# Asymptotics

## Unit aims

This unit aims to enhance students' ability to solve the type of equations that arise from applications of mathematics to natural and technological problems by giving a grounding in perturbation techniques. Emphasis is placed on methods of developing asymptotic solutions.

## Unit description

For most equations that arise in modelling applications it is unlikely that exact solutions can be found. Even convergent series approximations are often not available, or they are of limited use if they converge very slowly. Instead, asymptotic expansions can yield good approximations. They are typically divergent if summed to infinity but a few terms can often give excellent and well defined approximations.

This unit introduces the basic ideas and shows how they can be applied to algebraic and differential equations, and to the evaluation of integrals. Usually some parameter or some coordinate value is small (or large), which leads to an expansion of a solution in this parameter. These perturbation expansions can be well behaved (regular) if the perturbation parameter goes to zero, or they can become singular. Most emphasis is placed on the latter, singular perturbations. Practical problems are used as illustrations. These techniques are especially useful when accurate numerical solutions are hard, or impossible, to obtain.

## Relation to other units

This unit is a sequel to Mathematical Methods, and develops further techniques useful throughout applied mathematics.

## Learning objectives

At the end of the unit, the students should be able to take a wide range of mathematical problems and modify the equations in order to find perturbation solutions for at least part of the parameter and coordinate range of interest.

## Transferable skills

Clear logical thinking; problem solving; analysing complex equations, or other mathematical expressions, to obtain the essential ingredients of solutions. Experience in solving a wide range of problems that may be related to other applications.

## Syllabus

There may be minor changes to this syllabus.

- Introduction: solutions of algebraic equations with a small parameter; regular and singular perturbations; convergent series and asymptotic series. Definitions and terminology.
- Local approximations to linear ODEs; irregular singular points.
- Approximation of integrals; Laplace's method, stationary phase method, method of steepest descents.
- Regular perturbations of ODEs, eigenvalue problems.
- Singular perturbations that lead to boundary layers; matched asymptotic expansions.
- Singular perturbations that lead to highly oscillatory functions; WKB approximation.
- The method of multiple scales for finding uniformly valid perturbation expansions.

## Reading and References

- C. M. Bender & S. A. Orszag,
*Advanced mathematical methods for scientists and engineers*, McGraw-Hill 1978, (reprinted by Springer), Queens Library: TA330 BEN is a comprehensive text containing most of the material of the course. - E. J. Hinch,
*Perturbation methods*, Cambridge University Press, 1991, Queens Library: QC20.7.P47 HIN is a succinct account of a large part of the course - E.T. Copson, Asymptotic Expansions, Cambridge University Press, 1965 (reprinted 2004), Queen's Library: QA312 COP. A classic book on asymptotic expansions.
- C. C. Lin & L. A. Segel,
*Mathematics applied to deterministic problems in the natural sciences*, Macmillan, (reprinted by SIAM) 1974, Queens Library: QA37.2 LIN. Part B of this book gives extended discussions that place parts of this course in context. A very readable book for the developing applied mathematician. - J. Kevorkian & J. D. Cole,
*Multiple scale and singular perturbation methods,*Springer, 1996, Queens Library: QA371 SPA is an advanced text, useful for reference. - N. Bleistein & R. A. Handelsman
*Asymptotic expansions of integrals*, Dover 1986, Queens Library: QA311 BLE is an advanced text, useful for reference.

**Unit code:** MATHM4700

**Level of study:** M/7

**Credit points:** 20 credit points

**Teaching block (weeks):** 2 (13-24)

**Lecturer:** Professor Francesco Mezzadri

## Pre-requisites

MATH30800 Mathematical Methods

## Co-requisites

None

## Methods of teaching

The primary content of the course is taught using lectures, with reference to texts and the use of problem sheets to reinforce the material presented.

## Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

- 100% from a 2 hour 30 minute exam in May/June

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.