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:
- Math-orientated
- Chaos theory
- Fractals
- Control theory
- Physics-orientated
- 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 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>
anomalous diffusion[?] --
fractional brownian motion[?] --
fractals and fractional calculus[?] --
extraordinary differential equations[?] --
partial fractional derivatives[?] --
fractional reaction-diffusion equations[?] --
fractional calculus in continuum mechanics[?]
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[?]
"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
"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