Fractional calculus

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[?]

Table of contents 1 History 2 Differintegrals 3 Forms of fractional calculus 4 Closely related topics 4.1 Resource URLS 4.2 Resource Books

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 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>

Forms of fractional calculus

• elementary 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[?]

Resource URLS

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[?]

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

"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