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
- Dynamic equations
- Formal definitions
- Operations on time scales
- Classification of points
- Continuity
- Derivative
- Integration
- Laplace transform and z-transform
- Partial differentiation
- Multiple integration
- Stochastic dynamic equations on time scales
- Measure theory on time scales
- Distributions on time scales
- Integral equations on time scales
- Fractional calculus on time scales
- See also
- References
- Further reading
- External links

Time-scale calculus was introduced in 1988 by the German mathematician Stefan Hilger.^{ [1] } 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.^{ [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 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 . 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 time scale .

A single point in a time scale is defined as:

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/jump operator)
- (backward shift/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 ,

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 of 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 .

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) 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. If (the integers), , , is the forward difference operator used in difference equations.

The *delta integral* is defined as the antiderivative with respect to the delta derivative. If has a continuous derivative one sets

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^{ [2] } to a modified Z-transform:

Partial differential equations and partial 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 natural measure ^{ [8] }^{ [9] } defined via

where denotes Lebesgue measure 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^{ [10] }

The Dirac delta and Kronecker delta are unified on time scales as the *Hilger delta*:^{ [11] }^{ [12] }

Integral equations and summation equations are unified as integral equations on time scales.^{[ citation needed ]}

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

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

In physics, the **Navier–Stokes equations** are a set of partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes.

In the mathematical field of differential geometry, the **Riemann curvature tensor** or **Riemann–Christoffel tensor** is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold. It is a local invariant of Riemannian metrics which measure the failure of second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is *flat*, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

In statistical mechanics, the **Fokker–Planck equation** is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. It is named after Adriaan Fokker and Max Planck, and is also known as the **Kolmogorov forward equation**, after Andrey Kolmogorov, who independently discovered the concept in 1931. When applied to particle position distributions, it is better known as the **Smoluchowski equation**, and in this context it is equivalent to the convection–diffusion equation. The case with zero diffusion is known in statistical mechanics as the Liouville equation. The Fokker–Planck equation is obtained from the master equation through Kramers–Moyal expansion.

In mathematics, the **Kronecker delta** is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:

In mathematics, **Itô's lemma** is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. It can be heuristically derived by forming the Taylor series expansion of the function up to its second derivatives and retaining terms up to first order in the time increment and second order in the Wiener process increment. The lemma is widely employed in mathematical finance, and its best known application is in the derivation of the Black–Scholes equation for option values.

**Linear elasticity** is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

The **Einstein–Hilbert action** in general relativity is the action that yields the Einstein field equations through the principle of least action. With the (− + + +) metric signature, the gravitational part of the action is given as

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 unstable stock prices or physical systems subject to thermal fluctuations. Typically, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process. However, other types of random behaviour are possible, such as jump processes. Random differential equations are conjugate to stochastic differential equations.

In theoretical physics, the **Rarita–Schwinger equation** is the relativistic field equation of spin-3/2 fermions. It is similar to the Dirac equation for spin-1/2 fermions. This equation was first introduced by William Rarita and Julian Schwinger in 1941.

**Stokes flow**, also named **creeping flow** or **creeping motion**, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature this type of flow occurs in the swimming of microorganisms and sperm and the flow of lava. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.

The **Maxwell stress tensor** is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impractically difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand.

In relativistic physics, the **electromagnetic stress–energy tensor** is the contribution to the stress–energy tensor due to the electromagnetic field. The stress–energy tensor describes the flow of energy and momentum in spacetime. The electromagnetic stress–energy tensor contains the negative of the classical Maxwell stress tensor that governs the electromagnetic interactions.

The **covariant formulation of classical electromagnetism** refers to ways of writing the laws of classical electromagnetism in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.

In physics, **Maxwell's equations in curved spacetime** govern the dynamics of the electromagnetic field in curved spacetime or where one uses an arbitrary coordinate system. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. But because general relativity dictates that the presence of electromagnetic fields induce curvature in spacetime, Maxwell's equations in flat spacetime should be viewed as a convenient approximation.

The **Newman–Penrose** (**NP**) **formalism** is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.

There are various **mathematical descriptions of the electromagnetic field** that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

The intent of this article is to highlight the important points of the **derivation of the Navier–Stokes equations** as well as its application and formulation for different families of fluids.

A **product integral** is any product-based counterpart of the usual sum-based integral of calculus. The first product integral was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations. Other examples of product integrals are the geometric integral, the bigeometric integral, and some other integrals of non-Newtonian calculus.

In mathematics, **Ricci calculus** constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be called the **absolute differential calculus**, developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century.

**Lagrangian field theory** is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

- ↑ Hilger, Stefan (1989).
*Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten*(PhD thesis). Universität Würzburg. OCLC 246538565. - 1 2 Martin Bohner & Allan Peterson (2001).
*Dynamic Equations on Time Scales*. Birkhäuser. ISBN 978-0-8176-4225-9. - ↑ Ahlbrandt, Calvin D.; Morian, Christina (2002). "Partial differential equations on time scales".
*Journal of Computational and Applied Mathematics*.**141**(1–2): 35–55. Bibcode:2002JCoAM.141...35A. doi: 10.1016/S0377-0427(01)00434-4 . - ↑ Jackson, B. (2006). "Partial dynamic equations on time scales".
*Journal of Computational and Applied Mathematics*.**186**(2): 391–415. Bibcode:2006JCoAM.186..391J. doi: 10.1016/j.cam.2005.02.011 . - ↑ Bohner, M.; Guseinov, G. S. (2004). "Partial differentiation on time scales" (PDF).
*Dynamic Systems and Applications*.**13**: 351–379. - ↑ Bohner, M; Guseinov, GS (2005). "Multiple integration on time scales".
*Dynamic Systems and Applications*. CiteSeerX 10.1.1.79.8824 . - ↑ Sanyal, Suman (2008).
*Stochastic Dynamic Equations*(PhD thesis). Missouri University of Science and Technology. ProQuest 304364901. - ↑ Guseinov, G. S. (2003). "Integration on time scales".
*J. Math. Anal. Appl*.**285**: 107–127. doi:10.1016/S0022-247X(03)00361-5. - ↑ Deniz, A. (2007).
*Measure theory on time scales*(PDF) (Master's thesis). İzmir Institute of Technology. - ↑ Eckhardt, J.; Teschl, G. (2012). "On the connection between the Hilger and Radon–Nikodym derivatives".
*J. Math. Anal. Appl*.**385**(2): 1184–1189. arXiv: 1102.2511 . doi:10.1016/j.jmaa.2011.07.041. - ↑ Davis, John M.; Gravagne, Ian A.; Jackson, Billy J.; Marks, Robert J. II; Ramos, Alice A. (2007). "The Laplace transform on time scales revisited".
*J. Math. Anal. Appl*.**332**(2): 1291–1307. Bibcode:2007JMAA..332.1291D. doi:10.1016/j.jmaa.2006.10.089. - ↑ Davis, John M.; Gravagne, Ian A.; Marks, Robert J. II (2010). "Bilateral Laplace Transforms on Time Scales: Convergence, Convolution, and the Characterization of Stationary Stochastic Time Series".
*Circuits, Systems and Signal Processing*.**29**(6): 1141–1165. doi:10.1007/s00034-010-9196-2. - ↑ Bastos, Nuno R. O.; Mozyrska, Dorota; Torres, Delfim F. M. (2011). "Fractional Derivatives and Integrals on Time Scales via the Inverse Generalized Laplace Transform".
*International Journal of Mathematics & Computation*.**11**(J11): 1–9. arXiv: 1012.1555 . Bibcode:2010arXiv1012.1555B.

- 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 of
*Journal of Computational and Applied Mathematics*(2002) - Dynamic Equations And Applications Special Issue of
*Advances in Difference Equations*(2006) - Dynamic Equations on Time Scales: Qualitative Analysis and Applications Special issue of
*Nonlinear Dynamics And Systems Theory*(2009)

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