*Course Syllabus for*
# Partial Differential Equations

Partiella differentialekvationer

## FMA145F, 7.5 credits

**Valid from:** Autumn 2018

**Decided by:** Professor Thomas Johansson

**Date of establishment:** 2018-04-24

## 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:** MATP16

**Teaching language:** English

## Aim

The aim of is to give a thorough introduction to the modern mathematical theory for the partial differential equations that are of greatest importance in physics. For this purpose an introduction to the theory of distributions and the theory of Sobolev spaces.

## Goals

*Knowledge and Understanding*

For a passing grade the doctoral student must

- be able to account for the basics of distribution theory and the theory of Sobolev spaces.
- be able to describe the three main classes of second order equations: elliptic, parabolic and hyperbolic, and describe the properties of their solutions.

*Competences and Skills*

For a passing grade the doctoral student must

- be able to formulate and prove the most important theorems.
- be able to use the method of characteristics to solve first order linear equations.

*Judgement and Approach*

For a passing grade the doctoral student must
be able to explain the concept "well-posed problem" and its importance when modelling with differential equations.

## Course Contents

Quasi-linear equations of the first order. Classification of second-order equations. The Cauchy-Kowalevski theorem. The Holmgren uniqueness theorem. The Laplace equation. The wave equation. The heat equation.

## Course Literature

Evans, Lawrence C.: Partial Differential Equations. American Mathematical Soc., 2010. ISBN 9780821849743.

**Types of instruction:** Lectures, seminars

**Examination formats:** Written exam, oral exam, written assignments

**Grading scale:** Failed, pass

**Examiner:**

## Admission Details

**Admission requirements:** For admission to the course, English B is required as well as at least 82.5 credits in mathematics in which should be included the courses MATC11 Analytic functions, 15 credits and MATM14 Ordinary Differential Equations, 7.5 credits or the equivalent.

## Course Occasion Information

**Course coordinators:**

**Web page:** https://liveatlund.lu.se/departments/Mathnfak/MATM19/Pages/default.aspx