Dynamic Equations on Time Scales, Colloquium at Dicle University, July 12, 2007

Martin Bohner, University of Missouri-Rolla, Department of Mathematics and Statistics

Time scales have been introduced in order to unify continuous and discrete analysis and in order to extend those theories to cases "in between". We will offer a brief introduction into the calculus involved, including the so-called delta derivative of a function on a time scale. This delta derivative is equal to the usual derivative if the time scale is the set of all real numbers, and it is equal to the usual forward difference operator if the time scale is the set of all integers. However, in general, a time scale may be any closed subset of the reals.

We present some basic facts concerning dynamic equations on time scales (those are differential and difference equations, resp., in the above two mentioned cases) and initial value problems involving them. We introduce the exponential function on a general time scale and use it to solve initial value problems involving first order linear dynamic equation. We also present a unification of the Laplace and Z-transform, which serves to solve any higher order linear dynamic equations with constant coefficients.

Throughout the talk, many examples of time scales will be offered. Among others, we will discuss the following examples:

  1. The two standard examples (the reals and the integers).
  2. The set of all integer multiples of a positive number (this time scale is interesting for numerical purposes).
  3. The set of all integer powers of a number bigger than one (this time scale gives rise to so-called q-difference equations).
  4. The union of closed intervals (this time scale is interesting in population dynamics; for example, it can model insect populations that are continuous while in season, die out in say winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population).

References

  1. Dynamic Equations on Time Scales: An Introduction with Applications, Martin Bohner and Allan Peterson, Birkhäuser, 2001.
  2. Advances in Dynamic Equations on Time Scales, Martin Bohner and Allan Peterson, Birkhäuser, 2003.