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Fractional calculus

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Fractional calculus is a part of mathematics dealing with generalisations of the derivative to derivatives of arbitrary 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

Table of contents
1 History
2 Differintegrals
3 Elementary topics
4 Forms of fractional calculus
5 Closely related topics
6 External Resources


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

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

(where is a complimentary function.)

Elementary topics

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

External Resources

External links

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

"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
Publisher: Academic Press; (October 1998)
ISBN: 0125588402

"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