21-632: Introduction to Differential Equations

Fall 2018

Lectures: MWF 3:30 - 4:20 pm in Wean Hall room 7218

Professor: Robert Pego

Assignments:

Recommended Text:

Potentially useful reference books available via SpringerLink:

Course Outline:

The theory of differential equations is a vast and rather diffuse collection of topics and theories, which includes some of the most highly active areas of mathematical research. This course aims to introduce some of the basic prototypes, methods, and themes that indicate how DEs arise in mathematical modeling and a variety of important kinds of phenomena appearing in the analysis and behavior of solutions.

  1. Overview, samples, glimpses. A tidbit of analysis for Poisson's equation.
  2. Origins of important PDEs. Balance laws of physics. Elastic bodies and the wave equation. Heat transfer and the heat equation. Random walks.
  3. The most important PDEs.
    1. Laplace's equation. Fundamental solution, mean value property, Green's function, energy estimates. Maximum principle, harmonic, subharmonic functions. Representation formulae, uniqueness, regularity.
    2. Heat equation. Fundamental solution, energy method, Duhamel's principle. Maximum principle, sub and supersolutions. Representation formulae, uniqueness, backwards uniqueness.
    3. Wave equation. Spherical means, method of descent, Duhamel's principle, energy methods. Representation formulae, uniqueness, domain of dependence.
  4. Essential theory of nonlinear dynamical systems (aka ODEs). Initial value problem, solution via integral inequalities. Flows. Dependence on parameters, continuation and blow-up criteria.
  5. First order nonlinear PDE. Characteristics. Conservation laws and shocks. Lax's formula, development of singularities, weak solutions, entropy condition. Rarefaction waves, wave interactions.

Prerequisites:   Rigorous calculus including integration in N dimensions. Helpful (perhaps concurrently): measure theory, functional analysis, basic complex

Grading: Based on approximately 6 homework sets and a midterm and final exam (schedule TBA).

Homework will be posted online at http://www.math.cmu.edu/~bobpego/21632/ .

Academic integrity requires that your tests and homework solutions are your independent work and not copied from other sources. On homework you are encouraged, however, to discuss with others and consult other resources to improve your understanding.

Make-up tests are only possible in the case of a documented medical excuse, a university-sanctioned absence (e.g., participation in a varsity sporting event), or a family emergency. Please contact me at the earliest time possible to schedule a make-up.