MAT 351 University of Toronto 2017-18
MAT 351 Partial Differential Equations 2017-18
Almut Burchard, Instructor
How to reach me: Almut Burchard,
BA 6234, 8-3318.
- almut @ math.toronto.edu ,
- Lectures MWF 11:10-12noon, RW 142
- Tutorials F 10:10-11am, RW 142
- Office hours W 5:30-6:30
Teaching assistant: Afroditi Talidou,
atalidou @ math.toronto.edu
Content: This is a first course in Partial
Differential Equations, intended for Mathematics students with interests
in analysis, mathematical physics, geometry, and optimization.
The examples to be discussed include first-order equations,
harmonic functions, the diffusion equation, the wave equation,
Schrodinger's equation, and eigenvalue problems. In addition
to the classical representation formulas for the solutions
of these equations, there are techniques that apply more
broadly: the notion of well-posedness, the method of
characteristics, energy methods, maximum and comparison principles,
fundamental solutions, Green's functions, Duhamel's principle,
Fourier series, the min-max characterization of eigenvalues,
Bessel functions, spherical harmonics, and distributions.
Nonlinear phenomena such as shock waves and solitary waves are also introduced.
"Partial Differential Equations: An Introduction",
by Walter Strauss.
Second edition, Wiley 2008. ISBN 978-0-470-45056-7
(*the older edition will do*)
From the author's preface: "This book provides an introduction
to the basic properties of partial differential equations (PDEs) and to the
techniques that have proved useful in analyzing them.
My purpose is to provide for the student a broad
perspective on the subject, to illustrate the rich variety
of phenomena encompassed by it, and to impart a working knowledge of the
most important techniques of analysis of the solutions of the equations."
I will occasionally use other books, including the
graduate-level textbook by L. Craig Evans,
the classical monograph of Fritz John, and the essay
on spherical harmonics in the book on Special Functions
by Andrews, Askew, and Roy.
- 15% : weekly homework sets
Drop two. Assignments are collected in tutorial.
For each missed Friday, the value of a late assignment is cut in half.
- 45% : 3 term tests
(November 24, January 26, March 9, in class).
- 40% : Final examination
3 hours, compehensive.
Remarks. I expect that you
participate in lectures and tutorials. Use any occasion to discuss
problems and assignments among yourselves,
with Afroditi, with me, and anyone who is willing; feel free to
also consult other sources (books, wikipedia, ...). But please write up your
assignments in your own words, and be ready to defend them!
First lecture (September 8)
Overview -- What is a PDE?
Week 1 (September 11-15)
Chapter 1 -- Well-posed problems. Method of characteristics
- M: Deriving a PDE from a conservation law,
using the divergence theorem
- W: The Fundamental Lemma.
First order linear equations. The method of characteristicsAssignment 1 (due September 22)
Week 2 (September 18-22)
Section 1.2 -- First-order linear and quasilinear equations
- M: Existence and uniqueness of solutions
of initial-value problems for ODE
- W: Examples: Euler-Cauchy PDE for homogeneous functions;
- F: Burger's equation. The formation of shocks
Assignment 2 (due September 29)
Week 3 (September 25-29)
Sections 1.2 and 14.2 -- Shocks and rarefaction waves
- M: Rankine-Hugoniot jump condition
- W: Non-uniqueness of weak solutions.
Lax' entropy condition
- F: Second-order linear equations: elliptic, hyerbolic, parabolic.
Assignment 3 (due October 6)
Week 4 (October 2-6)
Chapter 2 -- Waves in one spatial dimension
- W: D'Alembert's formula
- M: Physical derivation of the wave equation.
- F: Energy and causality
(due October 13)
Week 5 (October 9-13)
Chapter 2 --- Heat (diffusion) equation
- M: Thanksgiving Holiday
Characteristics and characteristic coordinates
for hyperbolic PDE in two variables
(due October 20)
The maximum principle for the heat equation
Week 6 (October 16-20)
Chapter 2 -- More about diffusion.
Fundamental solution; boundary conditions
- M: The maximum principle in any dimension
- W: Solution formula for the heat equation
(due October 27)
- F: Dissipation of energy.
The method of reflections
Week 7 (October 23-27)
Chapter 3 -- Duhamel's principle for inhomogeneous equations
- M: Transport and diffusion
with source terms
- W: Waves with source terms
(due November 3)
- F: Separation of variables
Week 8 (October 30 - November 3)
Chapter 4 -- Boundary-value problems
- M: Eigenvalue problems. The role of boundary conditions
- W: General boundary conditions.
When is the second derivative operator Hermitioan?
Assignment 8 (due November
- F: Robin boundary conditions
Reading week (November 6-10)
Week 9 (November 13-17)
Chapter 5 -- Fourier series
- M: Hilbert spaces. Inner product and orthogonality
- W: Projection onto finite-dimensional subspaces.
- F: Orhonormal bases. Parseval's identity.
handout (no assignment)
Week 10 (November 20-24)
Chapter 5 -- Fourier series, cont'd
- M: Differentiating and integrating Fourier series
- W: Discussion of old tests, and review
- F: First midterm test
Old tests: 2009,
Week 11 (November 27 - December 1)
Chapter 5 -- Fourier series, cont'd
- M: Mean value property and strong maximum principle
- W: Discussion of midterm test
- F: Poisson's problem on the disc. Separation of variables
Assignment 9 (due
Week 12 (December 4-8)
Chapter 6 -- Harmonic functions
- M: Poisson's formula on the disc
- W: Proof of Poisson's formula.
Mean value property
Fall exams, Christmas break (December 9 - January 2)
Week 13 (January 1-5)
Chapter 6 -- Applications of Poisson's formula
- F: Completeness of the Fourier basis.
Strong maximum principle
Assignment 10 (due
Week 14 (January 8-13)
Chapter 7 -- Harmonic functions in higher dimensions
- M: Mean Value Property
- W: Green's identities. The Dirichlet principle
- F: The Fundamental solution of the Laplacian
Assignment 11 (due
Week 15 (January 15-19)
Chapter 7 -- Green's function and Poisson kernel
- M: Definition of the Green's function
- W: The Poisson problem
- F: Method of reflections: the half-space
Week 16 (January 22-26)
Section 14.3 -- Variational problems
- F: Second midterm test
Week 17 (January 29 - February 2)
Chapter 9 -- Waves in higher dimensions
Week 18 (February 5-9)
Chapter 9 -- Waves in 3-dimensional space and in the plane
Week 19 (February 12-16)
Chapter 10 -- Boundary-value problems in higher dimensions
Reading week (February 19-23)
Week 20 (February 26 - March 2)
Chapter 10 -- The Dirichlet problem on the ball.
Week 21 (March 5-9)
Chapter 11 -- Eigenvalues of the Laplacian
- F: Third midterm test
Week 22 (March 12-16)
Chapter 11 -- Eigenvalue asymptotics of the Laplacian
Week 23 (March 19-23)
Chapter 12 -- Distributions
Week 24 (March 26-30)
Chapter 12 -- The Fourier transform
- F: Good Friday (no class)
Last lecture (April 2)
Chapter 12 -- The Fourier transform (applications)
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