The Nature of Mathematical Modeling

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Cambridge University Press, 1999 - Mathematics - 344 pages
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This 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
Bibliography
330
Index
340
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