ACM10070 Mathematical Modelling in the Sciences

Academic Year 2018/2019

This module is an introduction to the mathematical modeling of phenomena in many branches of the physical and biological science. The models are formulated in terms of difference and differential equations and the students are introduced to the basic properties of these equations and certain analytical (i.e. "pen and paper") techniques for solving them.

[Mathematical background] Linear and non-linear ODEs, the order of a differential equation, separability of ODEs, homogeneity and inhomogeneity of ODEs, [Solution methods] substitution, separation of variables, integrating-factor technique, the exponential substitution for second-order linear homogeneous problems, the criterion for the existence and uniqueness of solutions, [Qualitative methods] Fixed points and bifurcations, [Modelling techniques] Dimensional analysis, the scientific method, “theory” versus “model”, [Applications] These include (but are not necessarily limited to) population models, fisheries models with harvesting, drug delivery, interest rates, epidemics, the fluid analogy of electrical circuits, RC and LRC circuits, [Discrete systems] discrete population dynamics, the Fibonacci sequence, properties of discrete maps (fixed points, orbits, stability), chaos in discrete maps, cellular automata