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