This is an introductory
course on PDEs that are central to the other CDT courses. The course emphasizes
rigorous treatment and analysis of PDEs through examples, representation
formulas, and properties that can be understood by using relatively elementary
mathematical tools and techniques. Topics will include: the heat equation, the
wave equation, Laplace's equation, conservation laws, and Hamilton-Jacobi
equations. Methods introduced through these topics will include: fundamental
solutions and Green's functions, energy principles, maximum principles,
separation of variables, Duhamel's principle, the method of characteristics,
transform methods, asymptotics, numerical methods,
and many more.
include undergraduate-level advanced calculus, linear algebra, and ODE, and
some exposure to complex analysis. Though this is an introductory course, it
will move quickly and require considerable mathematical maturity.
Learning Outcomes:Students will learn basic rigorous
treatment and analysis of partial differential equations with emphasis on
prototypical linear/nonlinear PDEs, as well as various techniques to represent
solutions of these PDEs.