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| :'''''Fractional integration''' redirects here. Not to be confused with [[Autoregressive fractionally integrated moving average]]
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| {{Calculus|expanded=Specialized calculi}}
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| In [[fractional calculus]], an area of [[applied mathematics]], the '''differintegral''' is a combined [[Differential operator|differentiation]]/[[integral operator|integration]] operator. Applied to a [[function (mathematics)|function]] ƒ, the ''q''-differintegral of ''f'', here denoted by
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| :<math>\mathbb{D}^qf</math>
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| is the fractional derivative (if ''q'' > 0) or fractional integral (if ''q'' < 0). If ''q'' = 0, then the ''q''-th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several legitimate definitions of the differintegral.
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| ==Standard definitions==
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| The three most common forms are:
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| *The [[Riemann–Liouville differintegral]]
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| :This is the simplest and easiest to use, and consequently it is the most often used. It is a generalization of the [[Cauchy formula for repeated integration]] to arbitrary order.
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| : <math>
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| \begin{align}
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| {}_a\mathbb{D}^q_tf(t) & = \frac{d^qf(t)}{d(t-a)^q} \\
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| & =\frac{1}{\Gamma(n-q)} \frac{d^n}{dt^n} \int_{a}^t (t-\tau)^{n-q-1}f(\tau)d\tau
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| \end{align}
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| </math> | |
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| *The [[Grunwald–Letnikov differintegral]]
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| :The Grunwald–Letnikov differintegral is a direct generalization of the definition of a [[derivative]]. It is more difficult to use than the Riemann–Liouville differintegral, but can sometimes be used to solve problems that the Riemann–Liouville cannot.
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| : <math>
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| \begin{align}
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| {}_a\mathbb{D}^q_tf(t) & = \frac{d^qf(t)}{d(t-a)^q} \\
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| & =\lim_{N \to \infty}\left[\frac{t-a}{N}\right]^{-q}\sum_{j=0}^{N-1}(-1)^j{q \choose j}f\left(t-j\left[\frac{t-a}{N}\right]\right)
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| \end{align}
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| </math>
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| *The [[Weyl differintegral]]
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| :This is formally similar to the Riemann–Liouville differintegral, but applies to [[periodic function]]s, with integral zero over a period.
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| ==Definitions via transforms==
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| Recall the [[continuous Fourier transform]], here denoted <math> \mathcal{F}</math> :
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| :<math> F(\omega) = \mathcal{F}\{f(t)\} = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t) e^{- i\omega t}\,dt </math>
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| Using the continuous Fourier transform, in Fourier space, differentiation transforms into a multiplication:
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| :<math>\mathcal{F}\left[\frac{df(t)}{dt}\right] = i \omega \mathcal{F}[f(t)]</math>
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| So,
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| :<math>\frac{d^nf(t)}{dt^n} = \mathcal{F}^{-1}\left\{(i \omega)^n\mathcal{F}[f(t)]\right\}</math>
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| which generalizes to | |
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| :<math>\mathbb{D}^qf(t)=\mathcal{F}^{-1}\left\{(i \omega)^q\mathcal{F}[f(t)]\right\}.</math>
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| Under the [[Laplace transform]], here denoted by <math> \mathcal{L}</math>, differentiation transforms into a multiplication
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| :<math>\mathcal{L}\left[\frac{df(t)}{dt}\right] = s\mathcal{L}[f(t)].</math>
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| Generalizing to arbitrary order and solving for ''D''<sup>''q''</sup>''f''(''t''), one obtains
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| :<math>\mathbb{D}^qf(t)=\mathcal{L}^{-1}\left\{s^q\mathcal{L}[f(t)]\right\}.</math>
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| ==Basic formal properties==
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| ''Linearity rules''
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| :<math>\mathbb{D}^q(f+g)=\mathbb{D}^q(f)+\mathbb{D}^q(g)</math>
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| :<math>\mathbb{D}^q(af)=a\mathbb{D}^q(f)</math>
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| ''Zero rule''
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| :<math>\mathbb{D}^0 f=f \, </math>
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| ''Product rule''
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| :<math>\mathbb{D}^q_t(fg)=\sum_{j=0}^{\infty} {q \choose j}\mathbb{D}^j_t(f)\mathbb{D}^{q-j}_t(g)</math>
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| In general, ''composition (or [[semigroup]]) rule''
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| :<math>\mathbb{D}^a\mathbb{D}^{b}f = \mathbb{D}^{a+b}f</math>
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| is '''not satisfied'''. See Property 2.4 (page 75) in the book A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. (Elsevier, 2006).
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| ==Some basic formulae==
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| :<math>\mathbb{D}^q(t^n)=\frac{\Gamma(n+1)}{\Gamma(n+1-q)}t^{n-q}</math>
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| :<math>\mathbb{D}^q(\sin(t))=\sin \left( t+\frac{q\pi}{2} \right) </math>
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| :<math>\mathbb{D}^q(e^{at})=a^q e^{at}</math>
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| == See also ==
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| * [[Fractional-order integrator]]
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| ==References==
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| * "An Introduction to the Fractional Calculus and Fractional Differential Equations", by Kenneth S. Miller, Bertram Ross (Editor), John Wiley & Sons; 1 edition (May 19, 1993). ISBN 0-471-58884-9.
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| * "The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order (Mathematics in Science and Engineering, V)", by Keith B. Oldham, Jerome Spanier, Academic Press; (November 1974). ISBN 0-12-525550-0.
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| * "Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications", (Mathematics in Science and Engineering, vol. 198), by Igor Podlubny, Academic Press (October 1998). ISBN 0-12-558840-2.
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| * "Fractals and Fractional Calculus in Continuum Mechanics", by A. Carpinteri (Editor), F. Mainardi (Editor), Springer-Verlag Telos; (January 1998). ISBN 3-211-82913-X.
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| * [http://www.worldscibooks.com/mathematics/p614.html Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models.] by F. Mainardi, Imperial College Press, 2010. 368 pages.
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| * [http://www.springer.com/physics/complexity/book/978-3-642-14002-0 Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media.] by V.E. Tarasov, Springer, 2010. 450 pages.
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| * [http://www.springer.com/physics/theoretical,+mathematical+%26+computational+physics/book/978-3-642-33910-3 Fractional Derivatives for Physicists and Engineers] by V.V. Uchaikin, Springer, Higher Education Press, 2012, 385 pages.
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| * "Physics of Fractal Operators", by Bruce J. West, Mauro Bologna, Paolo Grigolini, Springer Verlag; (January 14, 2003). ISBN 0-387-95554-2
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| * ''Operator of fractional derivative in the complex plane'', by Petr Zavada, Commun.Math.Phys.192, pp. 261–285,1998. {{doi|10.1007/s002200050299}} (available [http://www.springerlink.com/content/2xbape94pk99k75a/ online] or as the [http://arxiv.org/abs/funct-an/9608002 arXiv preprint])
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| * ''Relativistic wave equations with fractional derivatives and pseudodifferential operators'', by Petr Zavada, Journal of Applied Mathematics, vol. 2, no. 4, pp. 163–197, 2002. {{doi|10.1155/S1110757X02110102}} (available [http://www.hindawi.com/GetArticle.aspx?doi=10.1155/S1110757X02110102&e=cta online] or as the [http://arxiv.org/abs/hep-th/0003126 arXiv preprint])
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| ==External links==
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| * [http://mathworld.wolfram.com/FractionalCalculus.html MathWorld – Fractional calculus]
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| *[http://mathworld.wolfram.com/FractionalDerivative.html MathWorld – Fractional derivative]
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| * Specialized journal: [http://www.diogenes.bg/fcaa/ Fractional Calculus and Applied Analysis]
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| * Specialized journal: [http://fds.ele-math.com/ Fractional Dynamic Systems (FDS)]
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| * Specialized journal: [http://www.nonlinearscience.com/journal_2218-3892.php Communications in Fractional Calculus] (ISSN 2218-3892)
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| * http://www.nasatech.com/Briefs/Oct02/LEW17139.html
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| * http://unr.edu/homepage/mcubed/FRG.html
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| * [http://www.tuke.sk/podlubny/fc_resources.html Igor Podlubny's collection of related books, articles, links, software, etc. ]
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| * Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation. [http://www.diogenes.bg/fcaa/ Fractional Calculus and Applied Analysis], vol. 5, no. 4, 2002, 367–386. (available as [http://www.tuke.sk/podlubny/pspdf/pifcaa_r.pdf original article], or [http://arxiv.org/abs/math.CA/0110241 preprint at Arxiv.org])
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| [[Category:Fractional calculus]]
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| [[Category:Generalizations of the derivative]]
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| [[Category:Linear operators in calculus]]
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