Differintegral Explained

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

Dqf

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 definitions of the differintegral.

Standard definitions

The four most common forms are:

n=\lceilq\rceil

. \begin^_a\mathbb^q_tf(t) & = \frac \\& =\frac \frac \int_^t (t-\tau)^f(\tau)d\tau\end

^_a\mathbb^q_tf(t) & = \frac \\& =\lim_\left[\frac{t-a}{N}\right]^\sum_^(-1)^jf\left(t-j\left[\frac{t-a}{N}\right]\right)\end

f(t)

is equal to zero. Moreover, a form of the Laplace transform allows to simply evaluate the initial conditions by computing finite, integer-order derivatives at point

a

. \begin^_a\mathbb^q_tf(t) & = \frac \\& =\frac \int_^t \fracd\tau\end

Definitions via transforms

The definitions of fractional derivatives given by Liouville, Fourier, and Grunwald and Letnikov coincide.[1] They can be represented via Laplace, Fourier transforms or via Newton series expansion.

Recall the continuous Fourier transform, here denoted

l{F}

: F(\omega) = \mathcal\ = \frac\int_^\infty f(t) e^\,dt

Using the continuous Fourier transform, in Fourier space, differentiation transforms into a multiplication:\mathcal\left[\frac{df(t)}{dt}\right] = i \omega \mathcal[f(t)]

So,\frac = \mathcal^\left\which generalizes to\mathbb^qf(t) = \mathcal^\left\.

Under the bilateral Laplace transform, here denoted by

l{L}

and defined as \mathcal[f(t)] =\int_^\infty e^ f(t)\, dt, differentiation transforms into a multiplication\mathcal\left[\frac{df(t)}{dt}\right] = s\mathcal[f(t)].

Generalizing to arbitrary order and solving for

Dqf(t)

, one obtains\mathbb^qf(t)=\mathcal^\left\.

Representation via Newton series is the Newton interpolation over consecutive integer orders:

\mathbb^qf(t) =\sum_^ \binom m \sum_^m\binom mk(-1)^f^(x).

For fractional derivative definitions described in this section, the following identities hold:

Dq(t

n)=\Gamma(n+1)
\Gamma(n+1-q)

tn-q

Dq(\sin(t))=\sin\left(t+

q\pi
2

\right)

Dq(eat)=aqeat

[2]

Basic formal properties

\mathbb^q(af) = a\mathbb^q(f)

In general, composition (or semigroup) rule is a desirable property, but is hard to achieve mathematically and hence is not always completely satisfied by each proposed operator;[3] this forms part of the decision making process on which one to choose:

See also

References

External links

Notes and References

  1. Book: Herrmann, Richard . Fractional Calculus: An Introduction for Physicists . 2011 . 9789814551076 .
  2. See Book: Herrmann, Richard . 16 . Fractional Calculus: An Introduction for Physicists . 2011 . 9789814551076 .
  3. See Book: 75 . 2. Fractional Integrals and Fractional Derivatives §2.1 Property 2.4 . https://books.google.com/books?id=uxANOU0H8IUC&pg=PA75 . A. A. . Kilbas . H. M. . Srivastava . J. J. . Trujillo . Theory and Applications of Fractional Differential Equations . Elsevier . 2006 . 9780444518323 .