APMA E6301-Analytic Methods for PDEs

Time: TR 1:10pm-2:25pm

Place: 1127 Mudd Building

Zoom Room: https://columbiauniversity.zoom.us/j/93232240116

Syllabus

Instructor: Dr. James Scott (he/him)

Affiliation: Applied Physics and Applied Mathematics, SEAS

Email: See "Home" on this website

Office Hours: M 1:00-2:00pm, R 2:30-3:30pm, or by appt

Office Location: 287D Engineering Terrace (accessible by going through the APAM offices)

Course Content: This course is an introduction to the theory of partial differential equations. The students to be introduced to the concepts and methods used to study the properties of solutions to PDEs that are popular and powerful in practice.

General topics: First-order quasilinear PDEs, Laplace's equation, the Heat equation, the Wave equation and Schr\"odinger's equation. Each of these are prototypes of broad classes of PDEs that appear across many areas of applied science.

Specific topics: Method of Characteristics, weak solutions. Potential theory (fundamental solutions, Green's functions,...), maximum principle. Hilbert space approach to solving boundary-value problems. Variational methods, energy methods. Regularity and speed of propagation.

Analytical tools/methods: Fourier analysis, functional analysis, asymptotic methods, oscillatory integrals.

Textbook: Partial Differential Equations. C. Lawrence Evans. 2nd Ed. Graduate Studies in Mathematics Volume 19, American Mathematical Society. ISBN: 978-0-8218-4974-3

Grades: The course grade is entirely based on homework assignments. Homework assignments will be assigned every two weeks (and one during the week of finals) and submitted via Gradescope (on Courseworks) by 11:59 pm on the due date. You are welcome to work together on homework. However, each student must turn in their own assignment. All grades are posted on Gradescope.

Uploading Files to Gradescope: See this PDF and this video.

Homework