Differential Operators and Function Spaces (DifFun)
Course content
Differential operators. Distribution theory, Fourier transform of distributions. Function spaces. Applications to concrete differential operator problems.
MSc Programme in Mathematics
MSc Programme in Statistics
MSc Programme in Mathematics with a minor subject
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
5 hours of lectures and 2 hours of exercises each week for 8 weeks
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.
- ECTS
- 7,5 ECTS
- Type of assessment
-
Written assignment, Two 7 days take home assignmentsOn-site written exam, 3 hours under invigilation
- Type of assessment details
- The two written 7 days take home 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
- Re-exam
-
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 assessment
The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended learning outcome.
Single subject courses (day)
- Category
- Hours
- Lectures
- 40
- Preparation
- 117
- Theory exercises
- 16
- Guidance
- 13
- Exam
- 20
- English
- 206
Kursusinformation
- Language
- English
- Course number
- NMAK10019U
- ECTS
- 7,5 ECTS
- Programme level
- Full Degree Master
- Duration
-
1 block
- Placement
- Block 3
- Schedulegroup
-
C
- Capacity
- No limitation – unless you register in the late-registration period (BSc and MSc) or as a credit or single subject student.
- Studyboard
- Study Board of Mathematics and Computer Science
Contracting department
- Department of Mathematical Sciences
Contracting faculty
- Faculty of Science
Course Coordinators
- Jan Philip Solovej (7-7b7774777e6d724875697c7036737d366c73)
- Søren Fournais (8-6f787e7b776a727c49766a7d7137747e376d74)
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