620-331 Applied Partial Differential Equations

One of 620-231 or 620-233; and one of 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, shocks; applications from traffic flow, sedimentation. Second order wave equation topics include d'Alembert's solution, method of characteristics, nonlinear wave equations, dispersive waves. Diffusion and conduction topics include Fick's and Fourier's laws, similarity solutions, Stefan problems, Laplace and Poisson equations for steady-state problems. Second order linear partial differential equations topics include classification, eigenfunction methods, integral transforms, Green's functions.

Up to 24 pages of written assignments, a 3-hour end-of-semester written examination and class tests totalling not more than 1.5 hours.