School of Computing and Mathematics

Faculty of Natural Sciences

For academic year: 2019/20 Last Updated: 11 November 2019

MAT-30003 - Partial Differential Equations

Mathematics Minor (Level 6)

None

MAT-20008 Differential Equations

None

This module provides an introduction to analytical methods for solving partial differential equations (PDEs). Throughout the module focuses on PDEs in two independent variables, although generalisation to three, or more, independent variables is briefly discussed. The module begins by introducing the method of characteristics for solving first order linear and quasi-linear PDEs. This is followed by the classification of linear second order PDEs into hyperbolic, parabolic or elliptic type, with a detailed treatment of the reduction to canonical form in each case. Thereafter, the module introduces the separation of variables technique and transform methods to solve the wave, diffusion, Laplace and Poisson equations. These PDEs are frequently encountered in many branches of applied mathematics, including fluid dynamics, mathematical biology, financial mathematics and electromagnetism. Finally the module introduces the technique of eigenfunction expansions for solving inhomogeneous PDEs, and the powerful technique of Green's functions.

The module develops the following Keele Graduate attributes:

1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought.

2. An appreciation of the development and value of Mathematics and the links between different areas of the subject.

4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand.

6. The ability to communicate clearly and effectively in written form.

The module develops the following Keele Graduate attributes:

1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought.

2. An appreciation of the development and value of Mathematics and the links between different areas of the subject.

4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand.

6. The ability to communicate clearly and effectively in written form.

The aim of this module is to provide an introduction to analytical methods for solving partial differential equations.

classify partial differential equations (PDEs) into linear, quasi-linear and non-linear types: 1,2

solve first order linear and quasi-linear PDEs by the method of characteristics: 1,2

classify linear second order PDEs into hyperbolic, parabolic and elliptic types, reduce to canonical form and solve by the method of characteristics: 1,2

solve linear second order PDEs by separation of variables, with applications to the wave, diffusion and Laplace¿s equations: 1,2

demonstrate knowledge of Sturm-Liouville theory and the application of generalised Fourier series to solve boundary value problems: 1,2

solve inhomogeneous linear second order PDEs using the method of eigenfunction expansion: 1,2

solve boundary value problems using Fourier transform techniques: 1,2

solve linear second order PDEs using Green¿s functions: 1,2

solve first order linear and quasi-linear PDEs by the method of characteristics: 1,2

classify linear second order PDEs into hyperbolic, parabolic and elliptic types, reduce to canonical form and solve by the method of characteristics: 1,2

solve linear second order PDEs by separation of variables, with applications to the wave, diffusion and Laplace¿s equations: 1,2

demonstrate knowledge of Sturm-Liouville theory and the application of generalised Fourier series to solve boundary value problems: 1,2

solve inhomogeneous linear second order PDEs using the method of eigenfunction expansion: 1,2

solve boundary value problems using Fourier transform techniques: 1,2

solve linear second order PDEs using Green¿s functions: 1,2

Lectures: 36 hours

Independent study: 112 hours

Unseen examination : 2 hours

Independent study: 112 hours

Unseen examination : 2 hours

None

Two class tests

Continuous assessment will consist of two class tests which are held in weeks 4 and 8 approximately, each lasting 40 minutes.

2 HOUR UNSEEN EXAM

The examination paper will consist of no less than five and not more than eight questions all of which are compulsory.