# Unit information: Asymptotics in 2022/23

Please note: It is possible that the information shown for future academic years may change due to developments in the relevant academic field. Optional unit availability varies depending on both staffing, student choice and timetabling constraints.

Unit name Asymptotics MATHM4700 20 M/7 Teaching Block 2 (weeks 13 - 24) Professor. Mezzadri Not open MATH30800 Mathematical Methods None None School of Mathematics Faculty of Science

## Unit Information

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.

## Your learning on this unit

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.

Syllabus

1. Perturbation methods for Algebraic Equations.

2. Basic Definitions and Asymptotic Relations.

3. Expansion of solutions of linear equations in a neighbourhood of an irregular singular point.

4. Asymptotic Expansion of integrals.

5. Regular perturbation of ODEs.

6. Fredholm Alternative.

7. Boundary Layers.

8. WKB approach.

9. Multiple Scales Method.

10. Averaging Method.

## How you will learn

The unit will be taught through a combination of

Face-to-face lectures

• Problem classes
• Online material

## How you will be assessed

90% Timed, open-book examination

10% coursework (two assessed homework sheets)

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.

## Resources

If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.

If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATHM4700).

How much time the unit requires
Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours of study to complete. Your total learning time is made up of contact time, directed learning tasks, independent learning and assessment activity.