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/MATHUA9263 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.
Week #  date  Topic  Assignements 
Week 1  01/26  General Introduction (General presentation of the Transport, Laplace, Heat and Wave equations + derivation through the laws of physics), Classification of PDEs Lecture 1, Recitation 1, Solutions, (HandWritten) Solutions 

Part I : Four Important PDEs  
Week 2  01/31, 02/02  First order PDEs + method of characteristics
(including transport equation and Burgers’ equation) Intro Method of char., Recitation 6, ((partial) solutions) Recitation 7 ((partial) solutions) FanLike 1, Shocks + FanLike 2, Non Linear, Existence, Uniqueness, Recap + Rankine Hugoniot 
Assignment 1 
Week 3  02/07, 02/09 
Laplace equation + Poisson equation (fundamental solutions + Green function
+ harmonic functions, maximum principle, estimates on derivatives and Liouville’s theorem) Handwritten Notes (I): Harmonic functions+Mean value formulas Handwritten Notes (II): Harnacks+ Fund. Sol.+ Newt. Potential/Solution Poisson Handwritten Notes (III): Solution Poisson + Green function Lecture 6 Recitation 4, Recitation 5 , Solution R4 handwritten Solution R5 handwritten 
Readings 
Week 4  02/14, 02/16  The heat equation (Fundamental solution + homogeneous and non homogeneous Cauchy problems, Duhamel’s principle) HandWritten Notes Heat : Intro, Weak Max Principle, Fundamental Solution, Cauchy/Duhamel Lecture 2, Lecture 3, recitation 2, (Partial) Solutions recitation 2 notebook for questions 1 and 2 (Fourier) , Handwritten Solutions R2 
Readings
Assign. 2, Assig. 1 due 
Week 5  02/21, 02/23  The heat equation (properties of solutions, including uniqueness on bounded domains and strong maximum principle) HandWritten Notes Heat : Max Principle Cauchy Lecture 4, Lecture 5 Recitation 3, Handwritten solutions R3 14/15 
Readings 
Week 6  02/28, 03/02  The wave equation (Introduction, physical interpretation, solution by spherical means) Recitation 8 , Solution R8(part I) Handwritten Notes 1 , Handwritten Notes 2 
Assign. 2 due, Project choice MidTerm Revisions 
Week 7  03/07, 03/09  The wave equation (Kirchoff and Poisson formulas, Non homogeneous Cauchy problem, uniqueness and energy methods) Revision + Midterm Recitation 9 , Handwritten Notes 3 
Readings 
Week 8  03/14, 03/16  Spring Break  Readings 
Part II : Additional Solution Methods  
Week 9  03/21, 03/23  Separation of variables  Readings 
Week 10  03/28, 03/30  Similarity solutions  
Week 11  04/04, 04/06 
Transform methods (Part I, Introduction/reminders on the Fourier transform, convolutions, Fourier inversion, Plancherel’s Theorem) 
Readings Assign. 3 
Week 12  04/11, 04/13 
Transform methods (Part II, including Fourier, Radon and Laplace, application to linear constant coefficients PDEs, Bessel potentials, Heat + Wave + Schrödinger) 
Readings 
Part III : General Theory for Linear PDEs  
Week 13  04/18, 04/20  Introduction to Sobolev spaces+ second order elliptic equations (Existence of weak solutions + regularity, part I) 

Week 14  04/25, 04/27 
Second order elliptic equations (Existence of weak solutions + regularity, part II) + second order hyperbolic and parabolic PDEs (Existence of weak solutions + regularity)  
Part IV : Non Linear PDEs  
Week 15  05/02, 05/04  Nonlinear PDEs (Intro to calculus of variations + HamiltonJacobi) 

Week 16  05/09, 05/11  Revisions + Final Exams (part I)  
Week 17  05/16, 05/18  Final Exams (part II) 
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.