## The Nature of Mathematical ModelingThis book first covers exact and approximate analytical techniques (ordinary differential and difference equations, partial differential equations, variational principles, stochastic processes); numerical methods (finite differences for ODE's and PDE's, finite elements, cellular automata); model inference based on observations (function fitting, data transforms, network architectures, search techniques, density estimation); as well as the special role of time in modeling (filtering and state estimation, hidden Markov processes, linear and nonlinear time series). Each of the topics in the book would be the worthy subject of a dedicated text, but only by presenting the material in this way is it possible to make so much material accessible to so many people. Each chapter presents a concise summary of the core results in an area, providing an orientation to what they can (and cannot) do, enough background to use them to solve typical problems, and pointers to access the literature for particular applications. |

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### Contents

Ordinary Differential and Difference Equations | 9 |

Partial Differential Equations | 24 |

Variational Principles | 34 |

Random Systems | 44 |

Ordinary Differential Equations | 67 |

Partial Differential Equations | 78 |

Finite Elements | 93 |

Cellular Automata and Lattice Gases | 102 |

Benchmarking | 257 |

Problem Solutions | 259 |

Partial Differential Equations | 266 |

Variational Principles | 269 |

Random Systems | 271 |

Ordinary Differential Equations | 276 |

Partial Differential Equations | 281 |

Finite Elements | 289 |

Function Fitting | 115 |

Transforms | 128 |

Architectures | 139 |

Optimization and Search | 156 |

Clustering and Density Estimation | 169 |

Filtering and State Estimation | 186 |

Linear and Nonlinear Time Series | 204 |

Graphical and Mathematical Software | 225 |

Problems | 249 |

Socket IO | 251 |

Parallel Programming | 254 |

Cellular Automata and Lattice Gases | 292 |

Function Fitting | 302 |

Transforms | 305 |

Architectures | 309 |

Optimization and Search | 315 |

Clustering and Density Estimation | 319 |

Filtering and State Estimation | 323 |

Linear and Nonlinear Time Series 325 | 327 |

330 | |

340 | |

### Common terms and phrases

algorithm apply approximation assume basis functions boundary conditions calculation called cellular automata chapter clusters coefficients complex coordinate covariance matrix data set defined degrees of freedom density estimation depends derivative diagonal diffusion equation dimension discrete eigenvalues embedding energy entropy error Euler example expansion exponential finite difference finite elements Fourier transform frequency Gaussian gives GLfloat global governing equations gradient gradient descent initial conditions input integral internal inverse iteration Kalman filter Laplace transform lattice gas linear measurements method minimize needed Neil Gershenfeld Neil Gershenfeld 9/1/97 noise nonlinear NPTS nsteps observable orthogonal oscillator output parameters partial differential equations particle polynomial possible PostScript predict printf probability distribution problem random number random variable requires result scalar scanf signal simple simulated annealing solution solved space square step stochastic techniques update variance vector wavelet xt\t zero