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

${\mathbb{𝔻}}^{q}f$

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, $n=\lceil q\rceil$. $$\begin{array}{rl}{}_a^{RL}{\mathbb{𝔻}}_t^{q}f(t)& =\frac{d^{q}f(t)}{d(t-a{)}^q}\\ & =\frac{1}{\mathrm{\Gamma }(n-q)}\frac{d^n}{d{t}^n}\underset{a}{\overset{t}{\int }}(t-\tau {)}^{n-q-1}f(\tau )d\tau \end{array}$$
• 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. $$\begin{array}{rl}{}_a^{GL}{\mathbb{𝔻}}_t^{q}f(t)& =\frac{d^{q}f(t)}{d(t-a{)}^q}\\ & =\underset{N\to \mathrm{\infty }}{\lim }{\left[\frac{t-a}{N}\right]}^{-q}\sum _{j=0}^{N-1}(-1{)}^j(\genfrac{}{}{0ex}{}{q}{j})f(t-j\left[\frac{t-a}{N}\right])\end{array}$$
• 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 $f(t)$ 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 $a$. $$\begin{array}{rl}{}_a^C{\mathbb{𝔻}}_t^{q}f(t)& =\frac{d^{q}f(t)}{d(t-a{)}^q}\\ & =\frac{1}{\mathrm{\Gamma }(n-q)}\underset{a}{\overset{t}{\int }}\frac{f^{(n)}(\tau )}{(t-\tau {)}^{q-n+1}}d\tau \end{array}$$

Definitions via transforms

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

Recall the continuous Fourier transform, here denoted ${ℱ}$: $$F(\omega )={ℱ}\{f(t)\}=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t)e^{-i\omega t}dt$$

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

So, $$\frac{d^{n}f(t)}{d{t}^n}={{ℱ}}^{-1}\{(i\omega {)}^n{ℱ}[f(t)]\}$$ which generalizes to $${\mathbb{𝔻}}^{q}f(t)={{ℱ}}^{-1}\{(i\omega {)}^q{ℱ}[f(t)]\}.$$

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

Generalizing to arbitrary order and solving for ${\mathbb{𝔻}}^{q}f(t)$, one obtains $${\mathbb{𝔻}}^{q}f(t)={{ℒ}}^{-1}\{s^q{ℒ}[f(t)]\}.$$

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

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

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

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

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

${\mathbb{𝔻}}^q(e^{at})=a^q{e}^{at}$

Basic formal properties

• Linearity rules $${\mathbb{𝔻}}^q(f+g)={\mathbb{𝔻}}^q(f)+{\mathbb{𝔻}}^q(g)$$

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

• Zero rule $${\mathbb{𝔻}}^{0}f=f$$
• Product rule $${\mathbb{𝔻}}_t^q(fg)=\sum _{j=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{q}{j}){\mathbb{𝔻}}_t^j(f){\mathbb{𝔻}}_t^{q-j}(g)$$

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; this forms part of the decision making process on which one to choose:

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

See also

• Fractional-order integrator

References

• Miller, Kenneth S. (1993). Ross, Bertram. ed. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley. ISBN 0-471-58884-9. 
• Oldham, Keith B.; Spanier, Jerome (1974). The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order. Mathematics in Science and Engineering. V. Academic Press. ISBN 0-12-525550-0. 
• Podlubny, Igor (1998). Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering. 198. Academic Press. ISBN 0-12-558840-2. 
• Carpinteri, A.; Mainardi, F., eds (1998). Fractals and Fractional Calculus in Continuum Mechanics. Springer-Verlag. ISBN 3-211-82913-X. 
• Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press. ISBN 978-1-84816-329-4. http://www.worldscibooks.com/mathematics/p614.html. 
• Tarasov, V.E. (2010). Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Nonlinear Physical Science. Springer. ISBN 978-3-642-14003-7. https://www.springer.com/physics/complexity/book/978-3-642-14003-7. 
• Uchaikin, V.V. (2012). Fractional Derivatives for Physicists and Engineers. Nonlinear Physical Science. Springer. ISBN 978-3-642-33910-3. Bibcode: 2013fdpe.book.....U. https://www.springer.com/physics/theoretical,+mathematical+%26+computational+physics/book/978-3-642-33910-3. 
• West, Bruce J.; Bologna, Mauro; Grigolini, Paolo (2003). Physics of Fractal Operators. Springer Verlag. ISBN 0-387-95554-2. https://books.google.com/books?id=EgyTpQZOga0C&pg=PR7. 

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://www.techbriefs.com/component/content/article/tb/techbriefs/information-sciences/2264. 
• 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". Fractional Calculus and Applied Analysis 5 (4): 367–386. Bibcode: 2001math.....10241P. http://www.tuke.sk/podlubny/pspdf/pifcaa_r.pdf. Retrieved 2004-05-18. 
• Zavada, P. (1998). "Operator of fractional derivative in the complex plane". Communications in Mathematical Physics 192 (2): 261–285. doi:10.1007/s002200050299. Bibcode: 1998CMaPh.192..261Z.