Course Syllabus for

# Partial Differential Equations with Distribution Theory Partiella differentialekvationer med distributionsteori

## FMA250F, 7.5 credits

Valid from: Autumn 2013
Decided by: FN1/Anders Gustafsson
Date of establishment: 2013-11-15

## General Information

Division: Mathematics
Course type: Course given jointly for second and third cycle
The course is also given at second-cycle level with course code: FMA250
Teaching language: English

## 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 theoretical foundation for concepts and methods for PDEs that have been introduced in earlier courses, and a greater ability to independently use these, and on the other hand to develop the theory further. Moreover, the course aims to give the analytical background to some frequently used numerical solution methods.

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• be able to explain the foundations of the theory at an oral examination.
• be able to explain the concept of a weak solution to a PDE, and its connection to distribution theory.

Competences and Skills

For a passing grade the doctoral 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.

## 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 solutions. Approximation methods. Integral equations, finite element methods. Geometrical methods. Characteristics. The study of some model equations.

## Course Literature

Renardy, M. & Rogers, Robert C.: An Introduction to Partial Differential Equations. Springer, 2004. ISBN 9780387004440.
Material distributed by the department.

## Instruction Details

Types of instruction: Lectures, laboratory exercises, exercises

## Examination Details

Examination formats: Written exam, oral exam, written assignments. Written and/or oral test, to be decided by the examiner. Written assignments.
Grading scale: Failed, pass
Examiner:

## Admission Details

Assumed prior knowledge: FMA021, first part of FMA260.

## Contact and Other Information

Course coordinators:
Web page: http://www.ctr.maths.lu.se/course/partdiff/