# 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

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

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.)

## 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 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
- Hardcover
- 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
- Hardcover
- 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