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
(November 24, January 26, March 9, in class). 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.
handout (no assignment)

Week 10 (November 20-24)
Chapter 5 -- Fourier series, cont'd
M: Differentiating and integrating Fourier series
W: Discussion of old tests, and review
F: First midterm test (10:10-12, in-class).
Old tests: 2009, 2010, 2011, 2016 (one-hour).

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 4-8)
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 1-5)
Chapter 6 -- Applications of Poisson's formula
F: Completeness of the Fourier basis. Strong maximum principle
Assignment 10 (due January 12))

Week 14 (January 8-13)
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 15-19)
Chapter 7 -- Green's function and Poisson kernel
M: Definition of the Green's function
W: The Poisson problem
F: Method of reflections: Half-space and ball
handout (no assignment)

Week 16 (January 22-26)
Section 14.3 -- Variational problems
M: The Euler-Lagrange equation of a minimization problem
W: The Neumann boundary condition
F: Second midterm test (10:10-12, in-class).
Old tests: 2010, 2011, 2012, 2017.

Week 17 (January 29 - February 2)
Chapter 9 -- Waves in higher dimensions
M: Radial solutions. The Euler-Darboux 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 5-9)
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 12-16)
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 19-23)

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 5-9)
Chapter 10 -- Spherical Harmonics
M: The Laplacian in three-dimensional spherical coordinates
W: Harmonic polynomials in n dimensions
F: Third midterm test (10:10-12, Bahen 1210).
Old tests: 2012, 2017.
Also look at second term tests from 2010 and 2011.

Week 22 (March 12-16)
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 19-23)
Chapter 12 -- Distributions Assignment 16 (due Monday, April 2)

Week 24 (March 26-30)
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: 9-12am, 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