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Theory of Partial Differential Equations
Math 7386
Louisiana State University
Fall Semester, 2012
Prof. Stephen Shipman
Place: Lockett 119
Time: 1:30-2:20 Monday, Wednesday, Friday
Office: Room 314 of Lockett Hall
Telephone: 225/578-1674
Email: shipman@math.lsu.edu
Office Hours: Monday 9:00-11:00, Wednesday 9:00-11:00, or by appointment
Course Description
The course will give an overview of important partial differential equations in mathematics and physics and pervading ideas and techniques as well as delve into the details of several key areas. It will cover linear equations, fundamental solutions, and the Sobolev functional spaces and their utility in solving elliptic boundary-value problems. Nonlinear first-order systems as the Hamilton-Jacobi equation and solution by reduction to ODEs (characteristics) will be covered. Other topics may include completely integrable systems, the Maxwell equations of electromagnetics, or nonlinear wave equations and shocks.
Prerequisite
The prerequisite is a good graduate analysis course.
Course Topics
- A zoological garden of partial differential equations. Interpretations, physical meaning, special solutions, dispersion relations, techniques of solution.
- Functional analysis preliminaries (Folland, Ch. 0). Distributions, the Fourier transform, and compact operators.
- Local existence of solutions. Solution of general first-order scalar equations by reduction to ODEs (method of characteristics); the general initial-value problem (Cauchy problem) and the Cauchy-Kovalevsky theorem; fundamental solutions.
- The Laplace equation ("simplest" elliptic equation). Boundary-value problems and eigenvalues.
- The diffusion (heat) equation ("simplest" parabolic equation). The heat kernel and initial-value problems
- The wave equation ("simplest" hyperbolic equation). Initial-value problems, the "light cone", harmonic means, the Radon transform.
- Mean-square (L2) Sobolev spaces, compact embedding theorems, boundary-trace theorems.
- Sobolev spaces in PDE theory: elliptic problems and weak formulations.
- Pseudo-differential operators.
- Weakly nonlinear equations.
- Perturbation methods.
- Fully nonlinear (wave) equations.
Literature
The main textbook for this course will be Introduction to Partial Differential Equations by G. Folland. Other sources will also be used.
There is a bibliography of relevant works below, with links to some PDF files of excerpts.
Assignments
I will assign problems periodically. At the end of the course, I may give a longer and somewhat comprehensive set of problems.
Students may discuss problems with each other and other people (including me, of course) and consult other literature; in fact students are encouraged to search the literature and discuss ideas. However, all work that is turned in must ultimately be that of the submitter alone. If a student receives aid on an assigned problem from discussions with people or other sources, he or she must begin from scratch in writing the solution so that the result is the product of his or her own understanding alone.
Evaluation
Evaluation of performance in the course is based on performance on the assignments
Notes and Problem Sets
Problems 1-4 Due Sept. 5
Problems 5-7 Due Sept. 21
Problems 8-11 Due Oct. 1
Problems 12-13 Due Oct. 10
Problems 14-17 Due Oct. 31
Bibliography
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E. A. Coddington and N. Levinson,
Theory of Ordinary Differential Equations,
McGraw-Hill, 1955.
Excerpt:
Dependence on initial conditions.
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Lawrence C. Evans,
Partial Differential Equations,
AMS Graduate Studies in Math., Vol. 19, 1998.
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Gerald B. Folland,
Introduction to Partial Differential Equations,
Second Edition, Princeton University Press, 1995.
Excerpt: Chapter 0.
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Fritz John,
Partial Differential Equations,
Springer-Verlag, Appl. Math. Sci. Vol. 1, Fourth Ed., 1982.
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G. B. Whitham,
Linear and Nonlinear Waves,
Wiley-Interscience, 1973.
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