- Time scale calculus
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 1988by 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.
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
calculuson 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.
A time scale or measure chain "T" is a
closed subsetof the real line"R".
: (forward shift operator):
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