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# Fractional calculus

Back to: Mathematics | Next topic: Differintegrals

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:

Math-oriented
Chaos theory
Fractals
Control theory

Physics-oriented
Electricity
Mechanics
Heat conduction
Viscoelasticity
Hydrogeology
Nonlinear geophysics

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

## 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 equivalent. (provided that they are initialized (used) properly.)

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

(where is a complimentary function.)

## 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

### 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
Hardcover
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
Hardcover