A REVIEW

Abstract-The decomposition method can be an effective procedure for solution of nonlinear and/or stochastic continuous-time dynamical systems without usual restrictive assumptions. This paper is intended as a convenient tutorial review of the method.?
1.
INTRODUCTION
To hope to attain the objective of solving frontier physical problems, we must be prepared to deal with dynamical systems modeled by differential, delay-differential, integro-differential and partial differential equations which may be strongly nonlinear and stochastic in parameters, inputs or specified conditions. The modeling of such systems is necessarily a compromise between a realistic representation of the attributes of interest and a model which will be tractable by available mathematics. Thus, simplifying assumptions are generally necessary. Linearization and perturbation are common assumptions which have been very useful. However, their use may mean that the problem being solved is no longer a proper representation of the physical problem whose solution is desired. Thus the result, however elegant with well-stated theorems, may not be physically realistic. That is, the mathematical results may deviate seriously from the actual results. There are other considerations as well. Closed-form analytical results are considered ideal and series solutions are considered approximations. However, all modeling is approximation and a closed-form solution to a linearized version, or one in which stochastic behavior is replaced by delta-correlated processes, is not to be preferred over a series solution of the actual problemespecially if the convergence is rapid. The decomposition method does not change the problem into a convenient one for use with linear theory. It therefore provides more realistic solutions. It provides series solutions which generally converge very rapidly in real physical problems. When solutions are computed numerically, the rapid convergence is obvious. The method makes unnecessary the massive computation of discretized methods for solution of partial differential equations. No linearization or perturbation is required. It provides an effective procedure for analytical solution of a wide and general class of dynamical systems representing real physical problems (as opposed to pathological mathematical systems). There are some quite significant advantages over methods which must assume linearity, “smallness”, deterministic behavior, stationarity, restricted kinds of stochastic behavior, uncoupled boundary conditions etc. It has reached a point where rigorous study of the foundations will be valuab1e.S The method has features in common with many other methods but it is distinctly different on close examination and one should not be misled by apparent simplicity into superficial conclusions. The only proof that really matters is that it solves the physical problems.
equations
and
of the reason
for the rapid
rate of
18
G. ADOMAN
A general
description
with reasonable
examples?
follows.
2. BASIC
METHODS
Beginning with an equation Fu(t) = g(t), where F represents a general nonlinear ordinary differential operator involving both linear and nonlinear terms, the linear term is decomposed into L + R, where L is easily invertible and R is the remainder of the linear operator. For convenience, L may be taken as the highest order derivative which avoids the difficult integrations which result when complicated Green’s functions are involved. Thus, the equation may be written Lu+Ru+Nu=g, where Nu represents the nonlinear terms. Solving for Lu, (2) is - L-‘NU. (3) (1)
Lu=g-Ru-Nu. Because L is invertible, an equivalent L-‘Lu expression
= L-‘g+L-‘Ru
If this corresponds to an initial value problem, the integral operator L -’ may be regarded as definite integrations from 0 to t. If L is a second-order operator, L-’ is a two-fold integration operator and L-‘Lu = u - u(O) - tu’(0). For boundary-value problems (and, if desired, for initial-value problems as well), indefinite integrations are used and the constants are evaluated from the given conditions. Solving equation (3) for u yields: u=A+Bt+L-‘g-L-‘Ru-Lp’Nu. The nonlinear term Nu will be equated to Z,“=,, A,, where the A, are special polynomials discussed; and u will be decomposed into Cz==, u, , with u0 identified as A + Bt + Lp’g: n;oun=uo-L-‘R Consequently, we can write u, = -L-‘Rueu2= -L-‘Ru, %+
Math/ Compur. Modelling, Vol. 13, No. 7, pp. 1743, Printed in Great Britain. All rights reserved
1990 Copyright 0
089%7177/90 $3.00 + 0.00 1990 Pergamon Press plc
I =
(4) to be
5 u,-L-’II=0 Nhomakorabeaf
n=O
A,,.
L-IA,, - L-IA,,
- L-IA
-L-‘Run
?I.
(5)
The polynomials A,, are generated for each nonlinearity so that A, depends only on u.. A, depends only on u. and ul, A, depends on uo, u,, u2 etc. [l]. All of the u, components are calculable, and It is now established that the series CT=, A, for Nu is equal to a generalized Taylor u =Z;Eou,. series about f(u,), that E;P=ou, is equal to a generalized Taylor series about the function u. and that the series terms approach zero as l/(mn)! if m is the order of the highest linear differential operator. Since the series converges and does so very rapidly, the n-term partial sum (P,,= C;:A ui [cp, = Cy,d ui] = u. It is important to emphasize that the can serve as a practical solution; lim,,, A, can be calculated for complicated nonlinearities of the form f(u, u’, . . .) or f(g(u)). As an example, the A, for exp(-xi) are A, = exp( -xi), A, = 2x,x, exp(-xi) A, = (2xixy - xf 2x,x,}exp( -xi) etc. For a simple example such as x = k + exp( -x2), the error in a four-term approximation (p4 is already <O.OOl%. Also, since we are not linearizing or assuming “weak nonlinearity”, the solutions tend to be much more physically correct that those obtained by other methods of approximation based on simplifying assumptions. Numerical values can be computed if desired, and it is then easy to see convergence as we calculate terms [l]. Since the solutions are analytic (and verifiable by substitution), physical insight into functional relationships follows.
A REVIEW OF THE DECOMPOSITION METHOD AND SOME RECENT RESULTS FOR NONLINEAR EQUATIONS
G. ADOMIAN
Center for Applied Mathematics, University of Georgia, Athens, GA 30602, U.S.A. (Received for publication March 1990) Communicated by X. J. R. Avula
tit cannot, however, be a replacement for the references. $In this connection, a convergence proof for ordinary differential convergence is to be published. 17