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2014 | OriginalPaper | Buchkapitel

1. Introduction to the Geometric Theory of ODEs with Applications to Chemical Processes

verfasst von : Dietrich Flockerzi

Erschienen in: Large-Scale Networks in Engineering and Life Sciences

Verlag: Springer International Publishing

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Abstract

We give an introduction to the geometric theory of ordinary differential equations (ODEs) tailored to applications to biochemical reaction networks and chemical separation processes. Quite often, the ordinary differential equations under investigation are “reduced” partial differential equations (PDEs) as in the search of traveling wave solutions. So, we also address ODE topics that have their origin in the PDE context.
We present the mathematical theory of invariant and integral manifolds, in particular, of center and slow manifolds, which reflect the splitting of variables and/or processes into slow and fast ones. The invariance of a smooth manifold is characterized by a quasilinear partial differential equation, and the widely used approximations of invariant manifolds are derived from such PDEs. So we also offer, to some extent, an introduction to quasilinear PDEs. The basic ideas and crucial tools are illustrated with numerous examples and exercises. Concerning the proofs, we confine ourselves to outline the crucial steps and refer, especially in the first three sections, to the literature.
The final Sects. 1.4 and 1.5 on reaction–separation processes and on chromatographic separation present new results, including their proofs. They are the outcome of many fruitful discussions with my colleagues Malte Kaspereit and Achim Kienle.

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Nächstes Kapitel Mathematical Modeling and Analysis of Nonlinear Time-Invariant RLC Circuits
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Metadaten
Titel
Introduction to the Geometric Theory of ODEs with Applications to Chemical Processes
verfasst von
Dietrich Flockerzi
Copyright-Jahr
2014
Buch
Large-Scale Networks in Engineering and Life Sciences

Print ISBN: 978-3-319-08436-7

Electronic ISBN: 978-3-319-08437-4

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-3-319-08437-4_1