Differintegral

In fractional calculus, an area of applied mathematics, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the q-differintegral of f, here denoted by

\mathbb{D}^qf

is the fractional derivative (if q>0) or fractional integral (if q<0). If q=0, then the q-th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several legitimate definitions of the differintegral.

Contents

Standard definitions

The three most common forms are:

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.
{}_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
The Grunwald-Letnikov differintegral is a direct generalization of the definition of a derivative. It is more difficult to use than the Riemann-Liouville differintegral, but can sometimes be used to solve problems that the Riemann-Liouville cannot.
{}_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)
This is formally similar to the Riemann-Liouville differintegral, but applies to periodic functions, with integral zero over a period.

Definitions via transforms

Recall the continuous Fourier transform, here denoted \mathcal{F} :

F(\omega) = \mathcal{F}\{f(t)\} = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t) e^{- i\omega t}\,dt

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

\mathcal{F}\left[\frac{df(t)}{dt}\right] = i \omega \mathcal{F}[f(t)]

So,

\frac{d^nf(t)}{dt^n} = \mathcal{F}^{-1}\left\{(i \omega)^n\mathcal{F}[f(t)]\right\}

which generalizes to

\mathbb{D}^qf(t)=\mathcal{F}^{-1}\left\{(i \omega)^q\mathcal{F}[f(t)]\right\}.

Under the Laplace transform, here denoted by \mathcal{L}, differentiation transforms into a multiplication

\mathcal{L}\left[\frac{df(t)}{dt}\right] = s\mathcal{L}[f(t)].

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

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

Basic formal properties

Linearity rules

\mathbb{D}^{q}(f%2Bg)=\mathbb{D}^{q}(f)%2B\mathbb{D}^{q}(g)
\mathbb{D}^{q}(af)=a\mathbb{D}^{q}(f)

Zero rule

\mathbb{D}^{0}f=f

Product rule

\mathbb{D}^q_t(fg)=\sum_{j=0}^{\infty} {q \choose j}\mathbb{D}^j_t(f)\mathbb{D}^{q-j}_t(g)

In general, composition (or semigroup) rule

\mathbb{D}^a\mathbb{D}^{b}f = \mathbb{D}^{a%2Bb}f

is not satisfied. See Property 2.4 (page 75) in the book A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. (Elsevier, 2006).

Some basic formulae

\mathbb{D}^{q}(t^n)=\frac{\Gamma(n%2B1)}{\Gamma(n%2B1-q)}t^{n-q}
\mathbb{D}^{q}(\sin(t))=\sin \left( t%2B\frac{q\pi}{2} \right)
\mathbb{D}^{q}(e^{at})=a^{q}e^{at}

See also

References

External links