- Time scale calculus
In

mathematics ,**time scale calculus**is a unification of the theory ofdifference equation s with that ofdifferential equation s [*[*] . Discovered in*http://www.newscientist.com/article/mg17924045.000-taming-natures-numbers.html Taming nature's numbers*] - New Scientist article1988 by the German mathematicianStefan 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 [*] . 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*http://www.springer.com/west/home/birkhauser?SGWID=4-40290-22-2117582-0 link*]real number s or set ofinteger s but to more general time scales such as acantor set .The three most popular examples of

calculus on time scales aredifferential calculus , difference calculus, andquantum calculus . Dynamic equations on a time scale have a potential for applications, such as inpopulation 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 aclosed subset of thereal line "R".Define:

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

Let "t" be an element of "T": "t" is::left dense if $ho(t)\; =t$,:right dense if $sigma(t)\; =t$,:left scattered if $ho(t)<\; t$,: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" (theinteger s), σ("t") = "t" + 1, μ("t")=1, "fdelta" = Δ"f" is theforward 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 thelaplace 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 acantor 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

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