MATHEMATICS 423:01 -- TENTATIVE SYLLABUS
Spring 2001
 Inst: B. Walsh --- Hill Center 728 (Busch) --- (732) 445-3733

This syllabus is entirely tentative, and is subject to change: this version contains changes from its predecessor (Ch. 3 is out and Ch. 7 is in). It may undergo further modification as the semester progresses, but a current version will always be available from the Section Home Page. Students who browse this location should be aware that this syllabus is not 100% determinative as currently posted.

This preliminary version of the syllabus contains no selections of textbook problems.   Once the course begins,  there will be a  list of problems that will be adjusted throughout the semester. Students are responsible for the content of all assigned problems; this content may occur in examination questions.

Links to Additional notes may appear on the section home page.  Some of these notes may contain homework problems; these will be indicated in the syllabus.

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Textbook: Partial Differential Equations, an introduction, by Walter A. Strauss
 
Chapter/Section Topics
Chapter 1 Where PDEs come from
1.1 What is a Partial Differential Equation?
1.2 First-order Linear Equations (Solution in the constant-coefficient case; the variable-coefficient case and characteristic curves.
1.3 Flows, Vibrations and Diffusions (Derivations of PDEs in various physical situations; e.g., the vibrating string, the vibrating drumhead, diffusion, heat flow, hydrogen atom).
1.4 Initial and boundary conditions (the Dirichlet, Neumann and Robin conditions and their significance for the vibrating string and diffusion equations.  Conditions at infinity.)
1.5 Well- (and ill-)Posed Problems.
1.6 Types of second-order equations.
Chapter 2 Waves and Diffusion
2.1 The Wave Equation (D'Alembert's solution on the line; the plucked string).
2.2 Causality and Energy.
2.3 The Diffusion (or Heat) Equation (the maximum principle; uniqueness for the Dirichlet problem).
2.4 Diffusion on the whole real line (the Gaussian or fundamental solution).
2.5 Comparison of waves and diffusion.
Chapter 4 Boundary Problems
4.1 Separation of Variables, the Dirichlet Condition (both for the wave and the diffusion equations).
- Read §§ 5.1, 5.2 and 5.3 now! §5.1 problems are assigned.
4.2 The Neumann Condition.
4.3 Robin's Conditions (cases in which zero is an eigenvalue and cases in which one eigenvalue is negative).
Chapter 5 Fourier Series
5.1 The Coefficients (or discrete Fourier transform): formulas for the coefficients, applications to the wave and the diffusion equations.
5.2 Even, Odd, Periodic and Complex-valued functions.
5.3 Orthogonality and "General Fourier Series" (orthogonal systems from symmetric boundary conditions; complex eigenvalues),
5.4 Completeness (three notions of convergence: pointwise, uniform and mean-square: convergence results for Fourier series and their generalizations).
5.5 Completeness and the Gibbs phenomenon.
5.6 Inhomogeneous Boundary Conditions.
Chapter 6 Harmonic Functions
6.1 The Laplace Equation (its physical significance, maximum principle, uniqueness of solutions of the Dirichlet Problem, invariance of the Laplace operator under rigid motions).
6.2 Rectangles and Cubes.
Chapter 7 Green's Identities and Green's Functions
7.1 Green's First Identity (and some consequences).
7.2 Green's Second Identity (and some consequences).
7.3 Green's Functions and the Dirichlet Problem.
7.4 Half-Spaces and Spheres.

last revised 1245 EST 02/15/2001