Instructor: Christian Ketterer
How to reach me: ckettere(at)math(dot)toronto(dot)edu
Delivery Mode: Online - Synchronous.
Time: Mo 14--15, We 12--14 (via Zoom, Zoom ID: https://utoronto.zoom.us/j/83830090753). The lecture will be recorded and then available online on MyMedia, the video platform of UofT.
Online Office Hours: We, 10am -- 12am (via Zoom, Zoom ID: https://utoronto.zoom.us/j/87881304580)
Course Description: 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. These methods are extremely useful in various mathematical fields far beyond the content of this lecture and belong to an analysist's modern tool box. Nonlinear phenomena such as shock waves and solitary waves are also introduced.
Prerequisites: MAT257Y1 or 85% in MAT237Y1, MAT267H1
Grading Scheme:
15%: weekly homework assignments
Drop two. Assignments have to be sent to the instructor.
45%: 3 term tests during class
Nov 4, Jan 27, Mar 24
40%: final exam
April 16
Weekly Homework Assignments:
Homework will posted on Mondays before class on Blackboard and this webpage, and will be due on the following Monday, before class.
Remarks: I expect that you participate actively in the lectures and the tutorials. That means to follow the lecture watchful and to ask questions. I encourage you to join me online during the office hours for questions and discussions with me and among yourselves. Please always write up your homework assignments by yourself and in your own words. It will be judged by the clarity of your mathematical presentation as well as by correctness and by completeness. Be ready to defend them in front of me or the TA.
Text:
W. Strauss: Partial differential equations. An introduction. Second edition. John Wiley & Sons, Ltd. Chichester, 2008. ISBN: 978-0-470-05456-7
R. Choksi: Partial Differential Equations: A First Course
| Assignment 0 | ||
| 1 | Mo, 9/14 | Basic notations and Definitions, Linear and nonlinear PDEs. [recording, slides] |
| 2 | We, 9/16 | Fundamental theorem of calculus of variations, Deriving PDEs via transformation formula and by divergence theorem. [recording, slides, whiteboard] |
| Assignment 1 | ||
| 3 | Mo, 9/21 | Initial and boundary conditions, well-posed problems. [recording, slides] |
| 4 | We, 9/23 | Introduction to method of characteristicsr, linear PDEs via method of characteristics. [recording, slides, whiteboard] |
| Assignment 2 | ||
| 5 | Mo, 9/28 | Examples, Transversality condition, Existence of local solutions. [recording, slides, whiteboard] |
| 6 | We, 9/30 | The viscid Burger's equation, the formation of shocks. [recording, slides, whiteboard] |
| Assignment 3 | ||
| 7 | Mo, 10/05 | Introduction linear second order PDEs, elliptic, parabolic and hyperbolic PDEs . [recording, slides, whiteboard] |
| 8 | We, 10/07 | Deriving the wave equation and its general solution, d'Alembert's formula. [recording, slides, whiteboard] |
| Assignment 3 1/2 | ||
| -- | Mo, 10/12 | Thanksgiving. |
| 9 | We, 10/14 | Causality, energy method, the diffusion equation in 1D: maximum principles. [recording, slides, whiteboard] |
| Assignment 5 | ||
| 10 | Mo, 10/19 | Fundamental solution for the diffusion equation on the real line. . [recording, slides, whiteboard] |
| 11 | We, 10/21 | Properties of the fundamental solution, distributions, uniqueness. [recording, slides, whiteboard] |
| Assignment 6 | ||
| 12 | Mo, 10/26 | Diffusion equation with source. [recording, slides, whiteboard] |
| 13 | We, 10/28 | Reflection principle for the diffusion equation, wave equation with source. [recording, slides, whiteboard] |
| 14 | Mo, 11/02 | Reflection principle for the wave equation, few more properties about diffusion [recording, slides, whiteboard] |
| We, 11/05 | midterm | |
| Reading Week | ||
| Assignment 7 | ||
| 15 | Mo, 11/16 | Separation of variables for the Wave equation: Dirichlet problem. [recording, slides, whiteboard] |
| 16 | We, 11/18 | Separation of variables: Neumann problem, diffusion equation. Introduction Fourier series. [recording, slides, whiteboard] |
| Assignment 8 | ||
| 17 | Mo, 11/23 | General Fourier series: orhtogonality, symmetric boundary conditions, notions of convergence [recording, slides, whiteboard] |
| 18 | We, 11/30 | Convergence of Fourier series [recording, slides, whiteboard] |
| 19 | Mo, 11/30 | Solution of the heat equation with Dirichlet boundary conditions and the circle [recording, slides, whiteboard] |
| 20 | We, 12/02 | Laplace and Poisson equation: Physical motivation, polar and spherical coordinates [recording, slides, whiteboard] |
| Assignment 9 | ||
| 21 | Mo, 12/07 | Harmonic functions, weak and strong maximum principle, mean value property[recording, slides, whiteboard] |
| 22 | We, 12/09 | Poisson formula for Laplace equation on a ball in 2D[recording, slides, whiteboard] |
| Assignment 10 | ||
| 23 | Mo, 01/11 | Application of the Poisson Formula: Mean square convergence of Fourier series[recording, slides] |
| 24 | We, 01/13 | Liouvill Theorem, Mean Value Property; Green's identities, Dirichlet principle[recording, part 1 recording, part 2, whiteboard] |
| Assignment 11 | ||
| 25 | Mo, 01/18 | General Representation formula, Fundamental solution[recording, whiteboard] |
| 26 | We, 01/20 | Green's function, Representation formula for Laplace and Poisson equation, Green function for the Half plane[recording, whiteboard] |
| 27 | Mo, 01/25 | Green function for a ball, higher dimensional Poisson formula, Additinal remarks[recording, whiteboard] |
| We, 01/27 | second term test | |
| 28 | Mo, 02/01 | Harnack inequality, Poisson kernel[recording, whiteboard] |
| 29 | We, 02/03 | Local estimates, Sobolev functions, weak solutions, regularity of weak solutions[recording, whiteboard] |
| Assignment 12 | ||
| Mo, 02/08 | No lecture | |
| 30 | We, 02/10 | Wave equation in higer dimensions, Causality, Preservation of energy, Kirchhoff's formula[recording, whiteboard] |
| Reading week | ||
| 31 | Mo, 02/22 | Additional remarks on weak solutions, Arzela-Ascoli theorem[recording, whiteboard] |
| 32 | We, 02/24 | Kirchhoff's formula (3D), method of spherical means, Darboux equations[recording, whiteboard] |
| Assignment 13 | ||
| 33 | Mo, 03/01 | Solution to the intial value problem for the wave equation[recording, whiteboard, whiteboard] |
| 34 | We, 03/03 | Equation with source term, relativistic geometry[recording, whiteboard] |
| Assignment 14 | ||
| 35 | Mo, 03/08 | Propagation of singularities of solutions to the wave equation; Schroedinger equation[recording, whiteboard] |
| 36 | We, 03/10 | Diffusion equation in higher dimensions, Harmonic oscilator, Hermite polynomials, (bounded) energy levels for the hydrogen atom[recording, whiteboard] |
| Assignment 15 | ||
| 37 | Mo, 03/15 | Fourier method, Eigenvalue problems on domains with Dirichlet boundary conditions[recording, whiteboard] |
| 38 | We, 03/17 | Vibrations of a drumhead: Bessel functions, Vibrations of ball in 3D: Legendre functions[recording, whiteboard] |
| 39 | Mo, 03/22 | More on Bessel and Legendre functions, Spherical harmonics[recording, whiteboard] |
| We, 03/24 | third term test | |
| Assignment 16 | ||
| 40 | Mo, 03/29 | More on spherical harmonics; the hydrogen atom revisited[recording, whiteboard] |
| 41 | We, 03/31 | General eigenvalue problems, Rayleigh quotient, Min-Max characterizaton of eigenvalues, Completeness[recording, whiteboard] |
| Assignment 17 | ||
| 42 | Mo, 04/05 | Assymptotic behaviour of eigenvalues, Weyl's theorem[recording, whiteboard] |
| 43 | We, 04/07 | Proof of Weyl's theorem; Fourier transform, properties[recording, whiteboard] |
| 44 | Mo, 04/12 | Applications of the Fourier transform[recording, whiteboard] |
| We, 04/16 | Final exam |