Aims & Learning Objectives:
Aims: To study, by example, how mathematical models are hypothesised, modified and elaborated. To study a classic example of mathematical modelling, that of fluid mechanics.
At the end of the course the student should be able to
* construct an initial mathematical model for a real world process and assess this model critically
* suggest alterations or elaborations of proposed model in light of discrepancies between model predictions and observed data or failures of the model to exhibit correct qualitative behaviour. The student will also be familiar with the equations of motion of an ideal inviscid fluid (Euler's equations, Bernoulli's equation) and how to solve these in certain idealised flow situations.
Modelling and the scientific method: Objectives of mathematical modelling; the iterative nature of modelling; falsifiability and predictive accuracy; Occam's razor, paradigms and model components; self-consistency and structural stability. The three stages of modelling:
(1) Model formulation, including the use of empirical information,
(2) model fitting, and
(3) model validation. Possible case studies and projects include: The dynamics of measles epidemics; population growth in the USA; prey-predator and competition models; modelling water pollution; assessment of heat loss prevention by double glazing; forest management. Fluids: Lagrangian and Eulerian specifications, material time derivative, acceleration, angular velocity. Mass conservation, incompressible flow, simple examples of potential flow.