MATH3365 Mathematical Methods

15 creditsClass Size: 150

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

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2021/22

Pre-requisite qualifications

MATH2375, or equivalent.

This module is mutually exclusive with

This module is not approved as a discovery module

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.
(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.

Learning outcomes
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.

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).

Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Lecture 17 1.00 17.00 Private study hours 133.00 Total Contact hours 17.00 Total hours (100hr per 10 credits) 150.00

Private study

Consolidation of course notes and background reading.

Opportunities for Formative Feedback

Weekly personal contact with lecturer in examples classes to discuss/provide assistance with regular question sheets.

Assessment of success on examples sheets.

Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Exams
 Exam type Exam duration % of formal assessment Open Book exam 2 hr 30 mins 100.00 Total percentage (Assessment Exams) 100.00

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