This module will introduce the learner to the Multivariate Calculus and then move on to the analytical methods used to solve partial differential equations. The Learner will study systems of ordinary differential equations and the concept of phase space. Finally the learner will then cover the mathematics of the computational techniques and numerical integration packages they may use in industrial design contexts for solving differential equations, so that they will grasp their importance and limitations.
Multivariate Calculus: The definition of the partial derivative. Changing variables using the chain rule. The types of partial differential equations, including elliptical, hyperbolic and parabolic examples. The standard methods of Solution of partial differential equations including the Method of separation and transform methods. Initial value problems in these solutions and the use of Fourier series. Non-linear partial differential equations, including solitons.
Systems of Ordinary Differential Equations: Systems of ODEs and the concept of phase space. The properties of linear systems and characterising their behaviour by eigenvalue methods; nodes, spirals, saddle points and circles in two-dimensional cases. Finding the curve in phase space. The analysis of non-linear systems with critical points and characterising their behaviour. Three dimensional examples.
Numerical Integration: Initial values and their use in numerical integration problems. Ordinary Differential Equations and their solution by Euler and Euler-Cauchy methods, Runga Kutta and shooting methods. The limitations of these methods and their treatment as discrete systems. Numerical integration of partial differential equations, finite element methods, their limitations.
Multivariate Calculus
The definition of the partial derivative. Changing variables using the chain rule. The types of partial differential equations, including elliptical, hyperbolic and parabolic examples. The standard methods of Solution of partial differential equations including the Method of separation and transform methods. Initial value problems in these solutions and the use of Fourier series. Non-linear partial differential equations, including solitons.
Systems of Ordinary Differential Equations
Systems of ODEs and the concept of phase space. The properties of linear systems and characterising their behaviour by eigenvalue methods; nodes, spirals, saddle points and circles in two-dimensional cases. Finding the curve in phase space. The analysis of non-linear systems with critical points and characterising their behaviour. Three dimensional examples.
Numerical Integration
Initial values and their use in numerical integration problems. Ordinary Differential Equations and their solution by Euler and Euler-Cauchy methods, Runga Kutta and shooting methods. The limitations of these methods and their treatment as discrete systems. Numerical integration of partial differential equations, finite element methods, their limitations.
The calculus element of this module is best thought by introducing the methods of solution for ODEs, Series, Transforms and so on in lectures, explaining how and why they work and then practising as many examples as feasibly possible in the time allowed. The Systems content is also best taught with examples provided via worksheets.
The numerical analysis content of this module is conveyed using lectures to explain and discuss the fundamental concepts upon which the methods used rest. These lecture discussions should draw out the importance of rigorously interpreting the inherent strengths and weaknesses of differing methods. Workshops are used to give students a chance to implement numerical methods and the computational steps needed and then discuss what the results mean, in particular their limitations. The examples used should be drawn from any and all disciplines where numerical analysis is used, emphasising its broad application and also showing the commonality of the analysis used.
| Module Content & Assessment | |
|---|---|
| Assessment Breakdown | % |
| Formal Examination | 70 |
| Other Assessment(s) | 30 |