ADVANCED TECHNIQUES IN MATHEMATICS - 2021/2

Module Overview

This module introduces a selection of mathematical techniques which are applicable in a wide range of scientific applications.

Module provider

Mathematics

Module Leader

PRINSLOO Andrea (Physics)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 7

Module cap (Maximum number of students): N/A

Module Availability

Semester 1

Prerequisites / Co-requisites

Pre-requisites for this module are:

• Ordinary Differential Equations (MAT2007);
• Linear PDEs (MAT2011).

Module content

Indicative content includes:

A) Sturm-Liouville Theory

• Orthogonal functions relative to a weight and generalised series expansions.


• Self-adjoint differential operators and self-adjoint ODEs.


• Sturm-Liouville ODEs.


• The variational principle and the Rayleigh-Ritz formula.


• Series solutions and Orthogonal polynomials.


• Application to solving separable linear PDEs.


• The Dirac delta and Green's functions.

B) 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.

C) Asymptotic Approximations

• Asymptotic expansions.


• Solving algebraic equations using regular and singular perturbative expansions.


• Solving differential equations using perturbative expansions.


• Asymptotic evaluation of integrals.

Assessment pattern

Alternative Assessment

N/A

Assessment Strategy

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 and theorems, 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.

The summative assessment for this module consists of:

• one two hour examination – worth 80% of the module mark;


• one in-semester test – worth 20% of the module mark.

The formative assessment and feedback for this module includes:

• written feedback on two marked pieces of unassessed coursework;


• verbal feedback at weekly office hours or tutorials.

Module aims

• Introduce students to the following areas of mathematics: A) Sturm-Liouville Theory, B) Integral Transforms, and C) Asymptotic Approximations
• Enable students to solve problems related to each of these areas.
• Illustrate how these areas of mathematics relate to real world problems in science and engineering.

Learning outcomes

Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• A thorough introduction to 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 – material will be presented on blackboards/whiteboards 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. Q & A opportunities will be provided at weekly office hours or tutorials. Students will be strongly encouraged to make use of the reading list for the module.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Reading list

https://readinglists.surrey.ac.uk
Upon accessing the reading list, please search for the module using the module code: MATM045

Programmes this module appears in

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 2021/2 academic year.