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Preface | |

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Ordinary Differential Equations in the Physical Sciences | |

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Introduction | |

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Definitions | |

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Exercises for Sec. 1.1 | |

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Graphical Solution of Initial-Value Problems | |

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Direction Fields; Existence and Uniqueness of Solutions | |

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Direction Fields for Second-Order ODEs: Phase-Space Portraits | |

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Exercises for Sec. 1.2 | |

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Analytic Solution of Initial-Value Problems via DSolve | |

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DSolve | |

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Exercises for Sec. 1.3 | |

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Numerical Solution of Initial-Value Problems | |

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NDSolve | |

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Error in Chaotic Systems | |

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Euler's Method | |

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The Predictor-Corrector Method of Order 2 | |

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Euler's Method for Systems of ODEs | |

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The Numerical N-Body Problem: An Introduction to Molecular Dynamics | |

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Exercises for Sec. 1.4 | |

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Boundary-Value Problems | |

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Introduction | |

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Numerical Solution of Boundary-Value Problems: The Shooting Method | |

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Exercises for Sec. 1.5 | |

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Linear ODEs | |

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The Principle of Superposition | |

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The General Solution to the Homogeneous Equation | |

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Linear Differential Operators and Linear Algebra | |

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Inhomogeneous Linear ODEs | |

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Exercises for Sec. 1.6 | |

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References | |

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Fourier Series and Transforms | |

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Fourier Representation of Periodic Functions | |

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Introduction | |

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Fourier Coefficients and Orthogonality Relations | |

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Triangle Wave | |

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Square Wave | |

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Uniform and Nonuniform Convergence | |

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Gibbs Phenomenon for the Square Wave | |

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Exponential Notation for Fourier Series | |

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Response of a Damped Oscillator to Periodic Forcing | |

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Fourier Analysis, Sound, and Hearing | |

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Exercises for Sec. 2.1 | |

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Fourier Representation of Functions Defined on a Finite Interval | |

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Periodic Extension of a Function | |

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Even Periodic Extension | |

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Odd Periodic Extension | |

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Solution of Boundary-Value Problems Using Fourier Series | |

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Exercises for Sec. 2.2 | |

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Fourier Transforms | |

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Fourier Representation of Functions on the Real Line | |

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Fourier sine and cosine Transforms | |

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Some Properties of Fourier Transforms | |

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The Dirac [delta]-Function | |

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Fast Fourier Transforms | |

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Response of a Damped Oscillator to General Forcing. Green's Function for the Oscillator | |

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Exercises for Sec. 2.3 | |

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Green's Functions | |

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Introduction | |

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Constructing the Green's Function from Homogeneous Solutions | |

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Discretized Green's Function I: Initial-Value Problems by Matrix Inversion | |

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Green's Function for Boundary-Value Problems | |

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Discretized Green's Functions II: Boundary-Value Problems by Matrix Inversion | |

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Exercises for Sec. 2.4 | |

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References | |

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Introduction to Linear Partial Differential Equations | |

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Separation of Variables and Fourier Series Methods in Solutions of the Wave and Heat Equations | |

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Derivation of the Wave Equation | |

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Solution of the Wave Equation Using Separation of Variables | |

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Derivation of the Heat Equation | |

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Solution of the Heat Equation Using Separation of Variables | |

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Exercises for Sec. 3.1 | |

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Laplace's Equation in Some Separable Geometries | |

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Existence and Uniqueness of the Solution | |

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Rectangular Geometry | |

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2D Cylindrical Geometry | |

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Spherical Geometry | |

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3D Cylindrical Geometry | |

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Exercises for Sec. 3.2 | |

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References | |

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Eigenmode Analysis | |

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Generalized Fourier Series | |

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Inner Products and Orthogonal Functions | |

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Series of Orthogonal Functions | |

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Eigenmodes of Hermitian Operators | |

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Eigenmodes of Non-Hermitian Operators | |

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Exercises for Sec. 4.1 | |

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Beyond Separation of Variables: The General Solution of the 1D Wave and Heat Equations | |

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Standard Form for the PDE | |

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Generalized Fourier Series Expansion for the Solution | |

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Exercises for Sec. 4.2 | |

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Poisson's Equation in Two and Three Dimensions | |

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Introduction. Uniqueness and Standard Form | |

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Green's Function | |

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Expansion of g and 0 in Eigenmodes of the Laplacian Operator | |

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Eigenmodes of [down triangle, open superscript 2] in Separable Geometries | |

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Exercises for Sec. 4.3 | |

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The Wave and Heat Equations in Two and Three Dimensions | |

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Oscillations of a Circular Drumhead | |

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Large-Scale Ocean Modes | |

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The Rate of Cooling of the Earth | |

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Exercises for Sec. 4.4 | |

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References | |

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Partial Differential Equations in Infinite Domains | |

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Fourier Transform Methods | |

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The Wave Equation in One Dimension | |

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Dispersion; Phase and Group Velocities | |

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Waves in Two and Three Dimensions | |

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Exercises for Sec. 5.1 | |

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The WKB Method | |

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WKB Analysis without Dispersion | |

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WKB with Dispersion: Geometrical Optics | |

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Exercises for Sec. 5.2 | |

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Wave Action (Electronic Version Only) | |

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The Eikonal Equation | |

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Conservation of Wave Action | |

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Exercises for Sec. 5.3 | |

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References | |

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Numerical Solution of Linear Partial Differential Equations | |

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The Galerkin Method | |

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Introduction | |

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Boundary-Value Problems | |

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Time-Dependent Problems | |

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Exercises for Sec. 6.1 | |

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Grid Methods | |

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Time-Dependent Problems | |

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Boundary-Value Problems | |

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Exercises for Sec. 6.2 | |

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Numerical Eigenmode Methods (Electronic Version Only) | |

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Introduction | |

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Grid-Method Eigenmodes | |

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Galerkin-Method Eigenmodes | |

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WKB Eigenmodes | |

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Exercises for Sec. 6.3 | |

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References | |

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Nonlinear Partial Differential Equations | |

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The Method of Characteristics for First-Order PDEs | |

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Characteristics | |

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Linear Cases | |

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Nonlinear Waves | |

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Exercises for Sec. 7.1 | |

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The KdV Equation | |

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Shallow-Water Waves with Dispersion | |

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Steady Solutions: Cnoidal Waves and Solitons | |

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Time-Dependent Solutions: The Galerkin Method | |

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Shock Waves: Burgers' Equation | |

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Exercises for Sec. 7.2 | |

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The Particle-in-Cell Method (Electronic Version Only) | |

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Galactic Dynamics | |

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Strategy of the PIC Method | |

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Leapfrog Method | |

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Force | |

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Examples | |

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Exercises for Sec. 7.3 | |

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References | |

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Introduction to Random Processes | |

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Random Walks | |

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Introduction | |

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The Statistics of Random Walks | |

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Exercises for Sec. 8.1 | |

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Thermal Equilibrium | |

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Random Walks with Arbitrary Steps | |

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Simulations | |

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Thermal Equilibrium | |

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Exercises for Sec. 8.2 | |

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The Rosenbluth-Teller-Metropolis Monte Carlo Method (Electronic Version Only) | |

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Theory | |

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Simulations | |

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Exercises for Sec. 8.3 | |

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References | |

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An Introduction to Mathematica (Electronic Version Only) | |

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Starting Mathematica | |

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Mathematica Calculations | |

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Arithmetic | |

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Exact vs. Approximate Results | |

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Some Intrinsic Functions | |

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Special Numbers | |

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Complex Arithmetic | |

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The Function N and Arbitrary-Precision Numbers | |

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Exercises for Sec. 9.2 | |

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The Mathematica Front End and Kernel | |

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Using Previous Results | |

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The % Symbol | |

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Variables | |

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Pallets and Keyboard Equivalents | |

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Lists, Vectors, and Matrices | |

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Defining Lists, Vectors, and Matrices | |

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Vectors and Matrix Operations | |

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Creating Lists, Vectors, and Matrices with the Table Command | |

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Operations on Lists | |

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Exercises for Sec. 9.5 | |

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Plotting Results | |

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The Plot Command | |

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The Show Command | |

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Plotting Several Curves on the Same Graph | |

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The ListPlot Function | |

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Parametric Plots | |

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3D Plots | |

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Animations | |

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Add-On Packages | |

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Exercises for Sec. 9.6 | |

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Help for Mathematica Users | |

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Computer Algebra | |

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Manipulating Expressions | |

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Replacement | |

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Defining Functions | |

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Applying Functions | |

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Delayed Evaluation of Functions | |

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Putting Conditions on Function Definitions | |

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Exercises for Sec. 9.8 | |

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Calculus | |

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Derivatives | |

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Power Series | |

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Integration | |

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Exercises for Sec. 9.9 | |

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Analytic Solution of Algebraic Equations | |

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Solve and NSolve | |

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Exercises for Sec. 9.10 | |

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Numerical Analysis | |

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Numerical Solution of Algebraic Equations | |

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Numerical Integration | |

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Interpolation | |

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Fitting | |

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Exercises for Sec. 9.11 | |

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Summary of Basic Mathematica Commands | |

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Elementary Functions | |

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Using Previous Results; Substitution and Defining Variables | |

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Lists, Tables, Vectors and Matrices | |

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Graphics | |

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Symbolic Mathematics | |

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References | |

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Finite-Differenced Derivatives | |

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Index | |