UA Math | Contact Information

Introduction to the mathematical modeling of multi-scale phenomena

This module combines class discussions with numerical explorations and visualizations of experiments, to help students discover basic ideas of multiscale modeling.


We start with two important ideas that are illustrated on a variety of examples. First, phenomena that appear similar in nature can occur at completely different scales. Understanding them requires a universal theory, which is in some way scale-free. Second, most systems involve more than one scale, and are often described by models that are developed at a specific scale. Modeling complex systems across scales is a current challenge of multiscale analysis.

Scales and scalings

This short sections introduces simple concepts of dimensional analysis and scalings. It also illustrates the existence of self-similar solutions to partial differential equations.


The last section is devoted to the concept of diffusion at the microscopic and macroscopic levels. It includes applications of random walks to foraging behaviors and to bacterial motion.

Lecture notes

Mathematical modeling

© 2008 Arizona Board of Regents | Design by Joceline Lega