MTH 622 : Partial Differential Equations - Winter 2022
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General information
General information
Instructor: Malgorzata Peszynska, Professor of Mathematics (Contact information including office hours on instructor's department website)
Class: Lecture: MWF 13:00-13:50pm, STAG 260.
Course information, syllabus and assignments: see CANVAS.
Schedule: [textbook] {assignments}

    == Week 1
  • 1/3/22: First day: syllabus and course overview. Discussion of last term vs course overview. Techniques discussed in Fall in MTH 621.
  • 1/5/22: Recall energy method in Dirichlet vs Neumann vs other boundary conditions for the heat equation. Worksheet on "which PDE might this be?".
    Worksheet 1/5/22.
  • 1/7/22: Dirichlet vs Neumann: Poincare-Friedrichs inequality for Dirichlet problem helps to show significant decay of energy for Dirichlet problem. Compare findings for stationary and nonstationary diffusion equations. [Chapters 6-7].
    HW1 due.

    == Week 2
  • 1/10/22: Analogy of solving PDEs such as heat equation and system of ODEs. Energy method, eigenvalues. Neumann boundary conditions.
    HW2 due 1/18/22.
  • 1/12/22: Robin boundary conditions. What they are, energy method and eigenvalues when seeking Fourier expansions. [Chapters 6-7].
  • 1/14/22: worksheet on the proof of Dirichlet theorem and Gibbs phenomenon.
    Workheet 2.

    == Week 3
  • 1/17/22: No class (MLK holiday).
  • 1/19/22: Working with first order evolution PDEs similarly to the systems of ODEs in an inner product space. Change of variables to the basis of eigenfunctions, solving the diagonal problem, and reverting back to the original variables.
  • 1/21/22: Worksheet on analogy between linear system of ODEs and heat equation. Finding adjoint PDE operators for Sturm-Liouville problem; symmetric boundary conditions.

    == Week 4
  • 1/24/22: Laplace and Poisson equations; introduction and review of relevant vector calculus. Divergence theorem! Harmonic functions. [Chapter 8.2-8.4]
  • 1/26/22: Properties of harmonic functions. Mean value property. Maximum principle. Uniqueness via energy method and maximum principle.
  • 1/28/22: Solving Laplace equation with Separation of Variables in special geometries. Polar coordinates for a circle, etc.

    == Week 5
  • 1/31/22: Fundamental solution and Green's function for Laplace equation/Poisson problem. Dimension d=1. [class notes]. [Chapter 9]
  • 2/2/22: Details of calculations of Green's function in d=1. State the form of fundamental solution in d=2,3.
    HW 3 due.
  • 2/4/22: Review for midterm.

    == Week 6:
  • 2/7/22: Midterm.
  • 2/9/22: Revisit fundamental solution \Phi and \bar{\Phi} in d=2,3. Representation formula with \bar{\Phi} for harmonic functions and non-harmonic functions. Also, define Green's function as the "fundamental solution that satisfies the boundary conditions". [Chapter 8.1; also, 9.1]
  • 2/11/22:

    == Week 7:
  • 2/14/22: Functionals (linear, continuous, bounded).
    HW4 due.
  • 2/16/22: Distributions. [Chapter 9.]
  • 2/18/22: Examples of finding distributional derivatives. Solving differential equations in the sense of distributions.

    == Week 8:
  • 2/21/22: Examples on solving DE in the sense of distributions. Definition of weak derivatives. Sobolev spaces. [Chapter 10].
  • 2/23/22: Weak derivatives as a special case of distributional derivatives. Space H1 as a Hilbert space.
    HW5 due.
  • 2/25/22: (Review) Normed spaces, inner product spaces. Banach and Hilbert spaces. Cauchy-Bunuakovsky-Schwarz inequality follows for every inner product space. How to prove that L2 is well defined as a normed/inner product spaces. Which normed spaces are inner product spaces.

    == Week 9:
  • 2/28/22: (online). Notes are here Lecture-2-28-22.pdf . Functionals and how to compute the first variation.
  • 3/2/22: Examples of minimization of functionals and Euler-Lagrange equations. Non-homogeneous Dirichlet boundary conditions. Essential vs natural boundary conditions.
  • 3/4/22: Riesz representation theorem: proof using minimization principle.

    == Week 10:
  • 3/7/22: Applyimg Riesz representation theorem to solution of PDEs. Working with the weak/variational formulation of Poisson's problem in H^1. Space H_0^1. ,br> HW6 due.
  • 3/9/22: Riesz representation theorem, other way. Review: wrap up elliptic BVP. Equivalence of (V) weak/variational approach and (M) minimization of Dirichlet functional. Classical (D) imples (V). (V) implies (D) if solution is smooth enough.
  • 3/11/22: Review: wrap up topics 621-622.
    Final exam 3/17/22.