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The application of centre-manifold theory to the evolution of system which vary slowly in space

Published online by Cambridge University Press:  17 February 2009

A. J. Roberts
A. J. Roberts
Affiliation:
Department of Applied Mathematics, University of Adelaide, G. P. O. Box 498, Adelaide, S. A. 5001, Australia.
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Abstract

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In many physical problems, the system tends quickly to a particular structure, which then evolves relatively slowly in space and time. Various methods exist to derive equations describing the slow evolution of the particular structure; for example, the method of multiple scales. However, the resulting equations are typically valid only for a limited range of the parameters. In order to extend the range of validity and to improve the accuracy, correction terms must be found for the equations. Here we describe a procedure, inspired by centre-manifold theory, which provides a systematic approach to calculating a sequence of successively more accurate approximations to the evolution of the principal structure in space and time.

The formal procedure described here raises a number of questions for future research. For example: what sort of error bounds can be obtained, do the approximations converge or are they strictly asymptotic, and what sort of boundary conditions are appropriate in a given problem?

Type
Research Article
Information
The ANZIAM Journal , Volume 29 , Issue 4 , April 1988 , pp. 480 - 500
Copyright
Copyright © Australian Mathematical Society 1988

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