| Preface |
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ix | |
| About the Authors |
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xii | |
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A First Numerical Problem |
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1 | (17) |
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1 | (1) |
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2 | (1) |
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Design and Construction of a Working Program: Codes and Pseudocodes |
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3 | (8) |
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11 | (1) |
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12 | (2) |
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Programming Guidelines and Philosophy |
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14 | (4) |
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Realistic Projectile Motion |
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18 | (30) |
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Bicycle Racing: The Effect of Air Resistance |
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18 | (7) |
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Projectile Motion: The Trajectory of a Cannon Shell |
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25 | (6) |
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Baseball: Motion of a Batted Ball |
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31 | (5) |
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Throwing a Baseball: The Effects of Spin |
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36 | (8) |
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44 | (4) |
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Oscillatory Motion and Chaos |
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48 | (46) |
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48 | (6) |
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Making the Pendulum More Interesting: Adding Dissipation, Non-linearity, and a Driving Force |
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54 | (4) |
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Chaos in the Driven Nonlinear Pendulum |
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58 | (8) |
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Routes to Chaos: Period Doubling |
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66 | (4) |
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The Logistic Map: Why the Period Doubles |
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70 | (5) |
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75 | (7) |
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82 | (6) |
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Behavior in the Frequency Domain: Chaos and Noise |
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88 | (6) |
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94 | (35) |
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94 | (7) |
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The Inverse-Square Law and the Stability of Planetary Orbits |
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101 | (6) |
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Precession of the Perihelion of Mercury |
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107 | (6) |
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The Three-Body Problem and the Effect of Jupiter on Earth |
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113 | (5) |
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Resonances in the Solar System: Kirkwood Gaps and Planetary Rings |
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118 | (5) |
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Chaotic Tumbling of Hyperion |
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123 | (6) |
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129 | (27) |
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Electric Potentials and Fields: Laplace's Equation |
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129 | (14) |
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Potentials and Fields Near Electric Charges |
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143 | (5) |
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Magnetic Field Produced by a Current |
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148 | (3) |
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Magnetic Field of a Solenoid: Inside and Out |
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151 | (5) |
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156 | (25) |
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156 | (9) |
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Frequency Spectrum of Waves on a String |
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165 | (4) |
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Motion of a (Somewhat) Realistic String |
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169 | (5) |
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Waves on a String (Again): Spectral Methods |
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174 | (7) |
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181 | (54) |
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Why Perform Simulations of Random Processes? |
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181 | (2) |
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183 | (5) |
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188 | (7) |
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Random Walks and Diffusion |
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195 | (6) |
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Diffusion, Entropy, and the Arrow of Time |
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201 | (5) |
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206 | (6) |
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Fractal Dimensionalities of Curves |
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212 | (6) |
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218 | (11) |
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229 | (6) |
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Statistical Mechanics, Phase Transitions, and the Ising Model |
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235 | (35) |
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The Ising Model and Statistical Mechanics |
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235 | (4) |
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239 | (5) |
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244 | (2) |
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The Ising Model and Second-Order Phase Transitions |
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246 | (13) |
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First-Order Phase Transitions |
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259 | (5) |
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264 | (6) |
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270 | (33) |
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Introduction to the Method: Properties of a Dilute Gas |
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270 | (15) |
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285 | (9) |
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Equipartition and the Fermi-Pasta-Ulam Problem |
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294 | (9) |
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303 | (54) |
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Time-Independent Schrodinger Equation: Some Preliminaries |
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303 | (4) |
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One Dimension: Shooting and Matching Methods |
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307 | (16) |
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323 | (3) |
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326 | (7) |
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Time-Dependent Schrodinger Equation: Direct Solutions |
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333 | (12) |
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Time-Dependent Schrodinger Equation in Two Dimensions |
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345 | (4) |
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349 | (8) |
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Vibrations, Waves, and the Physics of Musical Instruments |
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357 | (32) |
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Plucking a String: Simulating a Guitar |
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357 | (5) |
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Striking a String: Pianos and Hammers |
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362 | (5) |
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Exciting a Vibrating System with Friction: Violins and Bows |
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367 | (5) |
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Vibrations of a Membrane: Normal Modes and Eigenvalue Problems |
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372 | (10) |
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382 | (7) |
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389 | (67) |
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389 | (16) |
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Earthquakes and Self-Organized Criticality |
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405 | (13) |
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Neural Networks and the Brain |
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418 | (18) |
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Real Neurons and Action Potentials |
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436 | (9) |
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445 | (11) |
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A Ordinary Differential Equations with Initial Values |
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456 | (13) |
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A.1 First-Order, Ordinary Differential Equations |
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456 | (4) |
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A.2 Second-Order, Ordinary Differential Equations |
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460 | (4) |
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A.3 Centered Difference Methods |
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464 | (3) |
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467 | (2) |
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B Root Finding and Optimization |
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469 | (10) |
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469 | (3) |
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472 | (1) |
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B.3 Stochastic Optimization |
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473 | (6) |
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479 | (14) |
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C.1 Theoretical Background |
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479 | (2) |
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C.2 Discrete Fourier Transform |
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481 | (2) |
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C.3 Fast Fourier Transform (FFT) |
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483 | (3) |
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C.4 Examples: Sampling Interval and Number of Data Points |
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486 | (2) |
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488 | (2) |
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490 | (3) |
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D Fitting Data to a Function |
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493 | (7) |
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493 | (1) |
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D.2 Method of Least Squares: Linear Regression for Two Variables |
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494 | (3) |
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D.3 Method of Least Squares: More General Cases |
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497 | (3) |
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500 | (12) |
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500 | (1) |
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E.2 Newton-Cotes Methods: Using Discrete Panels to Approximate an Integral |
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500 | (4) |
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E.3 Gaussian Quadrature: Beyond Classic Methods of Numerical Integration |
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504 | (2) |
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E.4 Monte Carlo Integration |
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506 | (6) |
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F Generation of Random Numbers |
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512 | (8) |
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F.1 Linear Congruential Generators |
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512 | (4) |
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F.2 Nonuniform Random Numbers |
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516 | (4) |
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G Statistical Tests of Hypotheses |
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520 | (7) |
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G.1 Central Limit Theorem and the Χ2 Distribution |
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521 | (2) |
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G.2 Χ2 Test of a Hypothesis |
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523 | (4) |
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527 | (14) |
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H.1 Solving A x = b, b, b ≠ 0 |
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528 | (7) |
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H.1.1 Gaussian Elimination |
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528 | (2) |
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H.1.2 Gauss-Jordan elimination |
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530 | (1) |
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531 | (2) |
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H.1.4 Relaxational method |
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533 | (2) |
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H.2 Eigenvalues and Eigenfunctions |
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535 | (6) |
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H.2.1 Approximate Solution of Eigensystems |
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537 | (4) |
| Index |
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541 | |