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 2020/21

Pre-requisite qualifications

MATH2375, or equivalent.

This module is mutually exclusive with

This module is not approved as a discovery module

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

To give an introduction to approximate methods of solution of ordinary and partial differential equations, and difference equations with a small parameter, since these represent the most important techniques for deriving analytical solutions when modelling real phenomena. Examples will be drawn from many areas of science and engineering.

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

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