APM 351 University of Toronto 201112
APM 351 Differential Equations of Mathematical Physics, 201112
Almut Burchard, Instructor
How to reach me: Almut Burchard, 215 Huron, # 1024, 64174.
 almut @ math ,
www.math.utoronto.ca/almut/
 Lectures MWF 9:1010am, SS 1074 .
 Office hours Wed 2:303:30, 5:156:30pm
Teaching assistant: Kyle Thompson,
kyle.thompson @ utoronto.ca .
Text:
"Partial Differential Equatoons: An Introduction",
by Walter Strauss.
Second edition, John Wiley 2008.ISBN 9780470450567

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."
We will also consult other sources, including the textbook
of Pinchover and Rubinstein (that was used last year), and
the classical monograph of Fritz John. I will post
additional notes on the web, as needed.
Prerequisites: Multivariable Calculus (with proofs),
MAT 267 (Ordinary Differential Equations),
Corequisites:
MAT 337/357 (Real Analysis)
Evaluation:
 20% : weekly homework sets (due Thursdays;
drop two)
 40% : 3 term tests
(November, January, March; closedbook, closednotes)
 40% : Final examination (comprehensive)
Remarks.
Do discuss lectures and homework problems among yourselves
and with me, and consult other sources.
But please write up your assignments yourself, in
your own words, and be ready to defend them!
Schedule:
Week 1 (September 1216)
Chapter 1  What is a PDE? Wellposed problems. Method of characteristics.
 M: Overview: What is a PDE?
 W: Wellposedness. First order linear equations.
 F: Initial and boundary conditions. The method
of characteristics for first order linear equations.
Assignment 1 (due September 22)
Week 2 (September 2923)
Chapter 1  Where do PDE come from?
 M: Flows, vibrations, and diffusions.
 W: The divergence theorem.
 F: Secondorder equations.
Assignment 2 (due September 29)
Week 3 (September 2630)
Chapter 2  Waves
 M: The wave equation. Characteristic coordinates.
 W: D'Alembert's formula. Examples.
 F: Causality and energy.
Assignment 3 (due October 6)
Week 4 (October 37)
Chapter 2  Diffusion
 M: The diffusion equation: Maximum principle.
 F: Construction of the fundamental solution.
 W: Using the fundamental solution to solve initialvalue problems.
Assignment 4 (due October 13)
Week 5 (October 1014)
Chapter 3  Reflections and sources
 M: Thanksgiving holiday
 W: Energy methods for the diffusion equation.
Diffusion and waves on the halfline.
 F: Reflections of waves.
Assignment 5 (due October 20)
Week 6 (October 1721)
Chapter 4  Separation of Variables
 M: Duhamel's formula for solving inhomogeneous equations.
 W: More about the heat equation. Nonuniqueness, infinite
speed of propagation.
 F: Separation of variables.
Assignment 6 (due October 27)
Week 7 (October 2428)
Chapter 5  Fourier series
 M: Sine, cosine, and exponential series.
The function space L^2.
 W: Orthogonality. Selfadjointness of $d_x^2$
with suitable boundary conditions.
 F: No lecture
Assignment 7
(due November 10)
Week 8 (October 31November 4)
Chapter 5  Fourier series, cont'd
 M: Orthogonal projections in L^2
 W: Bessel's inequality and Parseval's identity.
 F: Question hour (inclass)
Friday November 4, 57pm Test 1 (
Announcement,
2009,
2010)
Week 9 (November 711)
Chapter 5  Fourier series
 M: Fall Break
 W: Mean square convergence and Completeness.
Uniform convergence.
 F: Applications of Fourier series.
Assignment 8 (due November 17)
Week 10 (November 1418)
Chapter 5  Fourier series
 M: Hilbert spaces
 W: Pointwise vs. uniform convergence.
 F: Applications of Fourier series.
Assignment 9 (due November 24)
Week 11 (November 2125)
Chapter 6  Laplace's equation and Poisson's equation
 M: Harmonic functions in one and two dimensions.
Polar coordinates.
 W: Poisson's formula for the disc
 F: The strong maximum principle
Assignment 10 (due December 1)
Week 12 (November 28December 2)
Chapter 7  Laplace's equation in three dimensions.
 M: Green's first identity. The mean value property
for harmonic functions.
 W: The strong maximum principle and the Dirichlet principle
 F: Green' second identity.
Week 13 (December 37)
Chapter 7  Green's functions
 M: The fundamental solution
 W: The Green's function for a domain
Assignment 11 (due January 12)
Week 14 (January 913)
Chapter 7  Green's functions
 M: The Green's function for the halfspace and ball
 W: Poisson's formula in R^3 and R^2
 F: Properties of harmonic functions. Dirichlet's principle
Assignment 12 (due January 19)
Week 14 (January 1620)
Excursion  Spherical harmonics
 M: Harmonic polynomials. The recursion formula
 W: Spherical harmonics.
 F: Question hour (inclass)
Friday January 20, 57pm Test 2
(GB 404 Galbraith Building)
(Announcement,
2010,
2011)
Week 15 (January 2327)
Chapter 9  Waves in higher dimensions

M: Energy and causality. Spacetime and the light cone.
 W: No lecture, no office hours.
 F: Radial solutions. Spherical means, Darboux' equation
Assignment 13 (due February 2)
Week 16 (January 30February 3)
Chapter 9  Waves in higher dimensions
 M: Three dimensions: Kirchhoff's formula. Huygens' principle
 W: Two dimensions:
Hadamard's method of descent and
Poisson's formula.
 F: Rays, singularities, and sources. Duhamel's formula.
Assignment 14 (due February 9)
Week 17 (February 610)
Chapter 10  Boundaryvalue problems in higher dimensions
 M: The diffusion and Schrödinger equations
 W: The ground state of the hydrogen atom
 F: Vibrations of a membrane
Assignment 15 (canceled)
Week 18 (February 1317)
Chapter 10  Boundaryvalue problems in higher dimensions
 M: Solid vibrations of a ball. Nodes. Bessel functions
 W: Spherical harmonics, revisited. Orthogonality relations
 F: The hydrogen atom
Assignment 16 (due March 1)
Reading Week (February 2024)
Week 19 (February 27March 2)
Chapter 10  Eigenvalue problems for Schröodinger operators
with radia potentials
 M: The Laplacian in spherical coordinates
 W: Separation of spherical variables
 F: No lecture
Assignment 17 (due March 8)
Week 20 (March 59 )
Chapter 11  General eigenvalue problems
 M: Rayleigh's principle
 W: Orthogonality of eigenfunctions
 F: Weyl's law
Assignment 18 (due March 15): Strauss
p. 304 #5; p. 309 #8b, 9.
Week 21 (March 1216)
Chapter 11  Eigenvalue problems for the Laplacian.
 M: Proof of completeness
 W: Excursion: The EulerLagrange equation
of a variational problem. Existence of solutions (sketch)
 F: Question hour (inclass)
Friday March 16
Test 3, 57pm, GB 404 (Galbraith Building)
Week 22 (March 1923)
Chapter 12  Distributions and transforms
 M: SturmLiousville probems. Orthogonality and completeness
 W: Distributions
 F: Weak solutions
Assignment 19 (due March 29)
Week 23 (March 2630)
Chapter 12  The Fourier transform
 M: The Fourier integral
 W: Parseval's identity and construction of the
L^2Fourier transform
 F: The method of characteristics, revisited
Assignment 20 (due April 5)
Week 24 (April 15)
Chapter 14  Nonlinear PDEs
 M: Burger's equation: Weak solutions, shock waves
 W: Nonuniqueness; entropy solutions
Last Handout
FINAL EXAM (April 11)
 W: 912am, SS1083
(Topics,
2010,
2011).
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