# MA261 Differential Equations: Modelling and Numerics

**Lecturer: **Andreas Dedner

**Term(s): **2

**Status for Mathematics students:
**

**Commitment: **10 x 3 hour lectures + 9 x 1 hour support classes

**Assessment: **Coursework (50%) and 1 hour written exam (50%). Part of the course work will require some programming. These assignments can be completed using either MATLAB or Python. Support in form of solution templates will be provided for MATLAB only.

**Prerequisites: **MA133 Differential Equations and MA124 Maths by Computer (or equivalent)

**Leads To: **MA398 Matrix Analysis and Algorithms, MA3H0 Numerical Analysis and PDE's

**Content:**

Concepts of Mathematical Modelling, e.g. conservation and dissipation principle, dimensional analysis, non dimensionalization, asymptotic expansion, introduction to calculus of variations, minimization, Hamiltonian dynamics, Lagrange multipliers, inverse and optimal control problems, gradient flow

Numerical approximations

Derivation of explicit and implicit Runge Kutta and multistep methods, Butcher tableau, Newtonâ€™s method, polynomial interpolation and quadrature, stability, consistency, and convergence analysis,

This module focuses on fundamental concepts of mathematical modelling involving ordinary differential equations and their numerical solution. Modelling concepts such as conservation and dissipation principles, calculus of variations, and non dimensionalisation will be covered using typical examples from physics, biology, and other areas of science and engineering. Basic numerical approximation methods will be presented for solving the resulting systems of differential equations like Runge-Kutta and multistep methods. 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.

**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 Differental Equations: Inital Value Problems, Springer (2010)**Witlski, M.Brown, Methods of Mathematical Modelling: Continous System and Differential Equations, Springer (2015)*

* 3. R. L. Burden and J. D Faires, **Numerical Analysis**, 8th edition**, Brooks-Cole Publishing (2004).*