Time scale calculus: Information from Answers.com

In mathematics, time scale calculus is a unification of the theory of difference equations with that of differential equations. Discovered in 1988 by the German mathematician Stefan Hilger^[1], 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.

1 Dynamic equations
2 Formal Definitions
3 Derivative
4 Laplace transform and z-transform
5 See also
6 Notes
7 References
8 Further reading

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^[2]. 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.

Formal Definitions

A time scale (or measure chain) is a closed subset of the real line $\mathbb{R}$ . The common notation for a general time scale is $\mathbb{T}$ .

The two most commonly-encountered examples of time scales are the real numbers $\mathbb{R}$ and the discrete time scale $h\mathbb{Z}$ . For illustrations and additional examples of time scales, see Example_time_scales.

A single point in a time scale is defined as:

$t:t\in\mathbb{T}$

Operations On Time Scales

The forward jump, backward jump, and graininess operators on a discrete time scale

The forward jump and backward jump operators represent the closest point in the time scale on the right and left of a given point $t$ , respectively. Formally:

$\sigma(t) = \inf\{s \in \mathbb{T} : s>t\}$ (forward shift operator / forward jump operator)

$\rho(t) = \sup\{s \in \mathbb{T} : s<t\}$ (backward shift operator / backward jump operator)

The graininess $μ$ is the distance from a point to the closest point on the right and is given by:

μ(t) = σ(t) - t

For a right-dense $t$ , $σ(t) = t$ and $μ(t) = 0$ .
For a left-dense $t$ , $ρ(t) = t$ .

Classification of Points

Several points on a time scale with different classifications

For any $t\in\mathbb{T}$ , $t$ 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 both left dense and right dense
isolated if both left scattered and right scattered

As illustrated by the figure at right:

Point $t 1$ is dense
Point $t 2$ is left dense and right scattered
Point $t 3$ is isolated
Point $t 4$ is left scattered and right dense

Continuity

Continuity has a slightly different definition when dealing with time scales. A time scale is said to be right-continuous at point $t$ if it is right dense at point $t$ . Similarly, a time scale is said to be left-continuous at point $t$ if it is left dense at point $t$ .

Derivative

Take a function:

$f: T \rightarrow \mathbb{R}$ ,

(where R could be any Banach space, but set it to be the real line for simplicity).

Definition: generalized derivative (Hilger derivative) or $f Δ (t)$

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

$|f(\sigma(t))-f(s)- f^{\Delta}(t)(\sigma(t)-s)|\le \varepsilon|\sigma(t)-s|$

for all $s$ in $U$ .

Take $\mathbb{T} =\mathbb{R}.$ Then $σ(t) = t$ , $μ(t) = 0$ , $f Δ = f'$ ; is the derivative used in standard calculus. If $\mathbb{T} = \mathbb{Z}$ (the integers), $σ(t) = t + 1$ , $μ(t) = 1$ , $f Δ = Δ f$ is the forward difference operator used in difference equations.

Laplace transform and z-transform

A laplace transform can be defined for functions on time scales, which uses the same table of transforms for any arbitrary time scale. This transform can be used to solve dynamic equations on time scales. If the time scale is the non-negative integers then the transform is equal to a modified Z-transform: Z'{x}(z)=(Z{x}(z+1))/(z+1) ^[2].

Notes

^ Hilger, Stefan (1998). Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Universität Würzburg.
^ ^a ^b Martin Bohner & Allan Peterson (2001). Dynamic Equations on Time Scales. Birkhäuser. ISBN 978-0-8176-4225-9. link

References

Dynamic equations on time scales: a survey, Ravi Agarwal, Martin Bohner, Donal O’Regan, Allan Peterson, Journal of Computational and Applied Mathematics 141 (2002) 1–26

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Time scale calculus

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