| |
| |
| Preface | |
| |
| |
| |
| Dimensional Analysis, Scaling, and Differential Equations | |
| |
| |
| |
| Dimensional Analysis | |
| |
| |
| |
| The Program of Applied Mathematics | |
| |
| |
| |
| Dimensional Methods | |
| |
| |
| |
| The Pi Theorem | |
| |
| |
| |
| Proof of the Pi Theorem | |
| |
| |
| |
| Scaling | |
| |
| |
| |
| Characteristic Scales | |
| |
| |
| |
| A Chemical Reactor Problem | |
| |
| |
| |
| The Projectile Problem | |
| |
| |
| |
| Differential Equations | |
| |
| |
| |
| Review of Elementary Methods | |
| |
| |
| |
| Stability and Bifurcation | |
| |
| |
| |
| Two-Dimensional Problems | |
| |
| |
| |
| Phase Plane Phenomena | |
| |
| |
| |
| Linear Systems | |
| |
| |
| |
| Nonlinear Systems | |
| |
| |
| |
| Bifurcation | |
| |
| |
| |
| Perturbation Methods | |
| |
| |
| |
| Regular Perturbation | |
| |
| |
| |
| Motion in a Resistive Medium | |
| |
| |
| |
| Nonlinear Oscillations | |
| |
| |
| |
| The Poincare-Lindstedt Method | |
| |
| |
| |
| Asymptotic Analysis | |
| |
| |
| |
| Singular Perturbation | |
| |
| |
| |
| Algebraic Equations | |
| |
| |
| |
| Differential Equations | |
| |
| |
| |
| Boundary Layers | |
| |
| |
| |
| Boundary Layer Analysis | |
| |
| |
| |
| Inner and Outer Approximations | |
| |
| |
| |
| Matching | |
| |
| |
| |
| Uniform Approximations | |
| |
| |
| |
| General Procedures | |
| |
| |
| |
| Initial Layers | |
| |
| |
| |
| Damped Spring-Mass System | |
| |
| |
| |
| Chemical Reaction Kinetics | |
| |
| |
| |
| The WKB Approximation | |
| |
| |
| |
| The Non-oscillatory Case | |
| |
| |
| |
| The Oscillatory Case | |
| |
| |
| |
| Asymptotic Expansion of Integrals | |
| |
| |
| |
| Laplace Integrals | |
| |
| |
| |
| Integration by Parts | |
| |
| |
| |
| Other Integrals | |
| |
| |
| |
| Calculus of Variations | |
| |
| |
| |
| Variational Problems | |
| |
| |
| |
| Functionals | |
| |
| |
| |
| Examples | |
| |
| |
| |
| Necessary Conditions for Extrema | |
| |
| |
| |
| Normed Linear Spaces | |
| |
| |
| |
| Derivatives of Functionals | |
| |
| |
| |
| Necessary Conditions | |
| |
| |
| |
| The Simplest Problem | |
| |
| |
| |
| The Euler Equation | |
| |
| |
| |
| Solved Examples | |
| |
| |
| |
| First Integrals | |
| |
| |
| |
| Generalizations | |
| |
| |
| |
| Higher Derivatives | |
| |
| |
| |
| Several Functions | |
| |
| |
| |
| Natural Boundary Conditions | |
| |
| |
| |
| The Canonical Formalism | |
| |
| |
| |
| Hamilton's Principle | |
| |
| |
| |
| Hamilton's Equations | |
| |
| |
| |
| The Inverse Problem | |
| |
| |
| |
| Isoperimetric Problems | |
| |
| |
| |
| Eigenvalue Problems, Integral Equations, and Green's Functions | |
| |
| |
| |
| Orthogonal Expansions | |
| |
| |
| |
| Orthogonality | |
| |
| |
| |
| Classical Fourier Series | |
| |
| |
| |
| Sturm-Liouville Problems | |
| |
| |
| |
| Integral Equations | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Volterra Equations | |
| |
| |
| |
| Fredholm Equations with Degenerate Kernels | |
| |
| |
| |
| Symmetric Kernels | |
| |
| |
| |
| Green's Functions | |
| |
| |
| |
| Inverses of Differential Operators | |
| |
| |
| |
| Physical Interpretation | |
| |
| |
| |
| Green's Function via Eigenfunctions | |
| |
| |
| |
| Distributions | |
| |
| |
| |
| Test Functions | |
| |
| |
| |
| Distributions | |
| |
| |
| |
| Distribution Solutions to Differential Equations | |
| |
| |
| |
| Discrete Models | |
| |
| |
| |
| One-Dimensional Models | |
| |
| |
| |
| Linear and Nonlinear Models | |
| |
| |
| |
| Equilibria, Stability, and Chaos | |
| |
| |
| |
| Systems of Difference Equations | |
| |
| |
| |
| Linear Models | |
| |
| |
| |
| Nonlinear Interactions | |
| |
| |
| |
| Stochastic Models | |
| |
| |
| |
| Elementary Probability | |
| |
| |
| |
| Stochastic Processes | |
| |
| |
| |
| Environmental and Demographic Models | |
| |
| |
| |
| Probability-Based Models | |
| |
| |
| |
| Markov Processes | |
| |
| |
| |
| Random Walks | |
| |
| |
| |
| The Poisson Process | |
| |
| |
| |
| Partial Differential Equations | |
| |
| |
| |
| Basic Concepts | |
| |
| |
| |
| Linearity and Superposition | |
| |
| |
| |
| Conservation Laws | |
| |
| |
| |
| One Dimension | |
| |
| |
| |
| Several Dimensions | |
| |
| |
| |
| Constitutive Relations | |
| |
| |
| |
| Probability and Diffusion | |
| |
| |
| |
| Boundary Conditions | |
| |
| |
| |
| Equilibrium Equations | |
| |
| |
| |
| Laplace's Equation | |
| |
| |
| |
| Basic Properties | |
| |
| |
| |
| Eigenfunction Expansions | |
| |
| |
| |
| Spectrum of the Laplacian | |
| |
| |
| |
| Evolution Problems | |
| |
| |
| |
| Integral Transforms | |
| |
| |
| |
| Laplace Transforms | |
| |
| |
| |
| Fourier Transforms | |
| |
| |
| |
| Stability of Solutions | |
| |
| |
| |
| Reaction-Diffusion Equations | |
| |
| |
| |
| Pattern Formation | |
| |
| |
| |
| Distributions | |
| |
| |
| |
| Elliptic Problems | |
| |
| |
| |
| Tempered Distributions | |
| |
| |
| |
| Diffusion Problems | |
| |
| |
| |
| Wave Phenomena | |
| |
| |
| |
| Wave Propagation | |
| |
| |
| |
| Waves | |
| |
| |
| |
| The Advection Equation | |
| |
| |
| |
| Nonlinear Waves | |
| |
| |
| |
| Nonlinear Advection | |
| |
| |
| |
| Traveling Wave Solutions | |
| |
| |
| |
| Conservation Laws | |
| |
| |
| |
| Quasi-linear Equations | |
| |
| |
| |
| Age-Structured Populations | |
| |
| |
| |
| The Wave Equation | |
| |
| |
| |
| The Acoustic Approximation | |
| |
| |
| |
| Solutions to the Wave Equation | |
| |
| |
| |
| Scattering and Inverse Problems | |
| |
| |
| |
| Mathematical Models of Continua | |
| |
| |
| |
| Kinematics | |
| |
| |
| |
| Mass Conservation | |
| |
| |
| |
| Momentum Conservation | |
| |
| |
| |
| Thermodynamics and Energy Conservation | |
| |
| |
| |
| Stress Waves in Solids | |
| |
| |
| |
| Gas Dynamics | |
| |
| |
| |
| Riemann's Method | |
| |
| |
| |
| The Rankine-Hugoniot Conditions | |
| |
| |
| |
| Fluid Motions in R[superscript 3] | |
| |
| |
| |
| Kinematics | |
| |
| |
| |
| Dynamics | |
| |
| |
| |
| Energy | |
| |
| |
| Index | |