Computer Algebra Recipes. An Introductory Guide to the Mathematical Models of Science

434 Pages·2006·5.213 MB·english

by  Richard H. Enns, George C. McGuire

Richard H. Enns
George C. McGuire
Computer Algebra Recipes
An Introductory Guide to the
Mathematical Models of Science
Springer
Richard H. Enns George C. McGuire
Simon Fraser University University College of Fraser Valley
Department of Physics Department of Physics
Bumaby, B.C. V5A 1S6 Abbotsford, BC V2S 7M9
Canada Canada
Cover design by Mary Burgess.
Library of Congress Control Number: 2005937149
ISBN-10 0-387-25767-5 e-ISBN 0-387-31262-5
ISBN-13 978-387-25767-9
Printed on acid-free paper.
©2006 Springer Science-f Business Media
Preface
A computer algebra system (CAS) not only has the ability to "crunch numbers"
and plot results, like traditional computing languages such as Fortran and C,
but it can also perform the symbolic manipulations and analytic derivations
required in most undergraduate and graduate science and engineering courses.
To introduce students in these disciplines to mathematical modeling and com
putation using a CAS, the authors have previously published Computer Algebra
Recipes: A Gourmet ^s Guide to the Mathematical Models of Science, based on
the Maple CAS. Judging by course evaluations and reader feedback, the re
sponse to this book and the CAS approach has been quite favorable. After
observing students' enthusiasm, their higher quality of work, their ability to
solve more realistic problems, and best of all, their ability to answer "what
if?" questions, we believe that the importance of using a CAS in learning and
exploring mathematically based science subjects cannot be overstated.
With the release of new, more powerful, versions of the Maple CAS since the
first edition was published and the accumulation of many insightful comments
and helpful suggestions from readers of the text, it seemed timely to produce a
second edition. However, incorporating the necessary changes and suggestions
would make an already lengthy book even longer, so the topics of the first
edition have been reorganized and expanded into two new standalone volumes
based on the expected mathematical level of the reader.
In this first volume, we assume the reader's familiarity with linear algebra,
vectors, and elementary calculus, and knowledge of (but not necessarily exper
tise at) linear ordinary diflPerential equations. The second volume {Computer
Algebra Recipes: An Advanced Guide to the Mathematical Models of Science)
deals with more advanced differential equation models, both ordinary and par
tial, and nonlinear as well as linear.
Each volume, which may be used either as a course text or for self-study, fea
tures an eclectic collection of Maple computer algebra worksheets or "recipes"
drawn from a wide variety of disciplines, including biology, economics, medicine,
engineering, game theory, physics, mathematics, and chemistry. These recipes
are systematically organized to illustrate graphical, analytical, and numerical
techniques applied to scientific modeling. No prior knowledge of Maple is as
sumed in either volume, the early recipes of each book introducing the reader to
the basic Maple syntax, and the subsequent recipes introducing further Maple
commands and structure on a need-to-know basis.
vi PREFACE
The recipes are fully annotated in the text and typically presented as "sto
ries" or in a historical context. Each recipe takes the reader from the analytic
formulation or statement of an interesting mathematical model to its analytic
or numerical solution, and to a graphical visualization of the answer where rel
evant. The graphical representations vary from static 2-dimensional pictures,
to contour and vector field plots, to 3-dimensional graphs that can be rotated,
and to animations of analytic and numerical solutions.
Every recipe is followed by a set of problems that readers can use to check
their understanding or develop the topic further. For your convenience in solv
ing these problems and to facilitate further exploration of a given topic, the
unannotated recipes for each volume are included on an accompanying CD.
Contents
Preface
INTRODUCTION 1
A. Computer Algebra Systems 1
B. Computer Algebra Recipes 3
C. Introductory Recipe: Bridge Design 101 5
D. Maple Help 9
E. How to Use This Text 10
I THE APPETIZERS 11
1 The Pictures of Science 13
1.1 Data and Function Plots 13
1.1.1 Correcting for Inflation 15
1.1.2 The Plummeting Badminton Bird 22
1.1.3 Minimizing the Travel Time 31
1.2 Log-Log (Power Law) Plots 36
1.2.1 Chimpanzee Brain Size 36
1.2.2 Scaling Arguments and Gulliver's Travels 40
1.3 Contour and Gradient Plots 46
1.3.1 The Secret Message 46
1.3.2 Designing a Ski Hih 49
1.4 Animated Plots 56
1.4.1 Waves Are Dynamic 56
1.4.2 The Sands of Time 59
1.4.3 These Arrows Are Useful 61
2 Deriving Model Equations 65
2.1 Linear Correlation 66
2.1.1 The Corn Palace 67
2.2 Least Squares Derivations 69
2.2.1 Will You Be Better Off Than Your Parents? 71
2.2.2 What Was the Heart Rate of a Brachiosaur? 76
viii CONTENTS
2.2.3 Senate Renewal 84
2.2.4 Bikini Sales and the Logistic Curve 87
2.2.5 Following the Dow Jones Index 91
2.2.6 Variation of "y" with Latitude 98
2.2.7 Finding Romeo a Juliet 103
2.3 Multiple Regression Equations 106
2.3.1 Real Estate Appraisals 107
2.3.2 And the Winner Is? 113
II THE ENTREES 119
3 Algebraic Models. Part I 121
3.1 Scalar Models 121
3.1.1 Bombs Versus Schools 122
3.1.2 Kirchhoff Rules the Electrical World 129
3.1.3 The Window Washer's Secret 136
3.1.4 The Science Student's Summer Job Interview 142
3.1.5 Envelope of Safety 148
3.1.6 Rainbow County 152
3.2 Integral Examples 156
3.2.1 The Great Pyramid of Cheops 156
3.2.2 Noah's Ark 162
4 Algebraic Models. Part II 173
4.1 Vector Models 173
4.1.1 Vectoria's Mathematical Heritage 174
4.1.2 Vectoria and Fowles's Fly 179
4.1.3 Ain't She Sweet 183
4.1.4 Born Curl-Free 188
4.1.5 Of Coordinates and Circulation Too 194
4.1.6 All Is Flux 199
4.2 Matrix Models 202
4.2.1 Secret Message Revisited 202
4.2.2 A Fishy Tale 205
4.2.3 Population Waves 208
5 Linear ODE Models 213
5.1 Phase-Plane Portraits 214
5.1.1 Tenure Policy at Erehwon University 216
5.1.2 Vectoria Investigates the RLC Circuit 221
5.2 First-Order ODE Models 229
5.2.1 There Goes Louie's Alibi 229
5.2.2 The Water Skier 238
5.2.3 Ready to Charge 242
CONTENTS ix
5.3 Second-Order ODE Models 245
5.3.1 Shrinking the Safety Envelope 245
5.3.2 Frank N. Stein Is Not Heartless 251
5.3.3 Halley's Comet 255
5.3.4 Wheel of misFortune 260
5.3.5 The Weedeater 266
5.3.6 Can an Unstable Spring Find Stability? 269
6 Difference Equation Models 271
6.1 Linear Models 272
6.1.1 Those Dratted Gnats 272
6.1.2 Gone Fishing 276
6.1.3 Fibonacci's Adam and Eve Rabbit 279
6.1.4 How Red Is Your Blood? 283
6.1.5 Fermi-Pasta-Ulam Is Not a Spaghetti Western 285
6.2 Nonhnear Models 292
6.2.1 Competition for Available Resources 292
6.2.2 The Logistic Map and Cobweb Diagrams 299
6.2.3 The Bouncing Bah Art Gallery 306
6.2.4 Onset of Chaos: A Model for the Outbreak of War . . .. 310
III THE DESSERTS 317
7 Monte Carlo Methods 319
7.1 Random Walks 321
7.1.1 The Soccer Fan's Drunken Walk 324
7.1.2 Blowin' in the Wind 329
7.1.3 Flight of Penelope Jitter Bug 333
7.1.4 That Meandering Perfume Molecule 335
7.2 Monte Carlo Integration 338
7.2.1 Numerical Integration Methods 339
7.2.2 Wait and Buy Later! 344
7.2.3 Wait and Buy Later! The Sequel 348
7.2.4 Estimating TT 353
7.2.5 Chariot of Fire and Destruction 355
7.3 Probability Distributions 361
7.3.1 Of Nuts and Bolts and Hospital Beds Too 361
7.3.2 The Ice Wines of Rainbow County 367
7.4 Monte Carlo Statistical Distributions 372
7.4.1 Estimating e 372
7.4.2 Vapor Deposition 376
CONTENTS
Fractal Patterns 381
8.1 Difference Equations 384
8.1.1 Wallpaper for the Mind 384
8.1.2 Sierpinski's Fractal Gasket 386
8.1.3 Barnsley's Fern 391
8.1.4 Douady's Rabbit and Other Fauna and Flora 396
8.1.5 The Rings of Saturn 400
8.2 Cellular Automata 408
8.2.1 A Navaho Rug Design 408
8.2.2 The One out of Eight Rule 411
8.3 Strange Attractors 414
8.3.1 Lorenz's Butterfly 414
Epilogue 416
Bibliography 417
Index 421
Part I
THE APPETIZERS
Some books are to be tasted, others to be swallowed,
and some few to be chewed and digested.
Francis Bacon, English monk (1561-1626)
/ did toy with the idea of doing a cook-book.
The recipes were to be the routine ones:
how to make dry toast, instant coffee,—
But as an added attraction...
my idea was to put a fried egg on the cover.
I think a lot of people who hate literature but
love fried eggs would buy it if the price was right.
Groucho Marx, American comic actor (1895-1977)
It^s red hot, mate. I hate to think of this sort of book
getting into the wrong hands. As soon as
I^ve finished this, I shall recommend they ban it.
Tony Hancock, British comedian (1924-1968)
INTRODUCTION
A. Computer Algebra Systems
The purpose of computing is insight, not numbers,
R.W. Hamming, Numerical Methods for Scientists and Engineers (1973)
Although modern scientific models are usually not difficult to understand qual
itatively, the task of deriving the relevant model equations and finding, visual
izing, and interpreting the associated solutions may be too demanding or too
tedious to realistically carry out without the aid of a computer. As a con
sequence, over the last several years a new branch of science, referred to as
computational science, has evolved to deal with this issue. Traditionally, the
approach of most computational science texts [PFTV89], [DeV94], [LP97] has
been to introduce engineering and science students to the art of creating ef
ficient computer programs in languages such as Fortran and C to carry out
a multitude of numerical tasks ranging from finding the solutions to ordinary
and partial differential equations (ODEs and PDEs) to performing Monte Carlo
simulations and so on. Although some scientists and engineers may still wish
to learn one or more of these programming languages for certain specialized
research objectives, an even more powerful general computer algebra approach
is being developed. This new approach is changing the way that complex math
ematical modeling problems are tackled by science and engineering students as
well as by practitioners in these fields. It not only allows the user to handle
such problems numerically, but also permits him or her to carry out analytic
differentiation, integration, and other symbolic manipulations, and easily create
a wide variety of two- and three-dimensional static, as well as animated, plots.
Computer software systems, such as Maple, that can carry out all of these di
verse functions in a unified and cohesive fashion, are referred to as symbolic
computation systems or computer algebra systems (CASs).
As personal computers become smaller, cheaper, faster, and possess greater
memory, it is already clear that CASs, which are also rapidly increasing in
sophistication and ease of use, are making many of the traditional topics and
approaches covered in existing computational science texts less relevant, since
many of the same tasks can be executed more easily with a CAS. With a
CAS not only can complex model equations be analytically derived, they can
be solved either analytically or numerically and then plotted in two or three