Theory of Partial Differential Equation

5 credits

Syllabus, Master's level, 1MA054

This course has been discontinued.

A revised version of the syllabus is available.

Code: 1MA054
Education cycle: Second cycle
Main field(s) of study and in-depth level: Mathematics A1F
Grading system: Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
Finalised by: The Faculty Board of Science and Technology, 15 March 2007
Responsible department: Department of Mathematics

Entry requirements

BSc, Introduction to Partial Differential Equations, Functional Analysis I

Learning outcomes

The course objective is to further develop the theory of hyperbolic, parabolic and elliptic partial differential equations in connection to physical, technical and scientific problems. Main themes are well-posedness for various initial and boundary value problems and properties of solutions of the wave equation, the heat equation and the Laplace equation. Functional analysis is used as a tool for proving existence of weak solutions and studying regularity of solutions. Important areas of application are found in numerical analysis, optimisation theory, control theory, signal processing, image analysis, mechanics, solid mechanics and quantum mechanics

In order to pass the course (grade 3) the student should be able to

give an account of important concepts and definitions in the area of the course;

exemplify and interpret important concepts in specific cases;

formulate important results and theorems covered by the course;

describe the main features of the proofs of important theorems;

express problems from relevant areas of applications in a mathematical form suitable for further analysis;

use the theory, methods and techniques of the course to solve mathematical problems;

present mathematical arguments to others.

Higher grades, 4 or 5, require a higher level of proficiency. The student should be able to solve problems of greater complexity, i.e. problems requiring a combination of ideas and methods for their solution, and be able to give a more detailed account of the proofs of important theorems and by examples and counter-examples be able to motivate the scope of various results. Requirements concerning the student's ability to present mathematical arguments and reasoning are greater.

Content

Characteristics. Symbols of partial differential operators. The maximum principle. Sobolev spaces. Linear elliptic equations. Energy methods for Cachy problems for parabolic and hyperbolic equations. Fredholm theory and eigen function expansions. Potential theory.

Instruction

Lectures and problem solving sessions.

Assessment

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.