PARTIAL DIFFERENTIAL EQUATIONS
Prerequisites: Students should have proficiency in multivariable calculus, linear algebra, and real analysis.
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
Class Resources:
Lectures and problem sets require password
Lecture 1
Lecture 2
Lecture 3
Lecture 4
Lecture 5
Lecture 6
Lecture 7
Lecture 8
Lecture 9
Lecture 10
Lecture 11
Lecture 12
Lecture 13
Lecture 14
Lecture 15
Lecture 16
Lecture 17
Lecture 18
Lecture 19
Lecture 20
Lecture 21
Lecture 22
Lecture 23
Lecture 24
Lecture 25
Lecture 26
Problem Set 1
Problem Set 2
Problem Set 3
Problem Set 4
Problem Set 5
Problem Set 6
Problem Set 7
Problem Set 8
Problem Set 9
Problem Set 10
