Mathematical Modelling

Real-life phenomena, from traffic flow to the effect of greenhouse gases, are usually too complex to be analyzed quantitatively without idealization and simplification. It is just not feasible to follow the individual motions of dollars into and out of all the banking systems of the world, or the paths of individual water molecules in the ocean, as ways to understand the economy or the world's climate. When we simplify things, for example by neglecting the gravitational pull of Jupiter on the motion of an airplane, we are making a model of reality.

In some sense, the construction of useful mathematical models, for such prosaic things as flow of fluids through porous media (such as diapers, as was done in a study by our department for Johnson and Johnson) or for the motion of wood particles in fluid for the pulp and paper industry (David Jeffrey studies this because there is great interest in mechanical pulping because it is so much cleaner and easier on the environment than chemical pulping), is the very essence of applied mathematics.

We offer a third year course in this---not first year because the rudiments of the usual language for modelling, namely the calculus, must be mastered first---but it is not until graduate school that modelling really comes into its own in our department. Whether the models are for the physics of elementary particles, really getting down to fundamentals, or for the motion of blood in arteries, or for the number of people infected with HIV, the creation of useful and accurate models is as much an art as a science---imagination, judgement, taste, reasoning, experience, and skill all play a part. It is this which distinguishes applied from pure mathematics, because pure mathematicians care little for where the mathematics comes from, or for what it means. For us, what the equations mean is the heart of the matter.