Programme/Approved Electives for 2022/23
Available as a Free Standing Elective
MAT-30003 Partial Differential Equations.
This module introduces students to the asymptotic techniques which are extremely important tools in dealing with mathematical problems arising in the real world. These 'raw' problems very rarely resemble simple models which can be solved exactly. Fortunately, very often 'raw' problems have either large or small parameters, or there is a method for determining an approximate solution which is sufficient for most practical purposes.The module will present an overview of the main analytical perturbation techniques based upon the use of a small / large parameter or small / large values of a coordinate.
The module aims to equip students with a conceptual framework and practical skills f¿or deriving asymptotic models. The module also aims to provide students with the ability to look for and exploit the presence of small/large parameters in a wide range of problems.
Intended Learning Outcomes
identify the presence of a small / large parameter in an unseen linear / non-linear ordinary or partial differential equation and determine the optimal asymptotic technique which successfully exploits the smallness of the identified parameter;: 1critically analyse asymptotic expansions and explain clearly in what respects they differ from the convergent series;: 1perform dimension analysis and utilise this together with scaling arguments to identify the key small parameters in a particular problem;: 1apply regular perturbation methods (including the method of multiple scales and the method of averaging) and interpret solutions from these;: 1identify and analyse situations where singular perturbations are required. In particular, select and apply suitable modifications of matched asymptotic expansion techniques to derive asymptotic solutions;: 1apply the WKB method in appropriate problems and construct asymptotic solutions to equations with slowly varying coefficients.: 11
48 hours classes.149 hours private study.3 hour unseen examination.
1: Exam weighted 100%
Description of Module Assessment
ExaminationA three hour, end of module examination.