Content: Mathematics arises all around us not only is nature but also in social structures. A fundamental notion is that of Mathematical Modelling in which natural questions are turned into mathematical problems. Two types of mathematical models are (i) those arising from the application of physical laws and (ii) those arising from the analysis of data. In this module we expose some fundamental aspects of mathematical modelling involving ordinary differential equations. For example, fundamental principles in science like conservation laws and force balances lead to initial value problems. These principles can also be extended in epidemiology for the modelling of the transmission of diseases.

Mathematical models, in general, are too complex to solve explicitly, so that approximation methods and computation are essential tools. Consequently, this module also investigates different methods for approximating the solution to ODEs. Of particular interest and value are their mathematical properties, particularly in respecting properties of the underlying model.

Topics include:

• Demonstration of fundamental principles in deriving models ( reaction kinetics and Hamiltonian principle, and fundamental role of dimensional analysis perturbation theory to simplify complex models
• Approximation by discretisation (Runge-Kutta and multistep), and the tools needed to analyse there
• Analysis of discretisation (stability and convergence)
• Examples of the use of these tools in applications.

Aims: By the end of the module the student should be able to:

• Understand the central concepts of mathematical modelling
• Be able to derive and analyse fundamental numerical methods
• Implement and test numerical methods using a scripting language

Books:

• F. F. Griffiths and D. J. Higham, Numerical Methods for Ordinary Differential Equations: Initial Value Problems, Springer (2010)
• Witlski, M. Brown, Methods of Mathematical Modelling: Continous System and Differential Equations, Springer (2015)
• R. L. Burden and J. D Faires, Numerical Analysis, 8th edition, Brooks-Cole Publishing (2004)