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Unit information: Asymptotics in 2020/21

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Unit name Asymptotics
Unit code MATHM4700
Credit points 20
Level of study M/7
Teaching block(s) Teaching Block 2 (weeks 13 - 24)
Unit director Professor. Mezzadri
Open unit status Not open
Pre-requisites

MATH30800 Mathematical Methods

Co-requisites

None

School/department School of Mathematics
Faculty Faculty of Science

Description

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

Intended learning outcomes

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.

Teaching details

The unit will be taught through a combination of

  • synchronous online and, if subsequently possible, face-to-face lectures
  • asynchronous online materials, including narrated presentations and worked examples
  • guided asynchronous independent activities such as problem sheets and/or other exercises
  • synchronous weekly group problem/example classes, workshops and/or tutorials
  • synchronous weekly group tutorials
  • synchronous weekly office hours

Assessment Details

100% Timed, open-book examination

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

Reading and References

Recommended

  • Carl M. Bender and Steven A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill 1978
  • Norman Bleistein and Richard A. Handelsman Asymptotic Expansions of Integrals, Dover, 1986
  • E.T. Copson, Asymptotic Expansions, Cambridge University Press, 1965
  • E. J. Hinch, Perturbation Methods, Cambridge University Press, 1991
  • J. Kevorkian and J.D. Cole, Multiple Scale and Singular Perturbation Methods ,Springer, 1996
  • C. C. Lin and Lee A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences, Macmillan, 1974

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