Module code: MATM045

Module provider


Module Leader


Number of Credits


ECT Credits



FHEQ Level 7

JACs code


Module cap (Maximum number of students)


Module Availability

Semester 1

Overall student workload

Independent Study Hours: 117

Lecture Hours: 36

Tutorial Hours: 20

Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test IN-SEMESTER TEST 20
Examination EXAMINATION 80

Alternative Assessment


Prerequisites / Co-requisites

MAT2007 Ordinary Differential Equations, MAT2011 Linear PDEs

Module overview

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

Module aims

Learning outcomes

Attributes Developed
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

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

Module content

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;

Orthogonal polynomials;

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

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.


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.


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 , 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


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 MMath/MPhys 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 2017/8 academic year.