620-331 Applied Partial Differential Equations

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

This subject introduces the various types of partial differential equations and their methods for solution, and how they arise in physical problems. It develops the idea of characteristics and propagation of information, the need for shocks and the concepts of diffusion and flux. Students should develop the ability to solve wave equations using the methods of characteristics and shocks; to solve diffusion equations and their steady state versions; and to solve second order linear partial differential equations using various methods, including eigenfunction methods, integral transforms, and Green's functions. This subject demonstrates the description of many physical processes (for example traffic flow, sedimentation, head transfer, fluid flow) as partial differential equations; and it shows the power of various mathematical techniques to solve real-world problems.

First order partial differential equations topics include continuity equation, wave equation, method of characteristics and shocks; and applications from traffic flow and sedimentation. Second order wave equation topics include d'Alembert's solution, method of characteristics, nonlinear wave equations and dispersive waves. Diffusion and conduction topics include Fick's and Fourier's laws, similarity solutions, Stefan problems and Laplace and Poisson equations for steady-state problems. Second order linear partial differential equations topics include classification, eigenfunction methods, integral transforms and Green's 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.