NMAK10019U Differential Operators and Function Spaces (DifFun)
MSc Programme in Mathematics
Differential operators. Distribution theory, Fourier transform of distributions. Function spaces. Applications to concrete differential operator problems.
- Linear differential equations and their 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 differential 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
- 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
- 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
Knowledge of the Fourier transform corresponding to FunkAn is desirable.
Academic qualifications equivalent to a BSc degree is recommended.
- Theory exercises
- 7,5 ECTS
- Type of assessment
- Written assignment, Two 7 days take home assignmentsWritten examination, 3 hours under invigilationThe two written 7 days take home assignments count each 20% toward the final grade. The final exam counts 60%
- All aids allowed
- Marking scale
- 7-point grading scale
- Censorship form
- External censorship
Written exam, 3 hours under invigilation. All aids allowed.
The final grade is the largest of the two numbers: 1) Written exam counts 100% and 2) Written exam counts 60% and the results of the two take home assignments count 20% each.
Criteria for exam assesment
The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome.