16:30
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.