PARTIAL DIFFERENTIAL EQUATIONS (Fall 2018)
Description: This course is an introduction to the theory of partial differential equations (PDE) and their wide-ranging applications. PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics. Rather than study specific equations, this course will emphasize phenomena that are general among PDEs, and provide tools to not only solve PDEs, but to understand qualitative properties of their solutions. These qualitative tools must also be emphasized even in numerical solutions of PDE, since without this qualitative understanding one may use numerical methods that result in extremely inaccurate (or completely wrong) solutions even if one decreases the step size mesh size.
On completion of this course, the student should be able to:
- solve first-order linear/nonlinear PDEs and understand their local existence theory
- understand weak derivatives, Sobolev spaces and their application to solving PDEs
- understand properties of certain linear second-order PDEs such as regularity, energy estimates, maximum principles, and propagation
- solve and understand PDEs on domains with boundaries
- use Duhamel’s principle to solve certain inhomogeneous PDEs
- understand PDE existence theory via duality
- understand the connection between PDE solutions and minimization problems of certain energy functionals
Prerequisites: Students should have proficiency in multivariable calculus, linear algebra, and real analysis.