# 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

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.

## 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!

## 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

Chapter 1

Inhomogeneous transport equation

Rankine's poem

Corrected script

## 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 4 – Solutions

Sheet 5 – Solutions

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