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

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 differintegralThis 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 .

• The Grunwald–Letnikov differintegralThe 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. $$\backslash begin$$

^_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

• 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 differintegralIn 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 .

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} :$$F(\backslash omega)\; =\; \backslash mathcal\backslash \; =\; \backslash frac\backslash int\_^\backslash infty\; f(t)\; e^\backslash ,dt$$

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

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

Under the bilateral Laplace transform, here denoted by

l{L} and defined as $\backslash mathcal[f(t)]\; =\backslash int\_^\backslash infty\; e^\; f(t)\backslash ,\; dt$, differentiation transforms into a multiplication$$\backslash mathcal\backslash left[\backslash frac\{df(t)\}\{dt\}\backslash right]\; =\; s\backslash mathcal[f(t)].$$

Generalizing to arbitrary order and solving for

D ^{qf(t)} , one obtains$$\backslash mathbb^qf(t)=\backslash mathcal^\backslash left\backslash .$$

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

$$\backslash mathbb^qf(t)\; =\backslash sum\_^\; \backslash binom\; m\; \backslash sum\_^m\backslash binom\; mk(-1)^f^(x).$$

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

D ^{q(t} ^{n) = (\Gamma(n +1 ))/(\Gamma(n +1- q))} t ^{n-q} D ^{q(\sin(t))=\sin} \left( t + (q\pi)/(2) \right) D ^{q(e} ^{at} ) = a ^{q} e ^{at} ^[2]

Basic formal properties

• Linearity rules $$\backslash mathbb^q(f+g)\; =\; \backslash mathbb^q(f)+\backslash mathbb^q(g)$$

$$\backslash mathbb^q(af)\; =\; a\backslash mathbb^q(f)$$

• Zero rule $$\backslash mathbb^0\; f\; =\; f$$
• Product rule $$\backslash mathbb^q\_t(fg)\; =\; \backslash sum\_^\; \backslash mathbb^j\_t(f)\backslash mathbb^\_t(g)$$

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:

• $\backslash mathbb^a\backslash mathbb^f\; =\; \backslash mathbb^f$ (ideally)
• $\backslash mathbb^a\backslash mathbb^f\; \backslash neq\; \backslash mathbb^f$ (in practice)

See also

• Fractional-order integrator

References

• Book: Miller, Kenneth S. . Bertram . Ross . An Introduction to the Fractional Calculus and Fractional Differential Equations . Wiley . 1993 . 0-471-58884-9 .
• Book: Keith B. . Oldham . Jerome . Spanier . The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order . Academic Press . Mathematics in Science and Engineering . V . 1974 . 0-12-525550-0 .
• Book: Podlubny, Igor . Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications . Academic Press . Mathematics in Science and Engineering . 198 . 1998 . 0-12-558840-2 .
• Book: A. . Carpinteri . F. . Mainardi . Fractals and Fractional Calculus in Continuum Mechanics . Springer-Verlag . 1998 . 3-211-82913-X .
• Book: Mainardi, F. . Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models . Imperial College Press . 2010 . 978-1-84816-329-4 . https://web.archive.org/web/20120519174508/http://www.worldscibooks.com/mathematics/p614.html . dead . 2012-05-19 .
• Book: Tarasov, V.E. . Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media . Springer . 2010 . 978-3-642-14003-7 . Nonlinear Physical Science .
• Book: Uchaikin, V.V. . Fractional Derivatives for Physicists and Engineers . Springer . 2012 . 978-3-642-33910-3 . Nonlinear Physical Science . 2013fdpe.book.....U .
• Book: Bruce J. . West . Mauro . Bologna . Paolo . Grigolini . Physics of Fractal Operators . Springer Verlag . 2003 . 0-387-95554-2 .

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
• Specialized journal: Journal of Fractional Calculus and Applications (JFCA)
• Web site: Carl F. . Lorenzo . Tom T. . Hartley . Initialized Fractional Calculus . 2002 . 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.
• I. . Podlubny . Geometric and physical interpretation of fractional integration and fractional differentiation . Fractional Calculus and Applied Analysis . 5 . 4 . 367–386 . 2002 . math.CA/0110241 . 2001math.....10241P . 2004-05-18 . 2006-04-07 . https://web.archive.org/web/20060407100616/http://www.tuke.sk/podlubny/pspdf/pifcaa_r.pdf . dead .
• P. . Zavada . Operator of fractional derivative in the complex plane . Communications in Mathematical Physics. 192 . 2. 261–285 . 1998 . 10.1007/s002200050299 . funct-an/9608002. 1998CMaPh.192..261Z . 1201395 .

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 .