Convergence of Adomian's method applied to nonlinear equations

Mathl. Comput. Modelling Vol. 20, No. 9, pp. 69-73, 1994 Copyright@1994Elsevier Science Ltd

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Convergence of Adomian’s Method Applied to Nonlinear Equations K.

ABBAOUI AND Y. CHERRUAULT UniversitBParis VI LaboratoireMEDIMAT

15 Rue de 1’Ecole de Mtiecine,

75270 Paris Gclex 06 France

(Received June 1994; accepted July 1994)

Abstract-h

this paper, we give a proof of Adomian’s method using only some properties of the nonlinear function, and we apply these results to some concrete problems.

Keywords-Decomposition tions.

methods,

Adomian’s polynomials,

Nonlinear equations,

Series solu-

1. INTRODUCTION Our aim consists

in solving

in

R, the equation

f(z)

= 0, where f is a nonlinear

function.

The method developed by Adomian [1,2] since the beginning of the 1980’s gives the solution as an infinite series quickly converging towards an accurate solution. Of course, this convergence needs some conditions and hypothesis on the nonlinear function f [3]. The principles of the technique are very simple. The difficulties consist in calculating some special polynomials called the Adomian polynomials and in proving the convergence of the series solution [3,4]. Using some new formulae [5,6] for calculating the Adomian polynomials, we prove the convergence of the series solution of a general nonlinear equation such that f(x) = 0.

2. THE DECOMPOSITION NONLINEAR The following

notations

METHOD APPLIED EQUATIONS

will be used throughout

this paper:

k! := kl! . . . k,!, l/cl:= kl + . . . + k,,

Ink,l := ICI + 21c2+ 2=(a,...,GL), k.t Zk = Zk’ 1 ..*&I 1 I+)

Let us consider

a nonlinear

(x0)

equation

=

TO

-$F(z)iZ,,,.

. . . + n/c,,

(2-l) (2.2) (2.3) (2.4) (2.5) (2.6)

in the form:

f(x) = 0,

(2.7) Typesetby AM-‘I$$

69

70

K. ABBAOUI AND Y. CHERRUAULT

which can be transformed in the following (canonical) z = F(z)

relationship:

+ c,

(2.8)

where F is a nonlinear function and c a constant. The Adomian method consists in calculating the solution in the series form: m x= xi. (2.9) c i-0

The nonlinear function becomes: F(x) = cAi,

(2.10)

i-0

where the Ai’s are polynomials [5] depending only on x0, x1, . . . , xi and they are called Adomian polynomials. They are obtained from the formulae [5]:

A,(xo,...

,x,)

c

=

F(lkl)

Ink,l=n

(x0)-7Xk k!

n # 0,

(2.11)

Ao (~0)= F (50). Putting

(2.9) and (2.10) into (2.8) leads to: Fxi=FAi+c. i=o i=o

(2.12)

Each term of the series x = Cr”_, xi can be calculated from the formulae: x0 = c, x1 = Ao, (2.13)

The exact solution of the equation (2.8) is now entirely determined. But in practice, all the terms of the series cannot be determined and so we use an approximation of the solution from the truncated series: n-1

(Pn

=

c

with lim (P,, = x.

xi,

(2.14)

i=o

3. CONVERGENCE THEOREM (1)

OF THE TECHNIQUE

3.1.

If F is Coo in the neighbourhood given by:

of x0, then the An(F(xo),

A, (F (x0) , F(l) (x0), . . . , F(*) (x0))

c

ckl,..,,k,,

F(‘)(xo),

. . . , Fcn)(xo))

are

= [F (zo)]~+‘-‘”

[F(l) (zo)] k’

. . . [,@’

(x0)]

kn 7

(3.1)

Ink,l=n

where:

72.I Ck+.,k,

A,(l,

=

k!(l!)“l

1,. . . , 1) =

. . . (n!)“-(n

c Ink,l=n

ck,,...,k,

+

=

1-

;I;;.

Ikl)!’

(3.2)

(3.3)

Adomian’sMethod

71

PROOF. (1) By induction, we prove that: A,,=

- ’ dn [F(x)];::, (n + l)! dx”

(3.4)

.

The formula giving the nth derivative associated to formula (3.4) leads to (3.1). (2)

c

Ckl,...,k,

n!

c

=

ICI!. . .

Jnk,(=n

(nk, I=n

(3.5)

kn!(l)kl . . . (n!)kn(n + 1 - [I%()! I

=

c 81+-~+L%=n

(3.6)

. .Yb,.!(n + 1 - T)!

ICI!... k,!&!

815B2<~~~5/-% I

=

c “I+,;..=”

(3.7)

VI!. . .%Jrl;n + 1 - r)! n

(n + l)! c r=l r!(n f 1 - T)!

=&

=& * c

(3.8)

n!

=

YI!-..%+I

71+...+gl=n

b + lJn

(3.9)

(n + l)! *

THEOREM 3.2. If from some step no IF@)(xo)l 5 M 5 l/e, n > no, then the series CzM_oxi solution of (2.8) given by the relationships (2.13) is absolutely convergent and we have: lxn+ll

(3.10)

= IA,1 5 e(n+l)Mn+l.

PROOF. We know that: A, (F (20) , F(‘)

(~0),

. ) dn)

(x0))

=

c

Ckl,...,k,

IF (x0)1

n+l-lkl

[ F(1)

( z,lk’... >

[F(qxo)]kn,

Ink,,l=n

where

I 71. ck I,..., k=n

k!(l!)“l

. . . (n!)“n(n + 1 - lkl)!’

Then,

IAn (F (x0),F(l)(x01, . . . ,I+) (30,)1 =

c

Ck+.,k,,

IF

(x0)1

n+l-lkl

[ F(1)

( z. )

I*‘...

[F’n’(xo)]kn~

5

Mn+‘A,(l

,...,

1).

Ink,l=n

The using of the classical Stirling formula leads to (3.10). From Theorem 3.2, we deducted the following result. COROLLARY 3.3. If F(xo)

= 0, then the series Cz”=, is convergent and is reduced to x = x0.

REMARKS 3.4. (1) The solution given by the Adomian relationships (2.13) is only a particular solution of (2.8). (2) The transformation of f(x) = 0 into x = F(x) + c is always realizable with an infinity of manners. It is better to choose the canonical form for which the technique converges. (3) Our results can be applied to systems of nonlinear equations.

72

ABBAOW AND Y.

K.

NUMERICALS

4. Let us consider o,IP

+

the algebraic n-1

an-12

+

CHERRIJAULT

APPLICATIONS

equation:

~~~+u~~+ao=O,a,,...,ao,a~#Orealnumbers,

n E N.

(4.1)

From (2.12), we deduce:

x=_--...__---_z

a0

&-I

a1

The A,(P)

= z;,

A1 (9)

= nzo

A3 (9)

Ad (P)

---_2

=-ao-...

al

-

7

2

a1

Ai

(xc”-‘)

-

i=O

z

g

Ai (2”).

(4.2)

a=0

are given by:

Ao (9)

A2 (P)

an n

n-1

a1

n-l

1 = -n(n 2 1

= -n(n

6 1 = -n(n 24

x The identification

~0, - 1)x,

n-2

- l)(n

2 X1 + n2;;1-l22,

(4.3)

- 2)x, n-35T + n(n - l)~;f-~2~2~

- l)(n

- 2)(n - 3)~ ;-“zt

+ $(?I

+ n~r;-~2~,

- l)(n

- 2)s;-32;Q

(xn),

i 2 1.

+ n(n - 1)X;-”

(;x;+x1x3) +nxg-‘x4. leads to: zs = -ao, a1 xi+1

=

-

?A1

(x ‘+l)

- ZAl

(4.4

EXAMPLE 4.1. x3 + 4x2 + 82 + 8 = 0. Equation

(4.5)

(2.12) involves x=

gxi

=

-1

-

ix2 - ix3

i=o (4.6) =-1-~~Ai(22)-~~Ai(x3), a=0 where the Ai(x2)

and Ai(x3)

2=0

are given by the relationships

(4.3). Then

we have:

2s = -1.0, x1 = -0.375, x2 = -0.234375, x3 = -0.1640625,

(4.7)

x4 = -0.1179199, x5 = -0.0835876. Making

the sum of the first six terms

gives

p6 = x0 + . . . + x5 = -1.98, which is a good approximation

of the root x = -2.

(4.8)

Adomian’s Method EXAMPLE

4.2.

Consider

73

the equation x = k + e-“,

k > 0.

(4.9)

We have x=

2xi=k+gAi(e-‘). i-0

The xi are identified

(4.10) i-0

by: x0

k

=

nn-l

xn = (-l)n+l Then

the solution

is x = k + F(-I)n+l f

for example,

Geenk,

(4.12)

n-1

for k = 2, the graphical

solution

(p7 = XI-J+

gives the solution

(4.11)

Te-nk.

is x = 2.120028239.

. . . + 56 =

The 7 terms

approximation

2.120013302

(4.13)

with an error of 0.0001%.

5. CONCLUSION In this work, we have studied a powerful numerical method for solving nonlinear equations of different kinds. We have given a sufficient condition of convergence of the decomposition series. The convergence result can be generalized to operator equation. This condition of convergence is easy to verify on concrete examples. With these complementary results, the decomposition methods have now proved their power for solving all kinds of nonlinear equations. They provide an accurate and easily computable solution of nonlinear equations. Furthermore, the convergence of the truncated series is generally very fast.

REFERENCES 1. G. Adomian, Nonlinear Stochastic Systems and Applications to Physics, Kluwer, (1989). 2. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, (1994). 3. Y. Cherruault and G. Adomian, Decomposition method: A new proof of convergence, Math. Comput. Modelling 18 (12), 103-106 (1993). 4. G. Adomian and R. Rach, Algebraic computation and the decomposition method, Kybernetes 15, 33-37 (1986).

5. K. Abbaoui and Y. Cherruault, Comput.

6. K. Abbaoui Comput.

Convergence of Adomian’s method applied to differential equations,

Modelling 28 (5), 103-110

and Y. Cherruault, (to appear).

Modelling

Mathl.

(1994).

New ideas for proving convergence

of decomposition

methods,

Mathl.