Time scale calculus

# Time scale calculus

Overview
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, time-scale calculus is a unification of the theory of difference equations with that of differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s, unifying integral and differential calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

with the calculus of finite differences, offering a formalism for studying hybrid discrete–continuous dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

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 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.

It was introduced in 1988 by the German mathematician Stefan Hilger.
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Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, time-scale calculus is a unification of the theory of difference equations with that of differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s, unifying integral and differential calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

with the calculus of finite differences, offering a formalism for studying hybrid discrete–continuous dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

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 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.

## History

It 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 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
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

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
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

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 number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s or set of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s but to more general time scales such as a Cantor set
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....

.

The three most popular examples of calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

on time scales are differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

, difference calculus, and quantum calculus
Quantum calculus
Quantum calculus is equivalent to traditional infinitesimal calculus without the notion of limits. It defines "q-calculus" and "h-calculus". h ostensibly stands for Planck's constant while q stands for quantum...

. Dynamic equations on a time scale have a potential for applications, such as in population dynamics
Population dynamics
Population dynamics is the branch of life sciences that studies short-term and long-term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes...

. For example, it can model insect populations that are continuous 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 nonoverlapping population.

## Formal definitions

A time scale (or measure chain) is a closed subset of the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

. The common notation for a general time scale is .

The two most commonly encountered examples of time scales are the real numbers and the discrete
Discrete time
Discrete time is the discontinuity of a function's time domain that results from sampling a variable at a finite interval. For example, consider a newspaper that reports the price of crude oil once every day at 6:00AM. The newspaper is described as sampling the cost at a frequency of once per 24...

time scale .

A single point in a time scale is defined as:

### 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 , respectively. Formally:
(forward shift operator / forward jump operator) (backward shift operator / backward jump operator)

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

For a right-dense , and .

For a left-dense ,

### Classification of points

For any , is:
• left dense if
• right dense if
• left scattered if
• right scattered if
• dense if both left dense and right dense
• isolated if both left scattered and right scattered

As illustrated by the figure at right:
• Point is dense
• Point is left dense and right scattered
• Point is isolated
• Point is left scattered and right dense

### Continuity

Continuity
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

on a time scale is redefined as equivalent to density. A time scale is said to be right-continuous at point if it is right dense at point . Similarly, a time scale is said to be left-continuous at point if it is left dense at point .

## Derivative

Take a function:,
(where R could be any Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

, but set it to be the real line for simplicity).

Definition: The delta derivative (also Hilger derivative) exists if and only if:

For every there exists a neighborhood of such that:
for all in .

Take Then , , ; is the derivative used in standard calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. If (the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s), , , 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 has a continuous derivative one sets

## 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-transform
In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation....

:

## Partial differentiation

Partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s and partial difference equations are unified as partial dynamic equations on time scales.

## Stochastic dynamic equations on time scales

Stochastic differential equation
Stochastic differential equation
A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....

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
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

defined via

where denotes Lebesgue measure
Lebesgue measure
In measure theory, 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 with the standard measure of length, area, or volume. In general, it is also called...

and is the backward shift operator defined on . The delta integral
turns out to be the usual Lebesgue–Stieltjes integral with respect to this measure

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

## Distributions on time scales

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

## Integral equations on time scales

Integral equation
Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...

s and summation equation
Summation equation
In mathematics, a summation equation or discrete integral equation is an equation in which an unknown function appears under a summation sign. The theories of summation equations and integral equations can be unified as integral equations on time scales using time scale calculus...

s are unified as integral equations on time scales.