Time-scale calculus

Time-scale calculus was introduced in 1988 by the German mathematician Stefan 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 }} 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.

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 | 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 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, 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.

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

The two most commonly encountered examples of time scales are the real numbers $\backslash mathbb\{R\}$ and the discrete time scale $h\backslash mathbb\{Z\}$.

A single point in a time scale is defined as:

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

File:Timescales jump operators.png

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:

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

:

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

:$\backslash mu(t)\; =\; \backslash sigma(t)\; -t.$

For a right-dense $t$, $\backslash sigma(t)=t$ and $\backslash mu(t)=0$.

For a left-dense $t$, $\backslash rho(t)=t.$

File:Timescales point classifications.png

For any $t\backslash in\backslash mathbb\{T\}$, $t$ is:

• left dense if $\backslash rho(t)\; =t$
• right dense if $\backslash sigma(t)\; =t$
• left scattered if
• right scattered if $\backslash sigma(t)\; >\; t$
• 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 of a time scale is redefined as equivalent to density. 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 $t$.

Take a function:

:$f:\; \backslash mathbb\{T\}\; \backslash to\; \backslash R,$

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

Definition: The delta derivative (also Hilger derivative) $f^\{\backslash Delta\}(t)$ exists if and only if:

For every $\backslash varepsilon\; >\; 0$ there exists a neighborhood $U$ of $t$ such that:

:$\backslash left|f(\backslash sigma(t))-f(s)-\; f^\{\backslash Delta\}(t)(\backslash sigma(t)-s)\backslash right|\; \backslash le\; \backslash varepsilon\backslash left|\backslash sigma(t)-s\backslash right|$

for all $s$ in $U$.

Take $\backslash mathbb\{T\}\; =\backslash mathbb\{R\}.$ Then $\backslash sigma(t)\; =\; t$, $\backslash mu(t)\; =\; 0$, $f^\{\backslash Delta\}\; =\; f\text{'}$; is the derivative used in standard calculus. If $\backslash mathbb\{T\}\; =\; \backslash mathbb\{Z\}$ (the integers), $\backslash sigma(t)\; =\; t\; +\; 1$, $\backslash mu(t)=1$, $f^\{\backslash Delta\}\; =\; \backslash Delta\; f$ is the forward difference operator used in difference equations.

The delta integral is defined as the antiderivative with respect to the delta derivative. If $F(t)$ has a continuous derivative $f(t)=F^\backslash Delta(t)$ one sets

:$\backslash int\_r^s\; f(t)\; \backslash Delta(t)\; =\; F(s)\; -\; F(r).$

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:

$$\backslash mathcal\{Z\}\text{'}\backslash \{x[z]\backslash \}\; =\; \backslash frac\{\backslash mathcal\{Z\}\backslash \{x[z+1]\backslash \}\}\{z+1\}$$

Partial differential equations and partial difference equations are unified as partial dynamic equations on time scales.{{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 }}{{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 }}{{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 }}

Multiple integration on time scales is treated in Bohner (2005).{{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 }}

Stochastic differential equations and stochastic difference equations can be generalized to stochastic dynamic equations on time scales.{{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}} }}

Associated with every time scale is a natural measure{{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 }}{{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 }} defined via

:$\backslash mu^\backslash Delta(A)\; =\; \backslash lambda(\backslash rho^\{-1\}(A)),$

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

:$\backslash int\_r^s\; f(t)\; \backslash Delta\; t\; =\; \backslash int\_\{[r,s)\}\; f(t)\; d\backslash mu^\backslash Delta(t)$

and the delta derivative turns out to be the Radon–Nikodym derivative with respect to this measure{{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 }}

:$f^\backslash Delta(t)\; =\; \backslash frac\{df\}\{d\backslash mu^\backslash Delta\}(t).$

The Dirac delta and Kronecker delta are unified on time scales as the Hilger delta:{{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 }}{{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 }}

: $\backslash delta\_\{a\}^\{\backslash mathbb\{H\}\}(t)\; =\; \backslash begin\{cases\}$

\dfrac{1}{\mu(a)}, & t = a \\

0, & t \neq a

\end{cases}

Fractional calculus on time scales is treated in Bastos, Mozyrska, and Torres.{{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 }}

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

{{reflist}}

• {{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)

• [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