Introduction to Partial Differential Equations (MAA 510)
The course “Introduction to Partial Differential Equations” provides students with a solid foundation in the theory of partial differential equations (PDEs). A PDE describes a function of multiple variables in terms of an equation on its partial derivatives. PDEs play a fundamental role in various fields: especially physics, but also economics and stochastics.
In this course we will
- Explore different types of PDEs of the first and second order (linear/non-linear; elliptic, parabolic, hyperbolic) through the use of important examples: transport equation, Burgers equation, Laplace equation, heat equation, wave equation. Emphasis will be given to how PDEs differ from single-variable differential equations (ODE/dynamical systems) and from one another.
- Use a variety of methods to solve PDEs: characteristics, fundamental solutions, Fourier analysis, separation and spectral analysis, coordinate transformations. These will lead to explicit formulae of example PDEs, as well as familarise students with the use of these methods in general.
- Study boundary and initial value problems: Students will see how boundary value problems and initial value problems restrict and shape the possible solutions of PDEs, and the strengths and limitations of their use as models of real-word phenomena.
This course should give students a firm basis for other course offered, such as Partial Differential Equations and Numerics of PDEs, as well as a variety of seminars.
Organisation
There are two lectures and one tutorial per week. Weekly exercise sheets are graded, and a minimum of 50% of the points are needed to gain entry to the exam (Zulassung). The final exam will be conducted orally.
The course will be conducted in English but I do not want language to be barrier to this course; if you are not comfortable in English, talk to me and we can make an arrangement. In particular, you can write your weekly exercise sheets and take the final exam in either English or German.
This course is suitable for both Bachelor and Master students. Students should have a solid background in analysis (Analysis I and II). It is a bonus if you have already taken Dynamical Systems but we only occasionally need to solve ODEs in this course (mostly in Chapter 1). There is a small amount of linear algebra, but this will be revised in the first tutorial.
Please feel free to contact me, Dr Ross Ogilvie, if you have any questions about the course via email r.ogilvie@uni-mannheim.de . My office hour is after the lecture on Wednesday. My office is in B6 Bauteil C (the one with the blue door), Level 4, Room C407, or just walk back with me after the lecture. If you have a particular question, I would appreciate if you sent me an email ahead of time: some of your questions are really tricky and I have to think about them!
The exams will take place on 16th December 2024 and 3rd February 2024. Specific time slots will be organised by email closer to the exam. Exam consultation (for last minute questions) will also be announced closer to the exam.
Lectures
Lectures and tutorials will take place in A 101 Kleiner Hörsaal B 6, 23–25 Bauteil A.
Monday, 15:30 – 17:00, Wednesday 13:45 – 15:15
Script (updated 9.12.24)
Chapter 1
Inhomogeneous transport equation
Burger's equation characteristics and solution graph
Chapter 2
Some mathematical physics. The problem of whether newtonian gravity has “nonsense” solutions is called the Painlevé conjecture (even though it's now solved). I couldn't find an animation of the configuration; let me know if you find one. Here is a talk by the discoverer of the first 4-body configuration with non-collision singularities. These are ODE systems, but again show that even a well-understood and long studied model of physics can have strange impossible behaviour. I also mentioned the Yang-Mills mass gap problem as another example where physicists have a model that they think describes the world, but its mathematical properties are unproven.
The divergence theorem is a theorem that appears at different levels of generality:
- For intervals in R: Fundamental theorem of calculus
- For curves with endpoints: Gradient theorem
- For regions of the plane: Green's theorem
- For surfaces with boundary in R^3: Kelvin-Stokes theorem
- For regions in R^3: Gauss' theorem
- For regions in R^n: Divergence theorem
- For smooth manifolds with boundary: Stokes-Cartan theorem
And there are even more generalisations, which handle non-smooth boundaries (by Whitney and Federer, among others). In a different direction altogether, it is also the idea behind Cauchy's theorem in complex analysis, which leads to Cauchy's integral formula, Cauchy-Pompeiu formula, and a whole slew of other named formulas.
Functions with compact support
Chapter 3
A Green's function is a model of a Faraday cage
Green's function of the unit ball
The test functions used in Section 3.5
Chapter 4
Special solutions to the heat equation
A short time existing solution
Lecture videos for Weeks 10 and 11
Here is a list to the playlist of the videos.
In Week 10 you should watch up to Theorem 4.12 in Section 4.3. We have already covered part of Section 4.1.
In Week 11 you should watch the remainder of Chapter 4.
In case it is hard to read the screen in the video, or if you find I scroll the screen around too much, I also have the “board” as a pdf:
Sections 3.5, 4.0, 4.1, 4.2, 4.3, 4.4, 4.5
Chapter 5
Tutorials
Time: Mondays 13:45 -15:15
The first tutorial will take place on Monday 2nd September, i.e. before the first lecture, but only handle background material (multivariable calculus aka Analysis II, contour diagrams, index notation).
A minimum of 50% of the exercise points is necessary to be admitted to the exam. Please email your exercises to r.ogilvie@uni-mannheim.de before Monday 9:00 on the day of the tutorial, so I (Ross) can mark them and give feedback. The first exercise sheet (the one for the background tutorial) has no points. So, for example, do Sheet 2 and email it to me by 9am on Monday 9th September; we will discuss it that afternoon.
Exercise Sheets:
Sheet 3 – Solutions – Ex 9a, Ex 9c, Variant of 9c
Sheet 4 – Solutions – Ex 10, Ex 11, Shock wave, Water shock wave
Sheet 6 – Solutions – Convolution
Sheet 7 – Solutions – Harmonic Polynomials
Sheet 9 – Solutions – Green's function for half plane
Sheet 10 – Solutions – Tutorial Whiteboard
Sheet 11 – Solutions – Tutorial Whiteboard
Sheet 12 – Solutions – Exercises 35 and 36, Non-zero boundary condition, Method of Images
Sheet 13 – Solutions – Plane Wave, Spherical Wave
Sheet 14 – Solutions – 2D wave solution, Energy of wave modes
All Sheets in one pdf – All Solutions