Singular perturbations. Introduction to system order reduction methods with applications. With a foreword by Robert O'Malley
type: Item
instance of: scholarly article
title: Singular perturbations. Introduction to system order reduction methods with applications. With a foreword by Robert O'Malley
zbMATH Open document ID: 1397.34003
DOI: 10.1007/978-3-319-09570-7
author: Elena Shchepakina
author: V. A. Sobolev
author: Michael P. Mortell
published in: Lecture Notes in Mathematics
publication date: 2014-09-05T00:00:00Z
review text: The review consist of the abstracts to each chapter. Chapter 1: Introduction -- Chapter 1 provides an easy introduction to perturbation methods. It begins with an algebraic equation and proceeds to a second-order ODE. The concept of an initial or boundary layer is introduced. This motivates the method of multiple scales. The idea of slow surfaces and slow integral manifolds is introduced and illustrative examples are given. Then a statement of Tikhonov's theorem is given which answers the question about the permissibility of the application of a ``degenerate'' system \((\varepsilon=0)\) as a zero-approximation to the full system. Chapter 2: Slow Integral Manifolds -- In the present chapter we use a method for the qualitative asymptotic analysis of singularly perturbed differential equations by reducing the order of the differential system under consideration. The method relies on the theory of integral manifolds. It essentially replaces the original system by another system on an integral manifold with a lower dimension that is equal to that of the slow subsystem. The emphasis in this chapter is on the study of autonomous systems. Chapter 3: The Book of Numbers -- In this chapter the first number in the title of a section denotes the dimension of the slow variable, the second one denotes the dimension of the fast variable. A series of examples, of increasing complexity, are given to illustrate the theoretical concepts. The main examples come from applications in enzyme kinetics. These examples illustrate the effectiveness of the order reduction method. Chapter 4: Representations of Slow Integral Manifolds -- In constructing the asymptotic expansions of slow integral manifolds it is assumed that the degenerate equation \((\varepsilon=0)\) allows one to find the slow surface explicitly. In many problems this is not possible due to the fact that the degenerate equation is either a high degree polynomial or transcendental. In this situation many authors suggest the use …
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34-02
Mathematics Subject Classification ID: 34C45
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34E17
Mathematics Subject Classification ID: 34H20
Mathematics Subject Classification ID: 92C45
Mathematics Subject Classification ID: 34C26
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 6340730
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-09570-7
OpenAlex ID: W4245296879
description: scientific article; zbMATH DE number 6340730