# Partial Differential Equations (PDE)

### Course content

A selection from the following list of subjects:

The classical PDEs:

- Laplace's equation

- The heat equation

- The wave equation

Second order linear elliptic PDEs:

- Existence of weak solutions

- Regularity

- Maximum principles

Second order linear parabolic PDEs:

- Existence of weak solutions

- Regularity

- Maximum principles

Second order linear hyperbolic PDEs:

- Existence of weak solutions

- Regularity

- Propagation of singularities

Nonlinear PDEs:

- The Calculus of Variations

- Fixed point methods

- Method of sub-/supersolutions

- Non-existence of solutions

Education

MSc Programme in Mathematics

Learning outcome

Knowledge:
The properties of the PDEs covered in the course

Competencies:

• Understand the characteristics of the different types of PDEs
• Understand concepts such as existence, uniqueness and regularity of solutions to PDEs
• Determine when a certain solution method applies

Skills:

• Solve classical PDEs
• Establish existence, uniqueness and regularity of solutions to certain PDEs

5 hours of lectures and 2 hours of exercises each week for 8 weeks

See Absalon for a list of course literature

A knowledge of Lebesgue measure theory and Banach/Hilbert spaces, corresponding to at least the contents of the following courses:

Analyse 0 (An0),
Analyse 1 (An2) and
Lebesgueintegralet og målteori (LIM) - alternatively Analyse 2 (An2) from previous years.

Additionally, it might be helpful to have had some exposure to the material from a more advanced Analysis course, f.ex. one of either FunkAn or DifFun.

Academic qualifications equivalent to a BSc degree is recommended.

Written
Individual
Collective
Continuous feedback during the course of the semester
ECTS
7,5 ECTS
Type of assessment
Written examination, 4 hours under invigilation
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Aid
All aids allowed
Marking scale
Censorship form
No external censorship
One internal examiner.
##### Criteria for exam assessment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

Single subject courses (day)

• Category
• Hours
• Lectures
• 40
• Preparation
• 146
• Exercises
• 16
• Exam
• 4
• English
• 206