The purpose of the module is to consolidate students’ knowledge of and skills in modelling, analysis and applications. The module therefore is situated on the interface between Pure and Applied Mathematics and encompasses three important themes relevant to investigating physical phenomena, which will be addressed in series.

Theme 1. Asymptotic Methods

Many mathematical problems contain a small or large parameter that may be exploited to produce approximations to integrals and solutions of differential equations, for example. This theme provides an introduction to asymptotic approximations and perturbation analysis and their applications. Such techniques are important in almost every branch of applied mathematics especially those where exact analytic solutions are not available and numerical solutions are difficult to obtain.

Theme 2. Integral Equations

Many mathematical problems, particularly in applied mathematics, can be formulated in two distinct but related ways, namely as differential equations or integral equations. In the integral equation approach the boundary conditions are incorporated within the formulation of the problem and this confers a valuable advantage to the approach. The integral approach leads naturally to the solution of the problem in terms of an infinite series, known as the Neumann expansion. Integral equations have played a significant role in the history of mathematics. The Laplace and Fourier transforms are examples of integral equations. Another interesting problem is Huygens’ tautochrone problem, which is a special case of Abel’s integral equation. This course is concerned for the most part with linear integral equations. This module will introduce different types of integral equations and develop methods for their analysis and solution.

Theme 3. Calculus of Variations

What is the shortest distance between two points on a surface?  What is the shape of maximum area for a given perimeter?  These are two questions of the many that can be answered using calculus of variations.  The central problem involves an integral containing an unknown function – for example the length of a curve can be expressed as an integral along that curve.  Calculus of variations provides techniques for investigating minima of such integral functionals, which usually represent some physically or geometrically meaningful quantity. One example of great importance in modern technology is the use of minimisation in studying complex patterns observed under some conditions in shape-memory alloys. The course will consider the classical ``indirect'' approach to minimisation problems, through finding solutions of some related differential equations. However, due to some inherent (and indeed physically relevant) limitations of this method, which will become evident during the course, one has to combine it with a ``direct’’ variational technique. The power of the direct method spreads far and wide across the modern applications of mathematics. In particular, it provides a key to various techniques for finding approximate solutions to differential equations. 

This module can be taken by any student who is prepared to solve some differential equations and manipulate integrals.  Although some of the problems studied are of a physical origin, these will be presented in a self-contained way and there are no applied mathematics pre-requisites.

• Obtain perturbation solutions to algebraic equations involving a small parameter.
• Construct perturbation solutions to linear and nonlinear boundary value problems for ODEs.
• Use Laplace's method and Watson's Lemma to construct asymptotic expansions of integrals involving a large parameter.
• Understand how solutions to initial value problems may depend on slow and fast time scales and to be able to apply multiple scale methods to such problems.
• Classify Fredholm and Volterra integral equations of the first and second kind.
• Reformulate certain first and second order differential equations in terms of an integral equation.
• Expand the solution of an integral equation in terms of the Neumann series and analyse the convergence of the sequence of approximations resulting from truncating this series.
• Express the solution of an integral equation in terms of a resolvent kernel and examine its properties.
• Understand the idea of the classical variational method.
• Derive and solve the Euler-Lagrange equations for a variety of types of problems.
• Apply the necessary and sufficient conditions for existence of a minimiser in some simple examples.
• Understand the significance of the direct variational method.

54 - 50 minute lectures.

Some lecture notes and supplementary handouts will be made available on Learning Central, but students will be expected to take notes of lectures. Computer demonstrations may be used in lectures to illustrate the material, but students will not be expected to perform numerical calculations using computers.

Students are also expected to undertake at least 100 hours private study including preparation of solutions for selected exercises.

Skills:

Analytical and logical skills associated with the analysis, solution and approximation of integral equations and variational problems.

Transferable Skills:

Mathematical modelling of various types of problem. Ability to construct analytical approximations to definite integrals and solutions of differential equations.

Ability to construct analytical solutions to integral equations.

Ability to write necessary conditions for minimisers of physically relevant integrals and to apply appropriate sufficient conditions.

Formative assessment is carried out by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures.  

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of four from six equally weighted questions.

• Introduction to perturbation theory: algebraic equations; ordinary differential equations.
• Asymptotic series and approximations
• Asymptotic expansion of integrals: Laplace’s method, Watson’s lemma.
• Method of multiple scales.
• Classification of integral equations. Linear integral equations: Fredholm and Volterra equations of the first and second kind. Kernels: symmetric, separable, singular. Singular integral equations. Nonlinear equations.
• Connection with differential equations. First and second order differential equations. Green’s function.
• Method of Successive Approximations. Neumann series. Convergence of the Neumann series for Fredholm and Volterra equations of the second kind. Iterates and the resolvent kernel.
• The Resolvent. Resolvent equation. Uniqueness theorem. Characteristic values and functions. The Neumann series revisited. Degenerate kernels.
• Introduction to and historical overview of calculus of variations.  Some classical problems: brachistochrone problem, minimum energy shapes, etc. 
• ``Indirect’’ method and its limitations: classical and weak solutions to the Euler-Lagrange equation, their regularity.
• Necessary and sufficient conditions for the existence of a minimiser of an integral functional.
• Direct method of calculus of variations: minimising sequences, compactness.

Perturbation Methods, E J Hinch, Cambridge University Press

Asymptotic Analysis, J D Murray, Clarendon Press, Oxford.

Linear integral equations: theory and technique, R. P. Kanwal, Academic Press.

Integral equations, B.L. Moiseiwitsch, Longman

Integral equations, F. Smithies, Cambridge University Press

Integral equations, F. G. Tricomi, Interscience.

Calculus of Variations, Gelfand I M and Fomin S V, Dover

Introduction to the Calculus of Variations, Dacorogna B, World Scientific

Not applicable.