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In mathematics, the combined differentiation/integration operator used in fractional calculus is called the differintegral, and it has a few different forms which are all equivalent. (provided that they are initialized(used) properly.)

It is noted:

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

and is most generally defined as:

<math>{}_a\mathbb{D}^q_t= \left\{\begin{matrix} \frac{d^q}{dx^q}, & \mathbb{R}(q)>0 \\ 1, & \mathbb{R}(q)=0 \\ \int^t_a(dx)^{-q}, & \mathbb{R}(q)<0 \end{matrix}\right.</math>

By far, the three most common forms are:

  • The Riemann-Liouville differintegral(RL)
This is the simplest and easiest to use, and consequently it is the most often used.

We first introduce the Riemann-Liouville fractional integral, which is a straight-forward generalization of the Cauchy formula for repeated integration[?]:

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

This gives us integration to an arbitrary order. To get differentation to an arbitrary order, we simply integrate to arbitrary order n-q, and differentiate the result to integer order n. (We choose n and q so that n is the smallest positive integer greater than or equal to q (that is, the ceiling of q)):


Thus, we have differentiated n-(n-q)=q times. The RL differintegral is thus defined as(the constant is brought to the front):

<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</math> definition

When we are taking the differintegral at the upper bound (t), it is usually written:

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

see for more info: Riemann-Liouville differintegral[?].

  • The Grunwald-Letnikov differintegral(GL). -This has an interesting form. It may provide some geometric insight into fractional calculus if we can develop a good intepretation of it. This also poses fewer restrictions on the function being differintegrated. It is a generalization of the infinite Riemann Sum[?] which defines integer-order integration in calculus. It uses the Binomial coefficient (generalized by the gamma function to arbitrary domain), commonly used in the branch of mathematics called counting[?].:

<math>{}_a\mathbb{D}^q_tf(t)=\frac{d^qf(t)}{d(t-a)^q}=\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

see for more info: Grunwald-Letnikov differintegral[?].

  • The Weyl differintegral. -This is very similiar to the Riemann-Louiville differintegral. The most important difference is that its upper bound is infinity.

see for more info: Weyl differintegral[?].

Table of contents

Definitions via transform

Any function can be defined in a space isomorphic to a space which it has been shown to be defined in. We therefore define the differintegral via its behavior in certain transformed spaces corresponding to some common transformations.

  • The Fourier Transform. -Firstly, we define differintegration in Fourier space (using the continuous Fourier transform, here denoted F). In Fourier space, differentation transforms into a simple translation:

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

This easily generalizes to:

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

Note, however, that there are no bounds of differintegration.

  • The Laplace Transform. Under the Laplace transform, (denoted here by L), differentation transforms to a simple translation:
<math>\mathcal{L}[\frac{df(t)}{dt}] = s^{-1}\mathcal{L}[f(t)]</math>

Generalizing to arbitrary order and solving for Dqf(t), one obtains:

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

Again, there are no bounds of differintegration.


Web Resources

Book resourcees "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
Publisher: Academic Press; (November 1974)
ASIN: 0125255500

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