Schedule for TATA27 Partial Differential Equations, spring 2021

Here's an outline of what we will be doing during the online teams sessions, what sections to read, what videos to watch, which exercises to work on, etc. The page will be updated as we go along.

Period vt1

1. Tuesday Jan 19, 10–12. Introduction.

Mainly to say hello and get started. If you have time before the seminar, watch the introduction video, read Section 1 in David Rule's lecture notes, and have a go at some of the warmup problems (Section 1 in the pdf file with the exercises). Otherwise, do it afterwards.

2. Tuesday Jan 26, 10–12. The method of characteristics.

We'll discuss the problems in Section 2 of the exercise file (and maybe some leftovers from Section 1). Before the seminar, read Section 2 in the lecture notes, watch the video, and attempt as many of the problems as you can.

3. Tuesday Feb 2, 10–12. The physical origin of some PDEs.

Read Sections 3 and 4 in the lecture notes, watch the video, work on the exercises.

4. Tuesday Feb 9, 10–12. The Laplace equation: The weak maximum principle. Poisson's formula (for a disk).

Read Sections 5.1, 5.2, 5.3.1 and 5.3.2 in the lecture notes, watch the videos, work on the exercises.

5. Tuesday Feb 16, 10–12. The Laplace equation (cont.): The mean value property. The strong maximum principle.

Read Sections 5.3.3, 5.4.1 and 5.4.2 in the lecture notes, watch the videos, work on the exercises.

6. Tuesday Feb 23, 10–12. The Laplace equation (cont.): Dirichlet's principle. The fundamental solution.

Read Sections 5.4.3 and 5.4.4 in the lecture notes, watch the videos, work on the exercises.

7. Tuesday March 2, 10–12. The Laplace equation (cont.): Green's functions.

Read Sections 5.4.5, 5.4.6 and 5.4.7 in the lecture notes, watch the videos, work on the exercises.

8. Friday March 5, 8–10. The wave equation in one dimension. (On the whole real line.)

Read Sections 6.1, 6.2 and 6.3 in the lecture notes, watch the video, work on the exercises.

9. Tuesday March 9, 10–12. The wave equation in one dimension (cont.). (On a half-line or finite interval.)

Read Section 6.4 in the lecture notes, watch the video, work on the exercises. You might need to refresh what you know about Fourier series and separation of variables; this is not covered in David's notes, but it's described in great detail in Strauss's book, Chapters 4 and 5 (and in many other sources).

Period vt2

10. Monday March 29, 13–15. The wave equation in higher dimensions.

Read Section 6.5 in the lecture notes, watch the videos, work on the exercises. (The second video is optional. That material isn't covered in the lecture notes; see section 2.4.1 in Evans for details.)

11. Wednesday April 7, 8–10. The heat equation on a bounded domain.

Read Sections 7.1, 7.2 and 7.3 in the lecture notes, watch the video, work on the exercises.

12. Monday April 12, 13–15. The heat equation on Rⁿ.

Read Sections 7.4 and 7.5 in the lecture notes, watch the video, work on the exercises.

13. Monday April 19, 13–15. Classification of second-order linear PDEs.

What you'll need is in the video and in the exercises. This material is not mentioned at all in David Rule's lecture notes. Strauss says a little about the constant-coefficient case in Section 1.6. In Evans, the information about different types of PDEs is spread out over many different places in the book, and most of it is much more advanced than what we need here, but, for example, characteristic coordinates for second-order hyperbolic PDEs in two variables is described 7.2.5. Similarly for Folland; there is some material in Chapter 1 about characteristic hypersurfaces and the Cauchy problem.

14. Monday April 26, 13–15. Generalized solutions.

Not covered in the lecture notes. What you'll need should hopefully be in the videos and the exercises. For more info, see Strauss (Sections 12.1 and 14.1) or Evans (Section 3.4 plus several other sections with the words weak solutions in the title). This question from Mathematics StackExchange may also be of interest: Why should I “believe in” weak solutions to PDEs?

15. Monday May 3, 13–15. Numerical methods.

Read Section 8 in the lecture notes, watch the videos, work on the exercises.

16. Monday May 10, 13–15. Separation of variables in higher dimensions.

Watch the video and work on the exercises. This material is not covered in the lecture notes; see Strauss (Chapters 10 and 11) if you need more info. The articles about drums that sound the same can be accessed via LiUB: Kac, Chapman. See also this MathOverflow question: Can you hear the shape of a drum by choosing where to drum it?

17. Monday May 17, 13–15. Dispersive waves and solitons.

Watch the video and work on the exercises. This material is not covered in the lecture notes; there is a little bit of stuff about water waves and solitons in Strauss (Sections 13.2, 14.2 and 14.5).

18. Monday May 24, 13–15. Leftovers.

No new material this week, just time to catch up and maybe ask some questions before the exam (Friday May 28).

Sidansvarig: Hans Lundmark
Senast uppdaterad: 2023-01-11