Time-scale calculus
In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid discrete–continuous dynamical systems. 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, 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 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 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. Formal definitions. A time scale (or measure chain) is a closed subset of the real line formula_1. The common notation for a general time scale is formula_2. The two most commonly encountered examples of time scales are the real numbers formula_1 and the discrete time scale formula_4. 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 formula_6, respectively. Formally: The "graininess" formula_9 is the distance from a point to the closest point on the right and is given by: For a right-dense formula_6, formula_12 and formula_13. For a left-dense formula_6, formula_15 Classification of points. For any formula_16, formula_6 is: As illustrated by the figure at right: Continuity. Continuity of a time scale is redefined as equivalent to density. A time scale is said to be "right-continuous at  point formula_6" if it is right dense at point formula_6. Similarly, a time scale is said to be "left-continuous at point formula_6" if it is left dense at point formula_6. Derivative. Take a function: (where R could be any Banach space, but is set to the real line for simplicity). Definition: The "delta derivative" (also Hilger derivative) formula_31 exists if and only if: For every formula_32 there exists a neighborhood formula_33 of formula_6 such that: for all formula_36 in formula_33. Take formula_38 Then formula_39, formula_40, formula_41; is the derivative used in standard calculus. If formula_42 (the integers), formula_43, formula_44 , formula_45 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 formula_46 has a continuous derivative formula_47 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: formula_49 Partial differentiation. Partial differential equations and  partial difference equations are unified as partial dynamic equations on time scales. Multiple integration. Multiple integration on time scales is treated in Bohner (2005). Stochastic dynamic equations on time scales. Stochastic differential equations 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 where formula_51 denotes Lebesgue measure and formula_52 is the backward shift operator defined on formula_1. 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 equations and summation equations are unified as integral equations on time scales. Fractional calculus on time scales. Fractional calculus on time scales is treated in Bastos, Mozyrska, and Torres.