Browse Prior Art Database

MULTI-TIME METHODS FOR SYSTEMS OF DIFFERENCE EQUATIONS

IP.com Disclosure Number: IPCOM000148716D
Original Publication Date: 1976-May-19
Included in the Prior Art Database: 2007-Mar-30

Publishing Venue

Software Patent Institute

Related People

Hoppensteadt, Frank C.: AUTHOR [+3]

Abstract

RC 6001 (#26026) 5/19/76Mathematics 24 pages MULTI-TIME METHODS FOR SYSTEMS OF DIFFERENCE EQUATIONSFrank C. HoppensteadtCourant Institute of Mathematical SciencesMew York UniversityNew York, New York 10012andWillard L. Miranker I B M Thomas J. Watson Research CenterYorktown Heiqhts, New York 10598Tylsed by Linda P. Rubin on MTSTABSTRACT: Systems of difference equations containing small nararneters are studied by a constructive perturbation scheme analoqous to the one developed by the authors for the study of differential equations. The method results in an aver- aginq procedure for difference equations, and it is parti- cularly well suited to certain hiqhly oscillatory, nonlinear systems. The method is applied to problems from population genetics, pattern recognition, and the mathematical analysis of stiff differential equations. LIhIITED DISTRIBUTION NOTICE This report has been submitted for publication elsewhere and has been issued as a Research Report for early disseminationof its contents. As a courtesy to the intended publisher, it should not be widely distributed until after the date of outside publication.

This text was extracted from a PDF file.
At least one non-text object (such as an image or picture) has been suppressed.
This is the abbreviated version, containing approximately 15% of the total text.

Page 1 of 26

RC 6001

(#26026) 5/19/76
Mathematics

24 pages

MULTI-TIME METHODS FOR SYSTEMS OF DIFFERENCE EQUATIONS
Frank C. Hoppensteadt
Courant Institute of Mathematical Sciences
Mew York University
New York, New York 10012
and
Willard L. Miranker I B M Thomas J. Watson Research Center
Yorktown Heiqhts, New York 10598
Tylsed by Linda P. Rubin on MTST
ABSTRACT:
Systems of difference equations containing small nararneters are studied by a constructive perturbation scheme analoqous to the one developed by the authors for the study of differential equations. The method results in an aver- aginq procedure for difference equations, and it is parti- cularly well suited to certain hiqhly oscillatory, nonlinear systems. The method is applied to problems from population genetics, pattern recognition, and the mathematical analysis of stiff differential equations.

[This page contains 1 picture or other non-text object]

Page 2 of 26

LIhIITED DISTRIBUTION NOTICE

This report has been submitted for publication elsewhere and has been issued as a Research Report for early dissemination
of its contents. As a courtesy to the intended publisher, it should not be widely distributed until after the date of outside publication.

Copies may be requested from:


IBM Thomas J. Fatson Research Center Post Office Box 2 18
Yorktown Heights, New York 10598

[This page contains 1 picture or other non-text object]

Page 3 of 26

Introduction

    Systems of difference equations which contain small parameters arise
in many applications of mathematics to physical and biological problems,
to problems in information processing as well as to many problems arising
in the numerical analysis of systems of differential equations. The
analysis of difference equations is usually difficult. However, we show
here that perturbation schemes can be derived which reduce the problems
to systems of differential equations, and thereby provide a procedure for
determining behavior of solutions to difference equations.

The following example illustrates the point of view taken toward more
PXP

general problems in later sections. Let x E E' and A,B r E . The
difference equation

X n+l = (A + cB)xn , x given,

0

Now suppose that E is a small parameter, A is invertible and A and B
commute. Then

*

For large values of n, say n=K[l/~]+p , (I+EA-~B)

and so the solution may be written as
x
n = exp[~'~~~nlx~(l+neO(~)f

,

which is a useful approximation when n = 0(1/~). Thus, the solution can
be expressed asymptotically for E%O , by the product of a factor which

* Here [I/&] indicates the largest integer 5 1/€ .

has the solution

5 exp [A-'BKI [I+O( €1 1

[This page contains 1 picture or other non-text object]

Page 4 of 26

-1

varies rapidly (A") with n and one that varies slowly (exp(A Ben)) with n.

    This result can be obtained by another method which is applicable to more general systems. First, we factor out the dominant fas...