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International Mathematical Forum, 1, 2006, no. 39, 1919-1924

A Maple program for computing

Adomian polynomials

Jafar Biazar 1 and Masumeh Pourabd

Department of Mathematics, Faculty of ScienceGuilan University

P.O. Box 1914, Rasht, Iran

Abstract

Adomian Decomposition method is a well known device for solv-

ing many functional equations such as differential equations, integralequations, integro-differential equations and this method is extendedfor solving systems of such equations by the firs author and some ofhis collaborators [4-7]. Adomian Decomposition method yields an an-alytical solution in terms of a rapidly convergent infinite power serieswith easily computable terms. During the computations one encountersespecial Polynomials so called Adomian polynomials. Many researchershave suggested different methods and algorithm for computing thesepolynomials. To end these attempts we are presetting a Maple programto compute easily these polynomials for functions with one, two or sev-eral variables. This Program also computes Adomian polynomials for asystem of functional equations as well.

Keywords: Adomian decomposition method, Adomian Polynomials, MaplePackage

1 Introduction

ADM well addressed in [1-3] is a powerful tool for solving many functionalequations in the following canonical form,

1

3

u= f+G(u) (1)

Adomian considers the solution as the summation of a series say,

u=

n=0

un (2)

1Corresponding author. E-mail address: biazar@guilan.ac.ir

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1920 Jafar Biazar and Masumeh Pourabd

AndG(u) as a series say

G(u) =

n=0

An(u0, u1, . . . , un) (3)

Adomian introduces these polynomials, for functions with one variable, by thefollowing formula [2]

An(u0, u1, . . . , un) = 1

n![G(

n=0

unn]=0 (4)

Many researches have been working to derive a simple procedure for computingAdomian polynomials. Wazwaz [9] suggestion is to substitute (2) into (1) andthen by manipulation terms, based on algebraic operations , trigonometricidentities and Taylor series as appropriate, all terms can be collected such thatthe subscripts of the components of in each terms is the same. With this step

preformed, the calculation of the Adomian polynomials is thus completed.Javadi has sujected another method [] these two methods are very difficult

for functions with more than one variable.Biazar and some of his collaborators have used an alternate Algorithm for

computing Adomian polynomials...[10], which is a straightforward methodand can be easely extended for functions with multivariable. The short comingof this algorithm is modified in an improvement to an alternate algorithm forcomputing Adomian polynomials in special cases [11].

For functional equations with several variables the following extension of(1) can be used.

An(u10, . . . , u1n, u20, . . . , u2n, um0, . . . , umn) = 1

n![G(

n=0

u1nn, . . . ,

n=0

umnn)]=0

(5)Where G(u1, . . . , un) is a functional depending onn variables, each of themis an unknown function which are considered as the summation of series say,uj =

n=0 ujnn j= 0, 1, 2, . . .

2 Maple program for computing Adomian Poly-

nomials

>restart;>with(student):INPUT>G[1]:=G(u[1],u[2],...,u[N]);G[2]:=G(u[1],u[2],...,u[N]);

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A Maple program for computing Adomian polynomials 1921

...G[k]:=G(u[1],u[2],...,u[N]);>m:=M:>n:=N:>k:=K:PROGRAM> for i from 1 by 1 while i for i from 1 by 1 while i

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1922 Jafar Biazar and Masumeh Pourabd

A1,2= xu2

1,1+ 2xu1,0u1,2

A1,3= 2xu1,0u1,3+ 2xu1,1u1,2

A1,4= 2xu1,0u1,4+ 2xu1,1u1,3+xu2

1,2

A1,5= 2xu1,2u1,3+ 2xu1,0u1,5+ 2xu1,1u1,4A1,6= 2xu1,1u1,5+ 2xu1,2u1,4+ 2xu1,0u1,6+xu

2

1,3

Example 2: Adomian polynomials for a function with two variables are com-puted in this example.

G1= u2(t)

2 +u1(t)

where m=6 and k=1 and n=2Adomian polynomials:

A1,0= u2

,0

2 +u1,0

A1,1= u2,0u1,0

(2 +u1,0)2+

u2,1

2 +u1,0

A1,2= u2,0u

2

1,1

(2 +u1,0)3

u2,0u1,2

(2 +u1,0)2

u2,1u1,1

(2 +u1,0)2+

u2,2

2 +u1,0

A1,3= u2,2u1,1

(2 +u1,0)2+

u2,3

(2 +u1,0)

u2,1u1,2

(2 +u1,0)2

u2,0u1,3

(2 +u1,0)2+

u2,1u2

1,1

(2 +u1,0)3

u2,0u3

1,1

(2 +u1,0)4 +

2u2,0u1,2u1,1

(2 +u1,0)3

A1,4= u2,1u1,3

(2 +u1,0)2

u2,3u1,1

(2 +u1,0)2+

u2,4

2 +u1,0

u2,1u3

1,1

(2 +u1,0)4+

u2,0u4

1,1

(2 +u1,0)5

3u2,0u2

1,1u1,2

(2 +u1,0)4

+2u2,0u1,3u1,1(2 +u1,0)3

+2u2,1u1,1u1,2

(2 +u1,0)3 +

u2,2u2

1,1

(2 +u1,0)3

u2,0u1,4

(2 +u1,0)2

u2,2u1,2

(2 +u1,0)2

+ u2,0u

2

1,2

(2 +u1,0)3

A1,5= u2,2u

3

1,1

(2 +u1,0)4

u2,0u5

1,1

(2 +u1,0)6

u2,4u1,1

(2 +u1,0)2+

u2,3u2

1,1

(2 +u1,0)3

u2,0u1,5

(2 +u1,0)2+

u2,1u2

1,2

(2 +u1,0)3

u2,2u1,3

(2 +u1,0)2

u2,1u1,4

(2 +u1,0)2+

4u2,0u3

1,1u1,2

(2 +u1,0)5

3u2,0u1,1u2

1,2

(2 +u1,0)4

3u2,1u1,2u2

1,1

(2 +u1,0)4 +

2u2,0u1,4u1,1(2 +u1,0)3

+2u2,0u1,3u1,2(2 +u1,0)3

+2u2,2u1,1u1,2(2 +u1,0)3

3u2,0u2

1,1u1,3

(2 +u1,0)4 +

2u2,1u1,3u1,1(2 +u1,0)3

u2,3u1,2

(2 +u1,0)2+

u2,5

(2 +u1,0)

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A Maple program for computing Adomian polynomials 1923

+ u2,1u

4

1,1

(2 +u1,0)5

A1,6= u2,4u1,2

(2 +u1,0)2+

u2,0u2

1,3

(2 +u1,0)3

u2,0u3

1,2

(2 +u1,0)4

u2,3u1,3

(2 +u1,0)2+

2u2,0u1,5u1,1(2 +u1,0)3

+2u2,0u1,4u1,2(2 +u1,0)3

+6u2,0u21,1u21,2

(2 +u1,0)5

3u2,2u1,2u21,1(2 +u1,0)4

+ u2,6

(2 +u1,0)

u2,1u1,5

(2 +u1,0)2

u2,2u1,4

(2 +u1,0)2+

u2,2u21,2

(2 +u1,0)3

3u2,0u1,4u2

1,1

(2 +u1,0)4 +

2u2,3u1,1u1,2(2 +u1,0)3

+4u2,0u

3

1,1u1,3

(2 +u1,0)5 +

2u2,1u1,2u1,3(2 +u1,0)3

5u2,0u4

1,1u1,2

(2 +u1,0)6 +

4u2,1u3

1,1u1,2

(2 +u1,0)5

3u2,1u1,1u2

1,2

(2 +u1,0)4 +

2u2,2u1,3u1,1(2 +u1,0)3

+ u2,0u

6

1,1

(2 +u1,0)7

u2,0u1,6

(2 +u1,0)2+

2u2,1u1,4u1,1(2 +u1,0)3

3u2,1u2

1,1u1,3

(2 +u1,0)4

u2,5u1,1

(2 +u1,0)2

u2,1u5

1,1

(2 +u1,0)6+

u2,2u4

1,1

(2 +u1,0)5+

u2,4u2

1,1

(2 +u1,0)3

u2,3u

3

1,1

(2 +u1,0)4

6u2,0u1,3u1,2u1,1(2 +u1,0)4

Example 3: A function with four variables and computing its polynomialsis considered here.

G1= u1(t)u2(t) +u3(t)2 +u4(t)u1(t)

wherem= 6 and k= 1 and n= 4Adomian polynomials:

A1,0= u1,0u2,0+u2

3,0+u4,0u1,0

A1,1= u1,0u2,1+u4,0u1,1+u1,1u2,0+u4,1u1,0+ 2u3,0u3,1

A1,2= u2

3,1 + u1,2u2,0 + u4,0u1,2 + u1,0u2,2 + u4,1u1,1 + u4,2u1,0 + u1,1u2,1 + 2u3,0u3,2

A1,3= u1,0u2,3+u1,2u2,1+u4,1u1,2+u1,1u2,2+u1,3u2,0+u4,2u1,1+2u3,1u3,2+u4,3u1,0

+2u3,0u3,3+u4,0u1,3

A1,4= u3,2 + u4,0u1,4 + 2u3,1u3,3 + u4,1u1,3 + u1,2u2,2 + u1,4u2,0 + u1,0u2,4 + u4,2u1,2

+u1,3u2,1+u4,3u1,1+u1,1u2,3+u4,4u1,0+2u3,0u3,4A1,5= u4,5u1,0+u1,0u2,5+u4,1u1,4

+2u3,1u3,4+ u1,2u2,3+ u1,5u2,0+ u4,0u1,5+ 2u3,0u3,5 + u4,2u1,3 + u4,3u1,2 + 2u3,2u3,3

+u1,1u2,4+u1,3u2,2+u1,4u2,1+u4,4u1,1

A1,6= u4,3u1,3+2u3,0u3,6+u4,5u1,1+2u3,2u3,4+u4,2u1,4+u4,0u1,6+u1,3u2,3+u4,4u1,2

+u4,1u1,5+ 2u3,1u3,5+u1,2u2,4+u4,6u1,0

+u1,6u2,0+u1,5u2,1+u1,1u2,5+u1,0u2,6+u1,4u2,2+u2

3,3

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1924 Jafar Biazar and Masumeh Pourabd

4 Discussion

Simplicity and efficiency of the algorithm presented in this article, are il-lustrated briefly in the examples. Datas of the algorithm,n the number ofunknowns,mthe number of Adomian polynomials and k , the number of func-

tionals are three advantages of the Algorithm which enable us to apply thisalgorithm to any functional , or any system of functional equations. Authorsare working on programs for solving systems of functional equations, such assystems of ordinary differential equations, systems of partial differential equa-tions, systems of Volterra integral equations of the first and second kind, andso on.

References

[1] G. Adomian, G.E. Adomian, A global method for solution of complex

systems, Math. Model., 5(1984), 521 - 568.

[2] G. Adomian, Nonlinear Stochastic Systems and Applications to Physics,Kluwer, 1989.

[3] G. Adomian, Solving Frontier Problems of Physics: the DecompositionMethod, Kluwer Academic Publishers, Dordecht, 1994.

[4] J. Biazar, Solution of systems of integral-differential equations by Ado-mian decomposition method,Applied Mathematics and Computation,168(2005), 1232 - 1238.

[5] J. Biazar, E. Babolian, R. Islam, Solution of the system of ordinary differ-ential equations by Adomian decomposition method,Applied Mathematicsand Computation, 147(2004), 713 - 719.

[6] J. Biazar, E. Babolian, R. Islam, Solution of a system of Volterra integralequations of the first kind by Adomian method, Applied Mathematics andComputation,139(2003), 249 - 258.

[7] E. Babolian, J. Biazar, Solution of a system of nonlinear Volterra integralequations of the second kind, Far East J. Math. Sci., 2(2000), no. 6, 935- 945.

Received: May 17, 2006