(Created 2009-08-11.)

PARTIAL DIFFERENTIAL EQUATIONS WITH DISTRIBUTION THEORY | FMA250 |

**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 concept of a weak solution to a PDE, and its connection to distribution theory.

*Skills and abilities*

For a passing grade the student must

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.