Sam Stechmann, Spring 2018:
(Stechmann, 2015) This course focuses on the derivation, nature, and solution of canonical partial differential equations (PDEs) of applied math, engineering, and physics. The course is divided into three parts: linear PDEs, nonlinear PDEs, and asymptotic methods. For linear PDEs, the emphasis will be on the canonical equations from each class of PDEs: the advection equation, Laplace's equation (elliptic), heat equation (parabolic), and wave equation (hyperbolic). Among the many applications of these equations are fluid dynamics, electromagnetism, and acoustics. A variety of solution techniques will be compared and contrasted, including separation of variables, Fourier transform, the method of characteristics, Green's functions, and similarity solutions. For the second part of the class, the focus shifts to nonlinear PDEs, including Hamilton-Jacobi equations and conservation laws. Shocks and weak solutions will be introduced in the context of Burgers' equation and traffic flow, and this will be followed by applications involving shallow water equations and compressible fluid flow. Asymptotic techniques will be used in many contexts to gain insight into the nature of solutions. In addition, multi-scale asymptotics will be introduced in the context of dispersive waves, where multiple space and time scales reveal the roles of dispersion and nonlinearity.