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Mathl. Comput. Modelling Vol. 18, No. 12, 103-106, 1993 pp.

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Decomposition Methods: New Proof of Convergence

Y. CHERRUAULT AND G. ADOMIAN University Paris VI-Medimat

15 Rue de 1’Ecole de MBdecine, 75270 Paris CCdex 06, France

(Received and accepted August 1993)

Abstract-we propose a new convergence proof of Adotnian’s technique based on properties of convergent series. Then we deduce some results about the speed of convergence of this method allowing us to solve nonlinear functional equations.

1. HYPOTHESIS AND GENERALITIES

First recall the main principles of Adomian’s method. Let us consider the general nonlinear functional equation:

U - N(v) = f, (I)

where N and f are, respectively, operator and function given in convenient spaces. We are looking for a function u satisfying equation (1). We suppose that N is such that (1) admits a unique solution [1,2] in some well-adapted space.

Adomian’s technique allows us to find the solution of (1) as an infinite series u = X=1 ui using the recurrent scheme written below:

%l = f,

~1 = Ao(uo),

(2) 21, = An-1 (210,. . . , un-1) ,

where the Ai’s are a special kind of polynomial (called Adomian’s polynomials [3,4]) calculated owing to the basic formula [5,6]:

Proofs of convergence are given in [1,2] and in the Ph.D. Thesis of Lionel Gabet [7]. These works mainly use the fixed point theorem or the properties of substituted series for the reference [2].

For the present work, we shall suppose that

(i) the solution ‘1~ of (1) can be found as a series of functions pi, i.e., u = x2”=, ZJ~. Further- more, this series is supposed absolutely convergent, i.e., C(uil < +oo.

(ii) the nonlinear function N(U) is developable in entire series with a convergence radius equal to infinity. In other words, we may write

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103

104 Y. CHERRUAULT AND G. ADOMIAN

N(u) = 5 A$;) Iu( < 00.

n=O

(4)

This last hypothesis is almost always satisfied in concrete physical problems.

2. THEOREM OF CONVERGENCE

We have the following result:

THEOREM 2.1. With the previous hypothesis (i) and (ii), the Adomian series u = CEO ui is a

solution of equation (l), when the ui ‘s satisfy relationships (2).

PROOF. Hypothesis (ii) guarantees that the series C N(“)(0)(un/n!) converges for any u. Fur-

thermore, we know that u = CEO ui is absolutely convergent and, therefore, can be substituted

in C N(“)(O)(u”/n!) [2]. Indeed, we have:

(5)

Because of the absolute convergence of C,“=, jupJ = U < 03, we can rearrange N(u) under the

form C, A,. In fact, we have:

Un = 2 Ui ( ) 71

=&Auol...,uq), i=o q=o

because u = C ui is absolutely convergent. The ynq depend only on UO, . . . , uq. Furthermore, we

have C, lxynsl < u”. Then the series (5) is absolutely convergent. because

N(u) = 2 n=O 1 --~%qbo:...>uq,] y!(o)

q=o =~~~,nq. (6)

Taking the absolute value of N(U) leads to

(N(u)1 < 2 iv1 U”, n=O

where the last series

Q$Kl U”

converges, owing to hypothesis (ii).

This implies the absolute convergence of the double series defining N(u), and thus, the series (6)

may be rearranged.

It can easily be proven that we have [8]:

2 ( -4, 2~01.. . ) un) = 2 (u - PO)” iv(“)(?Q) = N(u), (7) n=O n=O 72.

which proves that the Adomian series c A, is a generalization of the Taylor series. It remains to be proven that the Ui’s satisfy the relationships (2).

Putting (7) into equation (1) leads to:

g ui - 2 A, = f. (8) n=O n=O

The relationship (8) is identically satisfied (in particular) if we have the relationships ug = f,

u1 = Ao,. . . , uI, = A,_1,. . . . This gives Adomian’s relationships (2). The theorem is proven. m

REMARK. In a previous paper [2], it has been proven that Adomian’s polynomials A, depend

only on UO, ~1, . . . ,u,. The same proof is valid for the 7nq.

Decomposition Methods 105

3. SPEED OF CONVERGENCE

In practice, Adomian’s method gives very good results even if we take a truncated series with

a small number of terms [6,7]. The reason of such a result is due to the analogy of the Adomian

series with the Taylor series. Indeed, we saw that

~ui&Ll,+!p gui n. ( ) (9) i=l n=O n=O i=O

If we consider a truncated series (

C,‘=, A”) + f for approximating 21 (

u N CrY, ui >

, we can

calculate the error as follows:

where we recall that U = CEO (~i(.

Let us suppose that N(“)(O) is bounded in norm by a constant k, independent of n, and that

U is also bounded (in norm) by M, then the error given is bounded by

(11)

and we have the following lemma:

LEMMA 1. With the following hypothesis: l\Ull 5 M, and I(N(rr)(0)l 5 k independent of n,

Cr=, ui is an approximation of the solution of the functional equation. The error when we

replace the complete series by the truncated series involving (p + 1) terms is equal to kMP/P!

REMARK. It is important to point out that this result can be improved when considering partial

differential equations containing nth order linear operators. Let us consider the following equation:

Lu+Nu=g. (12)

From the previous developments, we deduce:

u = u. - L-l c F N@) (uo). 71.

(13)

If L is nth order, each L-’ adds n-fold integrations from 0 to t or P/n!. Let us choose for

convenience N(u) = u2,uc = 1, then

tn u1 = -L-‘A0 = -L-lN(uo) = --

n!

uz = -L-l (2uoT.Q) = -2 2.

More generally, we have urn N Tmn/(mn)! and the previous lemma leads to an error equal to

kMP”/(Pn)!.

4. CONCLUSIONS

The decomposition technique developed by G. Adomian in the eighties is of great interest for

solving nonlinear functional equations of different kinds (algebraic, differential, partial differential,

integral,. . . ). It allows us to obtain the exact solution as an infinite series of functions. We

avoid discretization in space and time induced by classical methods of resolution of functional

106 Y. CHERRUAULT AND G. ADOMIAN

equations. Convergence of Adomian’s technique is ensured with weak hypothesis on the nonlinear operator and on the functional equation. Some papers (1,8] use fixed point theorems for proving convergence; in our approach, we have avoided this type of hypothesis which is difficult to satisfy and to verify in physical problems. One can object that it is practically difficult to find the exact sum of an Adomian series. Indeed, we shall only be able to calculate a finite number of the series’ terms. But this is not a great inconvenience because the Adomian series is quickly convergent and a truncation error can be calculated. So the truncated series (generally involving few terms) will be a good approximation of the solution. This approximation depends explicitly on time and on space variables. This is a great advantage in comparison with classical discretization techniques which will only permit us to calculate the approximated solution for some values of time and space variables. The Adomian solution will also take into account the parameters and functions contained in the functional equation unlike discretization techniques. Many problems related to this method remain opened for pure applied mathematicians and especially for doctoral students looking for dissertation topics [7]. We shall quote, for instance, applications to identification of models (in physics, chemistry, biology,. . ) and to optimal control problems coming from industrial problems. A coupling of Adomian’s method with global optimization methods such as (91 could be especially interesting and promising.

REFERENCES

1. 2.

3. 4. 5.

6.

7.

8. 9.

Y. Cherruault, Convergence of Adomian’s method, Kybernetes 18 (2), 31-38 (1989). Y. Cherruault, G. Saccomandi and B. Some, New results for convergence of Adomian’s method applied to integral equations, Math. Comput. Modelling 16 (2), 85-93 (1992). G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics, Kluwer, (1989). G. Adomian, Solving frontier problems of physics: The decomposition method, (1993) (to appear). G. Adomian, R.C. Rach and R.E. Meyers, An efficient methodolgy for the physical sciences, Kybernetes 20 (7), 24-34 (1991). G. Adomian, An analytical solution of the stochastic Navier-Stokes problem, Foundations of Physics 21 (7), 831-843 (1991). L. Gabet, Modelisation de la diffusion des medicaments a travers les capillaires et dans les tissus a la suite d’une injection et Esquisse d’une therorie decompositionnelle et applications aux equations aux dorivees partielles, These de 1’Ecole Centrale de Paris, (July 1, 1992). L. Gabet, Esquisse d’une theorie decompostionnelle, Bairo @AN (to appear). Y. Cherruault, New deterministic methods for global optimization and application to biomedicine, Int. J. Biomed. Corn@ 2’7, 215-229 (1991).