# Computer Algebra Recipes: An Advanced Guide to Scientific Modeling

Springer Science & Business Media, Dec 31, 2007 - Mathematics - 374 pages

Modern computer algebra systems are revolutionizing the teaching and learning of mathematically intensive subjects in science and engineering, enabling students to explore increasingly complex and computationally intensive models that provide analytic solutions, animated numerical solutions, and complex two- and three-dimensional graphic displays.

This self-contained text benefits from a spiral structure that regularly revisits the general topics of graphics, symbolic computation, and numerical simulation with increasing intricacy at each turn. The text is built around a large number of computer algebra worksheets or "recipes" that have been designed using MAPLE to provide tools for problem solving and to stimulate critical thinking. No prior knowledge of MAPLE is assumed. All relevant commands are introduced on a need-to-know basis and are indexed for easy reference. Each recipe is associated with a scientific model or method and an interesting or amusing story designed to both entertain and enhance concept comprehension and retention.

Aimed at third- and fourth-year undergraduates in science and engineering, the text contains numerous examples in disciplines that will challenge students progressing in mathematics, physics, engineering, game theory, and physical chemistry. Computer Algebra Recipes: An Advanced Guide to Mathematical Modeling can serve as an effective computational science text, with a set of problems following each section of recipes to enable readers to apply and confirm their understanding. The book may also be used as a reference, for self-study, or as the basis of an on-line course.

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### Contents

 INTRODUCTION 1 B Computer Algebra Recipes 2 Boys Will Be Boys 3 D Maple Help 8 E How to Use This Text 9 THE APPETIZERS 11 PhasePlane Portraits 13 111 Romeo and Juliet 18
 Linear PDE Models Part 1 207 512 Play It Sam 211 513 Three Easy Pieces 215 514 Complex Yet Simple 220 52 Diffusion and Laplaces Equation Models 223 522 Aussie Barbecue 227 523 Bennys Solution 231 524 Hugo and the Atomic Bomb 236

 112 Theres No Damping Vectorias Romantic Heart 23 113 Van der Pols Limit Cycle 28 12 ThreeDimensional Autonomous Systems 32 121 The PeriodDoubling Route to Chaos 33 122 The Oregonator 40 123 Rösslers Strange Attractor 44 PhasePlane Analysis 47 211 Foxes Munch Rabbits 51 212 The Mona Lisa of Nonlinear Science 58 213 Mike Creates a HigherOrder Fixed Point 67 214 The Gnus and Sung of Erehwon 73 215 A Plethora of Points 78 22 ThreeDimensional Autonomous Systems 82 23 Numerical Solution of ODEs 88 231 Finite Difference Approximations 89 The Sequel 91 233 Glycolytic Oscillator 96 234 Fox Rabies Epidemic 101 THE ENTREES 107 Linear ODE Models 109 31 FirstOrder Models 110 312 Greg Arious Nerds Problem 115 32 SecondOrder Models 118 322 Meet Mr Laplace 121 323 Jennifers Formidable Series 126 33 Special Function Models 130 331 Jennifer Introduces a Special Family 131 332 The Vibrating Bungee Cord 137 333 Mathieus Spring 142 334 QuantumMechanical Tunneling 144 Nonlinear ODE Models 149 41 FirstOrder Models 150 412 The Struggle for Existence 152 413 The Bad Bird Equation 161 42 SecondOrder Models 164 422 Oh What Sounds We Hear 168 423 Vectoria Feels the Force and Hits the Bottle 175 424 Golf Is Such an Uplifting Experience 179 43 Variational Calculus Models 185 432 Queen Dido Wasnt a Dodo 191 433 The Human Fly Plans His Escape Route 195 434 This Would Be a Great Amusement Park Ride 201
 525 Hugo Prepares for His Job Interview 241 Linear PDE Models Part 2 247 612 Homers Jiggle Test 251 613 Vectorias Second Problem 254 614 Sound of Music? 257 62 Semiinfinite and Infinite Domains 261 622 Assignment Complete 263 623 Radioactive Contamination 266 624 Play It Sam Revisited 270 63 Numerical Simulation of PDEs 274 631 Freeing Excalibur the Numerical Way 275 632 Enjoy the KleinGordon Vibes 278 633 Vectorias Secret 281 THE DESSERTS 285 The Hunt for Solitons 287 71 The Graphical Hunt for Solitons 290 712 In Search of Bright Solitons 293 713 Can Three Solitons Live Together? 296 72 Analytic Soliton Solutions 299 721 Follow That Wave 300 722 Looking for a Kinky Solution 304 723 We Have Solitons 306 73 Simulating Soliton Collisions 308 732 Are Diamonds a Kinks Best Friend? 312 Nonlinear Diagnostic Tools 319 811 A Rattler Signals Chaos 320 812 Hamiltonian Chaos 323 82 The Power Spectrum 329 821 Frank N Steins Heartbeat 332 822 The Rattler Returns 334 83 The Bifurcation Diagram 337 831 Pitchforks and Other Bifurcations 338 84 The Lyapunov Exponent 342 841 Mr Lyapunov Agrees 343 85 Reconstructing an Attractor 345 851 Putting Humpty Dumpty Together Again 346 852 Random Is Random 349 853 Butterfly Reconstruction 351 Epilogue 354 Bibliography 355 Index 361 Copyright

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