Differintegral

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

${\displaystyle \mathbb {D} ^{q}f}$

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.

Standard definitions[]

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

  

Definitions via transforms[]

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,

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

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

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

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

Basic formal properties[]

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]

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

See also[]

• Fractional-order integrator

References[]

External links[]

• 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.