Programme/Approved Electives for 2021/22
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: 2solve boundary value problems using Fourier transform techniques: 2solve linear second order PDEs using Green┐s functions: 2
Lectures: 36 hoursIndependent study: 112 hoursUnseen examination : 2 hours
1: Exercise weighted 20%
Description of Module Assessment
One mid-semester take-home assessmentThis single take-home assessment will be held around week 6.
2: Open Book Examination weighted 80%
2 hour Take-home examThe examination paper will consist of no less than five and not more than eight questions all of which are compulsory.