NMAK10019U Differential Operators and Function Spaces (DifFun)

Volume 2026/2027
Education

MSc Programme in Mathematics
MSc Programme in Statistics
MSc Programme in Mathematics with a minor subject

Content

Differential operators. Distribution theory, Fourier transform of distributions. Function spaces. Applications to concrete differential operator problems.

Learning Outcome

Knowledge:

  • 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

 

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

See Absalon for course literature.

Literature may include:

Springer Graduate Text in Mathematics: Gerd Grubb, Distributions and Operators.

A knowledge of Banach and Hilbert spaces corresponding to AdVec or similar.
Knowledge of Functional Analysis is not necessary, but may be helpful.
Academic qualifications equivalent to a BSc degree is recommended.
5 hours of lectures and 2 hours of exercises each week for 8 weeks
  • Category
  • Hours
  • Lectures
  • 40
  • Preparation
  • 117
  • Theory exercises
  • 16
  • Guidance
  • 13
  • Exam
  • 20
  • Total
  • 206
Written
Continuous feedback during the course of the semester
Credit
7.5 ECTS
Type of assessment
Continuous assessment
On-site written exam, 3 hours under invigilation
Type of assessment details
Continuous assessment: Two written 7 days take home assignments (in week 3 and 6) count each 10% toward the final grade.
The on-site written exam counts 80% towards the final grade.
Aid
Only certain aids allowed (see description below)

All aids allowed for the take home assignments.
All aids allowed except Generative AI and internet access for the written exam.

Marking scale
7-point grading scale
Censorship form
External censorship
Re-exam

Same as ordinary

On-site Written exam, 3 hours under invigilation. All aids allowed except GAI and internet  (80% of the final grade)  

Written assignment 7. Days. All aids allowed (10% of the final grade)

Written assignment, 7. Days. All aids allowed (10% of the final grade)

 

If the student has not passed the two written assignments for the ordinary exam, the student must submit the missing written assignments in the reexam week.

 

If the student has not passed the onsite Written exam, the student must participate in the on-site written exam for the reexamination.

 

Criteria for exam assesment

The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended learning outcome.