Mathematics 232: Introduction to Partial Differential Equations (Spring 2002)

Instructor

Andrea Bertozzi

Description

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. This course is a prerequisite for Math 281 (Hyperbolic Partial Differential Equations) and Math 282 (Elliptic Partial Differential Equations).

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.

    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.

    N.B.: Undergraduates who have done well in 133 should consider taking this course.

    Text(s)

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

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

    Course Website

    For more information see http://www.math.duke.edu/~bertozzi/PDEI/PDEI.html

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    Last modified: 18 October 2001