NMAK10019U Differential operators and function spaces (DifFun)

Volume 2013/2014
Education
MSc programme in Mathematics
Content
Differential operators. Distribution theory, Fourier transform of distributions. Function spaces. Applications to concrete differential operator problems.
Learning Outcome
Knowledge:

  • Linear differential equations and thei relevant side conditions (e.g. boundary, initial)
  • Concept of ellipticity
  • Distributions and their convergence properties
  • Multiplication by smooth functions and derivatives of distributions
  • Fourier transform of distributions
  • Function classes such as Sobolev spaces or Lp spaces and the action on differnetial operators and  the Fourier transform on these
  • Unbounded operators on Hilbert spaces
  • Solution methods for differential equations such as methods based on the Fourier transform or a variational approach

Competences:

  • Understand the different realizations of differential operators on relevant function spaces
  • Understand concepts such as existence uniqueness and regularity of solutions to differential equations within the relevant function spaces
  • Determine when a certain solution method applies 
  • Calculate with distributions (derivatives, multiplication, ...)
  • Calculate Fourier transform of distributions, and functions in different function classes
  • Know the relations (inclusions) of relevant function spaces
Skills:
  • Solve classical differential equations
  • Establish existence, uniqueness and regularity of solutions to certain differential equations
  • Describe the different realizations of concrete differential operators on Hilbert spaces
  • Calculate properties (e.g., domain, spectra) of realizations of differential operators
A knowledge of complex analysis, Banach and Hilbert spaces and the Fourier transform corresponding to KomAn and An2.
5 hours of lectures and 2 hours of exercises each week for 8 weeks
  • Category
  • Hours
  • Exam
  • 20
  • Guidance
  • 13
  • Lectures
  • 40
  • Preparation
  • 117
  • Theory exercises
  • 16
  • Total
  • 206
Credit
7,5 ECTS
Type of assessment
Written assignment, Two 1 week take home
Written examination, 3 hours under invigilation
The 2 written assignments count each 20% toward the final grade. The final exam
counts 60%
Aid
All aids allowed
Marking scale
7-point grading scale
Censorship form
External censorship
The external examinator is used at the final exam only
Criteria for exam assesment

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