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Asymptotics (MATH M4700)

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Administrative Information

  1. Unit number and title: MATH M4700 Asymptotics
  2. Level: M/7
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 1997/98
  6. Unit Organiser: Rich Kerswell, FRS
  7. Lecturer: Francesco Mezzadri & Rich Kerswell
  8. Teaching block: 1
  9. Prerequisites: MATH 30800 Mathematical Methods

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.

General Description of the Unit

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 Level H/6 Mathematical Methods, and develops further techniques useful throughout applied mathematics.

Teaching Methods

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.

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.

Assessment Methods

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.

Award of Credit Points

Credit points are gained by:

  • either passing the examination (i.e. gaining a final assessment mark of 50 or over);
  • or getting a final assessment mark of 30 or over and also handing in satisfactory attempts to at least five homework assignments.

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.

Texts

  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.
  2. 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
  3. E.T. Copson, Asymptotic Expansions, Cambridge University Press, 1965 (reprinted 2004), Queen's Library: QA312 COP. A classic book on asymptotic expansions.
  4. 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.
  5. J. Kevorkian & J. D. Cole, Multiple scale and singular perturbation methods, Springer, 1996. Queens Library: QA371 SPA
    is an advanced text, useful for reference.
  6. N. Bleistein & R. A. Handelsman Asymptotic expansions of integrals, Dover 1986. Queens Library: QA311 BLE
    is an advanced text, useful for reference.

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.