**Instructor:** Prof. Robert (Bob) Haslhofer

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

**Website:** http://www.math.toronto.edu/roberth/D46.html

**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