New Ideas for Proving Convergence of Decomposition Methods

1. I N T R O D U C T I O N In the eighties, G. Adomian [1] proposed a new and fruitful method for solving exactly nonlinear functional equations of various kinds (algebraic, differential, partial differential, integral, . . . ). The technique uses a decomposition of the nonlinear operator as a series of functions. Each term of this series is a generalized polynomial called Adomian's polynomial. The Adomian technique is very simple in its principles. The difficulties consist in calculating the Adomian's polynomials and in proving the convergence of the introduced series. Some at tempts to prove convergence have been given in [2-5]. None of these proofs tried to demonstrate directly the convergence of the series solution. In [5], a proof of convergence of the series solution is given but it uses a very strong hypothesis on the nonlinear operator. In the following, we shall prove convergence of the series solution owing to a new formula giving the Adomian's polynomials. This formula will express the series solution as a function of the first term of the series (this te rm is always known). Let us first recall the basic ideas of the decomposition method of Adomian [1,3,4]. Consider the general functional equation u = N(u) + f , (1.1) where N is a nonlinear operator from a Hilbert space H into H, and where f is a known function. We are looking for a solution u of (1.1) belonging to H. We shall suppose that (1.1) admits a unique solution. If (1.1) does not possess a unique solution, the decomposition method will give a solution among many (possible) other solutions. The decomposition method consists in looking for a solution having the series form o o = Z us. (1.2) i=0 The nonlinear operator N is decomposed as Typeset by A,~-TEX 103 104 K. ABBAOUI AND Y. CHEP~UAULT oo N(u) = Z An, n ~ O where the An are functions called the Adomian's polynomials. Adomian [1], the An were obtained by writing (1.3) In the first approach given by v = i ~ )d u4, N(v) = N i=0)~4u4 = nfo:k nan. (1.4) We remark that the An are formally obtained from the relationship [1,3,4,6] n! An = ~ N )~4 u4 , n = 0, 1, 2 , . . . . (1.5) 4=0 ~ = 0 These definitions are only formal, and nothing is proved or supposed about the convergence of the series ~ u4 and ~ An. Putting (1.2) and (1.3) into (1.1) leads to the relationship o¢~ oo u4 = ~ A4 + f, (1.6) 4=0 4----0 and the Adomian's method consists in identifying the u~ by means of the formulae UO ---f , U l = Ao, ?22 ~ A1,