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Schedule for TATA27 Partial Differential Equations, spring 2023

The course will be given in a flipped classroom style, meaning that you are expected to prepare for each lesson by reading the relevant material and watching the video lectures (see the outline below), and attempting a few problems in advance. There will be no regular lectures in the classroom, except maybe for short summaries or digressions. Instead, the classroom time is spent on discussing the theory, working on problems, clarifying details from the video lectures if needed, and so on.

Period vt1

1. Tuesday Jan 17, 10–12. Introduction.
Mainly to say hello, and to get started with Section 1 in the pdf file with the exercises. This time you don't really have to prepare anything, since the exercises are just "warmup problems" meant to refresh old stuff from your calculus courses. But if you have time before the lesson, you can watch the introduction video, read Section 1 in David Rule's lecture notes, and have a go at some of the problems.
2. Tuesday Jan 24, 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). Starting from this lesson, you are expected to come prepared! Before the lesson, read Section 2 in the lecture notes, watch the video, and attempt as many of the problems as you can.
3. Tuesday Jan 31, 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 7, 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. Wednesday Feb 15, 15–17. 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 21, 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. Friday Feb 24, 8–10. 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. Tuesday Feb 28, 10–12. 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. Friday March 3, 8–10. 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).
10. Tuesday March 7, 10–12. 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.)

Period vt2

11. Monday March 27, 13–15. 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 3, 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. Friday April 14, 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. Wednesday April 19, 8–10. 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 April 24, 13–15. Numerical methods.
Read Section 8 in the lecture notes, watch the videos, work on the exercises.
16. Friday April 28, 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. Wednesday May 3, 8–10. 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 8, 13–15. Harmonic polynomials and spherical harmonics.
Watch the video and work on the exercises. The material for this lecture is taken from Chapter 5 of the marvellous book Harmonic Function Theory by Sheldon Axler, Paul Bourdon and Wade Ramey, which can be freely downloaded from Axler's website. The traditional Swedish term for spherical harmonic is klotytefunktion (from German Kugelflächenfunktion).
19. Monday May 15, 13–15. The Schrödinger equation.
Read Section 9 in the lecture notes, watch the video, work on the exercises. The video covers only section 9.1 (the harmonic oscillator), and in quite a different way. There is some material expanding upon Section 9.2 (the hydrogen atom) in the exercises. For additional information, see (more or less) any book about quantum mechanics. Remark: In the lecture notes, David uses an unconventional sign convention for the potential function V; almost everyone else has the opposite sign (so David's V would be −V in other sources, including the video).
20. Monday May 22, 13–15. Leftovers.
No new material this time, just time to catch up and maybe ask some questions before the exam (which takes place on Tuesday May 30).

Sidansvarig: Hans Lundmark
Senast uppdaterad: 2023-05-12