620-331 Applied Partial Differential Equations

Either 620-231 or 620-233; and either 620-232 or 620-234.

This subject illustrates how partial differential equations (PDE's) of first and second order arise in mathematical modelling of the real world. It introduces basic techniques for solving these PDE's such as eigenfunction expansions, Green's functions, similarity solutions, method of images, and addresses general features of the solutions. The subject also covers certain topics in ordinary differential equations (ODE's). Topics covered include:

• First-order non-linear PDE's: characteristics, fans, shocks and applications.

• Classification of linear second order PDE's in two variables, canonical forms, initial and boundary conditions.

• The wave equation, d'Alembert's solution.

• Laplace's equation, Poisson's equation, harmonic functions, maximum and minimum principles.

• The heat equation, convective diffusion equation, Burgers' equation and the Hopf-Cole transformation.

• Sturm-Liouville equation, properties of eigenfunctions and eigenvalues.

• Series solutions of ODE's, ordinary points, regular singular points, Bessel and Legendre functions.

A 45-minute written test held mid-semester (either 0% or 20%); a 3-hour written examination in the examination period (80% or 100%). The relative weighting of the examination and the mid-semester test will be chosen so as to maximise the student's final mark.