1-s2.0-S0096300302001741-main

8/10/2019 1-s2.0-S0096300302001741-main

1/7

An alternate algorithm for computingAdomian polynomials in special cases

J. Biazar 1, E. Babolian 2, G. Kember, A. Nouri, R. Islam *

Faculty of Engineering, Dalhousie University, D510, 1360 Barrington Street, P.O. Box 1000,

Halifax, NS, Canada B3J 2X4

Abstract

In this article we introduce an alternate algorithm for computing Adomian poly-

nomials, and present some examples to show the simplicity and efficiency of the new

method.

2002 Published by Elsevier Science Inc.

Keyword: Decomposition method

1. Introduction

At the beginning of the 1980s Adomian developed a very powerful advice

for solving functional equations in the form:

u

f

G

u

1

In the functional space F [13].

The solution u is considered as the sum of a series,

u X1i0

ui 2

*Corresponding author.

E-mail addresses: [email protected](J. Biazar), [email protected] (R. Islam).1 Permanent address: Guilan University, Rasht, Iran.2 University for Teacher Education, Iran.

0096-3003/02/$ - see front matter 2002 Published by Elsevier Science Inc.

PII: S 0 0 9 6 - 3 0 0 3 ( 0 2 ) 0 0 1 7 4 - 1

Applied Mathematics and Computation 138 (2003) 523529

www.elsevier.com/locate/amc
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/

8/10/2019 1-s2.0-S0096300302001741-main

2/7

and Gu as the sum of the series

Gu X1i0

An 3

and the method consists of the following scheme:

u0 fun1 Anu0; u1;. . . ; un n 0; 1; 2;. . .

4

where Ans called Adomian polynomials, are defined as [4,5]:

An 1n!

d

n

dknG

X1i0

uiki !" #

k05

Wazwaz suggested the following algorithm for finding Adomian polynomials

An [6]:

Substituteu P1i0uiinto (1), SinceA0is always determined independent ofthe other polynomials An, nP 1, So that A0 is always defined by

A0 Gu0 6In view of (6), this algorithm suggests that we first separate A0 Gu0 fromother terms of the expansion of the nonlinear termGu. WithA0identified, theremaining terms ofGu can be expanded by using algebraic operations, trig-onometric identities and Taylor series as appropriate. Having established this

expansion, we then collect all terms of the expansion obtained such that the

sum of the subscripts of the components ofu in each terms is the same. With

this step preformed, the calculation of the Adomian polynomials is thus

completed.

We have suggested an alternate algorithm, which seems much simpler than

what Wazwaz suggested.

2. The alternate algorithm

For calculating Adomian polynomials, let us use the parameter k, and

consider

ukX1n0

unkn 7

and

Gku X1n0

Anu0; u1;. . . ; unkn 8

524 J. Biazar et al. / Appl. Math. Comput. 138 (2003) 523529

8/10/2019 1-s2.0-S0096300302001741-main

3/7

Using this presentation for uk, we try to rewrite Gku in terms of k. So wederive a power series in terms k, and in the comparison with (8), the coefficient

ofkn will determine Anu0; u1;. . . ; un.

Example 1. Wazwaz considered the following homogeneous nonlinear prob-

lem [6].

ut u2ux 0 ux; 0 3x 9The equivalent canonical of the equation is:

ux; t 3x Z t

0 u2

uxdt

Then the solution by Adomian decomposition method consists of the following

scheme:

u0 3x

un1 Z t

0

Anu0; u1;. . . ; undt n 0; 1; 2;. . .10

To find Ans, let ukP1

n0unkn, and computing Gu u2ux, we will have:

Gku u20u0x u20u1x 2u0u1u0xk u20u2x 2u0u1u1x 2u0u2u0x

u21u0xk2 u20u3x 2u0u1u2x 2u0u2u1x u21u1x 2u0u3u0x

2u1u2u0xk3

and in comparison with (8) we have:

A0u0 u20u0x

A1u0; u1 u20u1x 2u0u1u0x

A2u0; u1; u2 u20u2x 2u0u1u1x 2u0u2u0x u21u0x

A3u0; u1; u2; u3 u20u3x 2u0u1u2x 2u0u2u1x u21u1x 2u0u3u0x 2u1u2u0x

.

.

.

J. Biazar et al. / Appl. Math. Comput. 138 (2003) 523529 525

8/10/2019 1-s2.0-S0096300302001741-main

4/7

and from (10) we have:

u0x; t 3xu1x; t

Z t0

A0u0dt 27x2t

u2x; t Z t

0

A1u0; u1dt 729x3t2

u3x; t Z t

0

A2u0; u1; u2dt 19683x4t3

.

.

.

So u 3x 27x2t 729x3t2 19683x4t3Based on this, the solution can be expressed in this form:

ux; t 3x; t 01

6tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 36xtp

1; t>0

(

Example 2.Consider the diffusivity equation in radial form:

o2p

or21

r

op

orop

ot 11pri; 0 r

To solve this second-order PDE, rewrite the equation as:

op

ot o

2p

or2

1

r

op

or

Let L oot

, with inverse L1 Rt0dt, so we have:

pr; t pt 0 Z t

0

o2p

or2 1r opordt

Using the new procedure we have:

Anp0;p1;. . . ;pn o2pn

or2 1

r

opn

or 12

and the solution by Adomian decomposition method consists of the following

scheme:

p0 r

pn1Z

t

0

o2pn

or2

1

ropn

or

dt n 0; 1; 2;. . . 13


8/10/2019 1-s2.0-S0096300302001741-main

5/7

A0p0 o2p0

or2 1

r

op0

or 1

r p1 t

r

A1p0;p1 tr3

p2 t2

2r3

.

.

.

An1p0;. . .;pn 12 32 2n 32

n 1! r2n1 tn1 pn 1

2 32 2n 32tnn! r2n1

So

p

r; t

r

X1

n1

12 32 2n 32tn

n! r

2n

1

:

The algorithm can be easily extended for calculating Adomian polynomials

in several variables. Consider the function fu1; u2;. . . ; um ofm variables. Letui

P1k0uikk

k, and then if we can derive fu1; u2;. . . ; um as a power series ink, then the coefficient ofkn will be Anu10;. . . ; u1n;. . . ; um0;. . . ; umn.

Example 3.Consider the following system of integral equations with the exact

solution fx x, gx x2.

fx x 15

x7 Z x0

x2tftgtdt

gx x2 78

x4 Z x

0

x tf2t gtdt:

Adomian decomposition method for this problem consists of the following

scheme:

f0 x 15x7

g0 x2 76x4

8>:fn1

Rx0

x2tAnf0;. . . ;fn;g0;. . . ;gndtgn1

Rx0x tBnf0;. . .;fn;g0;. . . ;gndt

(

For more details for the solution of a system of integral equations

by Adomian decomposition method, see [7]. To derive polynomials An andBn,

let:

f X1n0

fnkn and g X1

n0gnk

n;


8/10/2019 1-s2.0-S0096300302001741-main

6/7

then

ftgt f0g0 f0g1 f1g0k f0g2 f1g1 f2g0k2

f0g3 f1g2 f2g1 fg0k3 f2t gt f20 g0 2f0f1 g1k 2f0f2 f21 g2k2

2f0f3 2f1f2 g3k3 and so

A0 f0g0 B0 f20 g0A1 f0g1 f1g0 B1 2f0f1 g1A2 f0g2 f1g1 f2g0 B2 2f0f2 f21 g2A3 f0g3 f1g2 f2g1 f3g0 B3 2f0f3 2f1f2 g3...

.

.

.

These polynomials are the same as those, can be derived by procedure pre-

sented in [5,8], for calculating Adomian polynomials for nonlinear operators.Some numerical results for f Pni0fi and g Pni0gi are presented in the

Table 1.

3. Discussion

This algorithm is easier than what Wazwaz suggested in [6], and does not

need the tiring manipulation with indexes, and can be used in more general

cases. With aide of mathematical packages like Derive, Mathematica, . . . this

method can be carried out easily. The computations associated with the ex-

amples discussed above were preformed by using Mathematica 4.

Table 1

Some numerical results for the Example 3

n 1 n 2f0:1 0.1 0.1f0:2 0.2 0.2f0:3 0.3 0.3f0:4 0.40000 0.4f0:5 0.49998 0.5g0:1 0.01 0.01g0:2 0.040000 0.04g0:3 0.090008 0.090000g0:4 0.160074 0.160000g

0:5

0.25043 0.25000


8/10/2019 1-s2.0-S0096300302001741-main

7/7

References

[1] G. Adomian, G.E. Adomian, A global method for solution of complex systems, Math. Model. 5

(1984) 521568.

[2] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer

Academic publishers, Dordrecht, 1994.

[3] G. Adomian, D. Sarafyan, Numerical solution of differential equations in the deterministic limit

of stochastic theory, Appl. Math. Comput. 8 (1981) 111119.

[4] L. Gabet, The theoretical foundation of Adomian method, Comput. Math. Appl. 27 (12) (1994)

4152.

[5] K. Abbaoi, Y. Cherruault, V. Seng, Practical formula for the calculus of multivariable Adomian

polynomials, Math. Comp. Model. 22 (1) (1995) 8993.

[6] A.M. Wazwaz, A new algorithm for calculating Adomian polynomials for non-linear operators,

Appl. Math. Comput. (111) (2000) 5369.[7] E. Babolian, J. Biazar, Solution of a system of nonlinear Volterra integral equations of the

second kind, Far East J. Math. Sci. 2 (6) (2000) 935945.

[8] V. Seng, K. Abbaoui, Y. Cherruault, Adomians polynomials for nonlinear operators, Math.

Comp. Model. 24 (1) (1996) 5965.


1-s2.0-S0096300302001741-main

Documents

Transcript of 1-s2.0-S0096300302001741-main