Differintegral

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)):

<math>{}_a\mathbb{D}^q_tf(x)=\frac{d^n}{dx^n}{}_a\mathbb{D}^{-(n-q)}_tf(x)</math>

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 1 Definitions via transform 2 History 3 Web Resources 4 Book resourcees

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 D^qf(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.

History

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

Hardcover

Publisher: Academic Press; (November 1974)

ASIN: 0125255500