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Methods of Small Parameter in Mathematical BiologySmall Parameter Methods: Basic Ideas

Methods of Small Parameter in Mathematical Biology: Small Parameter Methods: Basic Ideas [We introduce basic ideas of asymptotic analysis and present a number of models which describe complex processes, the components of which occur at significantly different rates. Such models in a natural way contain a small parameter which is the ratio of the slow and the fast rates, thus lending themselves to asymptotic analysis. We discuss, among others, classical models of fluid dynamics and kinetic theory, population problems with fast migrations, epidemiological problems concerning diseases with quick turnover, models of enzyme kinetics and Brownian motion with fast direction changes. We also discuss initial and boundary layer phenomena using a simplified fluid dynamics equation as an example. In the conclusion of the chapter we discuss a model of enzyme kinetics and show in detail the application of the Hilbert expansion method to derive (formally) the Michaelis–Menten model; a rigorous derivation is referred to Chap. 3.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Methods of Small Parameter in Mathematical BiologySmall Parameter Methods: Basic Ideas

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Publisher
Springer International Publishing
Copyright
© Springer International Publishing Switzerland 2014
ISBN
978-3-319-05139-0
Pages
1 –44
DOI
10.1007/978-3-319-05140-6_1
Publisher site
See Chapter on Publisher Site

Abstract

[We introduce basic ideas of asymptotic analysis and present a number of models which describe complex processes, the components of which occur at significantly different rates. Such models in a natural way contain a small parameter which is the ratio of the slow and the fast rates, thus lending themselves to asymptotic analysis. We discuss, among others, classical models of fluid dynamics and kinetic theory, population problems with fast migrations, epidemiological problems concerning diseases with quick turnover, models of enzyme kinetics and Brownian motion with fast direction changes. We also discuss initial and boundary layer phenomena using a simplified fluid dynamics equation as an example. In the conclusion of the chapter we discuss a model of enzyme kinetics and show in detail the application of the Hilbert expansion method to derive (formally) the Michaelis–Menten model; a rigorous derivation is referred to Chap. 3.]

Published: Mar 12, 2014

Keywords: Asymptotic analysis; Models with small parameters; Hilbert expansion

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