# 2017/18 Taught Postgraduate Module Catalogue

### 20 creditsClass Size: 40

Module manager: Dr Sandro Azaele
Email: S.Azaele@leeds.ac.uk

Taught: Semester 1 View Timetable

Year running 2017/18

### Pre-requisite qualifications

MATH2375 or equivalent.

### This module is mutually exclusive with

 MATH3365 Mathematical Methods

This module is approved as an Elective

### Module summary

This module aims to describe how precise approximations - ie ones in which the error is both understood and controllable - can be obtained using analytical (rather than numerical) techniques.

### Objectives

On completion of this module, students should be armed with numerous mathematical, rather than computational, techniques for solving a wide variety of initial-value and boundary-value problems that arise in the modelling of realistic phenomena in diverse scientific areas.

In particular, students will be able to solve frequently occurring small-parameter problems using a combination of asymptotic methods such as matching, multiple scales (in space and time), and series approximations, and the advanced topics should form a solid foundation for potential research students.

### Syllabus

The governing equations of mathematical models often involve features that make it impossible to obtain their exact solution, eg:
- the occurrence of a complicated algebraic equation
- the occurrence of a complicated integral
- varying coefficients in a differential equation
- an awkwardly shaped boundary
- a non-linear term in a differential equation.

When a large or small parameter occurs in a mathematical model of a process there are various methods of constructing perturbation expansions for the solution of the governing equations. Often the terms in the perturbation expansions are governed by simpler equations for which exact solution techniques are available.

Even if exact solutions cannot be obtained, the numerical methods used to solve the perturbation equations approximately are often easier to construct than the numerical approximation for the original governing equations.

Moreover, analytic perturbation approximations often constitute a powerful validation of any numerical model that might be employed.

Topics include:
- Asymptotic approximations
- Algebraic equations
- Iintegrals
- Rregular perturbations in PDEs
- Boundary layers
- Matched asymptotic expansions
- Strained co-ordinates
- Multiple scales
- Accelerated convergence
- Asymptotic expansion of integrals
- Approximate solution of difference equations.

- Choice from exponential asymptotics
- WKB theory and
- Integral-equation methods.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 44 1.00 44.00 Private study hours 156.00 Total Contact hours 44.00 Total hours (100hr per 10 credits) 200.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Worksheets (with feedback and model solutions).

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated