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Small Parameter Methods: Basic Ideas

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Methods of Small Parameter in Mathematical Biology

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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.

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Authors and Affiliations

  1. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

    Jacek Banasiak

  2. Institute of Mathematics, Technical University of Łódź, Łódź, Poland

    Jacek Banasiak

  3. Institute of Applied Mathematics and Mechanics Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland

    Mirosław Lachowicz

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  1. Jacek Banasiak
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  2. Mirosław Lachowicz
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Banasiak, J., Lachowicz, M. (2014). Small Parameter Methods: Basic Ideas. In: Methods of Small Parameter in Mathematical Biology. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-05140-6_1

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