# 2022/23 Taught Postgraduate Module Catalogue

### 20 creditsClass Size: 40

Module manager: Dr Stephen Griffiths
Email: s.d.griffiths@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2022/23

### Pre-requisite qualifications

MATH2375 or equivalent.

### This module is mutually exclusive with

 MATH3365 Mathematical Methods

This module is approved as an Elective

### Module summary

Mathematical modelling of phenomena in diverse scientific areas often leads to an idealised problem without an exact solution in closed form. This may be due to the occurrence of a complicated integral or algebraic equation, or the presence of a nonlinear term or varying coefficients in a differential equation. Although these problems may be solved numerically, it is often possible to construct approximate solutions if the problem contains a small or large parameter. These approximate solutions, which are typically based upon finding analytical solutions of a series of yet simpler problems, lead to so-called perturbation expansions or asymptotic expansions. In addition to providing useful insight into the nature of the exact solution, these approximate solutions can also be used to validate any numerical solutions that might have been obtained.

### Objectives

(a) To equip students with standard mathematical, rather than computational, techniques for solving a wide range of problems that arise in the modelling of phenomena in diverse areas of science and engineering. The advanced topics form a solid foundation for potential research students in contemporary applied mathematics.
(b) To introduce approximate methods for (i) finding roots of nonlinear equations, (ii) the evaluation of integrals,
(iii) the solution of differential equations.
(c) To understand the nature and properties of (convergent or divergent) asymptotic series at a more fundamental level, along with associated mathematical notation.

### 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:
1. Asymptotic expansions: order symbols, series, gauge functions, optimal truncation, accelerated convergence.
2. Roots of equations: regular and singular roots of algebraic and transcendental equations.
3. Asymptotic expansion of integrals.
4. Boundary-value problems: regular and singular problems, boundary layers, matching.
5. Initial-value problems: secularities, and their removal (e.g., by strained coordinates, multiple scales).
6. A range of advanced topics, perhaps drawn from WKB theory for differential equations, dominant balance
for differential equations, perturbation theory for eigenvalue problems, further integration techniques (e.g.,
Watson’s Lemma, method of stationary phase).

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 33 1.00 33.00 Private study hours 167.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 200.00

### Private study

Studying and revising of course material (online lecture videos).
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