Time scale calculus

In mathematics, time scale calculus is a unification of the theory of difference equations with that of differential equations [ [http://www.newscientist.com/article/mg17924045.000-taming-natures-numbers.html Taming nature's numbers] - New Scientist article] . Discovered in 1988 by the German mathematician Stefan Hilger, it has applications in any field that requires simultaneous modelling of discrete and continuous data. It gives a new definition of a derivative such that if you differentiate a function which acts on the real numbers then the definition is equivalent to standard differentiation, but if you use a function acting on the integers then it is equivalent to the forward difference operator.

Dynamic equations

Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts [cite book | author=Martin Bohner & Allan Peterson | title=Dynamic Equations on Time Scales | publisher=Birkhäuser | year=2001 | id=ISBN 978-0-8176-4225-9 [http://www.springer.com/west/home/birkhauser?SGWID=4-40290-22-2117582-0 link] ] . The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice — once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale (also known as a time-set), which may be an arbitrary closed subset of the reals. In this way, results apply not only to the set of real numbers or set of integers but to more general time scales such as a cantor set.

The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus. Dynamic equations on a time scale have a potential for applications, such as 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.

Precise definition

A time scale or measure chain "T" is a closed subset of the real line "R".

Define:

:$sigma(t)\; =\; inf\{s\; in\; T|s>t\}$ (forward shift operator):

Let "t" be an element of "T": "t" is::left dense if $ho(t)\; =t$,:right dense if $sigma(t)\; =t$,:left scattered if ,:right scattered if $sigma(t)\; >\; t$,:dense if left dense or right dense.

Define the "graininess" μ of a measure chain "T" by::$mu(t)\; =\; sigma(t)\; -t$.

Take a function::$f:\; T\; ightarrow\; mathbb\{R\}$, (where R could be any Banach space, but set it to be the real line for simplicity).

Definition: "generalised derivative" or "fdelta"("t")

For every ε > 0 there exists a neighbourhood "U" of "t" such that::$|f(sigma(t))-f(s)-mbox\{fdelta\}(t)(sigma(t)-s)|le\; varepsilon|sigma(t)-s|$for all "s" in "U".

Take "T" = "R". Then σ("t") = "t",μ("t") = 0, "fdelta" = "f"′ is the derivative used in standard calculus. If "T" = "Z" (the integers), σ("t") = "t" + 1, μ("t")=1, "fdelta" = Δ"f" is the forward difference operator used in difference equations.

Laplace transform and z-transform

By modifying the z-transform slightly you get a z*-transform for difference equations which uses the same table of transforms as the laplace transform for differential equations. This transform now applies to dynamic equations on all time-scales, not just integers or reals. [cite book | author=Martin Bohner & Allan Peterson | title=Dynamic Equations on Time Scales | publisher=Birkhäuser | year=2001 | id=ISBN 978-0-8176-4225-9 [http://www.springer.com/west/home/birkhauser?SGWID=4-40290-22-2117582-0 link] ] .

ee also

* Analysis on fractals for dynamic equations on a cantor set.

References

Further reading

* [http://web.umr.edu/~bohner/tisc.html Special Issue] of Journal of Computational and Applied Mathematics
* [http://www.timescales.org Time Scale Calculus] - Baylor University site
* [http://www.hindawi.com/journals/ade/volume-2006/si.1.html Dynamic Equations And Applications] - Special Issue of Advances in Difference Equations