Partial Differential Equations (NYU Paris, Spring 2022)

Partial Differential Equations are ubiquitous in science and engineering and are often useful in economics and finance. Because of their many connections to several mathematical subfields, they also represent a theoretical interest on their own (e.g. the spectral theory Laplace Beltrami operators as well as the scattering theory for wave equations are intimately tied to the study of automorphic forms in number theory). PDEs have been essential to the understanding of most physical phenomena such as sound, heat, electrostatics, electrodynamics, general relativity, fluid dynamics as well as the dynamics of financial markets (Black–Scholes).

As indicated by Sergiu Klainerman in his note “PDE as a unified subject” (Visions in Mathematics – Towards 2000), it however becomes increasingly difficult to view PDEs as a subject in its own right: “The deeper one digs into the study of one PDE, the more one has to take advantage of the particular features of the equation and therefore the corresponding results may make sense only as contributions to the particular field to which that PDE is relevant. Thus each major equation seems to generate isolated islands of mathematical activity. Moreover, a particular PDE may be studied from largely different points of view by an applied mathematician, a physicist, a geometer or an analyst”

As Klainerman later indicates, there are however a few powerful general ideas that reveal useful in a variety of contexts (some of the general principles listed by Klainerman include the notion of well/ill posedness, a priori estimates and continuity arguments, regularity theory, variational methods, Energy estimates,…).

In this course, we will study the general results as well as more particular notions that are useful to the study of the most important PDEs.

We will emphasize simple models (heat flow, vibrating strings and membranes). We will cover standard topics such as the method of separation of variables, the method of characteristics, Fourier series, orthogonal functions and the Fourier transform. Non homogeneous problems will be carefully introduced including Green functions for the Laplace, heat and wave equations.

Finally, for some of the equations, we will also study how to compute numerical solutions through finite differences and possibly the finite element method.

The assignments will include (but not be limited to) paper readings, pen and paper exercises as well as numerical simulations.

The class will follow the structure 

1. Lectures (introduction of the new material that will be needed during the lab sessions and for the assignements)

2. Programming (lab) sessions, (you have the opportunity to apply what you have learned during the lecture, and you can ask all the questions you want to make sure you understand everything before the assignement)

3. Assignments (You are given a new problem and you are evaluated on your ability to use the course material to solve this new problem)

Schedule and Classroom

Lecture: Monday/Wednesday, 7.15pm – 8.30pm (Paris Time), Room 410
Recitations: Monday (C03) 5.30am – 7.00pm (Paris time) . Room 410
Office hour: Tuesday 4.00pm (Paris time)

Assignments policy

Except if explicitely stated otherwise, assignments are due at the beginning of each class.

Current (temporary) version of the notes:   see below as well as the list of sections for the Final

Practice (theory) Questions for each exam can be found by clicking on those exams below

Exams: Enter password: Midterm

Enter password: Final Group 1

Enter password: Final Group 2

Enter password: Final Group 3

Exam : 60% of the grade (30% Midterm (Material), 30% Final(Material))

Assignments : 30 % of the grade (Tentative schedule below)

The Github page for the class will be hosted at https://github.com/acosse/MATH-UA-9263 and will essentially be used to post numerical verifications for the recitations.

Tentative schedule:

Legend: Lab sessions are in green, Homeworks are in red (right side of the table), dates related to the project are in orange.

• Lawrence C. Evans, Partial differential equations

• András Vasy, Partial Differential Equations: An Accessible Route through Theory and Applications

• Walter A. Strauss, Partial Differential Equations, an Introduction

• Sandro Salsa, Partial Differential Equations in Action: From Modelling to Theory

• Stanley J. Farlow, Partial Differential Equations for Scientists and Engineers

• Richard Haberman, Applied Partial Differential Equations, with Fourier Series and Boundary Values Problems,

Lab Sessions and programming policy

The lab sessions will require you to do some programming. It is strongly recommended to use python as it is more flexible and will be useful to you when moving to pytorch later on for more advanced machine learning methods requiring GPU processing.

Downloading and getting started with Python.

• Start by downloading anaconda: https://www.anaconda.com/download/#macos 
• If you don’t have a text editor yet, you can download sublime text (see interesting keyboard shortcuts here)