Singular Perturbation Theory
This book is a rigorous presentation of the method of matched asymptotic expansions, the primary tool for attacking singular perturbation problems. A knowledge of conventional asymptotic analysis is assumed. The first chapter introduces the th
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Lindsay A. Skinner
Singular Perturbation Theory
Lindsay A. Skinner Department of Mathematical Sciences University of Wisconsin - Milwaukee Milwaukee, Wisconsin 53201 USA [email protected]
e-ISBN 978-1-4419-9958-0 ISBN 978-1-4419-9957-3 DOI 10.1007/978-1-4419-9958-0 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011928077 Mathematical Subject Classification 2010: 34E05, 34E10, 34E15, 34E20 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
There are many fine books on singular perturbation theory and how to solve singular perturbation problems. Many readers of this book will already be familiar with one or more of them. What distinguishes this book from the others is its rigorous development and rigorous application of the method of matched asymptotic expansions. The point of view is that certain functions have a certain structure for which this method is valid and these are precisely the kinds of functions that arise in a wide variety of differential equation and integration problems. This book is intended to serve primarily as a supplement or follow-up to a typical first year graduate course in asymptotic and perturbation analysis. Hopefully it will also prove to be a valuable companion to all those who do, or wish to do, rigorous work in the field. The basic theory for the book is presented in Chapter 1. Then there are four chapters in which this theory is applied to a sequence of ordinary differential equation problems. There are a number of previously unpublished results. One of these is the uniformly valid expansion at the end of Chapter 4 for a problem involving logarithms once studied by L. E. Fraenkel. Another is the unexpectedly simple uniformly valid Bessel function expansion established at the end of Chapter 3. All the differential equations chosen for study in the text are linear, but this is not because of any limitation of the theory. Indeed, much has been done, and much more can be done, in applying the theory to nonlinear problems, as noted in Exercise 3.2, for example. Another unique feature of this book is its inclusion of several Maple programs for computing the terms of the various asymptotic expansions that arise in solving the problems. Developi
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