Syllabus academic year 2008/2009
(Created 2008-07-17.)
 PARTIAL DIFFERENTIAL EQUATIONS WITH DISTRIBUTION THEORY FMA250

Higher education credits: 7,5. Grading scale: TH. Level: A (Second level). Language of instruction: The course might be given in English. Optional for: D4, E4, F4, F4tmb, F4tvb, Pi3, Pi3bs. Course coordinator: Director of Studies, Lars-Christer Böiers, Lars_Christer.Boiers@math.lth.se, Matematik. Recommended prerequisits: FMA021, FMA120, FMA140, first part of FMA260. Assessment: Written and/or oral test, to be decided by the examiner. Written assignments. Home page: http://www.maths.lth.se/matematiklth/vitahyllan/vitahyllan.html.

Aim
The probably largest class of mathematical models among technichal systems is based on partial differential equations (PDE). An indispensable tool in the modern theory for these equations is distribution theory.

The aim of the course is on the one hand to give a more stable foundation for concepts and methods from earlier courses, and on the other hand a greater ability independently to use these as well as other methods in the field. Furthermore, the course aims to give the analytical background to some frequently used numerical solution methods.

Knowledge and understanding
For a passing grade the student must

be able to explain the foundations of the theory in an oral examination.

be able to explain the concept of a weak solution to a PDE, and its connection to distribution theory.

Skills and abilities
For a passing grade the student must

with access to literature independenly be able to integrate methods and views from different parts of the course in order to solve problems and answer questions within the framework of the course.

in writing and orally, with proper terminology and clear logic be able to explain the solution to a mathematical problem within the course.

Contents
Distribution theory: derivatives, convergence, fundamental solutions, Green's functions, the Fourier transform, the Laplace and the wave operators.

Partial differential equations: spectral methods, eigenfunction expansions, weak solution. Approximation methods. Integral equations, finite element methods, wavelets. Geometrical methods. Characteristics. The study of some model equations.

Literature
Renardy and Rogers: An Introduction to Partial Differential Equations, 2nd ed. Springer . ISBN 0-387-00444-0.
Some further material and completions.