Differintegral

The four 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. Here, ${\displaystyle n=\lceil q\rceil }$.

${\displaystyle {\begin{aligned}{}_{a}^{RL}\mathbb {D} _{t}^{q}f(t)&={\frac {d^{q}f(t)}{d(t-a)^{q}}}\\&={\frac {1}{\Gamma (n-q)}}{\frac {d^{n}}{dt^{n}}}\int _{a}^{t}(t-\tau )^{n-q-1}f(\tau )d\tau \end{aligned}}}$

• The Grunwald–Letnikov differintegral

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.

${\displaystyle {\begin{aligned}{}_{a}^{GL}\mathbb {D} _{t}^{q}f(t)&={\frac {d^{q}f(t)}{d(t-a)^{q}}}\\&=\lim _{N\to \infty }\left[{\frac {t-a}{N}}\right]^{-q}\sum _{j=0}^{N-1}(-1)^{j}{q \choose j}f\left(t-j\left[{\frac {t-a}{N}}\right]\right)\end{aligned}}}$

• The Weyl differintegral

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

• The Caputo differintegral

In opposite to the Riemann-Liouville differintegral, Caputo derivative of a constant ${\displaystyle f(t)}$ 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 ${\displaystyle a}$.

${\displaystyle {\begin{aligned}{}_{a}^{C}\mathbb {D} _{t}^{q}f(t)&={\frac {d^{q}f(t)}{d(t-a)^{q}}}\\&={\frac {1}{\Gamma (n-q)}}\int _{a}^{t}{\frac {f^{(n)}(\tau )}{(t-\tau )^{q-n+1}}}d\tau \end{aligned}}}$

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

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

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

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

So,

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

which generalizes to

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

Under the bilateral Laplace transform, here denoted by ${\displaystyle {\mathcal {L}}}$ and defined as ${\displaystyle {\mathcal {L}}[f(t)]=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt}$, differentiation transforms into a multiplication

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

Generalizing to arbitrary order and solving for D^qf(t), one obtains

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

Linearity rules

${\displaystyle \mathbb {D} ^{q}(f+g)=\mathbb {D} ^{q}(f)+\mathbb {D} ^{q}(g)}$

${\displaystyle \mathbb {D} ^{q}(af)=a\mathbb {D} ^{q}(f)}$

Zero rule

${\displaystyle \mathbb {D} ^{0}f=f\,}$

Product rule

${\displaystyle \mathbb {D} _{t}^{q}(fg)=\sum _{j=0}^{\infty }{q \choose j}\mathbb {D} _{t}^{j}(f)\mathbb {D} _{t}^{q-j}(g)}$

In general, composition (or semigroup) rule is not satisfied[1]:

${\displaystyle \mathbb {D} ^{a}\mathbb {D} ^{b}f\neq \mathbb {D} ^{a+b}f}$

${\displaystyle \mathbb {D} ^{q}(t^{n})={\frac {\Gamma (n+1)}{\Gamma (n+1-q)}}t^{n-q}}$

${\displaystyle \mathbb {D} ^{q}(\sin(t))=\sin \left(t+{\frac {q\pi }{2}}\right)}$

${\displaystyle \mathbb {D} ^{q}(e^{at})=a^{q}e^{at}}$

• Fractional-order integrator

• MathWorld – Fractional calculus
• MathWorld – Fractional derivative
• Specialized journal: Fractional Calculus and Applied Analysis (1998-2014) and Fractional Calculus and Applied Analysis (from 2015)
• Specialized journal: Fractional Differential Equations (FDE)
• Specialized journal: Communications in Fractional Calculus (ISSN 2218-3892)
• Specialized journal: Journal of Fractional Calculus and Applications (JFCA)
• Lorenzo, Carl F.; Hartley, Tom T. (2002). "Initialized Fractional Calculus". Information Technology. Tech Briefs Media Group.
• https://web.archive.org/web/20040502170831/http://unr.edu/homepage/mcubed/FRG.html
• Igor Podlubny's collection of related books, articles, links, software, etc.
• Podlubny, I. (2002). "Geometric and physical interpretation of fractional integration and fractional differentiation" (PDF). Fractional Calculus and Applied Analysis. 5 (4): 367–386. arXiv:math.CA/0110241. Bibcode:2001math.....10241P.
• Zavada, P. (1998). "Operator of fractional derivative in the complex plane". Communications in Mathematical Physics. 192 (2): 261–285. arXiv:funct-an/9608002. Bibcode:1998CMaPh.192..261Z. doi:10.1007/s002200050299.