The purpose of this double 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 introduces 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 equation approach leads naturally to the solution of the problem in terms of an infinite series, known as the Neumann series. 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 themeis 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 themewill consider the classical approach to minimisation problems, through finding solutions of some related differential equations.

This module can be taken by any student who is prepared to solve 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 and asymptotic expansions to solutions of 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.

•           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.

•           Derive and solve the Euler-Lagrange equations for a variety of minimisation problems.

•           Apply the necessary and sufficient conditions for existence of a minimiser in some simple examples.

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

Skills that will be practised and developed:

Analytical and logical skills associated with the analysis, solution and approximation of differential equations, 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.

·         Introduction to perturbation theory: algebraic equations; ordinary differential equations.

·         Asymptotic series and approximations

·         Boundary layers and matched asymptotic expansions

·         Asymptotic expansion of integrals: Laplace’s method, Watson’s lemma.

·         Poincaré-Linstedt method

·         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

·         Method of Successive Approximations. Neumann series. Convergence of the Neumann series for Fredholm and Volterra equations of the second kind. Iterated and resolvent kernels.

·         Degenerate kernels and the Fredholm alternative

·         Eigenvalues and eigenfunctions

·         Introduction to and historical overview of calculus of variations.  Some classical problems: brachistochrone problem, minimum energy shapes, etc

·         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.