Computer Algebra Recipes. An Advanced Guide to Scientific Modeling

Transcription

1 Computer Algebra Recipes An Advanced Guide to Scientific Modeling

2 Richard H. Enns George C. McGuire Computer Algebra Recipes An Advanced Guide to Scientific Modeling

3 Richard H. Enns Simon Fraser University Department of Physics Burnaby, B.C. V5A 1S6 Canada George C. McGuire University College of Fraser Valley Department of Physics Abbotsford, BC V2S 7M9 Canada Cover design by Mary Burgess. Library of Congress Control Number: ISBN-10: e-isbn-10: ISBN-13: e-isbn-13: Printed on acid-free paper. 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 springer.com (EB)

4 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

5 Contents PREFACE v INTRODUCTION 1 A. Computer Algebra Systems B. Computer Algebra Recipes C. Introductory Recipe: Boys Will Be Boys D. Maple Help E. How to Use This Text I THE APPETIZERS 11 1 Phase-Plane Portraits Phase-Plane Portraits Romeo and Juliet There's No Damping Vectoria's Romantic Heart Van der Pol's Limit Cycle Three-Dimensional Autonomous Systems The Period-Doubling Route to Chaos The Oregonator RÄossler's Strange Attractor Phase-Plane Analysis Phase-Plane Analysis Foxes Munch Rabbits The Mona Lisa of Nonlinear Science Mike Creates a Higher-Order Fixed Point The Gnus and Sung of Erehwon A Plethora of Points Three-Dimensional Autonomous Systems Lorenz's Butter y Numerical Solution of ODEs vii

6 viii CONTENTS Finite Di erence Approximations Rabbits and Foxes: The Sequel Glycolytic Oscillator Fox Rabies Epidemic II THE ENTREES Linear ODE Models First-Order Models How's Your Blood Pressure? Greg Arious Nerd's Problem Second-Order Models Daniel Encounters Resistance Meet Mr. Laplace Jennifer's Formidable Series Special Function Models Jennifer Introduces a Special Family The Vibrating Bungee Cord Mathieu's Spring Quantum-Mechanical Tunneling Nonlinear ODE Models First-Order Models An Irreversible Reaction The Struggle for Existence The Bad Bird Equation Second-Order Models Patches Gives Chase Oh What Sounds We Hear! Vectoria Feels the Force and Hits the Bottle Golf Is Such an \Uplifting" Experience Variational Calculus Models Dress Design, the Erehwonese Way Queen Dido Wasn't a Dodo The Human Fly Plans His Escape Route This Would Be a Great Amusement Park Ride Linear PDE Models. Part Checking Solutions The Palace of the Governors Play It, Sam Three Easy Pieces Complex, Yet Simple Di usion and Laplace's Equation Models

7 CONTENTS ix Freeing Excalibur Aussie Barbecue Benny's Solution Hugo and the Atomic Bomb Hugo Prepares for His Job Interview Linear PDE Models. Part Wave Equation Models Vectoria Encounters Simon Legree Homer's Jiggle Test Vectoria's Second Problem Sound of Music? Semi-in nite and In nite Domains Vectoria's Fourth Problem Assignment Complete! Radioactive Contamination \Play It, Sam" Revisited Numerical Simulation of PDEs Freeing Excalibur the Numerical Way Enjoy the Klein{Gordon Vibes Vectoria's Secret III THE DESSERTS The Hunt for Solitons The Graphical Hunt for Solitons Of Kinks and Antikinks In Search of Bright Solitons Can Three Solitons Live Together? Analytic Soliton Solutions Follow That Wave! Looking for a Kinky Solution We Have Solitons! Simulating Soliton Collisions To Be or Not to Be a Soliton Are Diamonds a Kink's Best Friend? Nonlinear Diagnostic Tools The Poincar e Section A Rattler Signals Chaos Hamiltonian Chaos The Power Spectrum Frank N. Stein's Heartbeat The Rattler Returns

8 x CONTENTS 8.3 The Bifurcation Diagram Pitchforks and Other Bifurcations The Lyapunov Exponent Mr. Lyapunov Agrees Reconstructing an Attractor Putting Humpty Dumpty Together Again Random Is Random Butter y Reconstruction Epilogue Bibliography 355 Index 361