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Numerical solutions for a generalized Ito system

by using Adomian decomposition method

Hassan A.Zedan1 and Eman Al-Aidrous2

1Department of Mathematics

King Abdul-Aziz University,P.O.Box:80203,

Jeddah 21589,Saudi Arabia

[email protected]

2Department of Mathematics, Girls Section

King Abdul-Aziz University,P.O.Box:80203,

Jeddah 21589,Saudi Arabia

[email protected]

ABSTRACT

In this paper, we used the Adomian decomposition method for solving a generalized Ito

system to construct the numerical solutions for this system. The solutions in the form of a

rapidly convergent power series with easily computable components, the accuracy of the

proposed numerical scheme is examined by comparison with other analytical and numericalresults.

Keywords: Adomian Decomposition Method, Generalized Ito System, Numerical Solu-

tions, Analytical Solutions.

2000 Mathematics Subject Classification: 49K05, 49K15, 49S05.

1 Introduction

In this paper, we implemented the Adomian decomposition method [1], [2] for finding the exact

and approximate solutions of a generalized Ito system [3] and [4]

ut = vx, (1.1)

vt = 2vxxx 6(uv)x 6(wp)x, (1.2)

wt = wxxx + 3uwx, (1.3)

pt = pxxx + 3upx. (1.4)

which is generalization of the well-known integrable Ito system [5]

ut = vx, (1.5)

vt = 2vxxx 6(uv)x. (1.6)

International Journal of Mathematics and Computation,ISSN 0974-570X (Online), ISSN 0974-5718 (Print),

Vol. 4; No. S09; September 2009,

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Hirota bilinear representation of the system (1.1)-(1.4) 3-soliton solutions were found in ref

[3] and [4] passes the Painleve test for integrability in five distinct cases. In recent years,

large amounts of effort have been directed towards finding exact solutions. Various meth-

ods for seeking explicit traveling solutions to nonlinear partial differential equations have been

proposed, such as Darboux transformation [6], [7], Hirota bilinear method [8], [9], Lie group

method [10], [11], the homogeneous balance method [12], the tanh method [13], the tanh-

coth method [14]. In the beginning of the 1980, a so-called Adomian decomposition method

(ADM) [15], [16], [1] Adomian has been used to solve effectively, easily, and accurately a large

class of linear and nonlinear equations, solutions partial, deterministic or stochastic differential

equations with approximates which converge rapidly.

Body Math Unlike classical techniques, the nonlinear equations are solved easily and elegantly

without transforming the equation by using the ADM. The technique has many advantages over

the classical techniques, mainly, it avoids linearization and perturbation in order to find explicit

solutions of a given nonlinear equations. Many authors have developed this method such as

[17], [18], [19], [20]Wazwaz11 and applied this method in numerous of nonlinear partial differ-

ential equations, for example [21] used the ADM to solve a generalized fifth order KdV equa-

tions, [22] applied it to find the solution the Kaup-Kupershmidt equation, [23] for approximating

the solution of the Korteweg-de Vries equation, [24] to construct the solution for the shallow

water equations, [25] solved the generalized Burgers-Huxley equation, [26] solved multi-term

linear and non-linear diffusionwave equations of fractional order by using Adomian decom-

position method, [27] used the ADM to solve coupled system of nonlinear physical equations,

coupled system of diffusion-reaction equation and integro-differential diffusion-reaction equa-

tion and [28] used the Adomian decomposition method for solving the Laplace equation with

Newmann and Dirichlet boundary conditions in annular domains.

Body Math In this paper, we would like to implement the Adomian decomposition method to a

generalized Ito system (1.1)-(1.4).

2 Analysis of the method (ADM)

Body Math In this section, we outline the main steps for applying our method in a generalized

Ito system (1.1)-(1.4). For this purposes, using Adomian decomposition method in order to

obtain a solution for this system. Let us consider the standard form of Eqs.(1.1)-(1.4) in an

operator form:

Body Math

Ltu = Lxv, (2.1)

Ltv = 2Lxxxv 6Lx(uv) 6Lx(wp), (2.2)

Ltw = Lxxxw + 3uLxw, (2.3)

Ltp = Lxxxp + 3uLxp. (2.4)

where the notation Lt = t , Lx = x , Lxx = 2

x2 and Lxxx = 3

x3 , symbolizes the lineardifferential operators. Assuming the inverse of the operatorL 1t exists and it can conveniently

be taken as the definite integral with respect to t from 0 to t, i.e.,

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Body Math

L1t =

t0

(.)dt,

to the system in Eqs.(2.1)-(2.4) yields:

u(x, t) = f1(x) + L1t [Lxv], (2.5)

v(x, t) = f2(x) + L1t [2Lxxxv 6(1 + 2) 6(3 + 4)], (2.6)

w(x, t) = f3(x) + L1t [Lxxxw + 35], (2.7)

p(x, t) = f4(x) + L1t [Lxxxp + 36]. (2.8)

Where

1(E1, E2) = E21E2, 2(E2) = E

32 ,

3(E2, n) = E2n, 4(E1) = E3

1 ,5(E1, E2) = E1E

22 , 6(E1, n) = E1n,

7(E1) = E21 , 8(E2) = E

22 .

are related to the nonlinear terms. The ADM [1] assume an infinite series of the unknown

functions u(x, t), v(x, t), w(x, t) and p(x, t) are given by:

u(x, t) =k=0

uk(x, t), v(x, t) =k=0

vk(x, t),

w(x, t) =k=0

wk(x, t), p(x, t) =k=0

pk(x, t). (2.9)

We can express 1, 2, 3, 4, 5and 6 in terms of the infinite series of Adomian polynomials

as follows:

1(u, v) =k=0

Ak, 2(u, v) =k=0

Bk,

3(w, p) =k=0

Ck, 4(w, p) =k=0

Dk,

5(u, w) =

k=0

Ek, 6(u, p) =

k=0

Fk. (2.10)

where An, Bn, Cn, Dn , En and Fn are called the appropriate Adomians polynomials. These

polynomials can be calculated for all forms of nonlinearity according to specific algorithms

constructed and given in [1]. For problem (2.5)-(2.8), we use the general form of formula for

Adomian polynomials An, Bn, Cn, Dn , En and Fn as :

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Ak(u0,...,uk, v0, . . ,vk) =1

k!

dk

dk[1(

k=0

kuk,

k=0

kvk)]=0

Bk(u0,...,uk, v0, . . ,vk) = 1k!

dk

dk[2(

k=0

kuk,

k=0

kvk)]=0

Ck(w0,...,wk, p0, . . ,pk) =1

k!

dk

dk[3(

k=0

kwk,

k=0

kpk)]=0

Dk(w0,...,wk, p0, . . ,pk) =1

k!

dk

dk[4(

k=0

kwk,

k=0

kpk)]=0

Ek(u0,...,uk, w0, . . ,wk) =1

k!

dk

dk[5(

k=0

kuk,

k=0

kwk)]=0

Fk(u0,...,uk, p0, . . ,pk) = 1k! dk

dk [6(

k=0

kuk,

k=0

kpk)]=0

; k 0. (2.11)

simple calculations gives us

Ak =k

j=0

uj .vx(kj), Bk =k

j=0

vj.ux(kj),

Ck =k

j=0

wj.px(kj), Dk =k

j=0

pj.wx(kj),

Ek =k

j=0

uj .wx(kj), Fk =k

j=0

uj.px(kj), (2.12)

Eqs.(2.11) are easy to set computer code to get many polynomials as we need in the calcu-

lation of the numerical as well as explicit solutions. The nonlinear system eqs.(2.5)-(2.8) is

constructed in the a form of the recursive relations given by:

u0(x, t) = f1(x), v0(x, t) = f2(x),

w0(x, t) = f3(x), p0(x, t) = f4(x)u(k+1)(x, t) = L

1t [Lxv],

v(k+1)(x, t) = L1t [2Lxxxv 6(Ak + Bk) 6(Ck + Dk)],

w(k+1)(x, t) = L1t [Lxxxw + 3Ek],

p(k+1)(x, t) = L1t [Lxxxp + 3Fk] ; k 1 (2.13)

where the functions f1(x), f2(x), f3(x) and f4(x) are getting from the initial conditions. we

construct the solution u(x, t), v(x, t), w(x, t) and p(x, t) as:

Limk

k = u(x, t), Limk

k = v(x, t),

Limk

k = w(x, t), Limk

k = p(x, t) (2.14)


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where

k(x, t) =k

n=0

uk(x, t), k(x, t) =k

n=0

vk(x, t),

k(x, t) =k

n=0

wk(x, t) , k(x, t) =k

n=0

pk(x, t); k 0 (2.15)

and the recurrence relation is given as in (2.13).The decomposition series (2.15) solutions

are generally converged very rapidly in real physical problem [1]. The convergence of the

decomposition series have been investigated by several authors. The theoretical treatment

of convergence of the decomposition method has been considered in the Literature [29]. For

example,in [20] the authors proposed a new convergence proof of Adomians technique based

on properties of convergent series. To demonstrate the convergence of the decomposition

method, the results of numerical example are presented in the following section,and presenteda few terms are required to obtain solutions.

Consider the initial conditions

u(x, t) = u(x, 0) = r1 22 tanh2(x) = f1(x),

v(x, t) = v(x, 0) = r2 + b2 tanh2(x) = f2(x),

w(x, t) = w(x, 0) = r3 + f1 tanh(x) = f3(x),

p(x, t) = p(x, 0) = t0 + t1 tanh(x) = f4(x). (2.16)

where r1 =b2 + 44

62, r2 =

b22 + 4f1t12 8b24

84, r3 =

f1t0

t1, , b2, t0, t1 and f1 are arbitrary

constants, substitute the initial conditions Eq.(2.16) into Eq.(2.13) gives some terms:

u1(x, t) = 2b2t sec h2(x) tanh(x),

u2(x, t) = (12b2r1 + 12f1t1 32b22 24r2

2)t22 sec h4(x)

+(6b2r1 + 6f1t1 16b22 12r2

2)t22 cosh(2x)sec h4(x)

+3(f1t0 + r3t1)t22 sinh(2x)sec h4(x)

v1(x, t) = (6f1t0 + 6r3t1)t sec h2(x)

(12b2r1 + 12f1t1 32b22 24r2

2)t sec h2(x) tanh(x),

v2(x, t) = 6(6b2r21 + b2r2 3f1r1t1 + 2(8b2r1 + 6r1r2 f1t1))t

22 sec h4(x)

+32(3b2r1 3f1t1 + 8b22 + 6r2

2)t24 sec h6(x)

3(f1t0 + r3t1)(3(r1 22) + (3r1 + 2

2) cosh(2x))t2 sec h4(x)tanh(x)

+12(6b2r21 + b2r2 3f1r1t1 + 2(8b2r1 + 6r1r2 f1t1)

2)t22 sec h2(x)tanh2(x)

2(9b22 156(b2r1 + f1t1)2 + 104(4b2 + 3r2)

4)t22 sec h4(x)tanh2(x)

+24(f1t0 r3t1)t24 sec h2(x)tanh3(x) + 4(3b2

2+ 12(b2r1 + f1t1)

2

8(4b2 + 3r2)4)t22 sec h2(x)tanh4(x),


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w1(x, t) = (3r1 22)f1t sec h

2(x),

w2(x, t) = 1

2

f1(6b2 + 9r21 48r1

2 + 284

+(3r1 22)2 cosh(2x))t22 sec h4(x) tanh(x)

p1(x, t) = (3r1 22)t1t sec h

2(x),

p2(x, t) = 1

2t1(6b2 + 9r

21 48r1

2 + 284

+(3r1 22)2 cosh(2x))t22 sec h4(x) tanh(x)

and soon to find the other components of u(x, t), v(x, t), w(x, t) and p(x, t) but we see it is

difficult to continuum in this manner to find the components any terms, so we see it is write

computer code for generating the adomian polynomials by using algorithm was obtained using

Mathematica software, so we only consider five terms in evaluating the approximate solutions

of a generalized Ito system, It achieves a high level of accuracy,

uapp(x, t) =5k=0 u(x, t), vapp(x, t) =

5k=0 v(x, t) ,

wapp(x, t) =5k=0 w(x, t), papp(x, t) =

5k=0 p(x, t) .

we can obtain the expression of u(x, t), v(x, t), w(x, t) and p(x, t) which is in a Taylor series,

then the closed form solutions yields as follows:

u(x, t) = r1 22 tanh2[(x

b2t

22)],

v(x, t) = r2 + b2 tanh2[(x

b2t

22)],

w(x, t) = r3 + f1 tanh[(xb2t

22)],

p(x, t) = t0 + t1 tanh[(xb2t

22)]. (2.17)

where r1 =

b2 + 44

62 , r2 =

b22 + 4f1t12 8b2

4

84 , r3 = f1t0

t1 , , b2, t0, t1 and f1 are arbitraryconstants. This result can be verified through substitution. Thus, we obtain the solutions of a

generalized Ito system.

In tables (1), (2), (3) and (4) show the absolute errors between the exact solution and the nu-

merical solution of a generalized Ito system (2.16) with initial conditions foru(x, t), v(x, t), w(x, t)

andp(x, t), respectively. The graphs of the analytic and numerical solutions foru(x, t), v(x, t), w(x, t)

and p(x, t) are depicted in Fig. 1, 2, 3 and 4, respectively.


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Table (1)

The absolute errors at some points for u(x, t) with initial conditions Eq.(2.16) when

= 0.5, b2 = 0.03, t0 = 0.4, t1 = 0.1, f1 = 0.6.

ti

xi 0.1 0.2 0.3 0.4 0.5

0.1 5.82867 1016 8.65974 1015 9.72555 1014 5.45175 1013 2.0802 1012

0.2 5.55112 1017 7.41074 1015 8.64309 1014 4.87471 1013 1.86146 1012

0.3 1.11022 1016 6.21725 1015 7.07767 1014 3.97349 1013 1.51851 1012

0.4 1.94289 1016 4.32987 1015 5.05707 1014 2.83745 1013 1.08777 1012

0.5 4.16334 1017 2.2482 1015 2.8047 1014 1.59248 1013 6.11344 1012

Table (2)

The absolute errors at some points for v(x, t) with initial conditions Eq.(2.16) when

= 0.5, b2 = 0.03, t0 = 0.4, t1 = 0.1, f1 = 0.6.ti

xi 0.1 0.2 0.3 0.4 0.5

0.1 1.11022 1016 8.04912 1016 5.19029 1015 3.21132 1014 1.23485 1013

0.2 3.60822 1016 3.88578 1016 4.77396 1015 2.80886 1014 1.05221 1013

0.3 3.88578 1016 8.60423 1016 3.69149 1015 2.12885 1014 8.7097 1014

0.4 1.38778 1016 5.82867 1016 3.58047 1015 1.53766 1014 6.57252 1014

0.5 1.94289 1016 2.77556 1016 2.55351 1015 1.23512 1014 3.4639 1014

Table (3)The absolute errors at some points for w(x, t) with initial conditions Eq.(2.16) when

= 0.5, b2 = 0.03, t0 = 0.4, t1 = 0.1, f1 = 0.6.

ti

xi 0.1 0.2 0.3 0.4 0.5

0.1 3.55271 1015 1.33227 1015 5.77316 1015 3.19744 1014 1.23013 1013

0.2 4.44089 1016 2.22045 1015 8.43769 1015 5.68434 1014 2.38476 1013

0.3 1.77636 1015 3.10862 1015 1.02141 1014 9.4591 1014 3.47722 1013

0.4 3.10862 1015 1.33227 1015 2.08722 1014 1.13243 1013 4.17444 1012

0.5 2

.22045

10

15

8.88178

10

16

2.08722

10

14

1.22125

10

13

4.70735

10

13

Table (4)

The absolute errors at some points forp(x, t) with initial conditions Eq.(2.16) when

= 0.5, b2 = 0.03, t0 = 0.4, t1 = 0.1, f1 = 0.6.

ti

xi 0.1 0.2 0.3 0.4 0.5

0.1 5.55112 1016 4.44089 1016 3.33067 1016 5.55112 1015 2.04836 1014

0.2 3.88578 1016 3.33067 1016 1.77636 1015 1.12133 1014 3.99125 1014

0.3 5.55112 1017 4.996 1016 2.72005 1015 1.57652 1014 5.68989 1014

0.4 4.996 1016 1.66533 1016 3.88578 1015 1.80966 1014 6.98885 1014

0.5 2.22045 1016 4.44089 1016 2.94209 1015 1.99285 1014 7.81042 1014


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