The Nature of Mathematical Modeling

Neil Gershenfeld


Surveys the range of levels of description useful for the mathematical description of real and virtual worlds, including: analytical solutions and approximations for difference and differential equations; finite difference, finite element and cellular automata numerical models; and stochastic processes, nonlinear function fitting and observational model inference. Emphasis on efficient practical implementation of these ideas.

A bouncing ball in many languages

Java Applet with sound link
Java Application with more physics link
Open GL 3D version link
X-Windows version running from a Unix server link
Postscript file to print link
Max/MSP with real-time sound synthesis link

Ordinary Differential Equations

A simple Harmonic Oscillator : Euler, Runge-Kutta and Numerov methods link
Fourth-order Runge-Kutta algorithm and Adaptative Variable Stepper link

Partial Differential Equations

Waves : a damped string model in a Java Applet link
Diffusion : a model in one dimension link

Cellular Automata and Lattice Gases

HPP and FHP Models of Gases or Fluids link

Random Systems & Random Number Generators

Random Walker in one dimension link


Inverse Discrete Wavelet Transformation link

Optimization & Search

Simulated Annealing link
Genetic Algorithm link

Clustering and Density Estimation

Cluster-Weighted Modeling link

Final class project

Modelization of the Mathematics for a Parametric EQ (dropped idea) link
Modelization of the Noise of Acoustic Instruments link