This module is not currently running in 2024 to 2025.
Over a century ago, the Norwegian mathematician Sophus Lie made a simple but profound observation: each well-known method for solving a class of ordinary differential equations (ODEs) uses a change of variables that exploits symmetries of the class. Lie went on to develop this idea into a systematic method for attacking the problem of solving unknown differential equations. Essentially, one can use mathematical tools to force a given differential equation to reveal whether or not it has certain symmetries – provided it has, they can be used to simplify or solve the equation. This module is designed to enable students to understand the mathematics behind Lie's methods and to become proficient in using these powerful tools.
Indicative content: symmetries of geometrical objects; symmetries of first-order ODEs; how to find Lie symmetries; differential invariants; reduction of order. At level 7, topics will be studied and assessed in greater depth.
42 hours
80% examination and 20% coursework.
P. E. Hydon, Symmetry Methods for Differential Equations, Cambridge University Press, (2000).
H. Stephani, Differential Equations: Their Solution Using Symmetries, Cambridge University Press, (1989).
G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations, Springer, (2002)
See the library reading list for this module (Canterbury)
The intended subject specific learning outcomes. On successfully completing the level 7 module students will be able to:
1 demonstrate systematic understanding of techniques for finding and using Lie point symmetries to obtain exact solutions of given equations.
2 demonstrate the capability to solve complex problems using a very good level of skill in calculation and manipulation of the material in the following areas: calculation of Lie point symmetry generators, canonical coordinates and differential invariants; identification of invariant solutions; successive reduction of order, where the Lie algebra is solvable; construction of the general solution of a given ordinary differential equation.
3 apply a range of concepts and principles in Lie symmetry methods in loosely defined contexts, showing good judgment in the selection and application of tools and techniques
The intended generic learning outcomes. On successfully completing the level 7 module students will be able to:
1 work competently and independently, be aware of their own strengths and understand when help is needed
2 demonstrate a high level of capability in developing and evaluating logical arguments
3 communicate arguments confidently with the effective and accurate conveyance of conclusions
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working
5 solve problems relating to qualitative and quantitative information
6 make effective use of information technology skills such as using online resources (Moodle).
7 communicate technical material effectively
8 demonstrate an increased level of skill in numeracy and computation
9 demonstrate the acquisition of the study skills needed for continuing professional development
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