MTH 621 : Partial Differential Equations - Fall 2021
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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 162.
Course information: see CANVAS.
Schedule: [textbook] {assignments}
  • 9/22: Introductions. Classification of ODEs and PDEs. [Chap.1 plus Appendices].
  • 9/24: Picard-Lindel\"of theorem. Examples.
  • 9/27: Proof of Picard thm (E by iteration, U and dependence on IC with Gronwall lemma. Regularity by bootstrap). {Extra 00: first day survey}
  • 9/29: First order PDEs [Chap.3]. Solution "algorithm".
  • 10/1: First order PDEs: justification of formal steps; how to recognize characteristic data. {Extra 1}{HW1 due}
  • 10/4: Where do the first order PDEs come from: transport problem. Non-smooth solutions: do they make sense?
  • 10/6: Generalize the transport equation. Burgers equation. Can the characteristics cross?
  • 10/8: Calculation of shock speed in Burgers eqn and beyond (Rankine-Hugoniot condition) {HW2 due}
  • 10/11: Second order PDEs in $\R^2$; classification and change of variables to the canonical form. [Chap 2.1-2.3 and Chap 4]
  • 10/13: Solving wave equation; homogeneous case. [Handout on d'Alembert solution]
  • 10/15: Domain of dependence and domain of influence. Solution to the inhomogeneous wave equation, two ways.
  • 10/18: Continuous dependence on data. "Energy is constant" calculation. Where the wave eqn comes from. {HW3 due}
  • 10/20: Worksheet on spring/masses and stress/strain in longitudinal vibrations of a bar. Solving wave equation by diagonalizing a system of first order PDEs. Solving wave equation as a system of two first order PDEs.
  • 10/22: The source operator method for solving inhomogeneous problem via the solution of homogeneous pbm with some nontrivial initial data. Duhamel principle for first order evolution equation.
  • 10/25: Source operator method for second order evolution eqn. How to solve the wave equation in quarter plane (in particular, why do you need the odd extensions of initial data). {HW4 due}
  • 10/27: Review.
  • 10/29: Midterm, in class.
  • 11/1: Diffusion equation on R [Chapter 5]. Special solutions.
  • 11/3: Fundamental solution and properties (proof using strong assumptions on I.C.). Solving inhomogeneous diffusion equation using the fundamental solution.
  • 11/5: Proof using weak(er) assumptions on the original function. (Weak) maximum principle.
  • 11/8: Using maximum and minimum principles to establish uniquness of solutions to the diffusion eqn. Energy method; use it also to establish uniquness. Derivation of the models of diffusion (with Fick's law) and heat conduction (with Fourier's law).
  • 11/10: More on applications. A different proof of maximum principle using convex functions. Solving diffusion equation by separation of variables and with Fourier series [Chapter 6]
  • 11/12: F-series for the case with homogeneous boundary conditions. {HW5 due}.
  • 11/15: Continue F-series for the heat equation. Ortogonality of sines and cosines in L^2. Connect to the idea of a basis in inner product space.
  • 11/17: Examples of Fourier series. Towards convergence of the F-series (recall pointwise, uniform, and mean-square). What is needed for integration and differentiation of series. Worksheet on rates of convergence of F-coefficients.
  • 11/19: Convergence theorems (state). Why Fourier series solution is good for the heat equation (what makes its convergence work?).
  • 11/22: Examples (graphical, via computer simulations) how Fourier series converges. Gibbs phenomenon around the points of discontinuity. Towards Bessel's inequality and Parseval's identity for an orthonormal sequence of functions.
  • 11/24: Parseval's identity example. Solution to the wave equation using SoV. (THX special: compare waves and diffusions).
  • 11/26: No class (THX holiday)
  • 11/29: Solving inhomogeneous heat equation using the SOV and interpretation with the source operator. Solving wave equation via SOV.
  • 12/1: Why F-coefficients are optimal. Proofs: Bessel's inequality, Riemann-Lebesque lemma; Dirichlet theorem.. beginning. {HW6 due}
  • 12/3: Review (review worksheet).
  • Office hours: Tuesday, Dec 7 3:00-5:00.
  • Final exam on Wednesday, Dec.8 at noon.