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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:

**Math-oriented****Physics-oriented**- Electricity
- Mechanics
- Heat conduction
- Viscoelasticity
- Hydrogeology
- Nonlinear geophysics

Table of contents |

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.)

## Elementary topics

- differintegral
- initialization of the differintegrals
- basic properties of the differintegral
- differintegration of some elementary functions
- basic rules of differintegration
- differintegration of some complex functions

## Forms of fractional calculus

- initialized fractional calculus
- local fractional derivative (LFD)

## 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

- http://mathworld.wolfram.com/FractionalCalculus.html
- http://www.diogenes.bg/fcaa/
- http://www.nasatech.com/Briefs/Oct02/LEW17139.html
- http://unr.edu/homepage/mcubed/FRG.html
- http://www.tuke.sk/podlubny/fc_resources.html

### 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

- by Keith B. Oldham, Jerome Spanier
- Hardcover
- Publisher: Academic Press; (November 1974)
- ASIN: 0125255500

- by Igor Podlubny
- Hardcover
- Publisher: Academic Press; (October 1998)
- ISBN: 0125588402

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

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