MAT D46 Partial Differential Equations (Winter 2022)

This course provides a first introduction to partial differential equations as they arise in physics, engineering, finance, optimization and geometry. It is also meant to introduce beautiful ideas and techniques which are part of most analysts' bag of tools.

Instructor: Prof. Robert (Bob) Haslhofer

Contact Information: roberth(at)math(dot)toronto(dot)edu


Lectures: Monday 2--5 in HL B110

Office hours: Monday 11--12 in IC477

Prerequisites: [MATA37 or MATA36] with grade of at least B+, and [MATB41], and [MATB44]

Textbook: W. Strauss: Partial Differential Equations - An Introduction, Wiley

Secondary references:
F. John: Partial Differential Equations (Applied Mathematical Sciences), Springer
L. Evans: Partial Differential Equations (only chapter 2), AMS

Midterm Exam (in class): Mon, Feb 7 from 3pm--5pm

Final Exam: Fr, Apr 29 from 2pm--4pm in IC204

Homework problem sets (please submit via crowdmark):
HW1: Problem Set 1 HW2: Problem Set 2 HW3: Problem Set 3 HW4: Problem Set 4 HW5: Problem Set 5

Grading Scheme: Active participation 10%, Homework 30%, Midterm 25%, Final exam 35%

Topics to be covered:

What is a PDE: Linear and nonlinear equations, first and second order equations, static equations and evolution equations, initial and boundary conditions, well-posed problems.

The transport equation: Derivation of the equation in physics and optimization, method of characteristics, conservation laws, the phenomenon of shocks.

Laplace equation and Poisson equation: Physical motivation, boundary value problem, separation of variables, the fundamental solution, mean value formula, maximum principle, smoothness of solutions.

The diffusion equation: derivation as a model for diffusion in physics, finance and geometry. Scaling properties, initial value problem, fundamental solution, solution via Fourier analysis, Duhamel's principle, maximum principle, smoothing effect.

The wave equation: derivation of the equation, the Cauchy problem, solution in one and two spatial dimensions, the principle of causality, energy conservation, formation of singularities.

Tentative schedule:

Jan 10
What is a PDE, First order linear PDEs (Strauss 1.1, 1.2) notes

Jan 17
Where PDEs come from, Initial and Boundary conditions (Strauss 1.3, 1.4) notes

Jan 24
The Wave Equation, Causality and Energy (Strauss 2.1, 2.2) notes

Jan 31
The Diffusion Equation, Diffusion on the whole line (Strauss 2.3, 2.4) notes

Feb 7
Comparison of Waves and Diffusion (Strauss 2.5), Midterm exam notes

Feb 14
Boundary value problems (Strauss 4.1, 4.2) notes

Feb 28
Fourier Series (Strauss 5.1, 5.2) notes

Mar 7
Harmonic functions (Strauss 6.1, 6.2) notes

Mar 14
Poisson's formula, mean value property, strong maximum principle (Strauss 6.3, 7.1) notes

Mar 21
Green's identities and Green's functions (Strauss 7.1 -- 7.4) notes

Mar 28
Maxwell's equations and Navier-Stokes equations (Strauss 13.1, 13.2) notes

April 4
Nonlinear PDEs: Lecture notes on nonlinear PDEs