Past Differential Equations and Applications Seminar

21 October 2004
Fordyce Davidson
When modelling a physical or biological system, it has to be decided what framework best captures the underlying properties of the system under investigation. Usually, either a continuous or a discrete approach is adopted and the evolution of the system variables can then be described by ordinary or partial differential equations or difference equations, as appropriate. It is sometimes the case, however, that the model variables evolve in space or time in a way which involves both discrete and continuous elements. This is best illustrated by a simple example. Suppose that the life span of a species of insect is one time unit and at the end of its life span, the insect mates, lays eggs and then dies. Suppose the eggs lie dormant for a further 1 time unit before hatching. The `time-scale' on which the insect population evolves is therefore best represented by a set of continuous intervals separated by discrete gaps. This concept of `time-scale' (or measure chain as it is referred to in a slightly wider context) can be extended to sets consisting of almost arbitrary combinations of intervals, discrete points and accumulation points, and `time-scale analysis' defines a calculus, on such sets. The standard `continuous' and `discrete' calculus then simply form special cases of this more general time scale calculus. In this talk, we will outline some of the basic properties of time scales and time scale calculus before discussing some if the technical problems that arise in deriving and analysing boundary value problems on time scales.
  • Differential Equations and Applications Seminar