Differintegral

In mathematics, the differintegral is the combined differentiation/integration operator used in fractional calculus. This operator is here denoted

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

See the page on fractional calculus for the general context.

Contents

1 Basic formal properties 2 Some basic formulae 3 Standard definitions 4 Definitions via transforms 5 External links 6 Resource books

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>

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(x+\frac{q\pi}{2})<math>

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

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)=\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> definition

• The Grunwald-Letnikov differintegral

<math>{}_a\mathbb{D}^q_tf(t)=\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> definition

• The Weyl differintegral

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

Definitions via transforms

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

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

This generalizes to

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

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

<math>\mathcal{L}[\frac{df(t)}{dt}] = 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> definition

External links

• MathWorld - Fractional calculus (http://mathworld.wolfram.com/FractionalCalculus.html)
• MathWorld - Fractional derivative (http://mathworld.wolfram.com/FractionalDerivative.html)
• Specialized journal: Fractional Calculus and Applied Analysis (http://www.diogenes.bg/fcaa/)
• http://www.nasatech.com/Briefs/Oct02/LEW17139.html
• http://unr.edu/homepage/mcubed/FRG.html
• Igor Podlubny's collection of related books, articles, links, software, etc.  (http://www.tuke.sk/podlubny/fc_resources.html)
• Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus and Applied Analysis (http://www.diogenes.bg/fcaa/), vol. 5, no. 4, 2002, 367–386. (available as original article (http://www.tuke.sk/podlubny/pspdf/pifcaa_r.pdf), or preprint at Arxiv.org (http://arxiv.org/abs/math.CA/0110241))

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)

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

Hardcover

Publisher: 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

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