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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

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