Partial Differential Equations

- Classical theory and representation formulas for solutions of transport, Laplace, heat and wave equations;
- Introduction to conservation laws;
- Sobolev spaces;
- Weak solutions of second order Elliptic equations

After successful completion of the module students are able to understand, apply and analyze basic methods to treat partial differential equations. In particular they distinguish different types of partial differential equation and understand their basic properties. The students understand the concepts of classical and weak (variational) solutions to elliptic partial differential equations including the questions on existence, uniqueness and well-posedness as well as on regularity of solutions. Moreover, they can analyze structural properties of such solutions, i.e. by applying maximum principles.

MA1001 Analysis 1, MA1002 Analysis 2, MA2003 Measure and Integration, MA2004 Vector Analysis
Bachelor 2019: MA0001 Analysis 1, MA0002 Analysis 2, MA0003 Analysis 3

The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should animate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress. At the beginning of the module, the practice sessions will be offered under guidance, but during the term the sessions will become more independent, and intensify learning individually as well as in small groups.

blackboard

L.C.Evans, Partial Differential Equations, Graduate Studies in Mathematics Vol. 19, AMS, 1998.

The module examination is based on a written exam (90 minutes). The students demonstrate that they have a profound understanding of theoretical concepts and methods to solve partial differential equations. On the basis of specific examples the students exhibit their ablilities to apply and analyze the learned concepts.