Programme/Approved Electives for 2022/23
Available as a Free Standing Elective
MAT-20008 Differential Equations
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 technique of Green's functions.
The aim of this module is to provide an introduction to analytical methods for solving partial differential equations.
Intended Learning Outcomes
classify partial differential equations (PDEs) into linear, quasi-linear and non-linear types: 1,2solve first order linear and quasi-linear PDEs by the method of characteristics: 1,2classify linear second order PDEs into hyperbolic, parabolic and elliptic types, reduce to canonical form and solve by the method of characteristics: 1,2solve linear second order PDEs by separation of variables, with applications to the wave, diffusion and Laplace┐s equations: 1,2demonstrate knowledge of Sturm-Liouville theory and the application of generalised Fourier series to solve boundary value problems: 1,2solve inhomogeneous linear second order PDEs using the method of eigenfunction expansion: 1,2solve boundary value problems using Fourier transform techniques: 1,2solve linear second order PDEs using Green┐s functions: 2
Lectures: 36 hoursIndependent study: 112 hoursUnseen examination : 2 hours
1: Class Test weighted 30%
Description of Module Assessment
Two class tests with one of them on MobiusThe two tests will be held in weeks 5 and 9, respectively. The first one will be conducted using Mobius.
2: Exam weighted 70%
2 hour closed-book examThe examination paper will consist of no less than five and not more than eight questions all of which are compulsory.