Differintegral

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In mathematics, the differintegral is the combined differentiation/integration operator used in fractional calculus. The operator does not define a separate function, but is a notation style for taking both the fractional derivative and the fractional integral of the same expression. This operator is here denoted

<math>\mathbb{D}^q_t.</math>

See the page on fractional calculus for the general context.

1 Standard definitions
2 Definitions via transforms
3 Basic formal properties
4 Some basic formulae
5 See also
6 References
7 External links

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Standard definitions

The three most common forms are:

The Riemann-Liouville differintegral

This is the simplest and easiest to use, and consequently it is the most often used. It is a generalization of the Cauchy formula for repeated integration to arbitrary order.

<math>{}_a\mathbb{D}^q_tf(t)</math>	<math>=\frac{d^qf(t)}{d(t-a)^q}</math>
	<math>=\frac{1}{\Gamma(n-q)} \frac{d^n}{dt^n} \int_{a}^{t}(t-\tau)^{n-q-1}f(\tau)d\tau</math>

The Grunwald-Letnikov differintegral

<math>{}_a\mathbb{D}^q_tf(t)</math>	<math>=\frac{d^qf(t)}{d(t-a)^q}</math>
	<math>=\lim_{N \to \infty}\left[\frac{t-a}{N}\right]^{-q}\sum_{j=0}^{N-1}(-1)^j{q \choose j}f\left(t-j\left[\frac{t-a}{N}\right]\right)</math>

The Weyl differintegral

This is formally similar to the Riemann-Liouville differintegral, but applies to periodic functions, with integral zero over a period.

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Definitions via transforms

Using the continuous Fourier transform, here denoted F: in Fourier space, differentiation transforms into a multiplication:

<math>\mathcal{F}\left[\frac{df(t)}{dt}\right] = it\mathcal{F}[f(t)]</math>

which generalizes to

<math>\mathbb{D}^qf(t)=\mathcal{F}^{-1}\left\{(it)^q\mathcal{F}[f(t)]\right\}.</math>

Under the Laplace transform, here denoted by L, differentiation transforms to a multiplication

<math>\mathcal{L}\left[\frac{df(t)}{dt}\right] = s^{-1}\mathcal{L}[f(t)].</math>

Generalizing to arbitrary order and solving for D^qf(t), one obtains

<math>\mathbb{D}^qf(t)=\mathcal{L}^{-1}\left\{s^{-q}\mathcal{L}[f(t)]\right\}.</math>

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Basic formal properties

Linearity rules

<math>\mathbb{D}^{q}(x+y)=\mathbb{D}^{q}(x)+\mathbb{D}^{q}(y)</math>

<math>\mathbb{D}^{q}(ax)=a\mathbb{D}^{q}(x)</math>

Composition (or semigroup) rule

<math>\mathbb{D}^a\mathbb{D}^{b}x = \mathbb{D}^{a+b}x</math>

Zero rule

<math>\mathbb{D}^{0}x=x</math>

Subclass rule

<math>\mathbb{D}^{a}x=d^{a}x</math> for a a natural number

Product rule of differintegration

<math>\mathbb{D}^q_t(xy)=\sum_{j=0}^{\infty} {q \choose j}\mathbb{D}^j_t(x)\mathbb{D}^{q-j}_t(y)</math>

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Some basic formulae

<math>\mathbb{D}^{q}(x^n)=\frac{\Gamma(n+1)}{\Gamma(n+1-q)}x^{n-q}</math>

<math>\mathbb{D}^{q}(\sin(x))=\sin \left( x+\frac{q\pi}{2} \right) </math>

<math>\mathbb{D}^{q}(e^{ax})=a^{q}e^{ax}</math>

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References

"An Introduction to the Fractional Calculus and Fractional Differential Equations", by Kenneth S. Miller, Bertram Ross (Editor), John Wiley & Sons; 1 edition (May 19, 1993). ISBN 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, Academic Press; (November 1974). ISBN 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, Academic Press (October 1998). ISBN 0125588402.

"Fractals and Fractional Calculus in Continuum Mechanics", by A. Carpinteri (Editor), F. Mainardi (Editor), Springer-Verlag Telos; (January 1998). ISBN 321182913X.