In this module students will learn techniques and constitutive laws that will enable them to convert physical phenomena into rigorous ordinary differential equations (ODEs). These equations are theoretically analysed with results backed up by illustrative numerical simulations. The results are then translated back into physical reality in order to check the relevance of the solution.

• Solve simple ODE systems, e.g. exponential growth and logistic equation.

• Apply multivariate Taylor series to linearise around specific points of a function.

• Convert systems of ODEs between Cartesian variables and polar variables.

• Define and use a set of constitutive modelling laws e.g. Newton’s laws of motion and Law of Mass Action.

• Non-dimensionalise a system of ODEs, specifying key parameter groups.

• Identify steady states of an ODE system and categorise them in terms of their stability characteristics.

• Specify how the existence and stability of the steady states depend on system parameters.

• Identify whether a system can exhibit hysteresis.

• Plot and interpret phase diagrams.

• Interpret mathematical results in terms of the physical underlying process.

27 fifty-minute lectures

Lecture notes with gaps will be provided in hard copy or via Learning Central. Students are expected to take additional notes during lectures.

Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.

Skills:

Problem formulation and modelling. Solution and qualitative analysis of ordinary differential equations using analytical techniques. Numerical simulation of simple ODE systems.

Transferable Skills:

Mathematical modelling of continuous time systems. Visualisation and interpretation of the results obtained from a mathematical model. Coding of dynamical systems.

Formative assessment is carried out by means of regular exercises.  These might include some exercises that involve the use of simple computer simulations to illustrate and complement theoretical results. Feedback to students on their solutions and their progress towards learning outcomes will be provided in lectures.  

Summative assessment is by written examination at the end of the module. The examination gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions

• Review of ODE theory, Taylor series and coordinate transformations.

• Introduction of a number of constitutive relations.

• Non dimensionalisation

• Derivation of steady states and stability of single ODEs and systems.

• Phase plane analysis.

• Applied examples combining examples from the rest of the course.

Lecture notes provided by the lecturer.

Lomen, D. and Lovelock, L. 1999. Differential equations. Wiley.

Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler. https://itunes.apple.com/us/itunes-u/id1102923623 or https://cosmolearning.org/courses/learn-differential-equations-tutorials-with-gilbert-strang-cleve-moler/