Partial Differential Equations
Syllabus, Master's level, 1MA216
- Code
- 1MA216
- Education cycle
- Second cycle
- Main field(s) of study and in-depth level
- Mathematics A1N
- Grading system
- Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
- Finalised by
- The Faculty Board of Science and Technology, 9 April 2015
- Responsible department
- Department of Mathematics
Entry requirements
120 credits including 90 credits of Mathematics including Real Analysis.
Learning outcomes
On completion of the course, the student should be able to
- account for central properties of solutions to Laplace's equation, the heat equation and wave equations;
- solve non-linear equations of the first order with the method with characteristics;
- account for Sobolev spaces and central properties of Sobolev functions like theorems concerning approximation, extension and traces as well as the Sobolev inequalities and theorems concerning compactness;
- account for existence - and uniqueness theorems of weak solutions to elliptic equations of the second order;
- account for regularity theory, maximum principles and eigenvalues/eigenfunctions for the elliptic equations of the order;
- account for the theory of the parabolic and hyperbolic equations of the second order;
- explain and use the theory of calculus of variations and, in particular, the concept of minimiser and to account for the regularity of minimisers;
Content
Laplace equation. Heat equation. The wave equation. Non-linear equations of the first order. The methods with characteristics. Some methods to design explicit solutions. Sobolev spaces and approximation theorems for Sobolev functions. Extension and trace theorems. The Sobolev inequalities and theorems concerning compactness. Existence and uniqueness of weak solutions to elliptic equations of second order. Regularity theory, maximum principles and eigenvalues/eigenfunctions for the elliptic equations of second order. The parabolic equations of second order.
The hyperbolic equations of second order. Calculus of variations. Existence of minimisers. Regularity theory.
Instruction
Lectures,problem solving sessions, student presentations.
Assessment
Assignments and oral examination.
Other directives
The course can not be included in higher education qualification together with Partial Differential Equations, introductory course, Partial Differential Equations, advanced course, or the equivalent.
Reading list
- Reading list valid from Autumn 2022, version 2
- Reading list valid from Autumn 2022, version 1
- Reading list valid from Autumn 2019
- Reading list valid from Spring 2019
- Reading list valid from Autumn 2016
- Reading list valid from Autumn 2015
- Reading list valid from Autumn 2013
- Reading list valid from Autumn 2012, version 3
- Reading list valid from Autumn 2012, version 2
- Reading list valid from Autumn 2012, version 1