Table Of ContentNonlinear Stochastic
Operator Equations
GEORGE ADOMIAN
CENTER FOR APPLIED MATHEMATICS
UNIVERSITY OF GEORGIA
ATHENS, GEORGIA
1986
( $)
ACADEMIC PRESS, INC.
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Library of Congress Cataloging in Publication Data
Adomian, G.
Nonlinear stochastic operator equations.
Includes index.
1. Stochastic systems. 2. Nonlinear theories.
3. Operator equations. I. Title.
QA402.A29 1986 003 85-26655
ISBN 0-12-044375-9 (alk. paper)
PRINTED IN THE UNITED STATES OF AMERICA
98765432 1
Dedicated to the Class of 1911
of the University of Georgia
Foreword
A previous volume, Stochastic Systems (1983), was the first systematic
book bringing nonlinear and stochastic equations within the reach of engi
neers and physicists concerned with the difficulties of real systems and fron
tier problems in which conventional techniques such as linearization and
perturbation are not sufficient or realistic.
This volume increases very substantially the scope of the work to cover
very wide classes of equations. It is a simple and powerful method applicable
to the solution of a multitude of problems in physics, engineering, operations
research, systems analysis, economics, biology, medicine, and other fields.
Forthcoming volumes will deal with applications to physics and engineer
ing. Previously, either such problems were not realistically formulated or the
approximations made and the methods used made the results either unreal
istic or misleading. This work will necessitate the rewriting of the mathemat
ics and physics books since mathematics deals largely with linear operator
theory and physics with perturbation. In my view, this overall work will have
a profound impact on mathematics and on the modeling of problems of
physics, engineering, economics, biology, and medicine and may well be
viewed in retrospect as one of the significant contributions of this century.
This work is a scientific breakthrough. The importance of it cannot be
overemphasized.
RICHARD E. BELLMAN
XI
In Memoriam
Richard Bellman's scientific contributions and service to mathematics
were cut short just before this book finally went to the publisher. My hope is
that the book will serve as another testimonial to his motivation and encour
agement of so many people.
G. ADOMIAN
Xll
Preface
This book addresses the need for realistic solutions of the nonlinear sto
chastic equations arising in the modeling of frontier problems in every area
of science. This includes linear or deterministic models or both as special
cases. What makes this methodology* different is the avoidance of perturba
tion, linearization, truncation, discretization, or the assumption of unphysi-
cal processes. Thus the solution sought is that of the problem at hand, rather
than one tailored to machine computation or the use of existing theorems of
mathematics. Although there is much that is new which should be of interest
to researchers, this material is still considered to be within the grasp of
students with the normal mathematical background of senior undergraduate
and graduate students in mathematical, physical, and engineering sciences.
* A short bibliography at the end of each chapter lists only those works that are felt to be
particularly relevant and necessary to this approach, rather than all work in the area.
xin
Acknowledgments
I would like to express my deep appreciation to Dr. Fred C. Davison,
President of the University of Georgia, and to Professor Richard Bellman for
support, encouragement, and confidence in my work.
I am appreciative also of the painstaking assistance and editing of Ran
dolph Rach.
Finally, I am grateful to Arlette Réveils, my secretary, who has "traveled"
with me cheerfully through complex equations and multidimensional spaces
for seventeen years.
xv
CHAPTER 1
Introduction
When confronted with a nonlinear systems engineering prob
lem, the first approach usually is to linearize, i.e., to try to avoid
the nonlinear aspects of the problem. It is indeed a happy
circumstance when a solution can be obtained in this way...
Wilson J. Rugh.
In a previous volume, "Stochastic Systems" [/], we introduced an approxi
mation method—we called it the decomposition method—for solving generic
operator equations of the form J^w = g, where the operator & may be
nonlinear (or linear as a special case) or stochastic (or deterministic as a
special case). It seems appropriate to call the method the "decomposition"
method, even though the word is used in other connections relating to the
theory of factorization and decomposition of dynamic systems. In Galois
theory, the representability of the roots of a polynomial in radicals is
considered in terms of factorizations of the group of automorphisms of the
decomposition field of the polynomial.
Decomposition has also been used in large-scale system theory where if
state spaces of subsystems are disjoint, decomposition into those subsystems
is an effective approach to simplify the system and if state spaces overlap, the
original system is expanded into a larger state space, including the original
system, such that overlapping subsystems appear as disjoint; then analysis
can be carried out using standard disjoint decomposition [2].
We discussed primarily differential equations but will now show that this
work is more general. Here $F may be a partial differential operator, a
differential operator, a differential-delay operator, or an algebraic operator.
The term g may be a function of t or x, or x and t, or x, y, z, i, or even x, y, z, i,
ω where ω e (Ω, F, μ) a. probability space. Thus the inhomogeneous term, as
well as initial or boundary conditions, and coefficients in the operator may be
stochastic processes. The boundary conditions may be simple or quite
complicated (nonlinear, coupled, or random). The equation may be linear or
nonlinear, and deterministic or stochastic.
1
2 1. INTRODUCTION
Basically two techniques are involved. First, any nonlinearities in the
equation to be solved are written in terms of the author's A polynomials, a
n
special set of polynomials which can be generated for the specific nonlinear-
ity. Second, the assumed solution u = 3F~ 1g is decomposed into components
to be determined such that the first component is the solution for the linear
deterministic part of ^, or of a suitable invertible part, including conditions
on u. The other components are then found in terms of preceding components.
In the stochastic case, the method yields a natural statistical separability so
that no truncations or closure approximations become necessary as discussed
in ref. 1.
Since the method is nonperturbative and does not resort to linearization or
assumptions of "small" or weak nonlinearity, "small" fluctuations, or special
processes, the solutions are not only quite general but more realistic as well.
This is true since we can solve a model, usually or often, much closer to the
physical model rather than a mathematized model to fit the mathematics at
hand.
Dynamical models whether for solitons, population problems, VLSI
devices, control systems, or the national economy are nonlinear and sto
chastic, in general. Special cases for which stochasticity vanishes or linearity
assumptions are adequate are precisely that, i.e., special cases included in the
general methodology. Thus, we can consider deterministic linear systems,
deterministic nonlinear systems, stochastic linear systems, and stochastic
nonlinear systems in a unified framework. Cases involving stochasticity
involve only an additional step of determining appropriate statistical mea
sures, and deterministic cases are not more difficult by embedding in a
general framework. The method was evolved to achieve statistical separ
ability and avoid truncations but is valuable in the deterministic case as well!
A tremendous variety of applications exist: plasmas, high-current beams,
propagation in random media, laser technology, turbulence, control theory
for nonlinear stochastic or adaptive systems, chemical and biological oscilla
tors, and many others. A number of important applications are dealt with
elsewhere [5].
Finally, we should mention that in some areas, the work is preliminary and
not a final document, in other words, it is an active research area. I hope it
will strike creative sparks in many dissertations.
REFERENCES
1. G. Adomian, "Stochastic Systems." Academic Press, New York, 1983.
2. M. Ikeda, D. D. Siljak, and E. E. White, IEEE Trans. Automatic Control AC-29, 244-249.
3. G. Adomian, Applications of Nonlinear Stochastic Systems Theory to Physics, in press.
CHAPTER 2
Operator Equations and the Decomposition Method
There must be a simpler way.
R. E. Bellman
2.1. MODELING, APPROXIMATION, AND REALITY
Frontier problems exist in physics, engineering, biology, medicine, astro
physics, and other disciplines and in applications ranging from nuclear
reactors, the fusion process, lasers, typhoons, tsunamis, computer chips,
radar, turbulence, internal waves, solitons, compartmental analyses in medi
cine, behavior of the national economy, and many more which require
sophisticated mathematical methods to even attempt to formulate, let alone
solve.
Their solutions—when appropriate models can be constructed—may
depend on equations for which present mathematical methods in general use
may require supercomputers and even so, possibly yield solutions which
simply are not correct or realistic. (A gigaflop computer could conceivably
give wrong answers much faster.)
The equations modeling these applications may be ordinary algebraic
(polynomial) equations starting from the quadratic equations familiar to
school students to high-order polynomials, transcendental and exponential
equations, differential equations or systems of such equations, delay-differen
tial equations, partial differential equations, or systems of partial differential
equations. The coefficients of these differential equations may be not only
space and time dependent but may be random processes. Finally, the
equations may be nonlinear.
If the latter two effects are relatively insignificant, then existing methods
are adequate. In general, these effects are not insignificant. Even mathemati
cians are sometimes unaware that the methodology in constant use for
3
