21632: Introduction to Differential Equations
Fall 2018
Lectures: MWF 3:30  4:20 pm in Wean Hall room 7218
Professor: Robert Pego
 Office: 6127 Wean Hall.
Email: rpego AT cmu.edu
Phone: (412)2682553.
Stop by during my office hours (initially set as 11am12pm MWF),
catch me after class, or arrange another time.
Assignments:

Homework 1, due Friday Sept. 21
pdf

Homework 2, due Wednesday Oct. 17
pdf

Homework 3, due Wednesday Nov. 7
pdf

Homework 4, due Wednesday Dec. 5
pdf
Recommended Text:
 Partial Differential Equations, by L. C. Evans,
published by the American Mathematical Society.
(Graduate Studies in Mathematics, 2nd ed. 2010. The 1st ed. could be ok too.)
Potentially useful reference books available via SpringerLink:
 Partial Differential Equations in Action
by Sandro Salsa, Springer, 3rd ed. 2016.
(Very good selection of models and variety of methods, complementary to Evans.)
 Differential Equations and Dynamical Systems
by L. Perko, Springer, 3rd ed. 2001.
(Contains a decent detailed treatment of existence & uniqueness,
though alas only in the autonomous case.)
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.
 Overview, samples, glimpses. A tidbit of analysis for Poisson's equation.

Origins of important PDEs. Balance laws of physics.
Elastic bodies and the wave equation. Heat transfer and the heat equation.
Random walks.

The most important PDEs.
 Laplace's equation.
Fundamental solution, mean value property, Green's function,
energy estimates. Maximum principle, harmonic, subharmonic functions.
Representation formulae, uniqueness, regularity.

Heat equation.
Fundamental solution, energy method, Duhamel's principle.
Maximum principle, sub and supersolutions.
Representation formulae, uniqueness, backwards uniqueness.

Wave equation.
Spherical means, method of descent, Duhamel's principle, energy methods.
Representation formulae, uniqueness, domain of dependence.
 Essential theory of nonlinear dynamical systems (aka ODEs).
Initial value problem, solution via integral inequalities.
Flows. Dependence on parameters, continuation and blowup criteria.

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.
Makeup tests are only possible in the case of a documented medical
excuse, a universitysanctioned absence (e.g., participation in a varsity
sporting event), or a family emergency. Please contact me at the earliest time
possible to schedule a makeup.