Program Overview

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Program Overview

The University of Copenhagen offers a Master's program in Mathematics with a minor subject, which includes a course on Partial Differential Equations (PDE).

Course Description

The course covers a selection of topics, including:

• 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

Learning Outcomes

Upon completing the course, students will have acquired:

• Knowledge: The properties of the PDEs covered in the course
• Skills:
  • Solve classical PDEs
  • Establish existence, uniqueness, and regularity of solutions to certain PDEs
• Competencies:
  • Understand the characteristic properties 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

Literature

The course literature is available on Absalon.

Recommended Academic Qualifications

Students are recommended to have a knowledge of real analysis, Lebesgue measure theory, L^p spaces, and basic theory of Banach/Hilbert spaces, corresponding to at least the contents of the following courses:

• Analyse 0 (An0)
• Analyse 1 (An1)
• Lebesgueintegralet og målteori (LIM)
• Advanced Vector Spaces (AdVec), which may be taken simultaneously with PDEs, or alternatively Functional Analysis (FunkAn) Having academic qualifications equivalent to a BSc degree is recommended.

Teaching and Learning Methods

The course consists of 5 hours of lectures and 2 hours of exercises each week for 8 weeks.

Remarks

After taking this course, students may naturally continue with the next course in the string, "PDE2", which is offered in the subsequent Block 2.

Workload

The workload for the course is:

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

Exam

The exam is an on-site written exam, 4 hours under invigilation, with all aids allowed except Generative AI and internet access. The marking scale is a 7-point grading scale, and there is no external censorship. One internal examiner will be present.

Re-exam

The re-exam is the same as the ordinary exam. If ten or fewer students have signed up for the re-exam, the type of assessment will be changed to a 30-minute oral exam with 30 minutes of preparation time. All aids are allowed during preparation time, but none are allowed during the examination. Several internal examiners will be present.

Criteria for Exam Assessment

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

Course Information

• Language: English
• Course code: NMAK16022U
• Credit: 7.5 ECTS
• Level: Full Degree Master
• Duration: 1 block
• Placement: Block 1
• Schedule: B
• Course capacity: No limitation – unless you register in the late-registration period (BSc and MSc) or as a credit or single subject student.

Study Board

The study board for this course is the Study Board of Mathematics and Computer Science.

Contracting Department

The contracting department for this course is the Department of Mathematical Sciences.

Contracting Faculty

The contracting faculty for this course is the Faculty of Science.

Course Coordinators

The course coordinators are Niels Martin Møller and Léo Morin.