PARTIAL DIFFERENTIAL EQUATIONS

MATH 719

Classical theory of partial differential equations, together with an introduction to the modern theory based on functional analysis. Familiarity with basic measure theory (e.g.MATH 629or721) or concurrent registration inMATH 721is strongly recommended.

MATH 322

Sturm-Liouville theory; Fourier series, including mean convergence; initial and boundary value problems for linear second order partial differential equations, including separation of variables and eigenfunction expansions; fundamental solutions and Green's functions in multiple dimensions.

MATH 704

Derivation, nature and solution of canonical partial differential equations of applied mathematics. Conservation laws, advection, diffusion. First order PDEs, characteristics, shocks. Traffic flow, eikonal and Hamilton-Jacobi equations. Higher order PDEs: classification, Fourier analysis, well-posedness. Series solutions and integral transforms. Green's functions and distributions. Similarity solutions. Asymptotics of Fourier integrals. Laplace's method, stationary phase. Ship waves. Perturbation methods. Knowledge of undergraduate linear algebra, analysis and complex analysis is strongly recommended.

COMPSCI/MATH 714

Development of finite difference methods for hyperbolic, parabolic and elliptic partial differential equations. Analysis of accuracy and stability of difference schemes. Direct and iterative methods for solving linear systems. Introduction to finite volume methods. Applications from science and engineering. Students are strongly encouraged to have programming skills (e.g.COMP SCI 200) and some undergraduate numerical analysis (e.g.MATH/​COMP SCI  514orCOMP SCI 412), analysis and differential equations (e.g.MATH 322andMATH 521) and linear algebra (e.g.MATH 341).

MATH 619

A rigorous introduction to the theoretical underpinnings of the basic methods and techniques in the modern theory of PDEs. It is aimed at math majors, but will also be useful to some students in the sciences, engineering and economics who feel the need for a deeper understanding of the theory of PDEs. The emphasis is on the exposure to a number of different methods of solution of PDEs and their connection to physical phenomena modeled by the equations. The goals include both learning to solve some basic types of PDEs as well as to understand the motivation behind and inner workings of the techniques involved.