Introductory description

The module introduces the fundamentals of mathematical modelling and scaling analysis, before discussing and analysing difference and differential equation models in the context of physics, chemistry, engineering as well as the life and social sciences.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1. Introduction to mathematical modelling: Mathematical modelling cycle, types of models (sto-chastic, deterministic, discrete, continuous, ….).
2. Scaling and dimensional analysis: Buckingham’s Pi Theorem, examples from chemical reac-tions and projectile motion, perturbation methods (time-permitting).
3. First order linear equations: first order linear equations, examples of existence and unique-ness, integration techniques (integrating factors, ..).
4. Second order equations: general homogeneous equations and linear second order equations with constant coefficients, reduction to 2x2 systems, sketching the flow under a vector field (2d only, phase diagrams).
5. Nonlinear equations and 2x2 systems: linear stability such as predator and prey models.
6. Difference equation: discrete population models such as the logistic model/fishery manage-ment, stability and instability of solutions (6+7 could be moved before 4).
7. Discretisation techniques: explicit and implicit Euler, connection to difference equations, sta-bility.

Learning outcomes

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

• understand the modelling cycle and be able to formulate and analyse simple models themselves
• use scaling, non-dimensionalisation and linear stability techniques to reveal and understand the main underlying dynamics/driving factors
• solve simple ODEs (first order and second order) and interpret their qualitative behavior
• solve simple difference equations and understand their connection to continuous ODEs
• understand the basic concepts of numerical approximation

Indicative reading list

Logan, David. A first course in differential equations. Springer, 2006.
Robinson, James C. An introduction to ordinary differential equations. Cambridge University Press, 2004.
Holmes, Mark H. Introduction to the foundations of applied mathematics. Springer, 2009.
Hermann, Martin, and Masoud Saravi. Nonlinear ordinary differential equations. Springer India, 2016.
Witelski, B. and Bowen, M., Methods of Mathematical Modelling: Continuous Systems and Differential Equations, Springer, 2015

Subject specific skills

The module introduces the fundamentals of mathematical modelling and scaling analysis, before discussing and analysing difference and differential equation models in the context of physics, chemistry, engineering as well as the life and social sciences.

Transferable skills

Students will acquire key modelling and problem solving skills which will empower them to address problems in a large range of scientific fields with confidence.