Computer Algebra Recipes

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Richard H. Enns Simon Fraser University Department of Physics Burnaby, B.C. V5A 1S6 Canada renns@sfu.ca Cover design by Mary Burgess. Library of Congress Control Number: 2006936017 ISBN-10: 0-387-25768-3 e-ISBN-10: 0-387-49333-6 ISBN-13: 978-387-25768-6 e-ISBN-13: 978-0-387-49333-6 Printed on acid-free paper. George C. McGuire University College of Fraser Valley Department of Physics Abbotsford, BC V2S 7M9 Canada george.mcguire@ucfv.ca c○2007 Springer Science+Business Media, LLC 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) and the author, 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. 987654321 springer.com (EB)
PREFACE A computer algebra system (CAS) not only has the number \crunching" and plotting capability of traditional computing languages such as Fortran and C, but also allows one to perform the symbolic manipulations and derivations required in most mathematically based science and engineering courses. To introduce students in these disciplines to CAS-based mathematical modeling and computation, the authors have previously developed and classroom tested the text Computer Algebra Recipes: A Gourmet's Guide to the Mathematical Models of Science [EM01] based on the Maple CAS. Judging by course evaluations and reader feedback, the response to this book and the computer algebra approach to modeling has been very favorable. With the release of several new versions of Maple since this text was published and the authors' accumulation of many insightful comments and helpful suggestions, a second up-dated edition seemed expedient. However, incorporating all the changes would make an already lengthy book even longer. So the topics of the Gourmet's Guide have been reorganized into two new stand-alone volumes, an already-published Introductory Guide [EM06] and this Advanced Guide. In this book, we explore mathematical models involving linear and nonlinear ordinary and partial di®erential equations (ODEs and PDEs). This volume, which may be used either as a course text or for self-study, features an eclectic collection of Maple computer algebra worksheets, or \recipes," that are systematically organized to illustrate graphical, analytical, and numerical techniques applied to ODE/PDE-based scienti¯c modeling. No prior knowledge of Maple is assumed, the early recipes introducing the reader to the basic Maple syntax, the subsequent recipes introducing further Maple commands and structure on a need-to-know basis. The recipes are fully annotated in the text and in most cases presented as \stories" or in a historical context. Each recipe typically takes the reader from the analytic formulation of an interesting mathematical model to its analytic or numerical solution and ¯nally to either a static or animated graphical visualization of the answer. Every recipe is followed by a set of problems that can be used to check one's understanding or develop the topic further. For your convenience, the recipes are included on a CD located in the inside back cover. v
Page 1: Richard H. Enns George C. McGuire C
Page 5 and 6: viii CONTENTS 2.3.1 Finite Di®eren
Page 7 and 8: x CONTENTS 8.3 The Bifurcation Diag
Page 9 and 10: 2 INTRODUCTION B. Computer Algebra
Page 11 and 12: 4 INTRODUCTION (a) At what angle Á
Page 13 and 14: 6 INTRODUCTION The negative answer
Page 15 and 16: 8 INTRODUCTION D. Maple Help We tea
Page 17 and 18: Part I THE APPETIZERS A man ceases
Page 19 and 20: 14 CHAPTER 1. PHASE-PLANE PORTRAITS
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48 CHAPTER 2. PHASE-PLANE ANALYSIS
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Chapter 3 Linear ODE Models Among a
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3.1. FIRST-ORDER MODELS 111 Therate
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3.1. FIRST-ORDER MODELS 113 Pressur
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3.1. FIRST-ORDER MODELS 115 3.1.2 G
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3.1. FIRST-ORDER MODELS 117 Evil Kn
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3.2. SECOND-ORDER MODELS 119 > tf:=
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3.2. SECOND-ORDER MODELS 121 3.2.2
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3.2. SECOND-ORDER MODELS 123 Loadin
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3.2. SECOND-ORDER MODELS 125 0.4 0.
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3.2. SECOND-ORDER MODELS 127 On ent
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3.2. SECOND-ORDER MODELS 129 Using
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3.3. SPECIAL FUNCTION MODELS 131 3.
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3.3. SPECIAL FUNCTION MODELS 133 1
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3.3. SPECIAL FUNCTION MODELS 137 3.
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3.3. SPECIAL FUNCTION MODELS 139 th
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3.3. SPECIAL FUNCTION MODELS 141 PR
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3.3. SPECIAL FUNCTION MODELS 143 ¡
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3.3. SPECIAL FUNCTION MODELS 145 μ
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3.3. SPECIAL FUNCTION MODELS 147 >
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150 CHAPTER 4. NONLINEAR ODE MODELS
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208 CHAPTER 5. LINEAR PDE MODELS. P
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Chapter 6 Linear PDE Models. Part 2
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6.1. WAVE EQUATION MODELS 249 > sol
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6.1. WAVE EQUATION MODELS 251 6.1.2
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6.1. WAVE EQUATION MODELS 253 The p
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6.1. WAVE EQUATION MODELS 255 Recog
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6.1. WAVE EQUATION MODELS 257 PROBL
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6.1. WAVE EQUATION MODELS 259 Then
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6.2. SEMI-INFINITE AND INFINITE DOM
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6.3. NUMERICAL SIMULATION OF PDES 2
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Part III THE DESSERTS The way a chi
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288 CHAPTER 7. THE HUNT FOR SOLITON
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320 CHAPTER 8. NONLINEAR DIAGNOSTIC
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356 BIBLIOGRAPHY [Dav62] H. T. Davi
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358 BIBLIOGRAPHY [Nyq28] H. Nyquist
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Index acoustical waveguide, 261 ade
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INDEX 363 eigenvalue, 130 Eiram Eir
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INDEX 365 Help, Full Text Search, 8
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INDEX 367 laplace, 121,267,268 lhs,
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INDEX 369 ¯sh harvesting, 100 ¯xe
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INDEX 371 quartic map, 345 Queen Di
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