Department of Mathematics
Spring 2004

APMA 935 - Mathematical Modelling

Week 00

Announcements:

05 January
syllabus (pdf)
info form (pdf), please bring completed form to first lecture

11 December
first lecture: 06 january, 9:30-11:30, SSC8666

Abstract:

22 October
abstract (link), (pdf)

A Mathematical View of Modern Science

Development of the theory of partial differential equations (PDEs) was one of the key influences behind the scientific revolution of the late 19th-century as it brought a powerful quantitative tool for the study of many problems in modern science. A complete applied mathematical study addresses two issues: translation of the scientific context with model equations (derivation & interpretation), and investigation of the mathematical properties of these equations (analysis, solution & computation). In this course, methods for the derivation, solution and computation of PDE models are discussed within the context of familiar examples from the physical sciences.

The lectures will discuss how the basic linear PDE trilogy, the diffusion, potential & wave equations, naturally arise in scientific theories such as probability, gravity and sound propagation. The analysis of these equations will also be revisited through the development of various solution techniques: eigenfunction expansions, Greens functions and integral transforms. Later lectures will present advanced examples involving systems of PDEs and nonlinearity.

Computer visualization will be an important accompaniment to the lectures and assigned work. Methods for numerical computing and graphics will be introduced through the use and modification of downloadable Matlab scripts.

The images below are illustrations of magnetic field lines, electron orbitals and polarized light waves. The mathematical theories associated with these phenomena involve the Poisson, Schrodinger and Maxwell equations.

Images & Interest:

  magnetic field lines
  electron orbitals
  polarized light wave