Math 232: Partial Differential Equations I.

Instructor: Andrea Bertozzi

Spring 2002, Duke University Dept. of Mathematics

Tues, Thurs, 3:50-5:05 pm in Physics 228E.

Introduction to the Theory of Partial Differential Equations

This will be a modern treatment of elementary PDEs. The course will focus mostly on analytical results with numerical computations used to illustrate the dynamics. Some numerical methods will be discussed in the context of their relationship to the mathematics.

Prerequisites: Either Math 231 or 204 is recommended. This course assumes having had a course like math 133 or 211 that covers things like Fourier Transform.

NB: undergraduates who have done well in 133 should consider taking this course.

Topics Covered

  • Some fundamental differences between PDEs and ODEs. Definition of well-posedness. Elementary function spaces and Fourier transform. Different basic types of equations (hyperbolic, elliptic, parabolic). Initial vs. boundary value problems. Issues with nonlinearity.
  • Introduction to parabolic equations. Review of heat equation on a line and on a periodic interval. Existence and uniqueness of solutions. The maximum principle. Higher order diffusion. Introduction to nonlinear parabolic equations, stability theory, and finite time singularities. Spectrum of the Laplacian and stability/monotonicity of single step numerical methods for parabolic equations.
  • Scalar hyperbolic conservation laws. Method of characteristics. Shocks. Effect of different types of diffusion. Upwind vs. downwind finite difference schemes and relationship to characteristics.
  • Elementary Elliptic PDE: Newtonian potential, Laplace's equation and smoothness of solutions, Weak solutions and Galerkin approximations. Finite element methods.
  • click here for a more detailed syllabus.

    Textbook: L. C. Evans, Partial Differential Equations, AMS Publications.

    Also: Haberman Chapter 4 "Vibrating strings and membranes", Strickwerda Chapter 2 "Analysis of finite difference schemes", and parts of Leveque's book.

    Handouts: (1) The heat equation: theory, stability, and monotonicity. (2) mollifiers and sobolev spaces, (3) singular integral operators, (4) "Contact line stability and undercomrpessive shocks in driven film flow" (PRL, Dec. 1998) This course will have a COMPUTATIONAL COMPONENT.