MAT 351 University of Toronto 2016-17
 

MAT 351 Partial Differential Equations 2016-17

Almut Burchard, Instructor

How to reach me: Almut Burchard, BA 6234, 8-3318.
almut @ math.toronto.edu , www.math.utoronto.ca/almut/
Lectures MWF 11:10-12noon, SS 1072
Tutorials W/F 10:10-11am, BA 3116
Office hours by appointment (Mondays)
Teaching assistant: Afroditi Talidou, atalidou @ math.toronto.edu
Text:   "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."
We will also consult other sources, including the graduate-level textbook by L. Craig Evans and the classical monograph of Fritz John. I will post additional notes, as needed.
Evaluation:
20% : weekly homework sets (due Thursdays; drop two)
40% : 3 term tests (Wednesday November 9 in-class; January, March; closed-book, closed-notes)
40% : Final examination (comprehensive)
Remarks. Do discuss homework problems among yourselves, with Afroditi, 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. Initial and boundary conditions
F: First order linear equations. The method of characteristics
Assignment 1 (due September 19)
Week 2 (September 19-23)
Section 1.2 -- The method of characteristics
M: Examples of well-posed and ill-posed problems
W: Derivation of transport equations
F: Burger's equation. The formation of shocks
Assignment 2 (due September 26)
Week 3 (September 26-29)
Sections 1.2 and 14.2 -- Scalar conservation laws
M: Weak solutions. Rankine-Hugoniot condition
W: Non-uniqueness of weak solutions. Lax' entropy condition
F: Construction of entropy solutions
Assignment 3 (due October 3)
Week 4 (October 3-7)
Chapter 2 Waves and diffusion in one spatial dimension
M: D'Alembert's formula
W: Energy and causality
F: The diffusion equation in one dimension
Assignment 4 (due October 17)
Week 5 (October 10-14)
Chapter 2 More about the diffusion equation. Even and odd reflections
M: Thanksgiving Holiday
W: The fundamental solution
F: Solving the initial-value problem on the real line
Week 6 (October 17-21)
Chapter 2 More about the diffusion equation. Even and odd reflections
M: Dissipation of energy
W: The maximum principle in any dimension
F: Boundary conditions. Even and odd reflections
Assignment 5 (front, back) (due October 31)
Week 7 (October 24-28)
Inhomogeneous equations
M: Duhamel's principle. Transport and diffusion with source terms
W: Waves with source terms
F: Separation of variables
Assignment 6 (due October 31)
Week 8 (October 31 - November 4)
Bounday-value problems
M: Eigenvalue problems. The role of boundary conditions
W: Robin boundary conditions. Oscillation and growth.
F: A first look at Fourier series.
Assignment 7 (due November 9)
Week 9 (November 7-11)
Fourier series
M: Reading break (no lecture)
W: First midterm test (Topics, 2009, 2010, 2011).
F: Fourier series
Week 10 (November 14-17)
Fourier series
M: Hilbert spaces. Inner product and orthogonality
W: Bessel's inequality
F: Orhonormal bases. Parseval's identity.
Assignment 8 (due November 21)
Week 11 (November 21-25)
Harmonic functions
M: Fourier series, continued.
W: Laplace's equation
F: The Poisson kernel on the disc
Assignment 9 (due November 28)
Week 12 (November 28 - December 2)
Harmonic functions
M: Mean value property and strong maximum principle
W: Smoothness and unique continuation
Assignment 10 (due December 5)
Week 13 (December 5-7, January 6)
Harmonic functions
M: Liouville's theorem
W: Completeness of the Fourier basis
F: Green's identities and the Fundamental Solution of Laplace's equation
Week 14 (January 9-13)
Harmonic functions in higher dimensions
M: Mean Value Property and Dirichlet's Principle
W: Green's functions
f: Green's functions, continued
Assignment 11 (due January 16)
Week 15 (January 16-20)
Green's function and Poisson kernel
M: Symmetry. Solving Poisson's problem
W: Method of reflections: the half-space
f: Reflection at a sphere: The ball
Assignment 12 (due January 3)
Week 16 (January 23-29)
The wave equation in higher dimensions
M: Green's functions are negative, Poisson kernels positive
W: Second midterm test 5-7pm (310 Exam Centre) (Topics, 2010, 2011, 2012).
Morning lecture replaced by an office hour
F: The wave equation: Energy and causality.
Week 17 (January 30 - February 3)
The wave equation (cont'd)
M: Kirchhoff's formula (statement)
W: The method of spherical means. Darboux's equation
F: Proof of Kirchhoff's formula
Assignment 13 (due February 6)
Week 18 (February 6-10)
Wave equations (conclusion)
M: Poisson's formula for the two-dimensional wave equation
W: Propagation of singularities on characteristic surfaces
F: Overviews: The Schr\"odinger eigenvalue problem
Assignment 14 (due February 13)
Week 19 (February 13-17)
Boundary-value problems in higher dimensions
M: Hermite polynomials
W: The Hydrogen atom
F: Other radial potentials. Separation of angular and radial variables
Assignment 15 (due February 27)
Week 20 (February 20-24)
The Dirichlet problem on the ball
M: Bessel functions
W: Spherical harmonics
F: Completeness of the spherical harnonics on the sphere
Assignment 16 (due March 6)
Week 21 (March 6-10)
Eigenvalues of the Laplacian
M: Dirichlet eigenvalues. The Rayleigh quotient
W: The Neumann problem. Min-max, max-min, and the Rayleigh-Ritz principle
F: Comparison principles. Statement of Weyl's law
Assignment 17 (due March 13)
Week 22 (March 13-17)
Eigenvalue asymptotics of the Laplacian
M: Eigenvalues of rectangles
W: Dirichlet-Neumann bracketing. Proof of Weyl's law. Completeness of the eigenfunctions
Second midterm test 5-7pm (MS 2173, 1 King's College Circle) (Announcement, 2012).
F: Sturm-Liouville problems
Week 23 (March 20-24)
Distributions
M: Distributions and test functions
W: Distributional derivatives. Weak solutions of PDE
F: The delta distribution. Green's functions, revisited
Assignment 18 (due March 27)
Week 24 (March 27-31)
The Fourier transform
M: The Fourier integral. Symmetries
W: The Fourier transform for square integrable functions. Plancherel's identity
F: Some applications
Assignment 19 (due April 3)
Last lecture (April 3)
Variational problems
M: Euler-Lagrange equation
W: no class
Last Office hours (April 3)
(Monday April 10, 2:10-6pm, in BA 6234)
Final Exam (Tuesday April 11, 9-12am, in BN 2S)
Clara Benson Building, 320 Huron Street
(Old exams 2010, 2011, 2012).

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