# ADVANCED TECHNIQUES IN MATHEMATICS - 2018/9

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 AH Dr (Maths)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 7

JACs code: G100

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) 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.

B) Orthogonal Functions and Sturm-Liouville Theory

Orthogonal functions relative to a weight and generalised series expansions;
Orthogonal polynomials;
Application to solving separable linear PDEs;
The Dirac delta and Green's functions.

C) 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.

D) Perturbation Theory

Asymptotic expansions;
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.

Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-Semester Test 20
Examination Examination 80

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) Variational Methods, B) Orthogonal Functions and Sturm-Liouville Theory, C) Integral Transforms, and D) Perturbation Theory.
• 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 In the area of Variational Methods, students will be able to do some or all of the following: 1) Derive the Euler-Lagrange equation using the calculus of variations; 2) Use variational methods to extremise integrals with and without constraints; 3) Use variational methods and the principal of least action to minimise an action functional in Classical Mechanics, and use Noether's Theorem to identify the symmetries of the action and calculate the associated conserved charges; 4) Use variational methods to extremise multiple integrals over multivariable functions; particularly, to solve for minimal surfaces; 5) Use variational methods and the principal of least action to minimise an action functional in Classical Field Theory. KCT
002 In the area of Orthogonal Functions and Sturm-Liouville Theory, students will be able to do some or all of the following: 1) Determine whether or not a linear differential operator subject to specific boundary conditions is self-adjoint; 2) Identify a regular, periodic or singular Sturm-Liouville problem, and hence quote the properties of its eigenfunctions and eigenvalues; 3) Transform a second order linear ODEs into Sturm-Liouville form and identify the weight function; 4) Find the solution of a regular, periodic or singular Sturm-Liouville problem in the form of a generalised series expansion over orthogonal functions; 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 in terms of generalised series expansions over orthogonal functions; 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
003 In the area of 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
004 In the area of Perturbation Theory, students will be able to do some or all of the following: 1) Express the solutions of an algebraic equation in the form of a regular or a singular perturbative expansion in a small parameter; 2) Express the solution of a differential equation in the form of a perturbative expansion in a small parameter, and determine whether or not the asymptotic nature of the expansion breaks either at a spatial boundary or at late times; 3) Use the method of matched asymptotic expansions to find the solution of a differential equation, with a boundary layer, in the form of two perturbative expansions inside and outside the boundary layer, which match in an intermediate region; 4) Use the method of multiple scales to find the solution of a differential equation, with fast and slow time scales, in the form of a perturbative expansion which is valid at late times. KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

Independent Study Hours: 117

Lecture Hours: 33

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.