Talk:Fractional calculus/alternative

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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
 * Chaos theory
 * Fractals
 * Control theory


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

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:

 $${}_a\mathbb{D}^q_tf(x)=\frac{1}{\Gamma(n-q)}\frac{d^n}{dx^n}\int_{a}^{t}(t-\tau)^{n-q-1}f(\tau)d\tau + \Psi(x)$$  definition

(where $$\Psi(t)$$ is a complementary 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

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