| Date |
Topics covered |
Section |
Homework problems |
| 9/1 |
What is a Partial Differential
Equation? |
1.1 | 10, 12 |
9/6
|
First-order Linear Equations
(Solution in the constant-coefficient
case; the variable-coefficient case and characteristic curves. |
1.2 | 3, 6, 7, 9 |
9/8
|
Flows, Vibrations and Diffusions
(Derivations of PDEs in various physical
situations; e.g., the vibrating string, the vibrating drumhead,
diffusion, heat flow, hydrogen atom). |
1.3 | 2, 4, 9 |
9/13
|
Initial and boundary conditions
(the Dirichlet, Neumann and Robin conditions
and their significance for the vibrating string and diffusion
equations.
Conditions at infinity.) |
1.4 |
| 9/15 |
Well- (and ill-)Posed
Problems. |
1.5 | 1, 2 |
9/20
|
Types of second-order
equations. |
1.6 | 2, 4 |
| 9/22 |
The Wave Equation (D'Alembert's
solution on the line; the plucked string). |
2.1 | 2, 4 |
| 9/27 |
Causality and Energy. |
2.2 |
| 9/29 |
The Diffusion (or
Heat)
Equation (the maximum principle; uniqueness
for the Dirichlet problem). |
2.3 | 1, 4, 5 |
| 10/4 |
Diffusion on the whole real line
(the Gaussian or fundamental solution). |
2.4 | 1, 3, 6, 9 |
| 10/6 |
First
Midterm--regular class hour and location; covers 1.1 through 2.3
|
| 10/11 |
Comparison of
waves and diffusion. |
2.5
|
10/13
|
Separation of Variables, the
Dirichlet Condition (both for the wave
and the diffusion equations). |
4.1
| 1, 2, 3, 6 |
10/18
|
The Neumann Condition. |
4.2
| 1, 2, 3 |
| 10/20 |
Robin's Conditions (cases in
which zero is an eigenvalue and cases
in which one eigenvalue is negative). |
4.3
| 1, 2, 7, 9, 17 |
| 10/25 |
The Coefficients (or
discrete Fourier transform): formulas for the
coefficients, applications to the wave and the diffusion
equations. |
5.1
| 1, 2, 4, 8 |
| 10/27 |
Even, Odd, Periodic and
Complex-valued functions. |
5.2
| 1, 4, 11, 15 |
| 11/1 |
Orthogonality and "General
Fourier Series" (orthogonal systems from
symmetric boundary conditions; complex eigenvalues) |
5.3
| 1, 2, 6, 12, 13 |
| 11/3 |
Completeness (three notions of
convergence: pointwise, uniform and
mean-square: convergence results for Fourier series and their
generalizations). |
5.4
| 1, 3, 11, 16 |
| 11/8 |
Completeness and the Gibbs
phenomenon. (Nice
applet illustration) |
5.5
| 1, 2, 11, 13 |
| 11/10 |
Second
Midterm--regular
class hour and location; covers 2.4 through 5.4 |
| 11/15 |
Inhomogeneous
Boundary Conditions. |
5.6
| 1, 5, 6, 13 |
| 11/17 |
The Laplace Equation (its
physical significance, maximum principle,
uniqueness of solutions of the Dirichlet Problem, invariance of the
Laplace
operator under rigid motions). |
6.1
| 1, 2, 4, 10 |
| 11/29 |
Rectangles and Cubes. |
6.2
| 2, 4, 7 |
| 12/1 |
Green's First Identity (and some
consequences). |
7.1
| 1, 4, 6 |
| 12/6 |
Green's Second Identity (and
some
consequences). |
7.2
| 1, 2 |
| 12/8 |
Green's Functions and the
Dirichlet Problem. |
7.3
| 1, 2 |
| 12/13 |
Half-Spaces and Spheres. |
7.4
| 1, 7, 8, 13 |