The Nature of Mathematical Modeling

Description

Cover Title page Preface 1 Introduction 1.1 Selected References Part One: Analytical Models 2 Ordinary Differential and Difference Equations 2.1 Linear Differentiai Equations 2.2 Systems of Differentiai Equations and Normal Modes 2.3 Laplace Transforms 2.4 Perturbation Expansions 2.5 Discrete Time Equations 2.6 z-Transforms 2.7 Selected References 2.8 Problems 3 Partial Differential Equations 3.1 The Origin of Partial Differentiai Equations 3.2 Linear Partial Differentiai Equations 3.3 Separation of Variables 3.3.1 Rectangular Coordinates 3.3.2 Cylindrical Coordinates 3.3.3 Spherical Coordinates 3.4 Transform Techniques 3.5 Selected References 3.6 Problems 4 Variational Principles 4.1 Variational Calculus 4.1.1 Euler's Equation 4.1.2 Integrals and Missing Variables 4.1.3 Constraints and Lagrange Multipliers 4.2 Variational Problems 4.2.1 Optics: Fermat's Principle 4.2.2 Analytical Mechanics: Hamilton's Principle 4.2.3 Symmetry; Noether's Theorem 4.3 Rigid Body Motion 4.4 Selected References 4.5 Problems 5 Random Systems 5.1 Random Variables 5.1.1 Joint Distributions 5.1.2 Characteristic Functions 5.2 Stochastic Processes 5.2.1 Distribution Evolution Equations 5.2.2 Stochastic Differentiai Equations 5.3 Random Number Generators 5.3.1 Linear Congruential 5.3.2 Linear Feedback 5.4 Selected References 5.5 Problems Part Two: Numerical Models 6 Finite Differences: Ordinary Differential Equations 6.1 Numerical Approximations 6.2 Runge-Kutta Methods 6.3 Beyond Runge-Kutta 6.4 Selected References 6.5 Problems 7 Finite Differences: Partial Differential Equations 7.1 Hyperbolic Equations: Waves 7.2 Parabolic Equations: Diffusion 7.3 Elliptic Equations: Boundary Values 7.4 Selected References 7.5 Problems 8 Finite Elements 8.1 Weighted Residuals 8.2 Rayleigh-Ritz Variational Methods 8.3 Selected References 8.4 Problems 9 Cellular Automata and Lattice Gases 9.1 Lattice Gases and Fluids 9.2 Cellular Automata and Computing 9.3 Selected References 9.4 Problems Part Three: Observational Models 10 Function Fitting 10.1 Model Estimation 10.2 Least Squares 10.3 Linear Least Squares 10.3.1 Singular Value Decomposition 10.4 Nonlinear Least Squares 10.4.1 Levenberg-Marquardt Method 10.5 Estimation, Fisher Information, and the Cramér-Rao Inequality 10.6 Selected References 10.7 Problems 11 Transforms 11.1 Orthogonal Transforms 11.2 Fourier Transforms 11.3 Wavelets 11.4 Principal Components 11.5 Selected References 11.6 Problems 12 Architectures 12.1 Polynomials 12.1.1 Padé Approximants 12.1.2 Splines 12.2 Orthogonal Functions 12.3 Radial Basis Functions 12.4 Overfitting 12.5 Curse of Dimensionality 12.6 Neural Networks 12.6.1 Back Propagation 12.7 Regularization 12.8 Selected References 12.9 Problems 13 Optimization and Search 13.1 Multidimensional Search 13.2 Local Minima 13.3 Simulated Annealing 13.4 Genetic Algorithms 13.5 The Blessing of Dimensionality 13.6 Selected References 13.7 Problems 14 Clustering and Density Estimation 14.1 Histogramming, Sorting, and Trees 14.2 Fitting Densities 14.3 Mixture Density Estimation and Expectation-Maximization 14.4 Cluster-Weighted Modeling 14.5 Selected References 14.6 Problems 15 Filtering and State Estimation 15.1 Matched Filters 15.2 Wiener Filters 15.3 Kalman Filters 15.4 Nonlinearity and Entrainment 15.5 Hidden Markov Models 15.6 Selected References 15.7 Problems 16 Linear and Nonlinear Time Series 16.1 Linear Time Series 16.2 The Breakdown of Linear Systems Theory 16.3 State-Space Reconstruction 16.4 Characterization 16.4.1 Dimensions 16.4.2 Lyapunov Exponents 16.4.3 Entropies 16.5 Forecasting 16.6 Selected References 16.7 Problems Appendix 1 Graphical and Mathematical Software A1.1 Math Packages A1.1.1 Programming Environments Al.l.2 Interactive Environments Al.2 Graphics Tools A1.2.1 Postscript Al.2.2 X Windows Al.2.3 OpenGL Al.2.4 Java Al.3 Problems Appendix 2 Network Programming A2.1 OSI, TCP/IP, and All That A2.2 Socket 1/0 A2.3 Parallel Programming Appendix 3 Benchmarking Appendix 4 Problem Solutions A4.1 Introduction A4.2 Ordinary Differentiai and Difference Equations A4.3 Partial Differentiai Equations A4.4 Variational Principles A4.5 Random Systems A4.6 Finite Differences: Ordinary Differentiai Equations A4.7 Finite Differences: Partial Differentiai Equations A4.8 Finite Elements A4.9 Cellular Automata and Lattice Gases A4.10 Function Fitting A4.11 Transforms A4.12 Architectures A4.13 Optimization and Search A4.14 Clustering and Density Estimation A4.15 Filtering and State Estimation A4.16 Linear and Nonlinear Time Series Bibliography Index