ADVANCED TECHNIQUES IN MATHEMATICS - 2017/8
Module code: MATM045
PRINSLOO AH Dr (Maths)
Number of Credits
FHEQ Level 7
Module cap (Maximum number of students)
Overall student workload
Independent Study Hours: 117
Lecture Hours: 36
Tutorial Hours: 20
|Assessment type||Unit of assessment||Weighting|
|School-timetabled exam/test||IN-SEMESTER TEST||20%|
Prerequisites / Co-requisites
MAT2007 Ordinary Differential Equations, MAT2011 Linear PDEs
This module introduces a selection of mathematical techniques which are applicable in a wide range of scientific applications.
|Use the method of Calculus of variations to find extremals of integrals with constraints.||KCT|
|Successfully identify and form expansions of functions using orthogonal functions such as Chebyshev polynomials.||KCT|
|Recognise which integral transform is appropriate to solve ODEs and PDEs and evaluate both the transform and inverse transforms using contour integration where necessary.||KC|
|Successfully identify and express solutions to algebraic and ordinary differential equations which contain a small parameter in the form of asymptotic solutions. Included in this will be determining where in the domain boundary-layers exist.OR Be able to find the fast and slow manifolds for a problem which has different time scales and solve problems involving multiple time scales.||KCT|
C - Cognitive/analytical
K - Subject knowledge
T - Transferable skills
P - Professional/Practical skills
Indicative content includes:
1. Variational Methods
Varying integrals and the Euler-Lagrange equation;
Varying integrals subject to constraints;
Application to Classical Mechanics: the principal of least action and symmetries;
Varying multiple integrals over multivariable functions;
Application to Classical Field Theory
2. Orthogonal Functions and Sturm-Liouville Theory
Orthogonal functions relative to a weight and generalised series expansions;
Self-adjoint differential operators and self-adjoint ODEs;
Application to solving separable linear PDEs;
The Dirac delta and Green's functions.
3. Integral Transforms
Fourier transforms, and inverse Fourier Transforms using residue calculus;
Application of Fourier Transforms to solving ODEs and PDEs;
Laplace transforms, and inverse Laplace transforms using residue calculus;
Application of Laplace transforms to solving ODEs.
4. Perturbation Theory
Solving algebraic equations using regular and singular perturbative expansions;
Solving differential equations using perturbative expansions: the method of matched asymptotics expansions and the method of multiple scales.
Methods of Teaching / Learning
The learning and teaching strategy is designed to provide:
A thorough advanced mathematical techniques which can be applied to problems which have a wide range of applications.
Experience (through demonstration) of critically assessing and determining the appropriate techniques to solve advanced mathematical problems.
The learning and teaching methods include:
3 hours of lectures per week x 11 weeks – module material will be presented on the blackboard/whiteboard and may be supplemented by module notes.
In addition to attending lectures and reading the module notes, students will learn by attempting a wide range of exercises and unassessed coursework problems.
Students will be strongly encouraged to use the books listed as background reading for the module.
The assessment strategy is designed to provide students with an opportunity to demonstrate:
Understanding of the methods and techniques required to solve problems in the topics listed in the module aims.
Subject knowledge through the recall of definitions , as well as the choice of technique required to solve particular problems.
Analytic ability through the solution of unseen and similar to seen problems in the test and exam.
Thus, the summative assessment for this module consists of:
One two hour examination at the end of Semester 1 – worth 80% of the module mark.
One in-semester test – worth 20% of the module mark.
Formative assessment and feedback
Students will receive written feedback in the form of two marked pieces of unassessed coursework. Additional verbal feedback will be provided by the lecturer at optional weekly tutorials or office-hours.
Reading list for ADVANCED TECHNIQUES IN MATHEMATICS : http://aspire.surrey.ac.uk/modules/matm045
Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2017/8 academic year.