Advanced Mathematical Methods

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This document only provides information for the academic year selected and does not form part of the student contract

School:

School of Computing and Engineering

Credit Rating:

Level (including FHEQ):

M (FHEQ Level 7)

Graded or Non Graded:

Graded

Version Valid From:

2022-09-01

Module Leader:

Ciprian Coman

Version Number

2024.02

Learning Methods

Guided Independent Study

Lecture

Practical Classes and Demonstrations

Synopsis

In this module you will develop and apply various methods used to derive, analyse and solve ordinary and partial differential equations, which often exhibit unusual properties rendering them beyond standard techniques. While the differential equations arise in various applications in this module you will focus on the methods used to … For more content click the Read More button below.

In this module you will develop and apply various methods used to derive, analyse and solve ordinary and partial differential equations, which often exhibit unusual properties rendering them beyond standard techniques. While the differential equations arise in various applications in this module you will focus on the methods used to solve them rather than the context. You will then apply the methods to specific differential equations found in applications with particular emphasis on active areas of research within the University.

Learning Strategy

Lectures will develop the techniques with many examples provided. Regular tutorials will give the students opportunity to practice and consolidate skills. Where possible mathematical software (e.g. Maple) will be used to illustrate solutions and allow students to focus on higher level concepts without being distracted by routine algebraic steps.

Outline Syllabus

• Methods of solving and analysing systems of linear and nonlinear ordinary differential equations (ODEs). Systems of ODEs, fundamental matrix, principal fundamental matrix. Matrix exponentials and their application to the solution of homogeneous linear systems of ODEs. The method of variation of parameters for systems of ODEs. Introduction to dynamical systems; phase portraits; equilibrium points & their classification; stability of trajectories. Linearisation of non-linear systems, Hartman-Grobman theorem, Poincare-Bendixson theorem and its applications to limit cycles.
• Special methods for periodic coefficients and solutions. Floquet theory (linear systems with time periodic coefficients), multipliers and characteristic exponents, Mathieu’s equation (characteristic curves). Applications to the stability of closed (phase-space) trajectories. Approximations of limit cycles (harmonic balance, averaging, Poincare-Lindstedt, etc).
• Laplace transforms and their applications to initial-value problems for ODEs and partial differential equations (PDEs). Inversion of Laplace transforms. Introduction to Fourier transforms and applications to PDEs.

Learning Outcomes

On successful completion of this module students will

Critically evaluate mathematical systems according to the types of solutions expected and the methods needed to solve them, making use of a comprehensive understanding of solution techniques.

Analyse and, where possible, solve systems of equations involving differential operators.

Formulate and solve complex problems such as determining periodic solutions and finding initial values.

Formative Assessment

Assessment 1: Written practice exercise

Summative Assessment

Assessment 1: Written Assignment

Description:The work should comprise a written element demonstrating the ability to solve a range of problems involving differential operators such as determining periodic solutions and finding solutions to initial-value problems by selecting an appropriate method and working through the steps of the solution.

Assessment Weight:100

Assessment Length:6000 Words (Word Count)

Module Learning Outcomes:MLO1, MLO2, MLO3

Individual or Group:Individual

Final Assessment Component:Yes

Marked Anonymously:No

Eligible for Tutor Reassessment:Yes

Assessment Criteria

• Methodology• Clarity of exposition• Correctness of results

Course and Module Catalogue