = =ɺ ɺ (2)

In Section 2, the solution existence of initial value problem for

the above systems is proved by using the method in reference

[30]. And in Section 3, the HAM and homotopy padé

approximation is exploited to the above systems. Section 4

gives an example for the above systems and obtains two types

of analytical approximation via two different sets of base

functions, and the solution is further modified by Padé

approximation. The conclusions are drawn in Section 5

finally.

2. The Existence Solution of Initial Value Problem

In this section, the solution existence of initial value problem

is proven, with the help of two theorems and a lemma.

Theorem 1 and Theorem 2 are both from the reference [30],

and Lemma 1 is proven in this section.

Theorem 1 (Existence and uniqueness): Let ( , )F t x be a vector

function in the region 0

0: ,t t a x x bΩ − ≤ − ≤ , and satisfies

i. ( , )F t x is continuous；

ii. ( , )F t x is locally Lipschitz in x , that is, for arbitrary

[1] [2]( , ),( , )t x t x ∈Ω , there exists a constant 0L > , such that

[1] [2] [1] [2]( , ) ( , )F t x F t x L x x− ≤ − , then the initial value

problem 0

0

( , )

( )

dxF t x

dt

x t x

= =

admits a unique solution for

0t t h− ≤ , where min( , )

bh a

M= , max ( , )M F t x

Ω= .

Lemma 1: Let ( , )F t x be a vector function in the region :Ω

0t t a− ≤ ,

0x x b− ≤ , if the partial derivatives

( 1,2, , )i

Fi n

x

∂ =∂

⋯ exists and continuous bounded, then

( , )F t x is locally Lipschitz in x ,where 1 2( , , , )T

nx x x x= ⋯ .

Proof Let

1 1 2

2 1 2

1 2

( , , , , )

( , , , , )( , )

( , , , , )

n

n

n n

f t x x x

f t x x xF t x

f t x x x

=

⋯

⋯

⋮

⋯

, (3)

and the vector norm is defined as 1-norm.

Suppose that the partial derivatives ( 1,2, , )i

Fi n

x

∂ =∂

⋯ exists

and continuous bounded. Then, for 1, 2, ,j n= ⋯ , the

International Journal of Mathematics and Computational Science Vol. 2, No. 2, 2016, pp. 43-54 45

continuous bounded partial derivatives of 1 2( , , , , )j nf t x x x⋯

denoted as ( 1, 2, , )j

i

fi n

x

∂=

∂⋯ are all exist.

Let

max , 1, 2, , , 1,2, ,j

i

f Li n j n

x nΩ

∂≤ = =

∂⋯ ⋯ , (4)

And [1] [2]( , ), ( , )t x t x∀ ∈Ω ,

[1] [1] [1] [1]

1 2 , , , nx x x x= ⋯ ,

[2] [2] [2] [2]

1 2 , , , nx x x x= ⋯ , let [2] [1] , 1,2, ,i i ix x x i n∆ = − = ⋯ ,

0 1, 1, 2, , .i

i nθ< < = ⋯

For arbitrary 1, 2, ,j n= ⋯ ,

[1] [2] [1] [2]

1

( , ) ( , ) ,n

j j i i

i

Lf t x f t x x x

n =− ≤ −∑ (5)

then,

[1] [2] [1] [2] [1] [2]

1

( , ) ( , ) ( , ) ( , ) .n

j j

j

F t x F t x f t x f t x L x x=

− = − ≤ −∑ (6)

That means, ( , )F t x is locally Lipschitz in x .

Theorem 2 (Continuation and continuity of the solution): Let

( , )F t x be a vector function in the region

0

0: ,t t a x x bΩ − ≤ − ≤ , if ( , )F t x is continuous in some

region G ⊇ Ω and locally Lipschitz in x , then the solution

0

0( , , )x x t t x= of ( , )dx

F t xdt

= with initial condition

0

0( )x t x=

can be continued, either to ( )or+∞ − ∞ , or to

make the point 0

0( , ( , , ))t x t t x

arbitrarily approach to the

boundary of G . And the solution 0

0( , , )x t t x is continuous

in its existence range.

Now, the existence solution of Eq. (1) with the initial

conditions (2) can be proofed.

Proof Let

1 3 2 4 1 2 3 4, , ( ) ( ( ), ( ), ( ), ( ))x x x x x t x t x t x t x t= = =ɺ ɺ , (7)

then

1 2 3 4( ) ( , , , )x t x x x x=ɺ ɺ ɺ ɺ ɺ ,

0

1 2 3 4 1 2 3(0) ( (0), (0), (0), (0)) ( , , ,0)x x x x x c c c x= = = . (8)

And the original equations are equivalent to

0

( ) ( , )

(0)

x t F t x

x x

= =

ɺ, (9)

where

1 1 2 3 4 3

2 1 2 3 4 4

3 2 2 3

3 1 2 3 4 1 1 2 1 2 3 1 2 4 2

3 2 2 3

4 1 2 3 4 1 1 2 1 2 3 1 2 4 2

( , , , , )

( , , , , )( , )

( , , , , )

( , , , , )

f t x x x x x

f t x x x x xF t x

f t x x x x a x a x x a x x a x

f t x x x x b x b x x b x x b x

= = + + + + + +

. (10)

Let us define the vector norm as 1-norm, then

0

1 1 2 2 3 3 4x x x c x c x c x− = − + − + − + . (11)

Because 1 3

,f x=2 4

,f x= 3 2 2 3

3 1 1 2 1 2 3 1 2 4 2 ,f a x a x x a x x a x= + + +3 2 2 3

4 1 1 2 1 2 3 1 2 4 2f b x b x x b x x b x= + + + 2 3

3 1 2 4 2b x x b x+ + are all

continuous in

5

1 2 3 4 1 1 2 2 3 3 4( , , , , ) , t x x x x R t a x c x c x c x bΩ = ∈ ≤ − + − + − + ≤ , (12)

( , )F t x is continuous in Ω .

What's more, the continuous bounded partial derivatives of

( , )F t x in Ω all exist as follow,

2 2 2 2

1 1 2 1 2 3 2 1 1 2 1 2 3 2

1

2 2 2 2

2 1 3 1 2 4 2 2 1 3 1 2 4 2

2

3

4

(0,0,3 2 ,3 2 ),

(0,0, 2 3 , 2 3 ),

(1,0,0,0),

(0,1,0,0).

Fa x a x x a x b x b x x b x

x

Fa x a x x a x b x b x x b x

x

F

x

F

x

∂ = + + + +∂∂ = + + + +∂∂ =∂∂ =∂

(13)

Thus ( , )F t x is locally Lipschitz in x .

That means, there exists an interval t h≤ , such that the

solution of initial value problem exists, where

min( , )b

h aM

= , (14)

3

1 2 3 4max ( , ) max( ) (2 )( ) ,M F t x f f f f b cµΩ Ω

= = + + + ≤ + + (15)

in which, 4

1

( )i i

i

a bµ=

= +∑ ,1 2 3

max , , c c c c= .

Let

3(2 )( )

b ba

M b cµ= ≥

+ +, (16)

and consider

30

(2 )( )

b

b b cµ∂ =∂ + +

, (17)

it can be got

46 J. M. Guo et al.: Construction of Analytic Solution for Coupled Cubic Nonlinear Systems

Using Homotopy Analysis Method

.2

cb = (18)

then,

3 2

/ 2 4.

(2 )( / 2 ) 27(2 )

b ca

M c c cµ µ= ≥ =

+ + + (19)

Therefore, for 2 2

4 4

27(2 ) 27(2 )t

c cµ µ− ≤ ≤

+ +, there exists

the unique initial value solution.

Furtherly, based on Theorem 2, the initial value solution can

be continued to ( )or+∞ − ∞ .

Well, the stability of the zero solution for the differential

equations ( ) ( , )x t F t x=ɺ

can be studied by using the

following Theorem 3 and Theorem 4, which are both from the

reference [30].

Theorem 3 For the differential equations

1 2( ) ( , ), ( , , , )T

nx t F t x x x x x= =ɺ ⋯ , if there exists a positive

definite function ( )V x defined on a neighborhood of the

origin， such that its total derivative 1

ni

i i

dxdV dV

dt dx dt=

=∑

is

negative semidefinite or identical to zero， then the zero

solution of the differential equations is stable.

Theorem 4 For the differential equations

1 2( ) ( , ), ( , , , )T

nx t F t x x x x x= =ɺ ⋯ , if there exists a function

( )V x defined on a neighborhood of the origin, and the

function ( )V x is not negative semidefinite (or positive

semidefinite), and (0) 0V = , such that its total derivative

1

ni

i i

dxdV dV

dt dx dt=

=∑

is positive definite (or negative definite)，

then the zero solution of the differential equations is unstable.

By using the above two theorems, some sufficient conditions

of the stability of the zero solution for the differential

equations 1 2( ) ( , ), ( , , , )T

nx t F t x x x x x= =ɺ ⋯ , where ( , )F t x is

defined as Eq. (10).

Conclusion 1 If the differential equations

1 2 3 4( , ), ( , , , )TF t x x x x x x=

defined as Eq. (10) satisfies

2 1 4 30, 0,a b a b+ = + = and

1 4 3 20, 0, 0a b a b> > + ≥ , then

the zero solution for the differential equations ( ) ( , )x t F t x=ɺ

is unstable.

Proof Set 1 3 2 4

( )V t x x x x= + , then

44 3 2 2

1 1 2 1 1 2 3 2 1 2

1

( ) ( )i

i i

dxdV dVa x a b x x a b x x

dt dx dt=

= = + + + +∑

2 4 2 2

4 3 1 2 4 2 3 4( ) .a b x x b x x x+ + + + + (20)

So when 2 1 4 3

0, 0,a b a b+ = + = and

1 4 3 20, 0, 0a b a b> > + ≥ , then

dV

dtis positive definite, and

( )V t is not negative semidefinite, that means, the zero solution

for the differential equations ( ) ( , )x t F t x=ɺ is unstable, based

on Theorem 4.

Conclusion 2 If the differential equations

1 2 3 4( , ), ( , , , )TF t x x x x x x=

defined as Eq. (10) satisfies

2 4 1 30,a a b b= = = = and

1 3 2 40, 0, 0, 0a a b b< < < < , then

the zero solution for the differential equations ( ) ( , )x t F t x=ɺ

is stable.

Proof When 2 4 1 3

0,a a b b= = = = and

1 3 2 40, 0, 0, 0a a b b< < < < , then 1 2 3 4( , ), ( , , , )TF t x x x x x x=

is defined as follows

3

4

3 2

1 1 3 1 2

2 3

2 1 2 4 2

( , ) .

x

xF t x

a x a x x

b x x b x

= + +

(21)

Set

4 2 2 4 2 21 3 4 31 3 1 2 2 3 4

2 2

( ) ( )2 2

a a b aV t x a x x x x x

b b

α α αα α− −= + − + + + , (22)

where 0α > , then ( )V t is positive definite.

And

43 2 2 33 4

1 1 3 1 2 3 3 1 2 2 4

1 2

3 2 2 333 1 1 3 1 2 4 2 1 2 4 2

2

2( 2 2 ) ( 2 )

22 ( ) ( )

0.

i

i i

dV dV dx a ba x a x x x a x x x x

dt dx dt b

ax a x a x x x b x x b x

b

αα α α

αα

=

= = − − + − −

+ + + +

=

∑

(23)

Based on Theorem 4, the zero solution for the differential

equations ( ) ( , )x t F t x=ɺ is unstable.

3. Application of the HAM and Homotopy Padé Approximation

In this section, the HAM and homotopy Padé approximation

are applied to solve the coupled cubic nonlinear systems (1)

with the initial conditions (2).

3.1. Application of the HAM

Following the form of Eq. (1), a nonlinear operator can be

International Journal of Mathematics and Computational Science Vol. 2, No. 2, 2016, pp. 43-54 47

constructed as

1

2

( ; )

( ; )

t qN

t q

ϕϕ

=

23 2 2 31

1 1 2 1 2 3 1 2 4 12

23 2 2 32

1 1 2 1 2 3 1 2 4 12

( ; )( ; ) ( ; ) ( ; ) ( ; ) ( ; ) ( ; )

,( ; )

( ; ) ( ; ) ( ; ) ( ; ) ( ; ) ( ; )

t qa t q a t q t q a t q t q a t q

t

t qb t q b t q t q b t q t q b t q

t

ϕ ϕ ϕ ϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕ ϕ ϕ ϕ

∂ − − − − ∂ ∂ − − − − ∂

(24)

and based on the HAM, the zeroth-order equation can be

defined as

1 1,0 1

2 2,0 2

( ; ) ( ) ( ; )(1 ) ( )[ ]

( ; ) ( ) ( ; )

t q x t t qq L q H t N

t q x t t q

ϕ ϕϕ ϕ

− − = −

ℏ , (25)

with the initial conditions

11 1 3

t 0

22 2

t 0

(t; )(0; ) ( ) ( )

t

(t; )(0; ) ( ) 0

t

qq c q c q

qq c q

ϕϕ

ϕϕ

=

=

∂= =

∂

∂= =

∂

, (26)

where, [0,1]q ∈

is an embedding parameter, ℏ is an

auxiliary parameter, ( )H t is an auxiliary function, L is an

auxiliary linear operator, and 1,0 ( )x t and 2,0 ( )x t are the

initial guess solutions.

When 0q = , Eq.(25)-(26) admit the solution

1 1,0(t; 0) (t)xϕ = , 2 2,0(t; 0) (t)xϕ = , (27)

and when 1q = , Eq.(25)-(26) are equivalent to the original

equations, and admit the solution

1 1(t;1) (t)xϕ = ,

2 2(t;1) (t)xϕ = . (28)

Hence, as q increases from 0 to 1, the solution ( )t;i

qϕ varies

from the initial guess solution ( ), 0t

ix to the exact solution

( )tix , where 1, 2i = .

Setting

( ) ( )1

1,

0

t;1t

!

m

m m

q

qx

m q

ϕ

=

∂=

∂, (29)

( ) ( )2

2,

0

t;1t

!

m

m m

q

qx

m q

ϕ

=

∂=

∂, (30)

then, for 1, 2i = , ( )t;i

qϕ can be expanded into the

following Taylor series with respect to q as

( ) ( ) ( )1 1, 0 1,

1

t; t t m

m

m

q x x qϕ+∞

== +∑ , (31)

( ) ( ) ( )2 2,0 2,

1

t; t tm

m

m

q x x qϕ+∞

=

= +∑ , (32)

Provided that the auxiliary linear operator L, initial guess

solution 1,0 ( ),x t 2,0 ( ),x t auxiliary parameter ℏ , and auxiliary

function ( )H t are properly chosen, the series expansion in

Eq.(31-32) converges at 1q = , thus

( ) ( ) ( )1 1,0 1,

1

t t tm

m

x x x+∞

=

= +∑ , (33)

( ) ( ) ( )2 2,0 2,

1

t t tm

m

x x x+∞

=

= +∑ . (34)

For convenience, the vector 1, nx

and 2, nx

are defined as

( ) ( ) ( ) 1, 1,0 1,1 1,t , t , , t

n nx x x x=

⋯ , (35)

( ) ( ) ( ) 2, 2,0 2,1 2,t , t , , t

n nx x x x=

⋯ . (36)

By differentiating the zeroth-order Eq. (25) ( 1)m m ≥ times

with respect to q, then dividing the equation by !m and setting

0q = , the mth-order equation can be yielded as

1, 1, 1 1, 1, 1

2, 2, 1 2, 2, 1

(t) χ (t) ( )( )

(t) χ (t) ( )

m m m m m

m m m m m

x x R xL H t

x x R x

− −

− −

− = −

ℏ , (37)

with the initial conditions

1, (0) 0mx = , 1, (0) 0mx =ɺ , 2, (0) 0mx = , 2, (0) 0mx =ɺ , ( 1m ≥ ), (38)

where

0, 1χ

1, 1m

m

m

≤= >

, (39)

and

1, 1, 1 1, 1 1 1, 1, 1,

1 0

2 2, 1, 1,

1 0

3 1, 2, 2,

1 0

4 2, 2, 2,

1 0

( ) ( ) [ ( )( )]

[ ( )( )]

[ ( )( )]

[ ( )( )],

s

m m m j i s i

j s m i

s

j i s i

j s m i

s

j i s i

j s m i

s

j i s i

j s m i

R x x t a x t x x

a x t x x

a x t x x

a x t x x

− − −+ = − =

−+ = − =

−+ = − =

−+ = − =

= −

−

−

−

∑ ∑

∑ ∑

∑ ∑

∑ ∑

ɺɺ

(40)

2, 2, 1 2, 1 1 1, 1, 1,

1 0

2 2, 1, 1,

1 0

3 1, 2, 2,

1 0

4 2, 2, 2,

1 0

( ) ( ) [ ( )( )]

[ ( )( )]

[ ( )( )]

[ ( )( )].

s

m m m j i s i

j s m i

s

j i s i

j s m i

s

j i s i

j s m i

s

j i s i

j s m i

R x x t b x t x x

b x t x x

b x t x x

b x t x x

− − −+ = − =

−+ = − =

−+ = − =

−+ = − =

= −

−

−

−

∑ ∑

∑ ∑

∑ ∑

∑ ∑

ɺɺ

(41)

48 J. M. Guo et al.: Construction of Analytic Solution for Coupled Cubic Nonlinear Systems

Using Homotopy Analysis Method

Based on Eq. (37) and the initial guess solution 1,0 ( )x t , 2,0 ( )x t ,

1, ( )( 1,2, )nx t n = ⋯ and 2, ( )( 1,2, )nx t n = ⋯ can be obtained

step by step. Thus, the mth-order analytical approximation can be

expressed in terms of the summation series

1 1, 1,0 1,1 1,

0

( ) ( ) ( ) ( ) ( )m

i m

i

x t x t x t x t x t=

≈ = + + +∑ ⋯ , (42)

2 2, 2,0 2,1 2,

0

( ) ( ) ( ) ( ) ( )m

i m

i

x t x t x t x t x t=

≈ = + + +∑ ⋯ . (43)

3.2. Application of Homotopy Padé Approximation

In order to achieve a broader region of convergence of the

HAM solution, the HAM can be combined with Padé

approximation. As a result, a [ , ]M N Padé approximation of

the HAM solution can be obtained.

Padé approximation expands a function as a ratio of two

power series. If the rational function is

0

1

( ) ,

1

Mk

k

k

Nk

k

k

a t

P t

b t

=

=

=+

∑

∑ (44)

Then, ( )P t is said to be a [ , ]M N Padé approximation of

the series

0

( ) ,k

k

k

f t c t∞

==∑ (45)

with

(0) (0), ( ) ( ) , 1,2, , .0 0

k k

k k

d dP f P t f t k M N

t tdt dt= = = +

= =⋯ (46)

Eq. (46) provides 1M N+ + equations for the unknowns

parameters 0, ,

Ma a⋯ and

0, ,

Nb b⋯ .

4. Example

Let us consider the following coupled cubic nonlinear system

221

1 22

2

3 222 1 22

2d x

x xdt

d xx x x

dt

= −

= − +

, (47)

with the initial conditions

1 1 2 2(0) 0, (0) 1, (0) 1, (0) 0x x x x= = = =ɺ ɺ , (48)

From Section 2, it is plenty of freedom to choose the auxiliary

parameter ℏ , auxiliary function ( )H t , auxiliary linear

operator L, and the initial guess solution 1,0 ( )x t , 2,0 ( )x t . For

convenience, a set of base functions must be proposed at first,

then all above will be chosen to suit for it. Based on different

sets of base functions, the solutions obtained by HAM will

have different effect. As an illustration, the analytical solution

will be expressed by the following two different set of base

functions.

4.1. To Be Expressed by the Set of Base Functions | = , , , , …

Suppose that the solution of the systems (47)-(48) can be

expressed by the set of base functions 0,1, 2,3, n

t n = ⋯ ,

then the analytical solution can be expressed as

1 1,

0

2 2,

0

( )

( )

n

n

n

n

n

n

x t a t

x t a t

+∞

=

+∞

=

= =

∑

∑. (49)

The initial guess solution is chosen as

1,0 2,0( ) , ( ) 1x t t x t= = , (50)

and the linear operator is expressed as

2

1

21

22 2

2

( ; )

( ; )

( ; ) ( ; )

t q

t q tL

t q t q

t

ϕϕϕ ϕ

∂ ∂ = ∂ ∂

. (51)

According to the rule of solution expression, the auxiliary

function is selected as ( ) 1H t = .

From Eq. (37)-(41), it implies that

( )1, 1, 1 1, 1, 1 1, 2,0 0

( ) ( ) ( )− −= + + +∫ ∫

ℏt

m m m m m m mx t x t R x ds d C t Cτ

χ τ , (52)

( )2, 2, 1 2, 2, 1 3, 4,0 0

( ) ( ) ( )− −= + + +∫ ∫

ℏt

m m m m m m mx t x t R x ds d C t Cτ

χ τ , (53)

where the constants 1, 2, 3, 4,, , ,m m m mC C C C can be obtained

from the initial conditions (37).

Therefore, the second-order analytical approximation is

obtained as

2 23 2 5 7

1

2 2( ) ( )

3 3 15 126x t t t t t≈ + + + −ℏ ℏ ℏ

ℏ , (54)

2 2 22 4 2 6 8

2

17( ) 1 ( ) ( )

2 24 6 360 672x t t t t t≈ + + + − − +ℏ ℏ ℏ ℏ

ℏ ℏ . (55)

In order to choose ℏ properly, it is necessary to consider the

International Journal of Mathematics and Computational Science Vol. 2, No. 2, 2016, pp. 43-54 49

impact of ℏ on the convergence of analytical approximation.

Figure 1 depicts the ℏ -curves for the 10th-order

approximation, it is evident that the region of admissible

values of ℏ is 1.5 0.5− < < −ℏ .

Figure 1. The ℏ -curves for the 10th-order HAM approximation by the set of base functions 0,1,2,3, = ⋯nt n .

Let 1= −ℏ , and accordingly the second-order analytical

approximation is

3 5 7

1

1 2 1( )

3 15 126x t t t t t≈ − + − , (56)

2 4 6 8

2

1 5 17 1( ) 1

2 24 360 672x t t t t t≈ − + − + . (57)

In Figure 2, it is shown that the second-order analytical

approximation of 1( )x t is much matched to that of

Runge-Kutta method in the interval of [-1, 1], and the same

conclusion is for 2( )x t . In Figure 3, it is shown that the

20th-order analytical approximation of 1( )x t is much

matched to that of Runge-Kutta method in the interval of

[-1.8,1.8], and the same conclusion is for 2( )x t . According to

the above two figures, it can be seen that as the order is higher,

the matched interval between the analytical approximation

with that of Runge-Kutta method will be broader. That means,

it is effective and feasible to use HAM with

0,1,2,3, n

t n = ⋯ as the set of base functions to solve the

system.

Figure 2. Comparison of the phase portrait curves of the second-order HAM approximation by the set of base functions 0,1,2,3, = ⋯nt n with Runge-Kutta

method .

50 J. M. Guo et al.: Construction of Analytic Solution for Coupled Cubic Nonlinear Systems

Using Homotopy Analysis Method

Figure 3. Comparison of the phase portrait curves of the 20th-order HAM approximation by the set of base functions 0,1,2,3, = ⋯nt n with Runge-Kutta

method .

4.2. To Be Expressed by the Set of Base Functions | = , , , ,…

Suppose that the solution of the system (47)-(48) can be

expressed by the set of base functions 0,1, 2,3, nt

e n− = ⋯ ,

then the analytical solution can be expressed as

1 1,

0

2 2,

0

( )

( )

nt

n

n

nt

n

n

x t a e

x t a e

+∞−

=

+∞−

=

= =

∑

∑. (58)

The initial guess solution is chosen as

2

1,0 1 3 3 2,0 2 2( ) , ( ) 2t t t

x t c c c e x t c e c e− − −= + − = − , (59)

and the linear operator is expressed as

2

1

21

22 2

2

( ; )

( ; )

( ; ) ( ; )

t q

t q tL

t q t q

t

ϕϕϕ ϕ

∂ ∂ = ∂ ∂

. (60)

According to the rule of solution expression, the auxiliary

function is selected as

0( )

0

t

t

eH t

e

−

−

=

. (61)

From Eq. (37)-(41), it implies that

( )1, 1, 1 1, 1, 1 0,0 0

2 3

1, 2, 3,

( ) ( ) ( )−

− −

− − −

= + +

+ + +

∫ ∫

ℏt

s

m m m m m m

t t t

m m m

x t x t e R x ds d C t

C e C e C e

τχ τ

, (62)

( )2, 2, 1 2, 2, 1 0,0 0

2 3

1, 2, 3,

( ) ( ) ( )−− −

− − −

= + +

+ + +

∫ ∫

ℏt

s

m m m m m m

t t t

m m m

x t x t e R x ds d D t

D e D e D e

τχ τ

. (63)

According to the rule of solution expression, 0,mC must be

equal to the opposite number of coefficient of the term t in

( )1, 1 1, 1, 10 0

( ) ( )t

s

m m m mx t e R x ds dτ

χ τ−− −+ ∫ ∫

ℏ , and , ( 1,2,3)i mC i =

must meet

2 3

0, 1, 2, 3,

t t t

m m m mC t C e C e C e− − −− = + + . (64)

By expressing 2

,t t

e e− −

, and 3te

− into the second-order power

series, Eq. (64) can be further expressed as

1, 0,

5

2m mC C= − , 2, 0,4m mC C= , 3, 0,

3

2m mC C= − . (65)

In the same way, 0,mD must be equal to the opposite number

of coefficient of the term t in

( )2, 1 2, 2, 10 0

( ) ( )t

s

m m m mx t e R x ds dτ

χ τ−− −+ ∫ ∫

ℏ , and , ( 1,2,3)i mD i =

must meet

1, 0, 2, 0, 3, 0,

5 3, 4 ,

2 2m m m m m mD D D D D D= − = = − . (66)

In order to choose ℏ properly, it is necessary to consider the

impact of ℏ on the convergence of analytical approximation.

Figure 4 depicts the ℏ -curves for the 10th-order

approximation, it is evident that the region of admissible

values of ℏ is 1.2 0.7− < < −ℏ .

International Journal of Mathematics and Computational Science Vol. 2, No. 2, 2016, pp. 43-54 51

Figure 4. The ℏ -curves for the 10th-order HAM approximation by the set of base functions 0,1,2,3, − = ⋯nte n .

Let 1= −ℏ , then the second-order analytical approximation is

obtained as

11 10 9 8

1

7 6 5 4 3

487235019163 85 2377 105484 9041( )

484090992000 53361 110250 893025 28800

21431 10039 2617 1526051 6891853

51450 22680 2250 705600 582120

− − − −

− − − − −

≈ − + − + −

+ − + −

t t t t

t t t t t

e e e ex t

e e e e e

2

0

2116889 1599709

3492720 3492720

− −

−

−t te e

, (67)

12 11 10 9

2

8 7 6 5

21478678031 73 37853 32777( )

268939440000 2352 11858 1102500 396900

78797 1399 45749 66699 23819

1411200 308700 113400 61250

− − − −

− − − −

≈ − + − + −

− − + −

t t t t

t t t t

e e e ex t

e e e e 4

3 2

9

352800

25012027 13641679 23247101

174636000 14553000 11642400

−

− − −

−

− +

t

t t t

e

e e e

. (68)

In Figure 5, it is shown that the second-order analytical

approximation of 1( )x t is much matched to that of

Runge-Kutta method in the interval of [-0.5, 0.75] , and the

same conclusion is for2( )x t . In Figure 6, it is shown that the

second-order analytical approximation of 1( )x t is much

matched to that of Runge-Kutta method in the interval of

[-0.75, 1.25] ,and the same conclusion is for2( )x t . According

to the above two figures, it can be seen that as the order is

higher, the matched interval between the analytical

approximation with that of Runge-Kutta method will be

broader. That means, it is effective and feasible to use HAM

with 0,1, 2,3, nt

e n− = ⋯ as the set of base functions to solve

the system.

Figure 5. Comparison of the phase portrait curves of the s econd-order HAM approximation by the set of base functions 0,1, 2,3, nt

e n− = ⋯ with

Runge-Kutta method .

52 J. M. Guo et al.: Construction of Analytic Solution for Coupled Cubic Nonlinear Systems

Using Homotopy Analysis Method

Figure 6. Comparison of the phase portrait curves of the 20th-order HAM approximation by the set of base functions 0,1, 2,3, nte n− = ⋯ with Runge-Kutta

method.

Furtherly, let's compare Figure 2 with Figure 5, and Figure 3

with Figure 6. Then, a conclusion can be come to that the

solution which is expressed by the set of base functions

0,1,2,3, n

t n = ⋯

is more efficient than the solution which

is expressed by the set of base functions 0,1, 2,3, nt

e n− = ⋯ ,

in this example.

Over all, it is full of freedom to choose a set of base

functions when using the HAM, and as the set of base

functions is chosen differently, the analytical

approximation that is acquired will has some different

effect. Therefore, it is an interesting and meaningful task to

get a more efficient analytical approximation by a better set

of base functions.

4.3. Homotopy Padé Approximation

The solution obtained by HAM can be further modified by

Padé approximation, and that may as well be compared with

the solution obtained by HAM. Let's take 0,1,2,3, n

t n = ⋯

as the set of base functions for example.

Figure 7 depicts the phase portrait curves of [10, 10]

homotopy padé approximation and that of Runge-Kutta

method, in which, the matched interval is at least expanded

into [-20, 20], compared with Figure 3. That means, the

matched interval between the homotopy Padé approximation

with Runge-Kutta method is broader than that between HAM

with Runge-Kutta method.

Figure 7. Comparison of the phase portrait curves of [10, 10] homotopy padé approximation with Runge-Kutta method.

Furtherly, in Figure 8, it is shown the absolute errors between

the 20th

-order HAM and Runge-Kutta method, both of them

agree well until 8t = . While, the [10, 10] homotopy padé

approximation and the solutions of Runge-Kutta method

virtually coalesce till 15t = , as shown in Figure 9.

International Journal of Mathematics and Computational Science Vol. 2, No. 2, 2016, pp. 43-54 53

Figure 8. The absolute errors between the 20th-order HAM and Runge-Kutta method with respect to t .

Figure 9. The absolute errors between [10, 10] homotopy padé approximation and Runge-Kutta method with respect to t .

Therefore, it is effective and feasible to use homotopy Padé

approximation to modify the solution obtained by HAM.

5. Conclusions

In summary, the coupled cubic nonlinear systems (1)-(2)

admits a unique initial value solution. And a kind of analytical

approximation of that systems can be obtained, via the HAM.

This method is effective and feasible, which can be seen from

the numerical simulation results in the paper. As an advantage

of the HAM, it is full of freedom to choose a set of base

functions to express the solution. Based on different sets of

base functions, the analytical approximations have some

different approximation effect, which can be seen in the above

example. That means, we can get a more efficient analytical

approximation by a better set of base functions. What is more,

by further optimizing the homotopy series solution with padé

approximation, homotopy padé approximation can be got to

achieve a broader region of convergence of the solution.

Acknowledgements

The authors Y. H. Qian and J. M. Guo gratefully acknowledge

the support of the National Natural Science Foundations of

China (NNSFC) through grant Nos. 11202189 and 11572288.

The author S. P. Chen gratefully acknowledge the support of

the National Natural Science Foundation of China (NNSFC)

through grant No. 11302184.

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