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.

Pre-requisite modules: MA1001, MA1006

• ·         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.

You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources to support the learning (e.g. videos, exercise sheets, lecture notes, quizzes)

Students are also expected to undertake at least 50 hours of self-guided study throughout the duration of the module, including preparation of formative assessments.

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.

• ·         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.