Home page for Math 644, Partial Differential Equations, Fall 2007
Instructor: Charles L. Epstein
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The Course
Partial Differential Equations are the language mathematics provides to
describe most models in physics, chemistry, biology, economics,
engineering, etc. Math 644 is a one semester introduction to this vast
subject. After a short dicussion of ordinary differential equations and
first order PDEs we consider the 3 fundamental, second order
constant coefficient equations, the wave equation, heat equation and
Laplace equation. These equations provide a foundation for most
applications of PDE and the rationale for much of the development of
the subject over the past three centuries. In addition to
elementary methods coming from Calculus, we will employ tools from
functional analysis and Fourier analysis. A good understanding of
advanced calculus is an essential prerequisite. Some familiarity with
Real Analysis and Complex Analyis will be very useful.
Coursework: Problem sets
will be assigned every other week.
The textbook is Partial Differential Equations, Basic Theory
by Michael E. Taylor. We will not follow the text closely, but it
is a very good general reference. Additional material may be
taken from
the other sources, for example:
- Partial Differential
Equations by Fritz John
- Introduction to Partial Differential Equations
by Gerald Folland
- Partial Differential Equations by Jeffrey Rauch
- Foundations of Potential Theory by O. Kellogg
Syllabus
- Review of Ordinary Differential Equations
- First Order PDE, Hamiltonian Systems, Characteristic surfaces
- The Physical Origins of Partial Differential Equations
- Basic Fourier Analysis
- The Equations of Mathematical Physics on Euclidean Space
- Laplace's equation: The Maximum Principle, Dirichlet and
Neumann problems, well and ill-posed problems, regularity in Sobolev
spaces, connections to analytic functions in dimension 2
- The Heat Equation: The Maximum Principle, Cauchy's problem
- The Wave Equation: Energy estimates, finite propagation speed,
the Cauchy problem, the Radon transform
- Sobolev spaces in bounded domains
- Boundary value problems for Laplace's equation
- Fundamental solutions and boundary integral methods for Laplace's
equation
- The class meets on TTh from 3:00-4:30 in DRL, room
4C8.
- My office hour is 3:30 to 5:00 on Mondays. Contact me by e-mail
if you
have any questions.
- My office in the Math Department is 4E7 DRL, tel. 8-8476.
- email: cle@math.upenn.edu.
Send e-mail if you have a question or need to contact me.
Announcements
Problem Sets
- Problem set 1 is due September 25, 2007.
- Problem set 2 is due
October 9, 2007.
- Problem set 3 is due
Octobe 23, 2007.
- Problem set 4 is due
November 6, 2007.
- Problem set 5 is due
November 20, 2007.
- Problem set 6 is due
December 6, 2007.
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