Unit information: Asymptotics in 2013/14

Unit name	Asymptotics
Unit code	MATHM4700
Credit points	20
Level of study	M/7
Teaching block(s)	Teaching Block 1 (weeks 1 - 12)
Unit director	Professor. Kerswell
Open unit status	Not open
Pre-requisites	MATH30800, MATH33000
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

The course treats asymptotic methods, which have their own fascination; asymptotic series are often divergent, yet can be more useful than convergent series as approximations (e.g. Stirling's formula for n! is the first term of one such series). Asymptotic approximations for various types of integral are discussed, both for large parameter values and large domains. Finally, we see how asymptotic methods can be used to find approximate solutions of ordinary differential equations in situations where other methods fail.

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.

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.
• Singular perturbations of partial differential equations.

Relation to Other Units

This unit is a sequel to Level H/6 Mathematical Methods, and develops further techniques useful throughout applied mathematics.

Intended Learning Outcomes

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.

Teaching Information

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. The unit consists of 30 lectures.

Assessment Information

The final assessment mark for Asymptotics is calculated from a 2½-hour written examination in April consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted in this examination.

Reading and References

1. 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.

1. 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

1. E.T. Copson, Asymptotic Expansions, Cambridge University Press, 1965 (reprinted 2004), Queen's Library: QA312 COP. A classic book on asymptotic expansions.
2. 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.

1. J. Kevorkian & J. D. Cole, Multiple scale and singular perturbation methods, Springer, 1996. Queens Library: QA371 SPA

is an advanced text, useful for reference.

1. N. Bleistein & R. A. Handelsman Asymptotic expansions of integrals, Dover 1986. Queens Library: QA311 BLE

is an advanced text, useful for reference.