MAT 351 University of Toronto 201718
MAT 351 Partial Differential Equations 201718
Almut Burchard, Instructor
How to reach me: Almut Burchard,
BA 6234, 83318.
 almut @ math.toronto.edu ,
www.math.utoronto.ca/almut/
 Lectures MWF 11:1012noon, RW 142
 Tutorials F 10:1011am, RW 142
 Office hours W 5:306:30
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 firstorder 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 wellposedness, the method of
characteristics, energy methods, maximum and comparison principles,
fundamental solutions, Green's functions, Duhamel's principle,
Fourier series, the minmax characterization of eigenvalues,
Bessel functions, spherical harmonics, and distributions.
Nonlinear phenomena such as shock waves and solitary waves are also introduced.
Text:
"Partial Differential Equations: An Introduction",
by Walter Strauss.
Second edition, Wiley 2008. ISBN 9780470450567
(*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
graduatelevel 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.
Evaluation:
 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).
Closedbook, closednotes.
 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!
Tentative Schedule
First lecture (September 8)
Overview  What is a PDE?
(handout)
Week 1 (September 1115)
Chapter 1  Wellposed problems. Method of characteristics
 M: Deriving a PDE from a conservation law,
using the divergence theorem
 W: The Fundamental Lemma.
Wellposed problems
 F:
First order linear equations. The method of characteristics
Assignment 1 (due September 22)
Week 2 (September 1822)
Section 1.2  Firstorder linear and quasilinear equations
 M: Existence and uniqueness of solutions
of initialvalue problems for ODE
 W: Examples: EulerCauchy PDE for homogeneous functions;
transportation equations.
 F: Burger's equation. The formation of shocks
Assignment 2 (due September 29)
Week 3 (September 2529)
Sections 1.2 and 14.2  Shocks and rarefaction waves
 M: RankineHugoniot jump condition
 W: Nonuniqueness of weak solutions.
Lax' entropy condition
 F: Secondorder linear equations: elliptic, hyerbolic, parabolic.
Assignment 3 (due October 6)
Week 4 (October 26)
Chapter 2  Waves in one spatial dimension
 W: D'Alembert's formula
 M: Physical derivation of the wave equation.
Characteristic coordinates
 F: Energy and causality
Assignment 4
(due October 13)
Week 5 (October 913)
Chapter 2  Heat (diffusion) equation
 M: Thanksgiving Holiday
 W:
Characteristics and characteristic coordinates
for hyperbolic PDE in two variables
 F:
The maximum principle for the heat equation
Assignment 5
(due October 20)
Week 6 (October 1620)
Chapter 2  More about diffusion.
Fundamental solution; boundary conditions
 M: The maximum principle in any dimension
 W: Solution formula for the heat equation
 F: Dissipation of energy.
The method of reflections
Assignment 6
(due October 27)
Week 7 (October 2327)
Chapter 3  Duhamel's principle for inhomogeneous equations
 M: Transport and diffusion
with source terms
 W: Waves with source terms
 F: Separation of variables
Assignment 7
(due November 3)
Week 8 (October 30  November 3)
Chapter 4  Boundaryvalue problems
 M: Eigenvalue problems. The role of boundary conditions
 W: General boundary conditions.
When is the second derivative operator Hermitioan?
 F: Robin boundary conditions
Assignment 8 (due November
17)
Reading week (November 610)
Week 9 (November 1317)
Chapter 5  Fourier series
 M: Hilbert spaces. Inner product and orthogonality
 W: Projection onto finitedimensional subspaces.
Bessel's inequality
 F: Orhonormal bases. Parseval's identity.
handout (no assignment)
Week 10 (November 2024)
Chapter 5  Fourier series, cont'd
 M: Differentiating and integrating Fourier series
 W: Discussion of old tests, and review
 F: First midterm test
(10:1012, inclass).
Old tests: 2009,
2010,
2011,
2016 (onehour).
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
January 5))
Week 12 (December 48)
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 15)
Chapter 6  Applications of Poisson's formula
 F: Completeness of the Fourier basis.
Strong maximum principle
Assignment 10 (due
January 12))
Week 14 (January 813)
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
January 19))
Week 15 (January 1519)
Chapter 7  Green's function and Poisson kernel
 M: Definition of the Green's function
 W: The Poisson problem
 F: Method of reflections: Halfspace and ball
handout (no assignment)
Week 16 (January 2226)
Section 14.3  Variational problems
 M: The EulerLagrange equation of
a minimization problem
 W: The Neumann boundary condition
 F: Second midterm test
(10:1012, inclass).
Old tests: 2010,
2011,
2012,
2017.
Week 17 (January 29  February 2)
Chapter 9  Waves in higher dimensions
 M: Radial solutions. The EulerDarboux equation
 W: The method of spherical means.
Kirchhoff's formula for waves in three dimensions
 F: Huygens' principle
Assignment 12 (due February 9)
Week 18 (February 59)
Chapter 9  Waves in higher dimensions, cont'd
 M: Poisson's formula for planar waves
 W: Energy and causality
 F: Rays and characteristic surfaces
Assignment 13 (due February 16)
Week 19 (February 1216)
Chapter 9  Some eigenvalue problems
 M: Propagation of singularities along characteristic surfaces
 W: Harmonic Oscillator. Hermite functions
 F: Radial eigenfunctions of the Hydrogen atom
Assignment 14
(due March 2)
Reading week (February 1923)
Week 20 (February 26  March 2)
Chapter 10  The Dirichlet eigenvalue problem on ball.
Bessel functions, spherical harmonics
 M: Dirichlet eigenvalues problem on the disc.
Separation of variables
 W: Bessel's equation
 F: Separation of radial and angular variables
in higher dimensions
Week 21 (March 59)
Chapter 10  Spherical Harmonics
 M: The Laplacian in threedimensional spherical
coordinates
 W: Harmonic polynomials in n dimensions
 F: Third midterm test
(10:1012, Bahen 1210).
Old tests: 2012,
2017.
Also look at second term tests from
2010 and 2011.
Week 22 (March 1216)
Chapter 11  Eigenvalues of the Laplacian on a domain
 M: Spherical harmonics form a
complete orthonormal basis for L^2 (sphere)
 W: Short preview of Weyl's law
 F: The Rayleigh quotient
Assignment 15
(due March 23)
Week 23 (March 1923)
Chapter 12  Distributions
Assignment 16
(due Monday, April 2)
Week 24 (March 2630)
Chapter 12  The Fourier transform
 F: Good Friday (no class)
Last lecture (April 2)
Chapter 12  The Fourier transform
 M: Distributional solutions of PDE
 W: Solving the heat equation by Fourier transform
Final Exam (April 30)
 M: 912am, in EX 300 (Exam Center, 255 McCaul Street)
 (Old exams 2010,
2011,
2012,
2017).
The University of Toronto is committed to accessibility. If you
require accommodations for a disability, or have any accessibility
concerns about the course, the classroom or
course materials, please contact Accessibility Services as
soon as possible: disability.services@utoronto.ca, or
http://studentlife.utoronto.ca/accessibility