Computer Algebra Recipes

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CONTENTS ix 5.2.1 Freeing Excalibur . . . . . . . . . . . . . . . . . . . . . . . 223 5.2.2 Aussie Barbecue . . . . . . . . . . . . . . . . . . . . . . . 227 5.2.3 Benny's Solution . . . . . . . . . . . . . . . . . . . . . . . 231 5.2.4 Hugo and the Atomic Bomb . . . . . . . . . . . . . . . . . 236 5.2.5 Hugo Prepares for His Job Interview . . . . . . . . . . . . 241 6 Linear PDE Models. Part 2 247 6.1 Wave Equation Models . . . . . . . . . . . . . . . . . . . . . . . . 247 6.1.1 Vectoria Encounters Simon Legree . . . . . . . . . . . . . 247 6.1.2 Homer's Jiggle Test . . . . . . . . . . . . . . . . . . . . . 251 6.1.3 Vectoria's Second Problem . . . . . . . . . . . . . . . . . 254 6.1.4 Sound of Music? . . . . . . . . . . . . . . . . . . . . . . . 257 6.2 Semi-in¯nite and In¯nite Domains . . . . . . . . . . . . . . . . . 261 6.2.1 Vectoria's Fourth Problem . . . . . . . . . . . . . . . . . . 261 6.2.2 Assignment Complete! . . . . . . . . . . . . . . . . . . . . 263 6.2.3 Radioactive Contamination . . . . . . . . . . . . . . . . . 266 6.2.4 \Play It, Sam" Revisited . . . . . . . . . . . . . . . . . . 270 6.3 Numerical Simulation of PDEs . . . . . . . . . . . . . . . . . . . 274 6.3.1 Freeing Excalibur the Numerical Way . . . . . . . . . . . 275 6.3.2 Enjoy the Klein{Gordon Vibes . . . . . . . . . . . . . . . 278 6.3.3 Vectoria's Secret . . . . . . . . . . . . . . . . . . . . . . . 281 III THE DESSERTS 285 7 The Hunt for Solitons 287 7.1 The Graphical Hunt for Solitons . . . . . . . . . . . . . . . . . . 290 7.1.1 Of Kinks and Antikinks . . . . . . . . . . . . . . . . . . . 290 7.1.2 In Search of Bright Solitons . . . . . . . . . . . . . . . . . 293 7.1.3 Can Three Solitons Live Together? . . . . . . . . . . . . . 296 7.2 Analytic Soliton Solutions . . . . . . . . . . . . . . . . . . . . . . 299 7.2.1 Follow That Wave! . . . . . . . . . . . . . . . . . . . . . . 300 7.2.2 Looking for a Kinky Solution . . . . . . . . . . . . . . . . 304 7.2.3 We Have Solitons! . . . . . . . . . . . . . . . . . . . . . . 306 7.3 Simulating Soliton Collisions . . . . . . . . . . . . . . . . . . . . 308 7.3.1 To Be or Not to Be a Soliton . . . . . . . . . . . . . . . . 308 7.3.2 Are Diamonds a Kink's Best Friend? . . . . . . . . . . . . 312 8 Nonlinear Diagnostic Tools 319 8.1 The Poincare Section . . . . . . . . . . . . . . . . . . . . . . . . . 319 8.1.1 A Rattler Signals Chaos . . . . . . . . . . . . . . . . . . . 320 8.1.2 Hamiltonian Chaos . . . . . . . . . . . . . . . . . . . . . . 323 8.2 The Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 329 8.2.1 Frank N. Stein's Heartbeat . . . . . . . . . . . . . . . . . 332 8.2.2 The Rattler Returns . . . . . . . . . . . . . . . . . . . . . 334
x CONTENTS 8.3 The Bifurcation Diagram . . . . . . . . . . . . . . . . . . . . . . 337 8.3.1 Pitchforks and Other Bifurcations . . . . . . . . . . . . . 338 8.4 The Lyapunov Exponent . . . . . . . . . . . . . . . . . . . . . . . 342 8.4.1 Mr. Lyapunov Agrees . . . . . . . . . . . . . . . . . . . . 343 8.5 Reconstructing an Attractor . . . . . . . . . . . . . . . . . . . . . 345 8.5.1 Putting Humpty Dumpty Together Again . . . . . . . . . 346 8.5.2 Random Is Random . . . . . . . . . . . . . . . . . . . . . 349 8.5.3 Butter°y Reconstruction . . . . . . . . . . . . . . . . . . . 351 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Bibliography 355 Index 361
Page 1 and 2: Richard H. Enns George C. McGuire C
Page 3 and 4: PREFACE A computer algebra system (
Page 5: viii CONTENTS 2.3.1 Finite Di®eren
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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52 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 135 Th
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3.3. SPECIAL FUNCTION MODELS 137 3.
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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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