In mathematics, time scale calculus is a unification of the theory of difference equations and standard calculus. 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. A precise definition follows at the end of this article: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ... For other uses, see Calculus (disambiguation). ... Year 1988 (MCMLXXXVIII) was a leap year starting on Friday (link displays 1988 Gregorian calendar). ...

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. 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, which may be an arbitrary closed subset of the reals. In this way, results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... The integers are commonly denoted by the above symbol. ...

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. Since then several authors have expounded on various aspects of this new theory. For other uses, see Calculus (disambiguation). ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... Quantum calculus is equivalent to traditional infinitesimal calculus without the notion of limits. ... Population dynamics is the study of marginal and long-term changes in the numbers, individual weights and age composition of individuals in one or several populations, and biological and environmental processes influencing those changes. ...

Precise definition

A time scale or measure chain T is a closed subset of the real line R. In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, the real line is simply the set of real numbers. ...

Define:

σ(t) = inf{s an element of T, s > t} (forward shift operator)

ρ(t) = sup{s an element of T, s < t} (backward shift operator)

Let t be an element of T: t is:

left dense if ρ(t) = t,

right dense if σ(t) = t,

left scattered if ρ(t) < t,

right scattered if σ(t) > t,

dense if left dense or right dense.

Define the graininess μ of a measure chain T by:

μ(t) = σ(t) − t.

Take a function:

f : T → R,

(where R could be any Banach space, but set it to be the real line for simplicity). In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

Definition: generalised derivative or fdelta(t)

For every ε > 0 there exists a neighbourhood U of t such that:

|f(σ(t)) − f(s) − fdelta(t)(σ(t) − s)| ≤ ε|σ(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 = Δf is the forward difference operator used in difference equations. For other uses, see Calculus (disambiguation). ... The integers are commonly denoted by the above symbol. ... In mathematics, a difference operator maps a function f(x) to another function f(x + a) &#8722; f(x + b). ...

Compare and Contrast

See also Discrete exterior calculus which also harmonizes the discrete and continuous. In mathematics, the discrete exterior calculus (DEC) or finite element exterior calculus (FEEC) is the extension to the method of finite elements of the exterior calculus of differentiable manifolds. ...

References

• Martin Bohner & Allan Peterson (2001). Dynamic Equations on Time Scales. Birkhäuser. ISBN 978-0-8176-4225-9.  link
• Special Issue of Journal of Computational and Applied Mathematics

External links

• Time Scale Calculus - Baylor University site
• Taming nature's numbers - New Scientist article
• Control systems on Time scales
• Publications of Martin Bohner - a large number of articles on Time scales.