Text Book: Partial Differential Equations: An Introduction, 3rd Edition, Walter A. Strauss, John Wiley & Sons
| Date | Topics covered | Section | Homework problems |
|---|---|---|---|
| 1/20 | What is a Partial Differential Equation? | 1.1 | 10, 12 |
| 1/24 | First-order Linear Equations (Solution in the constant-coefficient case; the variable-coefficient case and characteristic curves. | 1.2 | 3, 6, 7, 9 |
| 1/27 | 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 |
| 1/31 | 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 | |
| 2/3 | Well- (and ill-)Posed Problems. | 1.5 | 1, 2 |
| 2/7 | Types of second-order equations. | 1.6 | 2, 4 |
| 2/10 | The Wave Equation (D'Alembert's solution on the line; the plucked string). | 2.1 | 2, 4 |
| 2/14 | Causality and Energy. | 2.2 | |
| 2/17 | The Diffusion (or Heat) Equation (the maximum principle; uniqueness for the Dirichlet problem). | 2.3 | 1, 4, 5 |
| 2/21 | Diffusion on the whole real line (the Gaussian or fundamental solution). | 2.4 | 1, 3, 6, 9 |
| 2/24 | First Midterm (regular class hour and location; covers 1.1 through 2.3) | ||
| 2/28 | Comparison of waves and diffusion. | 2.5 | |
| 3/3 | Separation of Variables, the Dirichlet Condition (both for the wave and the diffusion equations). | 4.1 | 1, 2, 3, 6 |
| 3/7 | The Neumann Condition. | 4.2 | 1, 2, 3 |
| 3/10 | 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 |
| 3/21 | The Coefficients (or discrete Fourier transform): formulas for the coefficients, applications to the wave and the diffusion equations. | 5.1 | 1, 2, 4, 8 |
| 3/24 | Even, Odd, Periodic and Complex-valued functions. | 5.2 | 1, 4, 11, 15 |
| 3/28 | Orthogonality and "General Fourier Series" (orthogonal systems from symmetric boundary conditions; complex eigenvalues) | 5.3 | 1, 2, 6, 12, 13 |
| 3/31 | Completeness (three notions of convergence: pointwise, uniform and mean-square: convergence results for Fourier series and their generalizations). | 5.4 | 1, 3, 11, 16 |
| 4/4 | Completeness and the Gibbs phenomenon. | 5.5 | 1, 2, 11, 13 |
| 4/7 | Second Midterm (regular class hour and location; covers 2.4 through 5.4) | ||
| 4/11 | Inhomogeneous Boundary Conditions. | 5.6 | 1, 5, 6, 13 |
| 4/14 | 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 |
| 4/18 | Rectangles and Cubes. | 6.2 | 2, 4, 7 |
| 4/21 | Green's First Identity (and some consequences). | 7.1 | 1, 4, 6 |
| 4/25 | Green's Second Identity (and some consequences). | 7.2 | 1, 2 |
| 4/28 | Green's Functions and the Dirichlet Problem. | 7.3 | 1, 2 |
| 5/2 | Half-Spaces and Spheres. | 7.4 | 1, 7, 8, 13 |
| 5/5 | Final exam, 8:00 to 11:00 am (location to be announced) | ||