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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, time-scale calculus is a unification of the theory of
difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s with that of
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s, unifying integral and differential
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
with the
calculus of finite differences A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
, offering a formalism for studying
hybrid system A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior – a system that can both ''flow'' (described by a differential equation) and ''jump'' (described by a state machine or automaton). Often, the te ...
s. 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 one differentiates a function defined on the real numbers then the definition is equivalent to standard differentiation, but if one uses a function defined on the integers then it is equivalent to the forward difference operator.


History

Time-scale calculus was introduced in 1988 by the German mathematician Stefan Hilger. However, similar ideas have been used before and go back at least to the introduction of the
Riemann–Stieltjes integral In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an inst ...
, 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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
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 (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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s or set of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s but to more general time scales such as a
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...
. The three most popular examples of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
on time scales are
differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve ...
, difference calculus, and quantum calculus. Dynamic equations on a time scale have a potential for applications such as in
population dynamics Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems. History Population dynamics has traditionally been the dominant branch of mathematical biology, which has a ...
. 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 In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
of the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
\mathbb. The common notation for a general time scale is \mathbb. The two most commonly encountered examples of time scales are the real numbers \mathbb and the
discrete time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
scale h\mathbb. A single point in a time scale is defined as: :t:t\in\mathbb


Operations on time scales

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\ (forward shift/jump operator) :\rho(t) = \sup\ (backward shift/jump operator) The ''graininess'' \mu is the distance from a point to the closest point on the right and is given by: :\mu(t) = \sigma(t) -t. For a right-dense t, \sigma(t)=t and \mu(t)=0.
For a left-dense t, \rho(t)=t.


Classification of points

For any t\in\mathbb, t is: * ''left dense'' if \rho(t) =t * ''right dense'' if \sigma(t) =t * ''left scattered'' if \rho(t)< t * ''right scattered'' if \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

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.


Derivative

Take a function: :f: \mathbb \to \R, (where R could be any
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, but is set to the real line for simplicity). Definition: The ''delta derivative'' (also Hilger derivative) f^(t) exists if and only if: For every \varepsilon > 0 there exists a neighborhood U of t such that: :\left, f(\sigma(t))-f(s)- f^(t)(\sigma(t)-s)\ \le \varepsilon\left, \sigma(t)-s\ for all s in U. Take \mathbb =\mathbb. Then \sigma(t) = t, \mu(t) = 0, f^ = f'; is the derivative used in standard
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
. If \mathbb = \mathbb (the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s), \sigma(t) = t + 1, \mu(t)=1, f^ = \Delta f 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 F(t) has a continuous derivative f(t)=F^\Delta(t) one sets :\int_r^s f(t) \Delta(t) = F(s) - F(r).


Laplace transform and z-transform

A
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
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 In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-tim ...
: \mathcal'\ = \frac


Partial differentiation

Partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
s and partial difference equations are unified as partial dynamic equations on time scales.


Multiple integration

Multiple integration In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
on time scales is treated in Bohner (2005).


Stochastic dynamic equations on time scales

Stochastic differential equation A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
s and stochastic difference equations can be generalized to stochastic dynamic equations on time scales.


Measure theory on time scales

Associated with every time scale is a natural measure defined via :\mu^\Delta(A) = \lambda(\rho^(A)), where \lambda denotes
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...
and \rho is the backward shift operator defined on \mathbb. The delta integral turns out to be the usual Lebesgue–Stieltjes integral with respect to this measure :\int_r^s f(t) \Delta t = \int_ f(t) d\mu^\Delta(t) and the delta derivative turns out to be the Radon–Nikodym derivative with respect to this measure :f^\Delta(t) = \frac(t).


Distributions on time scales

The Dirac delta and
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
are unified on time scales as the ''Hilger delta'': : \delta_^(t) = \begin \dfrac, & t = a \\ 0, & t \neq a \end


Integral equations on time scales

Integral equations and summation equations are unified as integral equations on time scales.


Fractional calculus on time scales

Fractional calculus Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D :D f(x) = \frac f(x)\,, and of the integration ...
on time scales is treated in Bastos, Mozyrska, and Torres.


See also

* Analysis on fractals for dynamic equations on a
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...
. *
Multiple-scale analysis In mathematics and physics, multiple-scale analysis (also called the method of multiple scales) comprises techniques used to construct uniformly valid approximations to the solutions of perturbation problems, both for small as well as large value ...
* Method of averaging *
Krylov–Bogoliubov averaging method The Krylov–Bogolyubov averaging method (Krylov–Bogolyubov method of averaging) is a mathematical method for approximate analysis of oscillating processes in non-linear mechanics. The method is based on the averaging principle when the exact diff ...


References


Further reading

*
Dynamic Equations on Time Scales
Special issue of ''Journal of Computational and Applied Mathematics'' (2002)

Special Issue of ''Advances in Difference Equations'' (2006)

Special issue of ''Nonlinear Dynamics And Systems Theory'' (2009)


External links


The Baylor University Time Scales Group

Timescalewiki.org
{{DEFAULTSORT:Time Scale Calculus Dynamical systems Calculus Recurrence relations