Introduction to the subject of Partial Differential Equations. Topics include classification of PDEs, first-order linear and non-linear PDEs, second order PDEs including the heat, wave and Laplace's equation. Solution methods such as the method of characteristics for hyberbolic equations, separation of variables and eigenfunction expansions, Laplace and Fourier transforms.
| Lesson | Date | Topics | Comments | Readings | Exercises |
|---|---|---|---|---|---|
| 1 | Sept 12 | Classification of PDEs First-order linear equations with constant coefficients |
Discuss syllabus, course expectations |
Evans' notes on PDEs Strauss 1.1 |
Strauss 1.1 2: (a), (b) 3: (a), (b), (d), (g) (h) Strauss 1.2: 1, 2 |
| 2 | Sept 14 | First order linear equations with variable coefficients. | Strauss 1.2 | Strauss 1.2 5, 6, 7, and 10. |
|
| 3 | Sept 15 | Conservation laws, nonlinear advection, method of characteristics |
Logan pg. 25 | Logan 1.2: 2, 3, 5, 6(c), 8. | |
| 4 | Sept 19 | Classification of Second order PDEs Reduction to standard form |
Example | Jeffrey 3.1 pg 103-117. | Jeffrey 3.1 pg 117 |
| 5 | Sept 20 | Cauchy problem for the wave equation, d'Alembert's formula | Assignment 1 solutions |
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| 6 | Sept 21 | Cauchy problem for the heat equation,
heat kernel, convolution solution |
Logan 2.1 pg.79-85. | Logan 2.1: 1 and 6. | |
| 7 | Sept 22 | Boundary value problems, heat equation Fourier series, eigenfunction expansions |
Logan 4.1 pg. 155-163 | Logan 4.1 pg. 166: 1, 5. | |
| 8 | Sept 23 | Boundary value problems, wave equation Fourier series, eigenfunction expansions |
Meeting in MLT at 1130am | Logan 4.1 pg. 163-165 | Logan 4.1 pg. 166: 4 |
| 9 | Sept 26 | Fourier Transforms, convolution applications to PDEs |
Assignment 2 | Logan 2.7 pg. 117-121 | Logan 2.7 pg. 122: 1, 2, 3(a), (b), 11, 15, 16, 20 |
| 10 | Sept 27 | Laplace's equation solution in upper half-plane, disk |
Assignment 1 due | Logan 2.7 pg. 122 Logan 4.4 pg. 186 |
Logan 4.4 pg. 195: 1, 2, 3, 5, 8 |
| 11 | Sept 28 | Laplace transforms applications to PDEs |
Last day of class | Logan 2.6 pg. 106-114 |
Logan 2.6 pg. 115: 2, 8, 13. |
| Oct 3 | Final | Final Exam |