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.

## Instruction Details

Types of instruction: Lectures, seminars

## Examination Details

Examination formats: Written exam, oral exam, written assignments