The University of Southampton
Courses

# MATH3083 Advanced Partial Differential Equations

## Module Overview

Partial Differential Equations (PDEs) occur frequently in many areas of mathematics. This module extends earlier work on PDEs by presenting a variety of more advanced solution techniques together with some of the underlying theory.

### Aims and Objectives

#### Module Aims

This module extends earlier work on PDEs by presenting a variety of more advanced solution techniques together with some of the underlying theory.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Demonstrate knowledge and understanding partial differential equations and how they relate to different modelling situations
• Solve linear ODEs and PDEs with the use of the Green's function method.
• Show logical thinking in problem solving.
• Understand the concept of well-posedness.
• Understand the concept of the symbol of a PDE and the resulting classification of PDEs.
• Understand the existence of weak solutions and shocks.
• Understand similarity solutions and their applications.

### Syllabus

Common linear PDEs and their boundary conditions Cauchy data and the Cauchy-Kowalewski expansion Weak solutions of linear PDEs Well-posedness Classification of PDEs and PDE systems from their principal symbol Scalar conservation laws and the Riemann problem Generalised functions and the delta-function Green's functions for ODEs Green's functions and applications for Laplace, Poisson and Helmholtz equations Green's functions and applications for the heat equation Green's functions and applications for the the wave equation

### Learning and Teaching

#### Teaching and learning methods

Teaching methods include: • 12 tutorials in the form of problem classes • Summary lecture notes, solutions to problems and mock examination paper available on the Blackboard site for the module • Chalk and talk lectures Learning activities include: • Individual Study • Working through problems set in lecture notes at the end of each chapter • Working through mock exam paper and the coursework assignment

TypeHours
Independent Study102
Tutorial12
Lecture36
Total study time150

HOWISON S.. Practical Applied Mathematics.

Lecture notes. The module is based on summary lecture notes which are provided. There is no recommended book, but some suggestions are contained in an information sheet handed out at the start of the module and available on blackboard.

OCKENDON J, HOWISON S, LACEY A & MOVCHAN A. Applied Partial Differential Equations.

LEVEQUE R.. Numerical Methods for Conservations Laws.

STRAUSS W.. Partial differential equations.

### Assessment

#### Summative

MethodPercentage contribution
Coursework 20%
Written exam 80%

#### Referral

MethodPercentage contribution
Written exam 100%

#### Repeat Information

Repeat type: Internal & External