Differintegral

"Fractional integration" redirects here and is not to be confused with Autoregressive fractionally integrated moving average.

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

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

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 ${\textstyle {\mathcal {L}}[f(t)]=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt}$, 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 ${\displaystyle \mathbb {D} ^{q}f(t)}$, one obtains $${\displaystyle \mathbb {D} ^{q}f(t)={\mathcal {L}}^{-1}\left\{s^{q}{\mathcal {L}}[f(t)]\right\}.}$$

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

$${\displaystyle \mathbb {D} ^{q}f(t)=\sum _{m=0}^{\infty }{\binom {q}{m}}\sum _{k=0}^{m}{\binom {m}{k}}(-1)^{m-k}f^{(k)}(x).}$$

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

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

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

${\displaystyle \mathbb {D} ^{q}(e^{at})=a^{q}e^{at}}$^[2]

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 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:

• ${\textstyle \mathbb {D} ^{a}\mathbb {D} ^{b}f=\mathbb {D} ^{a+b}f}$ (ideally)
• ${\textstyle \mathbb {D} ^{a}\mathbb {D} ^{b}f\neq \mathbb {D} ^{a+b}f}$ (in practice)

See also

• Fractional-order integrator