# MAGIC058: Theory of Partial Differential Equations

## Course details

A core MAGIC course

### Semester

Spring 2014
Monday, January 20th to Friday, March 28th

### Hours

Live lecture hours
20
Recorded lecture hours
0
0

Mondays
09:05 - 09:55
Fridays
09:05 - 09:55

## Description

This course is intended to provide an introduction to the theory of partial differential equations (pdes).
Definitions and examples are given of first, second and higher order pdes and also systems of first order pdes.
The symbol of a pde or system of pdes is defined and the concepts of elliptic, parabolic, hyperbolic (and in the case of systems, mixed) pdes is given. Characteristic directions are defined and properties developed. Well- and ill-posedness is explained and the importance of existence and uniqueness (or otherwise) is discussed. This section of the course (approximately 10 lectures) leads up to a statement of the Cauchy-Kowalewskaya theorem.
Concepts from Functional Analysis required for the theory of pdes are briefly explained. The important ideas are those of functionals, distributions and various spaces of functions, e.g. metric, normed linear, Banach, inner product and Hilbert. Online material ("Crash courses in...") may be used as a resource to help in understanding some basic Functional Analysis. A brief introduction is given to Sobolev Spaces - these are the spaces which are suitable for the study of pdes. The concept of a weak formulation and weak solution of a pde is given. This section of the course leads up to the existence and uniqueness of linear second order elliptic pdes.

### Prerequisites

I have tried to make this course as self contained as possible. I am assuming some familiarity with Partial Differential Equations (from undergraduate methods courses) but such knowledge is not necessary to take the course. I am also assuming some familiarity with elementary Real Analysis (say from first and second year undergraduate courses).
One cannot get very far with the theory of partial differential equations without using Functional Analysis. I am not assuming knowledge of Functional Analysis (as applied mathematics students may not have taken such options). I will introduce, informally and without proofs, some aspects of Functional Analysis which are necessary to read and understand books on the theory of partial differential equations. I will provide some online notes ("Crash Courses in...") as a resource.
The material in the Crash Courses will not be explicitly assessed (the assessment will be solely on the material presented in the lectures) but someone with no knowledge of Functional Analysis would be well advised to read the Crash Courses on an informal level to enhance their understanding (e.g. words that are used in the lectures may be defined and explained in the Crash Courses).

### Syllabus

• Notation, definitions. Symbol of a pde and of systems.
• Single pdes of orders 1, 2 and higher. Systems of first order pdes.
• Examples from applied mathematics.
• Characteristics.
• Existence, uniqueness and continuous dependence on the data: well- and ill-posedness.
• Hyperbolic pdes and systems.
• Cauchy-Kowalewskaya theorem.
• Brief exposition of necessary Functional Analysis, e.g. functionals, distributions, differentiation, Sobolev spaces).
• Weak and strong solutions.
• Linear elliptic pde's, coercivity/energy estimates; Lax-Milgram lemma, Garding's inequality, existence and uniqueness of weak solutions.

## Lecturer

• DH

### Dr David Harris

University
University of Manchester

## Bibliography

### Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

• An Introduction to Partial Differential Equations, 2nd. edition (M. Renardy and R.C. Rogers, )

## Assessment

### Description

The assessment for this course is via a single take-home Assessment paper which is online now. The deadline for completing and submitting your answers online is April 27th. Your script should be handwritten.

### Assessment not available

Assessments are only visible to those being assessed for the course.