# ADVANCED TECHNIQUES IN MATHEMATICS - 2021/2

Module code: MATM045

## Module Overview

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

### Module provider

Mathematics

PRINSLOO Andrea (Physics)

## Overall student workload

Independent Learning Hours: 90

Seminar Hours: 22

Guided Learning: 5

Captured Content: 33

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

Assessment type Unit of assessment Weighting
Online Scheduled Summative Class Test Online Test 20
Examination Online Online Exam 80

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

 Attributes Developed 001 Sturm-Liouville Theory: students will be able to do some or all of the following: 1) Identify a regular, periodic or singular Sturm-Liouville problem. 2) Derive a Sturm-Liouville problem from variational principles. 3) Transform a second order linear ODE into Sturm-Liouville form. 4) Solve a Sturm-Liouville problem using a series expansion. 5) Derive the recurrence relations for orthogonal polynomial solutions of Sturm-Liouville problems, including Chebyshev, Legendre, Laguerre and Hermite polynomials. 6) Apply Sturm-Liouville theory to solve separable linear PDEs; particularly, solve the Laplace equation in terms of spherical harmonics. 7) Use Sturm-Liouville theory to solve inhomogeneous ODEs with boundary conditions by finding a Green's function. KCT 002 Integral Transforms: students will be able to do some or all of the following: 1) Quote and derive the properties of Fourier transforms and inverse Fourier transforms, and Laplace transforms. 2) Calculate inverse Fourier transforms and inverse Laplace transforms using residue calculus and Jordan's Lemma. 3) Recognise which integral transform is appropriate to solve a linear ODE or PDE. Use a Fourier transform or Laplace transform to find a solution to the ODE or PDE. KCT 003 Asymptotic Approximations: students will be able to do some or all of the following: 1) Solve an algebraic equation using a regular or a singular perturbative expansion. 2) Use the Method of Matched Asymptotic Expansions to solve perturbatively a differential equation with a boundary layer. 3) Use the Method of Multiple Time Scales to solve perturbatively a differential equation with fast and slow time scales. KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## 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.

Upon accessing the reading list, please search for the module using the module code: MATM045

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics MMath 1 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics MSc 1 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MPhys 1 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MMath 1 Optional A weighted aggregate mark of 50% is required to pass the module

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.