Inmathematics,time-scale calculus is a unification of the theory ofdifference equations with that ofdifferential equations, unifyingintegral anddifferential calculus with thecalculus of finite differences, offering a formalism for studyinghybrid systems. It has applications in any field that requires simultaneous modelling ofdiscrete and continuous data. It gives a new definition of aderivative such that if one differentiates a function defined on thereal numbers then the definition is equivalent to standard differentiation, but if one uses a function defined on theintegers then it is equivalent to theforward difference operator.

Time-scale calculus was introduced in 1988 by the German mathematicianStefan Hilger.^[1] However, similar ideas have been used before and go back at least to the introduction of theRiemann–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 theircontinuous 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 unknownfunction 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 theset ofreal numbers or set ofintegers but to more general time scales such as aCantor set.

The three most popular examples ofcalculus on time scales aredifferential calculus,difference calculus, andquantum calculus. Dynamic equations on a time scale have a potential for applications such as inpopulation 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.

Atime scale (ormeasure chain) is aclosed subset of thereal line${\displaystyle \mathbb{R}}$. The common notation for a general time scale is${\displaystyle \mathbb{T}}$.

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

A single point in a time scale is defined as:

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

Theforward jump andbackward jump operators represent the closest point in the time scale on the right and left of a given point${\displaystyle t}$, respectively. Formally:

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

(backward shift/jump operator)

Thegraininess${\displaystyle \mu }$ is the distance from a point to the closest point on the right and is given by:

${\displaystyle \mu (t)=\sigma (t)-t.}$

For a right-dense${\displaystyle t}$,${\displaystyle \sigma (t)=t}$ and${\displaystyle \mu (t)=0}$.
For a left-dense${\displaystyle t}$,${\displaystyle \rho (t)=t.}$

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

• left dense if${\displaystyle \rho (t)=t}$
• right dense if${\displaystyle \sigma (t)=t}$
• left scattered if
• right scattered if${\displaystyle \sigma (t)>t}$
• dense if both left dense and right dense
• isolated if both left scattered and right scattered

As illustrated by the figure at right:

• Point${\displaystyle t_1}$ isdense
• Point${\displaystyle t_2}$ isleft dense andright scattered
• Point${\displaystyle t_3}$ isisolated
• Point${\displaystyle t_4}$ isleft scattered andright dense

Continuity of a time scale is redefined as equivalent to density. A time scale is said to beright-continuous at point${\displaystyle t}$ if it is right dense at point${\displaystyle t}$. Similarly, a time scale is said to beleft-continuous at point${\displaystyle t}$ if it is left dense at point${\displaystyle t}$.

Take a function:

${\displaystyle f:\mathbb{T}\to \mathbb{R},}$

(whereR could be anyBanach space, but is set to the real line for simplicity).

Definition: Thedelta derivative (also Hilger derivative)${\displaystyle f^{\mathrm{\Delta }}(t)}$ exists if and only if:

For every${\displaystyle \epsilon >0}$ there exists a neighborhood${\displaystyle U}$ of${\displaystyle t}$ such that:

${\displaystyle \left|f(\sigma (t))-f(s)-f^{\mathrm{\Delta }}(t)(\sigma (t)-s)\right|\le \epsilon \left|\sigma (t)-s\right|}$

for all${\displaystyle s}$ in${\displaystyle U}$.

Take${\displaystyle \mathbb{T}=\mathbb{R}.}$ Then${\displaystyle \sigma (t)=t}$,${\displaystyle \mu (t)=0}$,${\displaystyle f^{\mathrm{\Delta }}=f^{\prime }}$; is the derivative used in standardcalculus. If${\displaystyle \mathbb{T}=\mathbb{Z}}$ (theintegers),${\displaystyle \sigma (t)=t+1}$,${\displaystyle \mu (t)=1}$,${\displaystyle f^{\mathrm{\Delta }}=\mathrm{\Delta }f}$ is theforward difference operator used in difference equations.

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

${\displaystyle {\int }_r^{s}f(t)\mathrm{\Delta }(t)=F(s)-F(r).}$

ALaplace 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^[2] to a modifiedZ-transform:$${\displaystyle {\mathcal{Z}}^{\prime }\{x[z]\}=\frac{\mathcal{Z}\{x[z+1]\}}{z+1}}$$

Partial differential equations andpartial difference equations are unified as partial dynamic equations on time scales.^[3][4][5]

Multiple integration on time scales is treated in Bohner (2005).^[6]

Stochastic differential equations and stochastic difference equations can be generalized to stochastic dynamic equations on time scales.^[7]

Associated with every time scale is a naturalmeasure^[8][9] defined via

${\displaystyle {\mu }^{\mathrm{\Delta }}(A)=\lambda ({\rho }^{-1}(A)),}$

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

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

and the delta derivative turns out to be theRadon–Nikodym derivative with respect to this measure^[10]

${\displaystyle f^{\mathrm{\Delta }}(t)=\frac{df}{d{\mu }^{\mathrm{\Delta }}}(t).}$

TheDirac delta andKronecker delta are unified on time scales as theHilger delta:^[11][12]

Fractional calculus on time scales is treated in Bastos, Mozyrska, and Torres.^[13]

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

• Agarwal, Ravi; Bohner, Martin; O’Regan, Donal; Peterson, Allan (2002)."Dynamic equations on time scales: a survey".Journal of Computational and Applied Mathematics.141 (1–2):1–26.Bibcode:2002JCoAM.141....1A.doi:10.1016/S0377-0427(01)00432-0.
• Dynamic Equations on Time Scales Special issue ofJournal of Computational and Applied Mathematics (2002)
• Dynamic Equations And Applications Special Issue ofAdvances in Difference Equations (2006)
• Dynamic Equations on Time Scales: Qualitative Analysis and Applications Special issue ofNonlinear Dynamics And Systems Theory (2009)

• The Baylor University Time Scales Group
• Timescalewiki.org

• Dynamical systems
• Calculus
• Recurrence relations

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