# School of Mathematics and Statistics

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2011/2012 Sem. 1

## Aims

To introduce students to asymptotic methods used in the construction of analytical approximations to definite integrals and solutions of differential equations.

## Objectives

By the end of the course students are expected to

- be able to use Laplace's method, the method of stationary phase and Watson's Lemma to construct asymptotic expansions of integrals involving a large parameter.

- understand when these methods can and cannot be applied.

- obtain perturbation solutions to algebraic equations involving a small parameter.

- construct perturbation solutions to linear and nonlinear boundary value problems for ODEs.

- identify a singular perturbation problem and to be able to apply the method of matched expansions to such problems.

- 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.

- be able to apply WKB methods to linear equations.

## Syllabus

Definitions of asymptotic series and expansions. Watson's lemma and applications.
Introduction to Laplace's method and method of stationary phase.
Regular and singular perturbation problems for algebraic and differential equations with a small parameter, including multiple scales and strained parameters and coordinates.
Boundary layers and the method of matched asymptotic expansions.
WKB method for linear equations with large parameters.

## Textbooks

Advanced Mathematical Methods for Scientists and Engineers:
C Bender & S Orszag;
Asymptotic Analysis: J D Murray, Springer;
Introduction to Perturbation Techniques: A H Nayfeh, Wiley.
Perturbation Methods for Engineers and Scientists: A W Bush; CRC Press

## Assessment

2 Hour Examination = 100%

MT3504.

## Availability

Academic year 2012/13 in semester 1 at 9

Dr S King