Asymptotology Ideas, Methods, and Applications
Asymptotic methods belong to the, perhaps, most romantic area of modern mathematics. They are widely known and have been used in me chanics, physics and other exact sciences for many, many decades. But more than this, asymptotic ideas are found in all br
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Mathematics and Its Applications
Managing Editor: M. HAZEWINKEL
Centre for Mathematics and Computer Science. Amsterdam, The Netherlands
Volume 551
Asymptotology Ideas, Methods, and Applications
by
Igor V. Andrianov Pridneprovie State Academy of Civil Engineering and Architecture ( PGASA), Dnepropetrovsk, Ukraine
Leonid I. Manevitch N.N. Semenov Institute for Chemical Physics of Russian Academy of Sciences, Moscow, Russia
with help from
Michiel Hazewinkel CWI (Center for Mathematics and Computer Science), Amsterdam, The Netherlands
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4613-4816-0 DOI 10.1007/978-1-4419-9162-1
ISBN 978-1-4419-9162-1 (eBook)
Printed on acid-free paper
All Rights Reserved © 2002 Springer Science+ Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2002 Softcover reprint of the hardcover 1st edition 2002 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.
We dedicate this book to our parents.
Contents
Foreword Preface Acknowledgments Synopsis
xi Xlll XV XVll
1. INTRODUCTION 1. General Remarks 2. What does "Popular Science Book" Mean? 3. Why "Asymptotology"?
1 1 2 4
2. WHAT ARE ASYMPTOTIC METHODS? 1. Reduction of the System's Dimension 2. Regular Asymptotics and Boundary Layers 3. Asymptotic Series 4. Averaging and Homogenization 5. Continuous limits 6. Local and Nonlocal Linearization 7. Estimation of Asymptotic Solution Errors 8. Summation Procedures 9. "Padeons" 10. How to Make Both Ends Meet 11. Renormalization 12. Asymptotics and Computers 13. Are Asymptotic Methods a Panacea?
7 8 9 11 16 19 22 23 25 26 27 32 34 36
3. A LITTLE MATHEMATICS Basic Formalism 1. A Simple Example 2. 3. Regular and Singular Asymptotics 4. Asymptotic Decomposition 5. Supplementary Asymptotics
41 42 46 47 56 58
Vll
vm
ASYMPTOTOLOGY: IDEAS, METHODS, AND APPLICATIONS
6. 7. 8. 9.
Continuous Approximation of a Chain of Masses Search for Small Parameters Newton Polyhedron Catastrophe Theory
4. HOW ASYMPTOTIC METHODS WORK Celestial Mechanics 1. Theory of Plates and Shells 2. Polymer Physics 3. Asymptotics and Engineering 4. Theory of Composite Materials 5. Biology 6. Et Cetera 7. Asymptotics and Art 8. Asymptotics in Pictures 9. 10. Formation of New Concepts 11. Is Understanding an Asymptotic Process? 5. ASYMPTOTIC METHODS AND PHYSICAL THEORIES Asymptotic Correspondence of Physical Theories l. Mechanics by Aristotle and Galileo--Newton 2. Newton Mechanics and Special Relativity Theory 3. Geometrical and Wave Optics 4. Classical and Quantum Mechanics 5. "Simple Theories" in Physics 6. ·'The Cube of Theories'' 7. Asymptotic W
