APM 351 University of Toronto 2011-12

# APM 351 Differential Equations of Mathematical Physics, 2011-12

## Almut Burchard, Instructor

How to reach me: Almut Burchard, 215 Huron, # 1024, 6-4174.
almut @ math , www.math.utoronto.ca/almut/
Lectures MWF 9:10-10am, SS 1074 .
Office hours Wed 2:30-3:30, 5:15-6:30pm
Teaching assistant: Kyle Thompson, kyle.thompson @ utoronto.ca .
Text:   "Partial Differential Equatoons: An Introduction", by Walter Strauss.
Second edition, John Wiley 2008.ISBN 978-0-470-45056-7
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),     Co-requisites: MAT 337/357 (Real Analysis)
Evaluation:
20% : weekly homework sets (due Thursdays; drop two)
40% : 3 term tests (November, January, March; closed-book, closed-notes)
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 12-16)
Chapter 1 -- What is a PDE? Well-posed 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 29-23)
Chapter 1 -- Where do PDE come from?
M: Flows, vibrations, and diffusions.
W: The divergence theorem.
F: Second-order equations.
Assignment 2 (due September 29)
Week 3 (September 26-30)
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 3-7)
Chapter 2 -- Diffusion
M: The diffusion equation: Maximum principle.
F: Construction of the fundamental solution.
W: Using the fundamental solution to solve initial-value problems.
Assignment 4 (due October 13)
Week 5 (October 10-14)
Chapter 3 -- Reflections and sources
M: Thanksgiving holiday
W: Energy methods for the diffusion equation. Diffusion and waves on the half-line.
F: Reflections of waves.
Assignment 5 (due October 20)
Week 6 (October 17-21)
Chapter 4 -- Separation of Variables
M: Duhamel's formula for solving inhomogeneous equations.
W: More about the heat equation. Non-uniqueness, infinite speed of propagation.
F: Separation of variables.
Assignment 6 (due October 27)
Week 7 (October 24-28)
Chapter 5 -- Fourier series
M: Sine, cosine, and exponential series. The function space L^2.
W: Orthogonality. Self-adjointness of \$-d_x^2\$ with suitable boundary conditions.
F: No lecture
Assignment 7 (due November 10)
Week 8 (October 31-November 4)
Chapter 5 -- Fourier series, cont'd
M: Orthogonal projections in L^2
W: Bessel's inequality and Parseval's identity.
F: Question hour (in-class)
Friday November 4, 5-7pm Test 1 ( Announcement, 2009, 2010)
Week 9 (November 7-11)
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 14-18)
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 21-25)
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 28-December 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 3-7)
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 9-13)
Chapter 7 -- Green's functions
M: The Green's function for the half-space 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 16-20)
Excursion -- Spherical harmonics
M: Harmonic polynomials. The recursion formula
W: Spherical harmonics.
F: Question hour (in-class)
Friday January 20, 5-7pm Test 2 (GB 404 Galbraith Building) (Announcement, 2010, 2011)
Week 15 (January 23-27)
Chapter 9 -- Waves in higher dimensions
M: Energy and causality. Space-time 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 30-February 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 6-10)
Chapter 10 -- Boundary-value 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 13-17)
Chapter 10 -- Boundary-value 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)
Week 19 (February 27-March 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 5-9 )
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 12-16)
Chapter 11 -- Eigenvalue problems for the Laplacian.
M: Proof of completeness
W: Excursion: The Euler-Lagrange equation of a variational problem. Existence of solutions (sketch)
F: Question hour (in-class)
Friday March 16 Test 3, 5-7pm, GB 404 (Galbraith Building)
Week 22 (March 19-23)
Chapter 12 -- Distributions and transforms
M: Sturm-Liousville probems. Orthogonality and completeness
W: Distributions
F: Weak solutions
Assignment 19 (due March 29)
Week 23 (March 26-30)
Chapter 12 -- The Fourier transform
M: The Fourier integral
W: Parseval's identity and construction of the L^2-Fourier transform
F: The method of characteristics, revisited
Assignment 20 (due April 5)
Week 24 (April 1-5)
Chapter 14 -- Nonlinear PDEs
M: Burger's equation: Weak solutions, shock waves
W: Nonuniqueness; entropy solutions
Last Handout
FINAL EXAM (April 11)
W: 9-12am, SS1083
(Topics, 2010, 2011).

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