MA261 Differential Equations: Modelling and Numerics
Lecturer: Andreas Dedner
Term(s): Term 2
Status for Mathematics students:
Commitment: 10 x 3 hour lectures + 9 x 1 hour support classes
Assessment: 100% Coursework. Part of this course work will require some programming using Python
Formal registration prerequisites: None
Assumed knowledge:

Basic knowledge on solving differential equations and the structure of solutions for systems of ODEs and DEs as provided e.g. by MA133 Differential Equations or MA113 Differential Equations A
 Programming in Python as provided e.g. by MA124 Maths by Computer
 Concepts like Taylor expansion and continuity of multivariable functions as discussed in MA259 Multivariable Calculus
Useful background: Good working knowledge in linear algebra and analysis
Synergies:
 MA117 Programming for Scientists
 MA269 Asymptotics and Integral Transforms
 MA256 Introduction to Systems Biology
 MA209 Variational Principles
Leads To: The following modules have this module listed as assumed knowledge or useful background:
 MA3J4 Mathematical Modelling with PDE
 MA398 Matrix Analysis and Algorithms
 MA3K1 Mathematics of Machine Learning
 MA3H0 Numerical Analysis and PDEs
 MA4M1 Epidemiology by Example
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 (RungeKutta 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, BrooksCole Publishing (2004)