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A COMPARISON OF CRITICAL TIME DEFINITIONS IN MULTILAYER DIFFUSION

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  20 March 2012

R. I. HICKSON ,
S. I. BARRY ,
H. S. SIDHU  and
G. N. MERCER
R. I. HICKSON*
Affiliation:
Applied and Industrial Mathematics Research Group, School of Physical, Environmental and Mathematical Sciences, University of New South Wales Canberra, Northcott Drive, Canberra, ACT 2600, Australia (email: R.Hickson@UNSWalumni.com, H.Sidhu@adfa.edu.au) National Centre for Epidemiology and Population Health, Australian National University, Canberra, ACT 0200, Australia (email: Steve.Barry@AirservicesAustralia.com, Geoff.Mercer@anu.edu.au)
S. I. BARRY
Affiliation:
National Centre for Epidemiology and Population Health, Australian National University, Canberra, ACT 0200, Australia (email: Steve.Barry@AirservicesAustralia.com, Geoff.Mercer@anu.edu.au)
H. S. SIDHU
Affiliation:
Applied and Industrial Mathematics Research Group, School of Physical, Environmental and Mathematical Sciences, University of New South Wales Canberra, Northcott Drive, Canberra, ACT 2600, Australia (email: R.Hickson@UNSWalumni.com, H.Sidhu@adfa.edu.au)
G. N. MERCER
Affiliation:
Applied and Industrial Mathematics Research Group, School of Physical, Environmental and Mathematical Sciences, University of New South Wales Canberra, Northcott Drive, Canberra, ACT 2600, Australia (email: R.Hickson@UNSWalumni.com, H.Sidhu@adfa.edu.au) National Centre for Epidemiology and Population Health, Australian National University, Canberra, ACT 0200, Australia (email: Steve.Barry@AirservicesAustralia.com, Geoff.Mercer@anu.edu.au)
*
✉For correspondence; e-mail: R.Hickson@UNSWalumni.com
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Abstract

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There are many ways to define how long diffusive processes take, and an appropriate “critical time” is highly dependent on the specific application. In particular, we are interested in diffusive processes through multilayered materials, which have applications to a wide range of areas. Here we perform a comprehensive comparison of six critical time definitions, outlining their strengths, weaknesses, and potential applications. A further four definitions are also briefly considered. Equivalences between appropriate definitions are determined in the asymptotic limit as the number of layers becomes large. Relatively simple approximations are obtained for the critical time definitions. The approximations are more accessible than inverting the analytical solution for time, and surprisingly accurate. The key definitions, their behaviour and approximations are summarized in tables.

Keywords

critical timemultilayer diffusionmean action time

MSC classification

Secondary: 35K05: Heat equation
Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 4 , April 2011 , pp. 333 - 358
Copyright
Copyright © Australian Mathematical Society 2012

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