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Fractional calculus

Fractional calculus is a part of mathematics dealing with generalisations of the derivative to derivatives of arbitary order (not necessarily an integer). The name "fractional calculus" is somewhat of a misnomer since the generalisations are by no means restricted to fractions, but the label persists for historical reasons.

The fractional derivative of a function to order a is often defined implicitly by the fourier transform. The fractional derivative in a point x is a local property only when a is an integer.

Applications of the fractional calculus includes partial differential equations, especially parabolic ones where it is sometimes useful to split a time-derivative into fractional time[?].

There are many well known fields of application where we can use the fractional calculus. Just a few of them are:

Chaos theory
Control theory

Heat conduction[?]
Nonlinear geophysics[?]

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History (fill this in (it started about 300 years ago.))

Differintegrals The combined differentation/integral operator used in fractional calculus is called the differintegral, and it has a couple of different forms which are all equavalent. (provided that they are initialized(used) properly.)

By far, the most common form is the Riemann-Louiville form:

<math>{}_a\mathbb{D}^q_tf(x)=\frac{1}{\Gamma(n-q)}\frac{d^n}{dx^n}\int_{a}^{t}(t-\tau)^{n-q-1}f(\tau)d\tau + \Psi(x)</math>

Forms of fractional calculus

Closely related topics anomalous diffusion[?] -- fractional brownian motion[?] -- fractals and fractional calculus[?] --

extraordinary differential equations[?] -- partial fractional derivatives[?] -- fractional reaction-diffusion equations[?] -- fractional calculus in continuum mechanics[?]

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Resource Books

"An Introduction to the Fractional Calculus and Fractional Differential Equations"

by Kenneth S. Miller, Bertram Ross (Editor)
Hardcover: 384 pages ; Dimensions (in inches): 1.00 x 9.75 x 6.50
Publisher: John Wiley & Sons; 1 edition (May 19, 1993)
ASIN: 0471588849

"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
Publisher: Academic Press; (November 1974)
ASIN: 0125255500

"Fractals and Fractional Calculus in Continuum Mechanics"

by A. Carpinteri (Editor), F. Mainardi (Editor)
Paperback: 348 pages
Publisher: Springer-Verlag Telos; (January 1998)
ISBN: 321182913X

"Physics of Fractal Operators"

by Bruce J. West, Mauro Bologna, Paolo Grigolini
Hardcover: 368 pages
Publisher: Springer Verlag; (January 14, 2003)
ISBN: 0387955542

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