Partial Differential Equations
This course is an introduction to partial differential equations (PDEs) and Fourier analysis.
- We will study the classical PDEs of mathematical physics: the heat equation, the wave equation, and Laplace’s equation.
- We will learn how to solve these equations using separation of variables and Fourier series.
- Complex variables will be introduced as tools to help us understand Fourier series.
- Properties of complex exponentials, sines, and cosines will be generalized to abstract properties of functions in \(L^2\) spaces. Accordingly, will will study the notion of Hilbert spaces.
- An introduction to Sturm-Liouville theory will be given, including eigenvalue problems and orthogonal functions.
- If time permits, we will study additional topics such as Fourier and Laplace transforms.
Course information
| Instructor: | John Myers |
| Office: | Marano 175 |
| Office hours: | 12-12:30 MWF, and by appointment |
| Syllabus: | link |
Schedule, slides, exercises, and info
| week | date | topics | slides | exercises | info |
|---|---|---|---|---|---|
| Finals week | 05.15 fri | ||||
| 05.13 wed | Final exam Shineman 178, 2-4pm |
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| 05.11 mon | |||||
| Week 15 | 05.08 fri | ||||
| 05.06 wed | |||||
| 05.04 mon | Sec 18: More examples of Sturm-Liouville problems | None | Coming soon | ||
| Week 14 | 05.01 fri | Exam 2 | |||
| 04.29 wed | Same as below ↓ | Same as below ↓ | Same as below ↓ | ||
| 04.27 mon | Sec 17: Sturm-Liouville problems, part 2 | Sec 17 Slides | Sec 17 Exercises | ||
| Week 13 | 04.24 fri | Continue Sec 16 | Same as below ↓ | Same as below ↓ | Quiz on Sec 16 |
| 04.22 wed | No class — Quest | ||||
| 04.20 mon | Same as below ↓ | Same as below ↓ | Same as below ↓ | ||
| Week 12 | 04.17 fri | Sec 16: Sturm-Liouville problems, part 1 | Sec 16 Slides | Sec 16 Exercises | Quiz on Sec 15 |
| 04.15 wed | Same as below ↓ | Same as below ↓ | Same as below ↓ | ||
| 04.13 mon | Sec 15: More on \(L^2\)-spaces | Sec 15 Slides | Sec 15 Exercises | ||
| Week 11 | 04.10 fri | Same as below ↓ | Same as below ↓ | Same as below ↓ | Quiz on Sec 13 and 14 |
| 04.08 wed | Sec 14: Convergence and completeness, part 2 | Sec 14 Slides | Same as below ↓ | ||
| Week 10 | 04.03 fri | Same as below ↓ | Same as below ↓ | Same as below ↓ | Quiz on Sec 12 |
| 04.01 wed | Sec 13: Convergence and completeness, part 1 | Sec 13 Slides | Sec 13 Exercises | ||
| 03.30 mon | Same as below ↓ | Same as below ↓ | Same as below ↓ | ||
| Week 9 | 03.27 fri | Sec 12: Functions and inner products | Sec 12 Slides | Sec 12 Exercises | Quiz on Sec 11 |
| 03.25 wed | Same as below ↓ | Same as below ↓ | Same as below ↓ | ||
| 03.23 mon | Sec 11: Vectors and inner products | Sec 11 Slides | Sec 11 Exercises | ||
| Week 8 | Spring break | ||||
| Week 7 | 03.13 fri | Exam 1 | |||
| 03.11 wed | Review for Exam 1 | ||||
| 03.09 mon | Same as below ↓ | Same as below ↓ | Same as below ↓ | ||
| Week 6 | 03.06 fri | Sec 10: Fourier series on different intervals | Sec 10 Slides | Sec 10 Exercises | Quiz on Secs 7-9 |
| 03.04 wed | Same as below ↓ | Same as below ↓ | Same as below ↓ | ||
| 03.02 mon | Sec 9: Differentiation and integration of Fourier series | Sec 9 Slides | Sec 9 Exercises | ||
| Week 5 | 02.27 fri | Sec 8: Convergence of Fourier series, part 2 | Sec 8 Slides | Sec 8 Exercises | Quiz on Sec 6 |
| 02.25 wed | Sec 7: Convergence of Fourier series, part 1 | Sec 7 Slides | Sec 7 Exercises | ||
| 02.23 mon | Same as below ↓ | Same as below ↓ | Same as below ↓ | ||
| Week 4 | 02.20 fri | Same as below ↓ | Same as below ↓ | Same as below ↓ | Quiz on Sec 4 and 5 |
| 02.18 wed | Sec 6: A first look at Fourier series, part 2 | Sec 6 Slides | Sec 6 Exercises | ||
| 02.16 mon | Sec 5: A first look at Fourier series, part 1 | Sec 5 Slides | Sec 5 Exercises | ||
| Week 3 | 02.13 fri | Same as below ↓ | Same as below ↓ | Same as below ↓ | Quiz on Sec 2 and 3 |
| 02.11 wed | Sec 4: General PDEs and boundary conditions | Sec 4 Slides | Sec 4 Exercises | ||
| 02.09 mon | Sec 3: The Laplacian and Laplace’s equation | Sec 3 Slides | Sec 3 Exercises | ||
| Week 2 | 02.06 fri | Same as below ↓ | Same as below ↓ | Same as below ↓ | Quiz on Sec 1 |
| 02.04 wed | Sec 2: Separation of variables | Sec 2 Slides | Sec 2 Exercises | ||
| 02.02 mon | Same as below ↓ | Same as below ↓ | Same as below ↓ | ||
| Week 1 | 01.30 fri | Sec 1: An introduction to PDEs | Sec 1 Slides | Same as below ↓ | |
| 01.28 wed | Another “remote day”: Watch the videos below if you haven’t already, continue working on Exercises 1-8. | Same as below ↓ | |||
| 01.26 mon | “Remote day”: Watch the videos here, here, and here. Begin working on Exercises 1-8 (see link to the right). | Sec 1 Exercises |