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Boundary conditions for approximate differential equations

Published online by Cambridge University Press:  17 February 2009

A. J. Roberts
A. J. Roberts
Affiliation:
Department of Applied Mathematics, The University of Adelaide, GPO Box 498, Adelaide 5001, South, Australia.
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Abstract

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A large number of mathematical models are expressed as differential equations. Such models are often derived through a slowly-varying approximation under the assumption that the domain of interest is arbitrarily large; however, typical solutions and the physical problem of interest possess finite domains. The issue is: what are the correct boundary conditions to be used at the edge of the domain for such model equations? Centre manifold theory [24] and its generalisations may be used to derive these sorts of approximations, and higher-order refinements, in an appealing and systematic fashion. Furthermore, the centre manifold approach permits the derivation of appropriate initial conditions and forcing for the models [25, 7]. Here I show how to derive asymptotically-correct boundary conditions for models which are based on the slowly-varying approximation. The dominant terms in the boundary conditions typically agree with those obtained through physical arguments. However, refined models of higher order require subtle corrections to the previously-deduced boundary conditions, and also require the provision of additional boundary conditions to form a complete model.

Type
Research Article
Information
The ANZIAM Journal , Volume 34 , Issue 1 , July 1992 , pp. 54 - 80
Copyright
Copyright © Australian Mathematical Society 1992

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