Editing Time-scale calculus

{{short description|Unification of discrete and continuous theories of calculus}}

In [[mathematics]], '''time-scale calculus''' is a unification of the theory of [[difference equation]]s with that of [[differential equation]]s, unifying [[integral]] and [[differential calculus]] with the [[calculus of finite differences]], offering a formalism for studying [[hybrid system]]s.  It has applications in any field that requires simultaneous modelling of [[Discrete time and continuous time|discrete and continuous]] data. It gives a new definition of a [[derivative]] such that if one differentiates a function defined on the [[real number]]s then the definition is equivalent to standard differentiation, but if one uses a function defined on the [[integer]]s then it is equivalent to the [[forward difference]] operator.

==History==

Time-scale calculus was introduced in 1988 by the German mathematician [[Stefan Hilger]].<ref name=hilger>{{cite thesis |type=PhD thesis | last = Hilger | first = Stefan | authorlink = Stefan Hilger |title = Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten |publisher = Universität Würzburg | year = 1989 |oclc=246538565 }}</ref> However, similar ideas have been used before and go back at least to the introduction of the [[Riemann–Stieltjes integral]], which unifies sums and integrals.

==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 function|continuous]] counterparts.<ref name=bp>{{cite book | author=Martin Bohner & Allan Peterson | title=Dynamic Equations on Time Scales | publisher=Birkhäuser | year=2001 | isbn=978-0-8176-4225-9 }}</ref>  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 (mathematics)|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 (mathematics)|set]] of [[real number]]s or set of [[integer]]s but to more general time scales such as a [[Cantor set]].

The three most popular examples of [[calculus]] on time scales are [[differential calculus]], [[finite differences|difference calculus]], and [[quantum calculus]]. Dynamic equations on a time scale have a potential for applications such as in [[population dynamics]]. For example, they can model insect populations that evolve continuously while in season, die out in winter while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a non-overlapping population.

==Formal definitions==
A '''time scale''' (or '''measure chain''') is a [[closed subset]] of the [[real line]] <math>\mathbb{R}</math>. The common notation for a general time scale is <math>\mathbb{T}</math>.

The two most commonly encountered examples of time scales are the real numbers <math>\mathbb{R}</math> and the [[discrete time]] scale <math>h\mathbb{Z}</math>.

A single point in a time scale is defined as:
:<math>t:t\in\mathbb{T}</math>

=== Operations on time scales===
[[File:Timescales jump operators.png|thumb|upright=2.0|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 <math>t</math>, respectively.  Formally:

:<math>\sigma(t) = \inf\{s \in \mathbb{T} : s>t\}</math> (forward shift/jump operator)
:<math>\rho(t) = \sup\{s \in \mathbb{T} : s<t\}</math> (backward shift/jump operator)

The ''graininess'' <math>\mu</math> is the distance from a point to the closest point on the right and is given by:
:<math>\mu(t) = \sigma(t) -t.</math>

For a right-dense <math>t</math>, <math>\sigma(t)=t</math> and <math>\mu(t)=0</math>.<br />
For a left-dense <math>t</math>, <math>\rho(t)=t.</math>

===Classification of points===
[[File:Timescales point classifications.png|thumb|upright=2.0|Several points on a time scale with different classifications]]
For any <math>t\in\mathbb{T}</math>, <math>t</math> is:
* ''left dense'' if <math>\rho(t) =t</math>
* ''right dense'' if <math>\sigma(t) =t</math>
* ''left scattered'' if <math>\rho(t)< t</math>
* ''right scattered'' if <math>\sigma(t) > t</math>
* ''dense'' if both left dense and right dense
* ''isolated'' if both left scattered and right scattered

As illustrated by the figure at right:
* Point <math>t_1</math> is ''dense''
* Point <math>t_2</math> is ''left dense'' and ''right scattered''
* Point <math>t_3</math> is ''isolated''
* Point <math>t_4</math> is ''left scattered'' and ''right dense''

===Continuity===
[[Continuous function|Continuity]] of a time scale is redefined as equivalent to density.  A time scale is said to be ''right-continuous at point <math>t</math>'' if it is right dense at point <math>t</math>.  Similarly, a time scale is said to be ''left-continuous at point <math>t</math>'' if it is left dense at point <math>t</math>.

==Derivative==
Take a function:
:<math>f: \mathbb{T} \to \R,</math>

(where '''R''' could be any [[Banach space]], but is set to  the real line for simplicity).

Definition: The ''delta derivative'' (also Hilger derivative) <math>f^{\Delta}(t)</math> exists if and only if:

For every <math>\varepsilon > 0</math> there exists a neighborhood <math>U</math> of <math>t</math> such that:
:<math>\left|f(\sigma(t))-f(s)- f^{\Delta}(t)(\sigma(t)-s)\right| \le \varepsilon\left|\sigma(t)-s\right|</math>
for all <math>s</math> in <math>U</math>.

Take <math>\mathbb{T} =\mathbb{R}.</math>  Then <math>\sigma(t) = t</math>, <math>\mu(t) = 0</math>, <math>f^{\Delta} = f'</math>; is the derivative used in standard [[calculus]]. If <math>\mathbb{T} = \mathbb{Z}</math> (the [[integer]]s), <math>\sigma(t)  = t + 1</math>, <math>\mu(t)=1</math>, <math>f^{\Delta} = \Delta f</math> is the [[forward difference operator]] used in difference equations.

==Integration==

The ''delta integral'' is defined as the antiderivative with respect to the delta derivative. If <math>F(t)</math> has a continuous derivative <math>f(t)=F^\Delta(t)</math> one sets

:<math>\int_r^s f(t) \Delta(t) = F(s) - F(r).</math>

==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<ref name=bp/> to a modified [[Z-transform]]:
<math display="block">\mathcal{Z}'\{x[z]\} = \frac{\mathcal{Z}\{x[z+1]\}}{z+1}</math>

==Partial differentiation==

[[Partial differential equation]]s and [[partial difference equation]]s are unified as partial dynamic equations on time scales.<ref>{{cite journal | doi = 10.1016/S0377-0427(01)00434-4 | volume=141 | issue=1–2 | title=Partial differential equations on time scales | year=2002 | journal=Journal of Computational and Applied Mathematics | pages=35–55 | last1 = Ahlbrandt | first1 = Calvin D. | last2 = Morian | first2 = Christina| bibcode=2002JCoAM.141...35A | doi-access=free }}</ref><ref>{{cite journal |title=Partial dynamic equations on time scales |first=B. |last=Jackson |journal=Journal of Computational and Applied Mathematics |year=2006 |volume=186 |issue=2 |pages=391–415 |doi=10.1016/j.cam.2005.02.011 |bibcode=2006JCoAM.186..391J |doi-access=free }}</ref><ref>{{cite journal |url=https://web.mst.edu/~bohner/papers/pdots.pdf |title=Partial differentiation on time scales |first1=M. |last1=Bohner |first2=G. S. |last2=Guseinov |journal=Dynamic Systems and Applications |volume=13 |year=2004 |pages=351–379 }}</ref>

==Multiple integration==

[[Multiple integration]] on time scales is treated in Bohner (2005).<ref>{{cite journal | citeseerx = 10.1.1.79.8824 | title = Multiple integration on time scales | first = M | last1 = Bohner | first2 = GS | last2 = Guseinov | journal = Dynamic Systems and Applications | year = 2005 }}</ref>

==Stochastic dynamic equations on time scales==

[[Stochastic differential equation]]s and stochastic difference equations can be generalized to stochastic dynamic equations on time scales.<ref>{{cite thesis |type=PhD thesis |url=https://scholarsmine.mst.edu/doctoral_dissertations/2276 |title=Stochastic Dynamic Equations |first=Suman |last=Sanyal |date=2008 |publisher=[[Missouri University of Science and Technology]] |id={{ProQuest|304364901}} }}</ref>

==Measure theory on time scales==

Associated with every time scale is a natural [[Measure (mathematics)|measure]]<ref>{{cite journal | doi = 10.1016/S0022-247X(03)00361-5 | title = Integration on time scales | first = G. S. | last = Guseinov | journal = J. Math. Anal. Appl. | volume = 285 | year = 2003 | pages = 107–127 | doi-access = free }}</ref><ref>{{cite thesis |type=Master's thesis | url = http://library.iyte.edu.tr/tezler/master/matematik/T000568.pdf | title = Measure theory on time scales | first = A. | last = Deniz | year = 2007 |publisher=[[İzmir Institute of Technology]] }}</ref> defined via

:<math>\mu^\Delta(A) = \lambda(\rho^{-1}(A)),</math>

where <math>\lambda</math> denotes [[Lebesgue measure]] and <math>\rho</math> is the backward shift operator defined on <math>\mathbb{R}</math>. The delta integral turns out to be the usual [[Lebesgue–Stieltjes integral]] with respect to this measure

:<math>\int_r^s f(t) \Delta t = \int_{[r,s)} f(t) d\mu^\Delta(t)</math>

and the delta derivative turns out to be the [[Radon–Nikodym derivative]] with respect to this measure<ref>{{cite journal | arxiv = 1102.2511 | title = On the connection between the Hilger and Radon–Nikodym derivatives | first1 = J. | last1 =Eckhardt  | authorlink2 = Gerald Teschl | first2 = G. | last2 = Teschl | journal = J. Math. Anal. Appl. | volume = 385 | issue = 2 | year = 2012 | pages = 1184–1189 | doi=10.1016/j.jmaa.2011.07.041| s2cid = 17178288 }}</ref>

:<math>f^\Delta(t) = \frac{df}{d\mu^\Delta}(t).</math>

==Distributions on time scales==

The [[Dirac delta]] and [[Kronecker delta]] are unified on time scales as the ''Hilger delta'':<ref>{{cite journal |first1=John M. |last1=Davis |first2=Ian A. |last2=Gravagne |first3=Billy J. |last3=Jackson |first4=Robert J. II |last4=Marks |first5=Alice A. |last5=Ramos |title=The Laplace transform on time scales revisited |journal=J. Math. Anal. Appl. |volume=332 |year=2007 |issue=2 |pages=1291–1307 |doi=10.1016/j.jmaa.2006.10.089 |bibcode=2007JMAA..332.1291D |doi-access=free }}</ref><ref>{{cite journal |first1=John M. |last1=Davis |first2=Ian A. |last2=Gravagne |first3=Robert J. II |last3=Marks |title=Bilateral Laplace Transforms on Time Scales: Convergence, Convolution, and the Characterization of Stationary Stochastic Time Series |journal=Circuits, Systems and Signal Processing |year=2010 |volume=29 |issue=6 |pages=1141–1165 |doi=10.1007/s00034-010-9196-2 |s2cid=16404013 }}</ref>

: <math>\delta_{a}^{\mathbb{H}}(t) = \begin{cases}
\dfrac{1}{\mu(a)}, & t = a \\
0, & t \neq a
\end{cases}</math>

==Fractional calculus on time scales==

[[Fractional calculus]] on time scales is treated in Bastos, Mozyrska, and Torres.<ref>{{cite journal | arxiv = 1012.1555 | title = Fractional Derivatives and Integrals on Time Scales via the Inverse Generalized Laplace Transform | first1 = Nuno R. O. | last1 = Bastos | first2 = Dorota | last2 = Mozyrska | first3 = Delfim F. M. | last3 = Torres | bibcode = 2010arXiv1012.1555B | year = 2011 |journal=International Journal of Mathematics & Computation |volume=11 |issue=J11 |pages=1–9 }}</ref>

==See also==

*[[Analysis on fractals]] for dynamic equations on a [[Cantor set]].
*[[Multiple-scale analysis]]
*[[Method of averaging]]
*[[Krylov–Bogoliubov averaging method]]

==References==
{{reflist}}

==Further reading==
*{{cite journal |title=Dynamic equations on time scales: a survey |first1=Ravi |last1=Agarwal |first2=Martin |last2=Bohner |first3=Donal |last3=O’Regan |first4=Allan |last4=Peterson |journal=Journal of Computational and Applied Mathematics |volume=141 |issue=1–2 |year=2002 |pages=1–26 |doi=10.1016/S0377-0427(01)00432-0 |bibcode=2002JCoAM.141....1A |doi-access=free }}
* [http://web.mst.edu/~bohner/tisc.html Dynamic Equations on Time Scales] Special issue of ''Journal of Computational and Applied Mathematics'' (2002)
* [http://www.hindawi.com/journals/ade/volume-2006/si.1.html Dynamic Equations And Applications] Special Issue of ''Advances in Difference Equations'' (2006)
* [http://www.e-ndst.kiev.ua/v9n1.htm Dynamic Equations on Time Scales: Qualitative Analysis and Applications] Special issue of ''Nonlinear Dynamics And Systems Theory'' (2009)

==External links==
* [http://www.timescales.org The Baylor University Time Scales Group]
* [http://timescalewiki.org/index.php/Main_Page Timescalewiki.org]

{{DEFAULTSORT:Time Scale Calculus}}
[[Category:Dynamical systems]]
[[Category:Calculus]]
[[Category:Recurrence relations]]