Week |
Monday |
Wednesday |
Friday |
|
| Sep 7-11 | Labour day. | What is a PDE? | First order linear equations. Weekly summary |
|
| Sep 14-18 | Flows, vibrations, and diffusions. | Initial and boundary conditions. Well-posed problems. |
Applying the method of characteristics. Assignment 1 |
|
| Sep 21-25 | The wave equation. | D'Alembert's formula. | Causality and energy. Weekly summary |
|
| Sep 28-Oct 2 | The diffusion equation. | Energy methods The maximum principle. |
The fundamental solution. Assignment 2 |
|
| Oct 5-9 | Diffusion on the half-line. | Reflections of waves. | Diffusion and waves with a source. Weekly summary |
|
| Oct 12-16 | Thanksgiving. | Proof that the representation formulas solve the heat and wave equation. |
Separation of variables. Assignment 3 |
|
| Oct 19-23 | Eigenfunctions. | Fourier sine and cosine series. Orthogonality in L^2 |
Full Fourier series for periodic functions. | |
| Oct 26-30 | Orthonormal bases. | Mean square convergence and completeness of L^2. |
Uniform convergence of Fourier series. Assignment 4 |
|
| Nov 2-6 | How to integrate or differentiate a Fourier series. |
Inhomogeneous boundary conditions. | Laplace's equation. Harmonic functions. |
|
| Nov 9-13 | Rectangles and cubes. | Poisson's formula. Assignment 5 |
Fall break. | |
| Nov 16-20 | No class. (To be rescheduled). |
Proof of Poisson's formula | Proof of Poisson's formuls. | |
| Nov 23-27 | Green's identities. Mean value property. Announcement of 1st midterm test. |
First Midterm Test. | Green's functions. | |
|
Nov 30-Dec 4 |
Green's function of the half-space. | Green's function of the sphere. | More properties of Green's functions. | |
| Dec 9 -Jan 3 |
Exam period. Winter break. |
Exam period. Winter break. |
Exam period. Winter break. |
|
|
Jan 4-8 |
The wave equation in space-time. Causality and energy. |
Spherical means and the Euler-Posson-Darboux equation. |
Kirchhoff's formula in 3-d. Huygen's principle. |
|
| Jan 11-15 | Poisson's formula in 2-d. | Rays, singularities, and sources. Assignment 6 |
Characteristic surfaces. Propagation of singularities. | |
| Jan 18-22 | Heat and Schrödinger equations on the whole space. |
Hermite polyomials. |
Radial eigenfunctions of the hydrogen atom. Assignment 7 |
|
| Jan 25-29 | Nodes. Bessel functions. Minimax characterization. |
The hydrogen atom. Fourier's method, revisited. |
Spherical Harmonics Summary |
|
| Feb. 1-5 | Non-radial eigenfunctions of the hydrogen atom. |
General eigenvalue problems. Rayleigh quotient. | Existence of minimizers. Assignment 8 |
|
| Feb 8-11 | The Rayleigh quotient. | Dirichlet and Neumann eigenvalues. | Minimax and Maximin principles. | |
| Feb 15-19 | Spring break. | Spring break. | Spring break. | |
| Feb 22-26 | Dirichlet-Neumann bracketing. Weyl's law. |
Completeness of Eigenfunctions. | Distributions. Announcement of 2nd midterm test. |
|
| Mar 1-5 | Green's functions, revisited. Second Midterm Test. |
Fourier transform. | No class. | |
| Mar 8-12 | More about Fourier transforms. Lieb & Loss, Chapter 5. |
Parseval's theorem. Inverse Fourier transform. |
Solving PDE by Fourier transform. Assignment 9 |
|
| Mar 15-19 | Nonlinear equations: Burger's equation. |
Shocks. Rankine-Hugoniot condition. |
Lax's entropy condition. Assignment 10 |
|
| Mar 22-26 | Calculus of Variations. Nonlinear equations in divergence form. |
Reaction-diffusion equations: Steady states and traveling waves. |
Reaction-diffusion equations: Linearization and Bifurcation. |
|
| Mar 29-Apr 1 | Comparison-principle for semilinear parabolic equations. |
The KdV equation. Solitons. |
Good Friday. | |
| Exam period | Final Exam: Apr 19, 2-5pm, in EX 320. Announcement |