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 , www.math.utoronto.ca/almut/
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.
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."
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.
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
(Date TBD in November, January, March; evening). Closed-book, closed-notes.
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 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. Wellposed problems
F: First order linear equations. The method of characteristics
Assignment 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; transportation equations.
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. Characteristic coordinates
F: Energy and causality
Assignment 4 (due October 13)
Week 5 (October 9-13)
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 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
F: Dissipation of energy. The method of reflections
Assignment 6 (due October 27)
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
F: Separation of variables
Assignment 7 (due November 3)
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?
F: Robin boundary conditions
Assignment 8 (due November 17)
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. Bessel's inequality
F: Orhonormal bases. Parseval's identity.
Week 10 (November 20-24)
Chapter 5 -- Fourier series, cont'd
F: First midterm test (10:10-12, in-class)
Week 11 (November 27 - December 1)
Chapter 6 -- Harmonic functions of two variables
Week 12 (December 4-8)
Chapter 6 -- Harmonic functions, cont'd
R: makeup Monday
F: study break
Fall exams, Christmas break (December 9 - January 2)
Week 13 (January 1-5)
Chapter 7 -- Harmonic functions in higher dimensions
Week 14 (January 8-13)
Sections 7.3 and 7.4 -- Green's function and Poisson kernel
Week 15 (January 15-19)
Chapter 9 -- Waves in higher dimensions
F: Family day (no class)
Week 16 (January 22-26)
Chapter 9 -- Waves in three-dimensional space
Week 17 (January 29 - February 2)
Chapter 9 -- Waves in the plane
Week 18 (February 5-9)
Chapter 10 -- Boundary-value problems in higher dimensions
Week 19 (February 12-16)
Chapter 10 -- The Dirichlet problem on the ball
Reading week (February 19-23)
Week 20 (February 26 - March 2)
Chapter 10 -- Spherical harmonics
Week 21 (March 5-9)
Chapter 11 -- Eigenvalues of the Laplacian
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)
Section 14.3 -- Variational problems
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