Differintegral#Basic formal properties

The four most common forms are:

• The Riemann–Liouville differintegral{{pb}}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, $n\; =\; \backslash lceil\; q\; \backslash rceil$. $$\backslash begin\{align\}$$

{}^{RL}_a\mathbb{D}^q_tf(t) & = \frac{d^qf(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{align}

• The Grunwald–Letnikov differintegral{{pb}}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. $$\backslash begin\{align\}$$

{}^{GL}_a\mathbb{D}^q_tf(t) & = \frac{d^qf(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{align}

• The Weyl differintegral{{pb}} This is formally similar to the Riemann–Liouville differintegral, but applies to periodic functions, with integral zero over a period.
• The Caputo differintegral{{pb}}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 $a$. $$\backslash begin\{align\}$$

{}^{C}_a\mathbb{D}^q_tf(t) & = \frac{d^qf(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{align}

The definitions of fractional derivatives given by Liouville, Fourier, and Grunwald and Letnikov coincide.{{cite book |url=https://books.google.com/books?id=mPXzp1f7ycMC&pg=PA11 |first=Richard |last=Herrmann|title=Fractional Calculus: An Introduction for Physicists |year=2011 |isbn=9789814551076 }} They can be represented via Laplace, Fourier transforms or via Newton series expansion.

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

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

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

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

So,

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

which generalizes to

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

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

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

Generalizing to arbitrary order and solving for $\backslash mathbb\{D\}^qf(t)$, one obtains

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

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

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

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

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

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

:$\backslash mathbb\{D\}^q(e^\{at\})=a^q\; e^\{at\}$See {{cite book |page=16 |url=https://books.google.com/books?id=mPXzp1f7ycMC&pg=PA11 |first=Richard |last=Herrmann|title=Fractional Calculus: An Introduction for Physicists | year=2011 |isbn=9789814551076 }}

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

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

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

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;See {{cite book |page=75 |chapter=2. Fractional Integrals and Fractional Derivatives §2.1 Property 2.4 |chapter-url=https://books.google.com/books?id=uxANOU0H8IUC&pg=PA75 |first1=A. A. |last1=Kilbas |first2=H. M. |last2=Srivastava |first3=J. J. |last3=Trujillo |title=Theory and Applications of Fractional Differential Equations |publisher=Elsevier |year=2006 |isbn=9780444518323 }} this forms part of the decision making process on which one to choose:

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

• Fractional-order integrator

{{Reflist}}

{{refbegin}}

• {{cite book |first=Kenneth S. |last=Miller |editor-first=Bertram |editor-last=Ross |title=An Introduction to the Fractional Calculus and Fractional Differential Equations |publisher=Wiley |year=1993 |isbn=0-471-58884-9 }}
• {{cite book |first1=Keith B. |last1=Oldham |first2=Jerome |last2=Spanier |title=The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order |publisher=Academic Press |series=Mathematics in Science and Engineering |volume=V |year=1974 |isbn=0-12-525550-0 }}
• {{cite book |first=Igor |last=Podlubny |title=Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications |publisher=Academic Press |series=Mathematics in Science and Engineering |volume=198 |year=1998 |isbn=0-12-558840-2 }}
• {{cite book |editor-first=A. |editor-last=Carpinteri |editor2-first=F. |editor2-last=Mainardi |title=Fractals and Fractional Calculus in Continuum Mechanics |publisher=Springer-Verlag |year=1998 |isbn=3-211-82913-X }}
• {{cite book |first=F. |last=Mainardi |title=Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models |publisher=Imperial College Press |year=2010 |isbn=978-1-84816-329-4 |url=http://www.worldscibooks.com/mathematics/p614.html |archive-url=https://web.archive.org/web/20120519174508/http://www.worldscibooks.com/mathematics/p614.html |url-status=dead |archive-date=2012-05-19 }}
• {{cite book |first=V.E. |last=Tarasov |title=Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media |publisher=Springer |year=2010 |isbn=978-3-642-14003-7 |url=https://www.springer.com/physics/complexity/book/978-3-642-14003-7|series=Nonlinear Physical Science }}
• {{cite book |first=V.V. |last=Uchaikin |title=Fractional Derivatives for Physicists and Engineers |publisher=Springer |year=2012 |isbn=978-3-642-33910-3 |url=https://www.springer.com/physics/theoretical,+mathematical+%26+computational+physics/book/978-3-642-33910-3|series=Nonlinear Physical Science |bibcode=2013fdpe.book.....U }}
• {{cite book |first1=Bruce J. |last1=West |first2=Mauro |last2=Bologna |first3=Paolo |last3=Grigolini |title=Physics of Fractal Operators |publisher=Springer Verlag |year=2003 |isbn=0-387-95554-2 |url=https://books.google.com/books?id=EgyTpQZOga0C&pg=PR7}}

{{refend}}

• [http://mathworld.wolfram.com/FractionalCalculus.html MathWorld – Fractional calculus]
• [http://mathworld.wolfram.com/FractionalDerivative.html MathWorld – Fractional derivative]
• Specialized journal: [http://www.diogenes.bg/fcaa/ Fractional Calculus and Applied Analysis (1998-2014)] and [http://www.degruyter.com/view/j/fca Fractional Calculus and Applied Analysis (from 2015)]
• Specialized journal: [https://archive.today/20120712033445/http://fde.ele-math.com/ Fractional Differential Equations (FDE)]
• Specialized journal: [https://web.archive.org/web/20180421124535/http://www.nonlinearscience.com/journal_2218-3892.php Communications in Fractional Calculus] ({{issn|2218-3892}})
• Specialized journal: [http://fcag-egypt.com/Journals/JFCA/ Journal of Fractional Calculus and Applications (JFCA)]
• {{cite web |first1=Carl F. |last1=Lorenzo |first2=Tom T. |last2=Hartley |title=Initialized Fractional Calculus |date=2002 |work=Information Technology |publisher=Tech Briefs Media Group |url=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
• [http://www.tuke.sk/podlubny/fc_resources.html Igor Podlubny's collection of related books, articles, links, software, etc. ]
• {{cite journal |first=I. |last=Podlubny |title=Geometric and physical interpretation of fractional integration and fractional differentiation |journal=Fractional Calculus and Applied Analysis |volume=5 |issue=4 |pages=367–386 |year=2002 |url=http://www.tuke.sk/podlubny/pspdf/pifcaa_r.pdf |arxiv=math.CA/0110241 |bibcode=2001math.....10241P |access-date=2004-05-18 |archive-date=2006-04-07 |archive-url=https://web.archive.org/web/20060407100616/http://www.tuke.sk/podlubny/pspdf/pifcaa_r.pdf |url-status=dead }}
• {{cite journal |first=P. |last=Zavada |title=Operator of fractional derivative in the complex plane |journal= Communications in Mathematical Physics|volume=192 |issue= 2|pages=261–285 |year=1998 |doi=10.1007/s002200050299 |arxiv=funct-an/9608002|bibcode=1998CMaPh.192..261Z |s2cid=1201395 }}

Category:Fractional calculus

Category:Generalizations of the derivative

Category:Linear operators in calculus