Yuchun Wang, Lixia Wang and Wenbin Zhang- Application of the Adomian Decomposition Method to Fully...

8/3/2019 Yuchun Wang, Lixia Wang and Wenbin Zhang- Application of the Adomian Decomposition Method to Fully Nonlinear Sine-Gordon Equation

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ISSN 1479-3889 (print), 1479-3897 (online)

International Journal of Nonlinear Science

Vol.2 (2006) No.1, pp. 29-38

Application of the Adomian Decomposition Method

to Fully Nonlinear Sine-Gordon Equation

Yuchun Wang 1, Lixia Wang, Wenbin ZhangNonlinear Scientific Research Center, Faculty of Science,

Jiangsu University, Zhenjiang, Jiangsu, 212013, P.R.China

(Received 2 February 2006, accepted 8 May 2006)

Abstract: In this paper, a type of fully nonlinear Sine-Gordon equations and the approxi-

mate Sine-Gordon equation (under the condition:

|up

|is very small) are studied.Through proper

transformation, some initial value problems of many equations with special nonlinear terms aresolved by the Adomian decomposition method and some exact solutions: kink solution ,com-

pacton solution, multi-compacton solution, and compacton-kink solution. Some new types of

solutions are also generated by combining different kinds of solutions.

Keywords: The fully nonlinear Sine-Gordon equation; the approximate S-G equation; themodified Adomian decomposition method;Adomian polynomials, compacton solution

1 Introduction

As is well known, various physical phenomena in engineering and physics may be described by non-

linear differential equations. In order to get some information about the physical systems, the exact and

approximate solutions of these equations must be given, especially some solitary solutions with physical

context. Compactons are a new class of localized solutions to families of fully nonlinear dispersive par-

tial differential equations. They are proved to collide elastically and vanish identically outside a finite core

region. two important features of compactons structures are observed:

(1)Compactons are solitons characterized by the absence of exponential wings or infinite tails.

(2)The width of compactons is independent of the amplitude.

Studying special solitons to nonlinear equations in mathematical physics has become more and more

attractive in solitary theory [1-8]. For example, in 1993, Rosenau and Hyman[1] introduced a class of

solitary waves with compact support in fully nonlinear KdV equation K(m,n):

ut + (um)x + (u

n)3x = 0, m > 0, 1 < n < 3

Yan Zhen-ya[4] studied nonlinear dispersive Boussinesq equation B(m,n) and obtained compacton so-

lutions and the solutions with solitary patterns having cups or infinite slopes. Unlike classical solitons, the

compactons are nonanalytic solutions. Four main methods of handling the compactons structures are the

psendo-spectral method, the tri-Hamiltonian operators, the dispersion-velocity method and the Adomian

decomposition method. The Adomian decomposition method for solving differential and integral equations,

linear or nonlinear, has been well developed [9]. Recently, A.M.Wazwaz [10-11] developed an efficient

modification of this method that will facilitate the calculation.

1Corresponding author. E-mail address: [email protected]

Copyright cWorld Academic Press, World Academic Union

IJNS.2006.08.15/029


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30 International Journal of Nonlinear Science, Vol.2(2006), No.1, pp.29-38

In this paper, we study the fully nonlinear Sine-Gordon equation and the approximate Sine-Gordon

equation:

(um)tt (un)xx + sin(up) = 0 (1)(um)tt (un)xx + (up) (u3p)/3! = 0 (2)

which is introduced in[12] by the modified Adomian decomposition method.

The rest of the paper is organized as follows. In section 2,the proposed method is introduced briefly.Twokinds of solitary solutions to equation(1)are obtained in section 3.In section 4,compacton solutions to equa-tion (2)are given and two special solitons are also discussed.In section 5,other exact solutions are obtainedby combining different kinds of solutions.Finally, section 6 is some discussions and conclusions.

2 The proposed method

For convenience, we will present a review of the standard Adomian decomposition method and the

modified decomposition method. We consider the differential equation:

Lu + Ru + N u = g(x) (3)

where L is the highest order derivative which is assumed to be easily invertible , R the linear differentialoperator of less order than L,N u the nonlinear terms, and g(x) the source term. Applying the inverseoperator L1 to both sides of (3),and using the given conditions, we obtain

u = f(x) L1(Ru) L1(N u) (4)where the function f(x) represents the terms arising from integrating the source term g(x),and from usingthe given conditions, all of which are assumed to be prescribed. The nonlinear operator N u = F(u) isusually represented by an infinite series of so-called Adomian polynomials

F(u) =

k=o

Ak

where the polynomials Ak are defined as follows:

Ak =1

k!

dk

dk

F

i=0

iui

=0

, k = 0, 1, 2 (5)

The unknown function u(x, t) is assumed to decomposed by a series of components

u(x, t) =

k=o

uk (x, t)

where the components u0, u1, u2

are usually determined recursively. We employ the recursive rela-

tion: u0 = f(x)uk+1 = L1(Ruk) L1(Ak), k 0 (6)

All of the components can be calculated by inserting (5) into (6),and the series solution of u(x, t) followsimmediately. The series solution may provide the solution in a closed form if an exact solution exists.

The modified decomposition method. The modified form was established based on the assumption that

the function f(x) can be divided into two parts, namely f0 (x) and f1 (x). Under this assumption, weset f(x) = f1 (x) + f0 (x). Consequently, the modified recursive relation

u0 = f0 (x)u1 = f1 (x) L1 (Ru0) L1 (A0)uk+2 = L1 (Ruk+1) L1 (Ak+1) , k 0

(7)

was developed.

IJNS email for contribution: [email protected]


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Y Wang, L Wang, W Zhang: Application of the Adomian Decomposition Method to Fully ... 31

3 Solitary solutions of Eq.(1)

Here, we rewrite the equation (1):

(um)tt (un)xx + sin (up) = 0 (8)where

m,n,pstand for nonlinear intensity, and

is arbitrary constant.

When m = n = p, (1) becomes

(um)tt (um)xx + sin (um) = 0 (9)After the transform: v = um, we get

(v)tt (v)xx + sin (v) = 0 (10)Now, we consider the equation (10) with two conditions:

v (x, 0) = 4 arctan(ex) , vt (x, 0) = 4

+ 1ex

1 + e2x

According to the above-mentioned method, we collect L = 2

t2, N v = sinv, and f0 (x) = 4 arctan (ex)

, f1 (x) = 4+1ex

1+e2xt, then we employ the recursive relation as follows:

v0 (x, t) = 4 arctan ex

v1 (x, t) =4+1ex

1+e2xt + L1 [(v0)xx] L1 (A0)

vk+2 = L1 [(vk+1)xx] L1 (Ak+1) , k 0

(11)

Adomian polynomials Ak are obtained

A0 = F (v0) = sin (4 arctan ex ) = sin (4 arctan ex)

A1 = v1F (v0) = cos (4 arctan e

x )

4 + 1ex

1 + e2xt + 2

2e3x (1 + )(1 + e2x)2

+ex (1 + )

1 + e2x

t2

A2 = v2F (v0) +

1

2v21F

(v0) = (12)

Substituting (12) into (11) gives

v0 (x, t) = 4 arctan ex ,

v1 (x, t) =4 + 1ex

1 + e2xt + 2

2e3x (1 + )(1 + e2x)2

+ex (1 + )

1 + e2x

t2 ,

Thus, this gives the solution to equation (10) in series form

v (x, t) = 4 arctan ex 4

+ 1ex

1 + e2xt + 2

2e3x (1 + )(1 + e2x)2

+ex (1 + )

1 + e2x

t2 + (13)

Using Taylor series into (13),we obtain the closed form solution

v (x, t) = 4 arctan e(x+1t)

IJNS homepage:http://www.nonlinearscience.org.uk/


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Then, we get a kink solitary wave solution to equation (9)

u (x, t) =

4 arctane(x+1t)

1m

which is shown in Fig.1(a) with = 3, m = 1, and in Fig.1(b) when t = 0.

Figure 1: (a)kink solution with = 3, m = 1; (b)plane graph

In addition, we can develop another exact solution for equation (9). Now, in view of the solution above,

we consider another initial value problem of equation(10):

v (x, 0) = 2 arctan

cosh

/3 (1 2)x

,

vt (x, 0) = 2

/3 (1 2) sinh

/3 (1 2)x

1 + cosh2

/3 (1 2)x

Using the manner discussed above, we have the components of the series

v0 (x, t) = 2 arctan

cosh

x

/1 23

,

v1 (x, t) =

2

/1 2 sinhx/12

3

3

1 + cosh2

x/12

3

t

v2 (x, t) =

2 cosh

x/12

3

3 (1 2)

cosh2x/12

3

22 cosh

x/12

3

sinh2

x/12

3

3 (1 2)

cosh2x/12

3

2 t2


v (x, t) = 2 arctan

cosh

x

/1 23

+

2

/1 2 sinhx/123

3

1 + cosh2

x/12

3

t



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+

2 cosh

x/12

3

3 (1 2)

cosh2

x/12

3

22 cosh

x/12

3

sinh2

x/12

3

3 (1 2)cosh2

x/12

3 2

t2 +

(14)Using Taylor series into (14),we obtain the closed form solution

v (x, t) = 2 arctan

cosh

3 (1 2) (x t)

Then, we get a new solitary solution

u (x, t) =

2 arctan

cosh

3 (1 2) (x t) 1

m

which is shown in Fig.2(a) with = 3, = 1/2, m = 1, and also in Fig.2(b)when t = 0.

Figure 2: (a)solitary solution with = 3, = 1/2, m = 1; (b)plane graph

4 Compacton solutions of Eq.(2)

(1)when m = p, n = 3p, equation (2) becomes

(up)tt

u3pxx

+ up 3!

u3p = 0

After the transform: v = um, we get

(v)tt v

3xx + v

3!

v3 = 0 (15)

Now, we consider the equation (15) under two conditions :

v (x, 0) =

9 D

2

6cos

54x

, vt (x, 0) = D

54 D2

6sin

54x

We collect f0 (x) =

9 D26

cos

54

x

, f1 (x) = Dt

54D26

sin

54

x

, and Adomian polynomi-

als Akand Bk are obtained

A0 = v30xx = (9

D2

6)3(cos3

54

x)xx , A1 = 3v1v20xx = ,

and

B0 =v3

0

3!=

6

9 D

2

6

3cos3

54x

,



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B1 = v1

v30

3!

+

1

2v21

v30

3!

= ,


v (x, t) =

9 D2

6 cos

54 x

+ Dt54 D26 sin54 x

+1

108

D2

9 D

2

6cos

54x

t2 + (16)

Using Taylor series into (16),we obtain the closed form solution

v (x, t) =

9 D

2

6cos

54(x Dt)

Then, we get a compacton solution

u (x, t) =

9 D2

6cos

54

(x Dt) 1p , 54

(x Dt) 2

0 , others

(17)

which is shown in Fig.3(a) with D = 3, = 6, p = 1, and in Fig.3(b)when t = 0.

Figure 3: (a)compacton solution with D = 3, = 6, p = 1; (b)plane graph

It is interesting to find that if we define (17) as below

u (x, t) = 9

D2

6

cos 54

(x

Dt)1

p

, 4n - 1

254 (x Dt) 4n + 12

0 , others(18)

where n = 0,1,2, ,we known (18) is a mult-compacton solution. For example,(a)when n = 0,(18) is the same to (17) with one peak.

(b)when n = 0, 1, (18) is a mult-compacton solution with two peaks.

(c)when n = 1, 0, 1, 2, (18) is a mult-compacton solution with four peaks ,which is shown in Fig.4(a)with D =

6, = 6, p = 1, it is also shown in Fig.4(b) when t = 0.

In addition, we consider another initial value problem of equation (15):

v (x, 0) =

9 D26

sin

54

x

, vt (x, 0) = D

54 D26

cos

54

x

Using the method discussed above, we get another compacton solution:



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Figure 4: (a)mult-compacton solution with four peaks with D =

6, = 6, p = 1;(b)plane graph

u (x, t) =

9 D2

6sin

54

(x Dt) 1p , 54

(x Dt) 0 , others

(19)

(2) when m = 3p,n = p, equation (2) becomes

u3p

tt (up)xx + up

3!u3p = 0 (20)

After the transform: v = um, we get

v3tt vxx + v

3!v3 = 0 (21)

Here, we collect L = 2

x2, and consider the initial value problem of equation (21):

v (0, t) =

9 1

6D2cos

54

t

, vx (0, t) =

54D2

324D2sin

54

t

Using the method discussed above, we get a compacton solution

u (x, t) =

9 1

6D2cos

54D2

(x Dt) 1

p

,

54D2(x Dt)

2

0 , others(22)

which is shown in Fig.5(a) with D = 3, = 6, p = 3, and in Fig.5(b)when t = 0.

Figure 5: (a)compacton solution with D = 3, = 6, p = 3; (b)plane graph

In addition, we consider another initial value problem of equation (21):

v (0, t) =

9 16D2

sin

54t

,vx (0, t) =

54D2324D2

cos

54t

Using the manner as discussed above, we get another compacton solution:



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u (x, t) =

9 1

6D2sin

54D2

(x Dt) 1

p

,

54D2(x Dt)

0 , others

(23)

It is interesting to find that if we define (23) as below:

u (x, t) =

9 16D2 ,54D2 (x Dt) < 29 1

6D2sin

54D2

(x Dt) 1

p

,2

54D2

(x Dt) 2

9 16D2

,

54D2

(x Dt) > 2

(24)

we known (24) is a compacton-kink solution, which is shown in Fig.6(a) with D = 3, = 6, p = 1, it isalso shown in Fig.6(b) when t = 0.

Figure 6: (a)compacton-kink solution with D = 3, = 6, p = 1; (b)plane graph

5 Other exact solutions

(1) When m = p, n = 3pIn section 4, two exact compacton solutions were given in the form

u (x, t) =

9 D2

6cos

54

(x Dt) 1p , 54

(x Dt) 2

0 , others

(25)

and

u (x, t) =

9 D2

6sin

54

(x Dt) 1p , 54

(x Dt) 0 , others

(26)

By combining the two results, we will find that

u (x, t) =

a

9 D26

cos

54

(x Dt) + b9 D26

sin

54

(x Dt) 1p ,54

(x Dt) 2

0 ,others

satisfies the equation(15),where a and b are constants ifa2 + b2 = 1.In addition, adding a constant to the argument in (25) and (26) will exhibit more exact solutions. In

other words, we have the exact solution

u (x, t) =

9 D2

6cos

54

(x Dt) + k 1p ,2 k

54(x Dt)

2 k

0 ,others



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and

u (x, t) =

9 D2

6sin

54

(x Dt) + k 1p , k 54

(x Dt) k0 ,others

where r is arbitrary real.(2) When m = 3p,n = p

As discussed before, two exact compacton solutions were given in the form

u (x, t) =

9 1

6D2cos

54D2

(x Dt) 1

p

,

54D2(x Dt)

2

0 , others(27)

and

u (x, t) =

9 1

6D2sin

54D2

(x Dt) 1

p

,

54D2

(x Dt)

0 , others(28)

We can obtain a new exact solution by combining the two results (27) and (28) ,and we find that

u (x, t) =

a

9 16D2

cos

54D2

(x Dt)

+ b

9 16D2

sin

54D2

(x Dt) 1

p

,54D2

(x Dt)

2

0 , others

satisfies the equation (20),where a and b are constants ifa2 + b2 = 1.

Moreover, adding a constant to the argument in (27) and (28) will exhibit more exact solutions. In other

words, we have the exact solution

u (x, t) =

9 1

6D2cos

54D2

(x Dt) + k 1

p

,2 k

54D2

(x Dt) 2 k

0 ,others

and

u (x, t) =

9 1

6D2sin

54D2

(x Dt) + k 1

p

, k

54D2

(x Dt) k0 ,others

where r is arbitrary real.

6 Conclusion

In this paper, the modified Adomian decomposition method was used to the fully nonlinear sine-gordon

equation and the approximate sine-gordon equation under initial conditions, and some exact solutions were

obtained. Meanwhile we developed new exact solutions by combining the two different results. It is worth-

while to point out that some new exact solutions may be obtained by collecting other appropriate initial

conditions.

Acknowledgements

Research was supported by the National Nature Science Foundation of China(NO.10071033)and Nature

Science Foundation of Jiangsu Province(NO.BK2002003).



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