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# Partial Differential Equations

## Module MA3005

This Module is offered by TUM Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

### Module version of SS 2021 (current)

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
SS 2021WS 2020/1SS 2019WS 2013/4SS 2013

### Basic Information

MA3005 is a semester module in English language at Master’s level which is offered in summer semester.

This Module is included in the following catalogues within the study programs in physics.

• Catalogue of non-physics elective courses
• Specialization Modules in Elite-Master Program Theoretical and Mathematical Physics (TMP)
270 h 90 h 9 CP

### Content, Learning Outcome and Preconditions

#### Content

- 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

#### Learning Outcome

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.

#### Preconditions

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

### Courses, Learning and Teaching Methods and Literature

#### Courses and Schedule

VO 4 Partial Differential Equations Friesecke, G. Mon, 12:15–13:45, MI HS3
Fri, 12:15–13:45, MI HS3
UE 2 Exercises for Partial Differential Equations Friesecke, G. dates in groups

#### Learning and Teaching Methods

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

#### Literature

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

### Module Exam

#### Description of exams and course work

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 abilities to apply and analyze the learned concepts.

#### Exam Repetition

The exam may be repeated at the end of the semester.

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