# MAGIC058: Theory of Partial Differential Equations

## Course details

A core MAGIC course

### Semester

Spring 2012
Monday, January 16th to Friday, March 23rd

### Hours

Live lecture hours
20
Recorded lecture hours
0
0

Mondays
11:05 - 11:55
Thursdays
11:05 - 11:55

## Description

Introduction to the theory of pdes for applied mathematics.

### Prerequisites

Undergraduate courses on real analysis and partial differential equations(methods courses) will be assumed without explicit mention. Functionalanalysis is more problematic (as applied mathematics students may not havetaken such options) but time constraints prevent assuming no priorknowledge. Probably the best way forward is to present some necessaryfunctional analysis briefly during the lectures and to provide ädditional"notes online and together with careful page references to books covering thematerial in the hope that students who have little or no functional analysiswill wish to learn more in ßelf-study" as a means to coming to a deeperunderstanding of the "theory" of PDEs.

### Syllabus

• Systems of first order pdes and single pdes of higher order, examples from continuum mechanics
• Symbol of a pde and of systems; characteristics; existence, uniqueness and continuous dependence on the data; well- and ill-posedness.
• (Brief exposition of necessary functional analysis, e.g. operator theory, distributions, Sobolev spaces).
• Weak and strong solutions.
• Maximum principles for elliptic and parabolic pde's, existence of solutions.
• Linear elliptic pde's, coercivity/energy estimates; Lax-Milgram lemma, Garding's inequality, existence and uniqueness of weak solutions.
• Evolutionary pde's - abstract parabolic initial value problems, energy methods, uniqueness and existence.
• Nonlinear elliptic pde's, monotone operators, existence of a weak solution.
• Systems of hyperbolic equations; Symmetrisable systems; well-posedness.
• Introduction to semi-group methods.

## Lecturer

• DH

### Dr David Harris

University
University of Manchester

## Bibliography

No bibliography has been specified for this course.

## Assessment

The assessment for this course will be released on Monday 21st September 2020 and is due in by Monday 9th April 2012 at 23:59.

There are two assessments.
They are equally weighted.
Assessment 1 was due April 9th.
Assessment 2 is due May 30th.

Please note that you are not registered for assessment on this course.