| Date | Section | Topics covered |
| Aug. 29 | 2.1, 2.2.1 |
Generalities, method of characteristics for constant velocity |
Aug. 31 | 2.2.2 |
Spatially dependent velocity |
Sept. 3 |
|
Labor Day |
Sept. 5 | 2.3 |
Nonlinear conservation laws |
Sept. 7 |
|
A MATLAB demonstration for spatially dependent velocity |
Sept. 10 | 2.3, 2.5.1 |
Characteristics for conservation laws; weak solutions |
Sept. 12 |
|
Classes cancelled |
Sept. 14 | 2.5.2, 2.5.3 |
Weak solutions conservation laws; Rankine Hugoniot condition; Riemann problem |
Sept. 17 | 2.5.4, 2.5.5 |
Shock waves and rarefaction waves, nonuniqueness of weak solutions, entropy conditions |
Sept. 19 | 3.1 |
The heat equation, Fourier's law; motivation of the maximum principle |
Sept. 21 | 3.2, 3.3 |
The maximum principle; towards the fundamental solution |
Sept. 24 | 3.3 |
Construction of the fundamental solution |
Sept. 26 | 3.3 |
The source function; well-posedness of the IVP for the heat equation |
Sept. 28 | 3.3 |
Uniqueness; sources on the line; boundary values on the half line |
Oct. 1 | 3.4 |
The error function; boundary values on the half line |
Oct. 3 | 3.4 |
Boundary values on the half line; separation of variables |
Oct. 5 | 4.1 |
Separation of variables for the heat equation; the Dirichlet problem |
Oct. 8 | 4.1 |
Separation of variables for the heat equation; the Neumann problem;
orthogonality |
Oct. 10 | 4.2 |
Examples of Fourier sine series; notions of convergence |
Oct. 12 | 4.2 |
Convergence in mean square and orthogonality |
Oct. 15 | 4.3 |
Symmetric boundary conditions |
Oct. 17 | 4.3 |
Symmetric boundary conditions |
Oct. 19 |
|
Test 1 |
Oct. 22 | 5.2 |
Derivation of the wave equation, D'Alembert's formula |
Oct. 24 | 5.3 |
D'Alembert's formula, conservation of energy |
Oct. 26 | 5.4 |
Dirichlet problem on the half line |
Oct. 29 | 5.4 |
Odd and even initial data; transmission problem |
Oct. 31 | 5.4 |
Solution of the transmission problem |
Nov. 2 | 5.5 |
Separation of variables for the Dirichlet problem; modes of vibration |
Nov. 5 | 5.5 |
Separation of variables for the Neumann problem; conservation of energy |
Nov. 7 | 5.5 |
Comparison of waves and diffusion; the diffusion equation in 2 and 3 dimensions |
Nov. 9 | 8.1 |
The heat equation in two dimensions; the fundamental solution |
Nov. 12 | 8.3 |
Separation of variables for the heat equation in two dimensions |
Nov. 14 | 8.3, A.4 |
Separation of variables for the heat equation in two dimensions;
the Laplace operator in polar coordinates |
Nov. 16 |
|
Test 2 - Chapters 4 and 5 |
Nov. 19 | A.4 |
Discussion of test problems, Laplace in polar coordinates |
Nov. 21 | 8.4 |
Separation of variables for the heat equation on the disk |
Nov. 23 | 8.4 |
Separation of variables for the heat equation on the disk |
Nov. 26 | 8.4 |
Bessel function expansions, orthogonality; equilibrium |
Nov. 28 | 9.2 |
The Dirichlet problem in the disk |
Nov. 30 | 9.2 |
Separation of variables for the Dirichlet problem in the disk |
Dec. 4 | 9.2 |
Fourier series solution for Laplace's equation |
Dec. 6 | 9.1-9.2 |
Poisson's formula on the disk, mean-value property |
Dec. 8 | 9.2 |
Maximum principle for Laplace's equation |
Dec. 11 | 9.1 |
Review |