4. September – Due to a mix-up in the room bookings, the second lecture is on Wednesday. See below.

30. August – The problem with the course in Portal2 has been fixed.

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

1. 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.

2. 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.

3. 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.

There are two lectures a week and one tutorial. 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. Language should be no barrier to this course; if you are not comfortable in English, talk to me and we can make an arrangement.

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 .

Mondays 15:30 -17:00 in Hörsaal C015 (A5 Bauteil C)

Wednesday, 13:45 – 15:15 in Kleiner Hörsaal A101 (B6 Bauteil A)

Lecture script (Taylor's Version)

Lecture 1: Transport Equation, Inhomo Transport Equation, Characteristics

Lecture 2: Burger Characteristics I and in 3D

Lecture 7: Projection onto the tangent plane

Lecture 8: Compact Support

Mondays 17:15 -18:30 in Hörsaal C015 (A5 Bauteil C)

The “rhythm” of the course is as follows. We have lectures Monday and Tuesday. Then you have the rest of the week to work on the exercises. Your exercises should be submitted by 9:00 on Monday morning, and we will discuss them in the tutorial on Monday afternoon. The cycle repeats. The naming scheme is that “Sheet n” will be discussed in the tutorial in week n, and deals with the material from the lectures of week (n-1).

The first tutorial is an exception. It will take place on Monday 4.9.2023 at 17:15 as scheduled. However it will be a revision on linear algebra and vector calculus. You do not have to submit Sheet 1 and it has no points.

**Exercise Sheets:**

Sheet 3 – Solutions – Burger's Equation, Exercise 8a, Exercise 8c, Variant of Exercise 8c

Sheet 4 – Solutions – Gas Shock Wave, Water Shock Wave, Self-driving traffic jam

Sheet 5 – Solutions – There are four bonus points for the following question: Suppose that Phi(x) = (x,λ(x)) is a regular parameterisation of a graph, x in R^{n-1}. Show that the area element obeys det(Phi'^T Phi') = 1 + |grad(λ)|^2. – The Black Spot

Sheet 6 – Solutions

Sheet 7 – Solutions

Sheet 8 – Solutions

Sheet 9 – Solutions

Sheet 10 – Solutions

Sheet 11 – Solutions

Sheet 12 – Solutions

Sheet 13 – Solutions

Sheet 14 – Solutions

All Sheets in one pdf – All Solutions

For your interest, here are two papers about PDE theory broadly: a historical perspective and on the unity of the subject.

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!

A number of textbooks are suitable for reference. These can be found at the end of the lecture script.

There are recordings of the lectures from the pandemic years, which can be found in this playlist. There is also a Repetition Course are at the end of the above playlist. The blank slides are here, for you to follow along with, and my annotated slides here. Please note that the course this year may deviate from previous years, particularly the chapter on the heat equation.