# MA2K4 Content

**Aims**: This module focuses on fundamental concepts of the analysis of numerical methods and mathematical modelling involving for example ordinary differential equations showcased using typical examples from physics, biology, and other areas of science and engineering. Basic numerical approximation methods will be presented including for example methods for solving systems of differential equations, the solution to algebraic equations, polynomial interpolation and .extrapolation, and quadrature. Concepts like stability, consistency, and convergence will be covered in this module, with the aim of introducing the approximation techniques used in tackling mathematical problems which do not yield to closed form analytic formulae. Other application will include for example graphics and visualization or medical imaging.

**Content**: Numerical approximations, Derivation of explicit and implicit Runge Kutta and multistep methods, Butcher tableau, Newtonâ€™s method, polynomial interpolation/extrapolation, linear regression, optimization, and quadrature, stability, consistency, and convergence analysis.

Derivation and analysis of numerical methods for complex real word application is a focus of this module. Concepts like stability, consistency, and convergence will be covered in this module, with the aim of introducing the approximation techniques used in tackling mathematical problems which do not yield to closed form analytic formulae.

Basic numerical approximation methods will be presented for solving complex systems of differential equations like Runge-Kutta and multistep methods. The derivation and implementation of these methods will require studying a wide range of other approximation problems like polynomial interpolation, approximations to derivatives and integrals (finite differences and quadratures) and the solution to algebraic problems. Examples for the application of these methods will be taken from physics, biology but also for example from computer graphics/visualization, medical imagi

**Objectives**: By the end of the module, students should be able to:

- derive and analyse fundamental numerical methods
- implement and test numerical methods using a scripting language
- design and evaluate problem solving strategies for real-world application

**Books: **

- D. Kincaid and W. Cheney,
*Numerical Analysis: Mathematics of Scientific Computing* - L. N. Trefethen,
*Approximation Theory and Approximation Practice* - E. Suli and D. F. Mayers,
*An Introduction to Numerical Analysis* - D. F. F. Griffiths and D. J. Higham,
*Numerical Methods for Ordinary Differential Equations* - T. Witelski, M. Bowen,
*Methods of Mathematical Modelling: Continuous System and Differential Equations* - R. L. Burden, J. D. Faires, A.M. Burden,
*Numerical Analysis*