Editing Fractional calculus

{{Short description|Branch of mathematical analysis}}
{{For|the associated operator|differintegral}}
{{Calculus}}
'''Fractional calculus''' is a branch of [[mathematical analysis]] that studies the several different possibilities of defining [[real number]] powers or [[complex number]] powers of the [[derivative|differentiation]] [[operator (mathematics)|operator]] <math>D</math> 
<math display="block">D f(x) = \frac{d}{dx} f(x)\,,</math>

and of the [[integral|integration]] operator <math>J</math> <ref group=Note>The symbol <math>J</math> is commonly used instead of the intuitive <math>I</math> in order to avoid confusion with other concepts identified by similar {{nowrap|<math>I</math>–like}} [[glyph]]s, such as [[identity (mathematics)|identities]].</ref>
<math display="block">J f(x) = \int_0^x f(s) \,ds\,,</math>

and developing a [[calculus]] for such operators generalizing the classical one.

In this context, the term ''powers'' refers to iterative application of a [[linear operator]] <math>D</math> to a [[function (mathematics)|function]] {{nowrap|<math>f</math>,}} that is, repeatedly [[function composition|composing]] <math>D</math> with itself, as in
<math display="block">\begin{align}
D^n(f) &= (\underbrace{D\circ D\circ D\circ\cdots \circ D}_n)(f) \\
       &= \underbrace{D(D(D(\cdots D}_n (f)\cdots))).
\end{align}</math>

For example, one may ask for a meaningful interpretation of
<math display="block">\sqrt{D} = D^{\scriptstyle{\frac12}}</math>

as an analogue of the [[functional square root]] for the differentiation operator, that is, an expression for some linear operator that, when applied {{em|twice}} to any function, will have the same effect as [[derivative|differentiation]]. More generally, one can look at the question of defining a linear operator
<math display="block">D^a</math>

for every real number <math>a</math> in such a way that, when <math>a</math> takes an [[integer]] value {{nowrap|<math>n\in\mathbb{Z}</math>,}} it coincides with the usual {{nowrap|<math>n</math>-fold}} differentiation <math>D</math> if {{nowrap|<math>n>0</math>,}} and with the {{nowrap|<math>n</math>-th}} power of <math>J</math> when {{nowrap|<math>n<0</math>.}}

One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator <math>D</math> is that the [[set (mathematics)|sets]] of operator powers <math>\{D^a\mid a\in\R\}</math> defined in this way are ''continuous'' [[semigroup]]s with parameter {{nowrap|<math>a</math>,}} of which the original ''discrete'' semigroup of <math>\{D^n\mid n\in\Z\}</math> for integer <math>n</math> is a [[denumerable set|denumerable]] subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics.

Fractional [[differential equation]]s, also known as extraordinary differential equations,<ref name=Zwillinger2014>{{cite book |author=Daniel Zwillinger |title=Handbook of Differential Equations |url=https://books.google.com/books?id=9QLjBQAAQBAJ |date=12 May 2014 |publisher=Elsevier Science |isbn=978-1-4832-2096-3}}</ref> are a generalization of differential equations through the application of fractional calculus.

== Historical notes ==
In [[applied mathematics]] and mathematical analysis, a '''fractional derivative''' is a derivative of any arbitrary order, real or complex. Its first appearance is in a letter written to [[Guillaume de l'Hôpital]] by [[Gottfried Wilhelm Leibniz]] in 1695.<ref name=Derivative>{{cite journal |last=Katugampola |first=Udita N. |date=15 October 2014 |title=A New Approach To Generalized Fractional Derivatives |url=https://www.emis.de/journals/BMAA/repository/docs/BMAA6-4-1.pdf |journal=Bulletin of Mathematical Analysis and Applications |volume=6 |issue=4 |pages=1–15 |arxiv=1106.0965 }}</ref> Around the same time, Leibniz wrote to [[Johann Bernoulli]] about derivatives of "general order".<ref name=":1">{{Cite book |last=Miller |first=Kenneth S. |title=An Introduction to the Fractional Calculus and Fractional Differential Equations |last2=Ross |first2=Bertram |date=1993 |publisher=Wiley |isbn=978-0-471-58884-9 |location=New York |pages=1–2}}</ref> In the correspondence between Leibniz and [[John Wallis]] in 1697, Wallis's infinite product for <math>\pi/2</math> is discussed. Leibniz suggested using differential calculus to achieve this result. Leibniz further used the notation <math>{d}^{1/2}{y}</math> to denote the derivative of order {{sfrac|1|2}}.<ref name=":1" /> 

Fractional calculus was introduced in one of [[Niels Henrik Abel]]'s early papers<ref>{{cite journal |title=Oplösning af et Par Opgaver ved Hjelp af bestemte Integraler (Solution de quelques problèmes à l'aide d'intégrales définies, Solution of a couple of problems by means of definite integrals) |year=1823 |journal=Magazin for Naturvidenskaberne |place=Kristiania (Oslo) |pages=55–68 |author=Niels Henrik Abel |url=https://abelprize.no/sites/default/files/2021-04/Magazin_for_Naturvidenskaberne_oplosning_av_et_par1_opt.pdf}}</ref> where all the elements can be found: the idea of fractional-order integration and differentiation, the mutually inverse relationship between them, the understanding that fractional-order differentiation and integration can be considered as the same generalized operation, and the unified notation for differentiation and integration of arbitrary real order.<ref>{{cite journal |doi=10.1515/fca-2017-0057 |title=Niels Henrik Abel and the birth of fractional calculus |year=2017 |journal=Fractional Calculus and Applied Analysis |pages=1068–1075 |last1=Podlubny |first1=Igor |last2=Magin |first2=Richard L. |last3=Trymorush |first3=Irina |volume=20 |issue=5 |arxiv=1802.05441 |s2cid=119664694}}</ref>
Independently, the foundations of the subject were laid by [[Liouville]] in a paper from 1832.<ref>{{Citation |last=Liouville |first=Joseph |author-link=Joseph Liouville |year=1832 |title=Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions |journal=Journal de l'École Polytechnique |volume=13 |pages=1–69 |location=Paris |url=https://gallica.bnf.fr/ark:/12148/bpt6k4336778/f2.item.r=Joseph%20Liouville}}.</ref><ref>{{Citation |last=Liouville |first=Joseph |author-link=Joseph Liouville |year=1832 |title=Mémoire sur le calcul des différentielles à indices quelconques |journal=Journal de l'École Polytechnique |volume=13 |pages=71–162 |location=Paris |url=https://gallica.bnf.fr/ark:/12148/bpt6k4336778/f72.image}}.</ref><ref>For the history of the subject, see the thesis (in French): Stéphane Dugowson, [http://s.dugowson.free.fr/recherche/dones/index.html ''Les différentielles métaphysiques''] (''histoire et philosophie de la généralisation de l'ordre de dérivation''), Thèse, Université Paris Nord (1994)</ref> [[Oliver Heaviside]] introduced the practical use of [[operational calculus|fractional differential operators]] in electrical transmission line analysis circa 1890.<ref>For a historical review of the subject up to the beginning of the 20th century, see: {{cite journal |doi=10.1016/0315-0860(77)90039-8 |title=The development of fractional calculus 1695–1900 |year=1977 |journal=Historia Mathematica |pages=75–89 |author=Bertram Ross |volume=4 |s2cid=122146887 |doi-access=}}</ref> The theory and applications of fractional calculus expanded greatly over the 19th and 20th centuries, and numerous contributors have given different definitions for fractional derivatives and integrals.<ref>{{cite journal |last1=Valério |first1=Duarte |last2=Machado |first2=José |last3=Kiryakova |first3=Virginia |author3-link=Virginia Kiryakova |date=2014-01-01 |title=Some pioneers of the applications of fractional calculus |journal=Fractional Calculus and Applied Analysis |volume=17 |issue=2 |pages=552–578 |doi=10.2478/s13540-014-0185-1 |hdl=10400.22/5491 |s2cid=121482200 |issn=1314-2224 |hdl-access=free}}</ref>

==Computing the fractional integral==
Let <math>f(x)</math> be a function defined for <math>x>0</math>. Form the definite integral from 0 to <math>x</math>. Call this
<math display="block">( J f ) ( x ) = \int_0^x f(t) \, dt \,.</math>

Repeating this process gives
<math display="block">\begin{align}
\left( J^2 f \right) (x) &= \int_0^x (Jf)(t) \,dt \\
&= \int_0^x \left(\int_0^t f(s) \,ds \right) dt \,,
\end{align}</math>

and this can be extended arbitrarily.

The [[Cauchy formula for repeated integration]], namely
<math display="block">\left(J^n f\right) ( x ) = \frac{1}{ (n-1) ! } \int_0^x \left(x-t\right)^{n-1} f(t) \, dt \,,</math>
leads in a straightforward way to a generalization for real {{mvar|n}}: using the [[gamma function]] to remove the discrete nature of the factorial function gives us a natural candidate for applications of the fractional integral operator as
<math display="block">\left(J^\alpha f\right) ( x ) = \frac{1}{ \Gamma ( \alpha ) } \int_0^x \left(x-t\right)^{\alpha-1} f(t) \, dt \,.</math>

This is in fact a well-defined operator.

It is straightforward to show that the {{mvar|J}} operator satisfies
<math display="block">\begin{align}
\left(J^\alpha\right) \left(J^\beta f\right)(x) &= \left(J^\beta\right) \left(J^\alpha f\right)(x) \\
   &= \left(J^{\alpha+\beta} f\right)(x) \\
   &= \frac{1}{ \Gamma ( \alpha + \beta) } \int_0^x \left(x-t\right)^{\alpha+\beta-1} f(t) \, dt \,.
\end{align}</math>

{{Collapse top|title=Proof of this identity}}
<math display="block">
\begin{align}
\left(J^\alpha\right) \left(J^\beta f\right)(x) & = \frac{1}{\Gamma(\alpha)} \int_0^x (x-t)^{\alpha-1} \left(J^\beta f\right)(t) \, dt \\
& = \frac{1}{\Gamma(\alpha) \Gamma(\beta)} \int_0^x \int_0^t \left(x-t\right)^{\alpha-1} \left(t-s\right)^{\beta-1} f(s) \, ds \, dt \\
& = \frac{1}{\Gamma(\alpha) \Gamma(\beta)} \int_0^x f(s) \left( \int_s^x \left(x-t\right)^{\alpha-1} \left(t-s\right)^{\beta-1} \, dt \right) \, ds
\end{align}
</math>

where in the last step we exchanged the order of integration and pulled out the {{math|''f''(''s'')}} factor from the {{mvar|t}} integration.

Changing variables to {{mvar|r}} defined by {{math|1=''t'' = ''s'' + (''x'' − ''s'')''r''}},
<math display="block">\left(J^\alpha\right) \left(J^\beta f\right)(x) = \frac{1}{\Gamma(\alpha) \Gamma(\beta)} \int_0^x \left(x-s\right)^{\alpha + \beta - 1} f(s) \left( \int_0^1 \left(1-r\right)^{\alpha-1} r^{\beta-1} \, dr \right)\, ds</math>

The inner integral is the [[beta function]] which satisfies the following property:
<math display="block">\int_0^1 \left(1-r\right)^{\alpha-1} r^{\beta-1} \, dr = B(\alpha, \beta) = \frac{\Gamma(\alpha)\,\Gamma(\beta)}{\Gamma(\alpha+\beta)}</math>

Substituting back into the equation:
<math display="block">\begin{align}
\left(J^\alpha\right) \left(J^\beta f\right)(x) &= \frac{1}{\Gamma(\alpha + \beta)} \int_0^x \left(x-s\right)^{\alpha + \beta - 1} f(s) \, ds \\
   &= \left(J^{\alpha + \beta} f\right)(x)
\end{align}</math>

Interchanging {{mvar|α}} and {{mvar|β}} shows that the order in which the {{mvar|J}} operator is applied is irrelevant and completes the proof.
{{Collapse bottom}}

This relationship is called the semigroup property of fractional [[differintegral]] operators. 

===Riemann–Liouville fractional integral===
The classical form of fractional calculus is given by the [[Riemann–Liouville integral]], which is essentially what has been described above. The theory of fractional integration for [[periodic function]]s (therefore including the "boundary condition" of repeating after a period) is given by the [[Weyl integral]]. It is defined on [[Fourier series]], and requires the constant Fourier coefficient to vanish (thus, it applies to functions on the [[unit circle]] whose integrals evaluate to zero). The Riemann–Liouville integral exists in two forms, upper and lower. Considering the interval {{closed-closed|''a'',''b''}}, the integrals are defined as
<math display="block">\begin{align}
\sideset{_a}{_t^{-\alpha}}D f(t) &= \sideset{_a}{_t^\alpha}I f(t) \\
   &=\frac{1}{\Gamma(\alpha)}\int_a^t \left(t-\tau\right)^{\alpha-1} f(\tau) \, d\tau \\
\sideset{_t}{_b^{-\alpha}}D f(t) &= \sideset{_t}{_b^\alpha}I f(t) \\
  &=\frac{1}{\Gamma(\alpha)}\int_t^b \left(\tau-t\right)^{\alpha-1} f(\tau) \, d\tau
\end{align}</math>

Where the former is valid for {{math|''t'' > ''a''}} and the latter is valid for {{math|''t'' < ''b''}}.<ref>{{cite book |last=Hermann |first=Richard |date=2014 |title=Fractional Calculus: An Introduction for Physicists |edition=2nd |location=New Jersey |publisher=World Scientific Publishing |page=46 |isbn=978-981-4551-07-6 |doi=10.1142/8934 |bibcode=2014fcip.book.....H}}</ref> 

It has been suggested<ref name=Mainardi/> that the integral on the positive real axis (i.e. <math>a = 0</math>) would be more appropriately named the Abel–Riemann integral, on the basis of history of discovery and use, and in the same vein the integral over the entire real line be named Liouville–Weyl integral.

By contrast the [[Grünwald–Letnikov derivative]] starts with the derivative instead of the integral.

===Hadamard fractional integral===
The ''Hadamard fractional integral'' was introduced by [[Jacques Hadamard]]<ref>{{cite journal |last=Hadamard |first=J. |date=1892 |title=Essai sur l'étude des fonctions données par leur développement de Taylor |url=http://sites.mathdoc.fr/JMPA/PDF/JMPA_1892_4_8_A4_0.pdf |journal=Journal de Mathématiques Pures et Appliquées |volume=4 |issue=8 |pages=101–186}}</ref> and is given by the following formula,
<math display="block">\sideset{_a}{_t^{-\alpha}}{\mathbf{D}} f(t) = \frac{1}{\Gamma(\alpha)} \int_a^t \left(\log\frac{t}{\tau} \right)^{\alpha -1} f(\tau)\frac{d\tau}{\tau}, \qquad t > a\,.</math>

===Atangana–Baleanu fractional integral (AB fractional integral)===
The Atangana–Baleanu fractional integral of a continuous function is defined as:
<math display="block">\sideset{_{\hphantom{A}a}^\operatorname{AB}}{_t^\alpha}I f(t)=\frac{1-\alpha}{\operatorname{AB}(\alpha)}f(t)+\frac{\alpha}{\operatorname{AB}(\alpha)\Gamma(\alpha)}\int_a^t \left(t-\tau\right)^{\alpha-1} f(\tau) \, d\tau </math>

==Fractional derivatives==
{{Distinguish|Fractal derivative}}

Unfortunately, the comparable process for the derivative operator {{mvar|D}} is significantly more complex, but it can be shown that {{mvar|D}} is neither [[commutative]] nor [[additive map|additive]] in general.<ref>{{cite book |last1=Kilbas |first1=A. Anatolii Aleksandrovich |url=https://books.google.com/books?id=LhkO83ZioQkC |title=Theory And Applications of Fractional Differential Equations |last2=Srivastava |first2=Hari Mohan |last3=Trujillo |first3=Juan J. |date=2006 |publisher=Elsevier |isbn=978-0-444-51832-3 |page=[{{google books|plainurl=yes|id=LhkO83ZioQkC|page=75}} 75 (Property 2.4)] |language=en}}</ref>

Unlike classical Newtonian derivatives, fractional derivatives can be defined in a variety of different ways that often do not all lead to the same result even for smooth functions. Some of these are defined via a fractional integral. Because of the incompatibility of definitions, it is frequently necessary to be explicit about which definition is used.
[[File:Fractionalderivative.gif|thumb|Fractional derivatives of a Gaussian, interpolating continuously between the function and its first derivative]]

===Riemann–Liouville fractional derivative===
The corresponding derivative is calculated using Lagrange's rule for differential operators. To find the {{mvar|α}}th order derivative, the {{mvar|n}}th order derivative of the integral of order {{math|(''n'' − ''α'')}} is computed, where {{mvar|n}} is the smallest integer greater than {{mvar|α}} (that is, {{math|''n'' {{=}} {{ceil|''α''}}}}). The Riemann–Liouville fractional derivative and integral has multiple applications such as in case of solutions to the equation in the case of multiple systems such as the tokamak systems, and Variable order fractional parameter.<ref name="Mostafanejad">{{Cite journal|doi = 10.1002/qua.26762|title = Fractional paradigms in quantum chemistry |year = 2021|last = Mostafanejad |first = Mohammad |journal = International Journal of Quantum Chemistry |volume = 121|issue = 20 |doi-access = free }}</ref><ref name="Al-Raeei">{{Cite journal|doi = 10.1016/j.chaos.2021.111209|title = Applying fractional quantum mechanics to systems with electrical screening effects |year = 2021|last = Al-Raeei|first = Marwan | url=https://www.sciencedirect.com/science/article/abs/pii/S0960077921005634 |journal = Chaos, Solitons & Fractals |volume = 150|issue = September|pages = 111209|bibcode = 2021CSF...15011209A }}</ref> Similar to the definitions for the Riemann–Liouville integral, the derivative has upper and lower variants.<ref>{{cite book |editor-last=Herrmann |editor-first=Richard |date=2014 |title=Fractional Calculus: An Introduction for Physicists |edition=2nd |location=New Jersey |publisher=World Scientific Publishing Co. |page=[https://books.google.com/books?id=60S7CgAAQBAJ&pg=PA54 54]{{Verify source |date=July 2020}}|isbn=978-981-4551-07-6|doi=10.1142/8934 |bibcode=2014fcip.book.....H}}</ref>
<math display="block">\begin{align}
  \sideset{_a}{_t^\alpha}D f(t) &= \frac{d^n}{dt^n} \sideset{_a}{_t^{-(n-\alpha)}}Df(t) \\
    &= \frac{d^n}{dt^n} \sideset{_a}{_t^{n-\alpha}}I f(t) \\
  \sideset{_t}{_b^\alpha}D f(t) &= \frac{d^n}{dt^n} \sideset{_t}{_b^{-(n-\alpha)}}Df(t) \\
    &= \frac{d^n}{dt^n} \sideset{_t}{_b^{n-\alpha}}I f(t)
\end{align}</math>

===Caputo fractional derivative===
{{main|Caputo fractional derivative}}
Another option for computing fractional derivatives is the [[Caputo fractional derivative]]. It was introduced by [[Michele Caputo]] in his 1967 paper.<ref>{{cite journal |last=Caputo |first=Michele |title=Linear model of dissipation whose ''Q'' is almost frequency independent. II |journal=Geophysical Journal International |year=1967 |volume=13 |issue=5 |pages=529–539 |doi=10.1111/j.1365-246x.1967.tb02303.x |bibcode=1967GeoJ...13..529C |doi-access=free}}.</ref> In contrast to the Riemann–Liouville fractional derivative, when solving differential equations using Caputo's definition, it is not necessary to define the fractional order initial conditions. Caputo's definition is illustrated as follows, where again {{math|1=''n'' = ⌈''α''⌉}}:
<math display="block">\sideset{^C}{_t^\alpha}D f(t)=\frac{1}{\Gamma(n-\alpha)} \int_0^t \frac{f^{(n)}(\tau)}{\left(t-\tau\right)^{\alpha+1-n}}\, d\tau.</math>

There is the Caputo fractional derivative defined as:
<math display="block">D^\nu f(t)=\frac{1}{\Gamma(n-\nu)} \int_0^t (t-u)^{(n-\nu-1)}f^{(n)}(u)\, du \qquad (n-1)<\nu<n</math>
which has the advantage that it is zero when {{math|''f''(''t'')}} is constant and its Laplace Transform is expressed by means of the initial values of the function and its derivative. Moreover, there is the Caputo fractional derivative of distributed order defined as
<math display="block">\begin{align}
\sideset{_a^b}{^nu}Df(t) &= \int_a^b \phi(\nu)\left[D^{(\nu)}f(t)\right]\,d\nu \\
   &= \int_a^b\left[\frac{\phi(\nu)}{\Gamma(1-\nu)}\int_0^t \left(t-u\right)^{-\nu}f'(u)\,du \right]\,d\nu
\end{align}</math>

where {{math|''ϕ''(''ν'')}} is a weight function and which is used to represent mathematically the presence of multiple memory formalisms.

===Caputo–Fabrizio fractional derivative===
In a paper of 2015, M. Caputo and M. Fabrizio presented a definition of fractional derivative with a non singular kernel, for a function <math>f(t)</math> of <math>C^1</math> given by:
<math display="block">\sideset{_{\hphantom{C}a}^\text{CF}}{_t^\alpha}Df(t)=\frac{1}{1-\alpha} \int_a^t f'(\tau) \ e^\left(-\alpha\frac{t-\tau}{1-\alpha}\right) \ d\tau,</math>

where {{nowrap|<math>a < 0, \alpha \in (0,1]</math>.}}<ref>{{cite journal |last1=Caputo |first1=Michele |last2=Fabrizio |first2=Mauro |date=2015 |title=A new Definition of Fractional Derivative without Singular Kernel |url=https://www.naturalspublishing.com/ContIss.asp?IssID=255 |journal=Progress in Fractional Differentiation and Applications |volume=1 |issue=2 |pages=73–85 |access-date=7 August 2020}}</ref>

===Atangana–Baleanu fractional derivative===
In 2016, Atangana and Baleanu suggested differential operators based on the generalized [[Mittag-Leffler function]] <math> E_{\alpha}</math>. The aim was to introduce fractional differential operators with non-singular nonlocal kernel. Their fractional differential operators are given below in Riemann–Liouville sense and Caputo sense respectively. For a function <math>f(t)</math> of <math>C^1</math> given by <ref name=Algahtani2016/><ref name="doiserbia.nb.rs">{{cite journal |last1=Atangana |first1=Abdon |last2=Baleanu |first2=Dumitru |date=2016 |title=New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model |url=http://www.doiserbia.nb.rs/Article.aspx?ID=0354-98361600018A |journal=Thermal Science |language=en |volume=20 |issue=2 |pages=763–769 |doi=10.2298/TSCI160111018A |arxiv=1602.03408 |issn=0354-9836 |doi-access=free}}</ref>
<math display="block">\sideset{_{\hphantom{AB}a}^{\text{ABC}}}{_t^\alpha}D f(t)=\frac{\operatorname{AB}(\alpha)}{1-\alpha} \int_a^t f'(\tau)E_{\alpha}\left(-\alpha\frac{(t-\tau)^{\alpha}}{1-\alpha}\right)d\tau,</math>

If the function is continuous, the Atangana–Baleanu derivative in Riemann–Liouville sense is given by:
<math display="block">\sideset{_{\hphantom{AB}a}^{\text{ABC}}}{_t^\alpha}D f(t)=\frac{\operatorname{AB}(\alpha)}{1-\alpha} \frac{d}{dt}\int_a^t f(\tau)E_{\alpha}\left(-\alpha\frac{(t-\tau)^{\alpha}}{1-\alpha}\right)d\tau,</math>

The kernel used in Atangana–Baleanu fractional derivative has some properties of a cumulative distribution function. For example, for all {{nowrap|<math>\alpha \in (0, 1]</math>,}} the function <math>E_\alpha</math> is increasing on the real line, converges to <math>0</math> in {{nowrap|<math>- \infty</math>,}} and {{nowrap|<math>E_\alpha (0) = 1</math>.}} Therefore, we have that, the function <math>x \mapsto 1-E_\alpha (-x^\alpha)</math> is the cumulative distribution function of a probability measure on the positive real numbers. The distribution is therefore defined, and any of its multiples is called a [[Mittag-Leffler distribution]] of order {{nowrap|<math>\alpha</math>.}} It is also very well-known that, all these probability distributions are [[absolute continuity|absolutely continuous]]. In particular, the function Mittag-Leffler has a particular case {{nowrap|<math>E_1</math>,}} which is the exponential function, the Mittag-Leffler distribution of order <math>1</math> is therefore an [[exponential distribution]]. However, for {{nowrap|<math>\alpha \in (0, 1)</math>,}} the Mittag-Leffler distributions are [[heavy-tailed distribution|heavy-tailed]]. Their Laplace transform is given by:
<math display="block">\mathbb{E} (e^{- \lambda X_\alpha}) = \frac{1}{1+\lambda^\alpha},</math>

This directly implies that, for {{nowrap|<math>\alpha \in (0, 1)</math>,}} the expectation is infinite. In addition, these distributions are [[geometric stable distribution]]s.

===Riesz derivative===
The Riesz derivative is defined as
<math display="block"> \mathcal{F} \left\{ \frac{\partial^\alpha u}{\partial \left|x\right|^\alpha} \right\}(k) = -\left|k\right|^{\alpha} \mathcal{F} \{u \}(k), </math>

where <math>\mathcal{F}</math> denotes the [[Fourier transform]].<ref>{{cite journal |last1=Chen |first1=YangQuan |last2=Li |first2=Changpin |last3=Ding |first3=Hengfei |date=22 May 2014 |title=High-Order Algorithms for Riesz Derivative and Their Applications |journal=[[Abstract and Applied Analysis]] |volume=2014 |pages=1–17 |language=en |doi=10.1155/2014/653797 |doi-access=free}}</ref><ref>{{cite journal |last=Bayın |first=Selçuk Ş. |date=5 December 2016 |title=Definition of the Riesz derivative and its application to space fractional quantum mechanics |journal=Journal of Mathematical Physics |volume=57 |issue=12 |pages=123501 |arxiv=1612.03046 |doi=10.1063/1.4968819 |bibcode=2016JMP....57l3501B |s2cid=119099201}}</ref>

=== Conformable fractional derivative ===
The conformable fractional derivative of a function <math>f</math> of order <math>\alpha</math> is given by<math display="block"> T_a(f)(t) = \lim_{\epsilon \rightarrow 0}\frac{f\left(t+\epsilon t^{1-\alpha}\right) - f(t)}{\epsilon} </math>Unlike other definitions of the fractional derivative, the conformable fractional derivative obeys the [[Product rule|product]] and [[quotient rule]] has analogs to [[Rolle's theorem]] and the [[mean value theorem]].<ref>{{Cite journal |last1=Khalil |first1=R. |last2=Al Horani |first2=M. |last3=Yousef |first3=A. |last4=Sababheh |first4=M. |date=2014-07-01 |title=A new definition of fractional derivative |url=https://www.sciencedirect.com/science/article/pii/S0377042714000065 |journal=Journal of Computational and Applied Mathematics |volume=264 |pages=65–70 |doi=10.1016/j.cam.2014.01.002 |issn=0377-0427|doi-access=free }}</ref><ref name=":0">{{Cite journal |last1=Gao |first1=Feng |last2=Chi |first2=Chunmei |date=2020 |title=Improvement on Conformable Fractional Derivative and Its Applications in Fractional Differential Equations |journal=Journal of Function Spaces |language=en |volume=2020 |issue=1 |pages=5852414 |doi=10.1155/2020/5852414 |doi-access=free |issn=2314-8888}}</ref> However, this fractional derivative produces significantly different results compared to the Riemann-Liouville and Caputo fractional derivative. In 2020, Feng Gao and Chunmei Chi defined the improved Caputo-type conformable fractional derivative, which more closely approximates the behavior of the Caputo fractional derivative:<ref name=":0" /><math display="block"> ^C_a\widetilde{T}_a(f)(t) = \lim_{\epsilon \rightarrow 0}\left[(1-\alpha)(f(t)-f(a))+\alpha\frac{f\left(t+\epsilon (t-a)^{1-\alpha}\right) - f(t)}{\epsilon}\right] </math>where <math>a</math> and <math>t</math> are [[real numbers]] and <math>a<t</math>. They also defined the improved Riemann-Liouville-type conformable fractional derivative to similarly approximate the Riemann-Liouville fractional derivative:<ref name=":0" />

<math display="block"> ^{RL}_a\widetilde{T}_a(f)(t) = \lim_{\epsilon \rightarrow 0}\left[(1-\alpha)f(t)+\alpha\frac{f\left(t+\epsilon (t-a)^{1-\alpha}\right) - f(t)}{\epsilon}\right] </math>where <math>a</math> and <math>t</math> are [[real numbers]] and <math>a<t</math>. Both improved conformable fractional derivatives have analogs to Rolle's theorem and the [[interior extremum theorem]].<ref>{{Cite journal |last=Hasanah |first=Dahliatul |date=2022-10-31 |title=On continuity properties of the improved conformable fractional derivatives |url=http://mail.fourier.or.id/index.php/FOURIER/article/view/176 |journal=Jurnal Fourier |language=en |volume=11 |issue=2 |pages=88–96 |doi=10.14421/fourier.2022.112.88-96 |issn=2541-5239|doi-access=free }}</ref>

=== Other types ===
Classical fractional derivatives include:
* [[Grünwald–Letnikov derivative]]<ref name=deOliveira2014>{{cite journal |last1=de Oliveira |first1=Edmundo Capelas |last2=Tenreiro Machado |first2=José António |date=2014-06-10 |title=A Review of Definitions for Fractional Derivatives and Integral |journal=Mathematical Problems in Engineering |volume=2014 |pages=1–6 |language=en |doi=10.1155/2014/238459 |doi-access=free|hdl=10400.22/5497 |hdl-access=free }}</ref><ref name=Aslan2015>{{cite journal |last=Aslan |first=İsmail |date=2015-01-15 |title=An analytic approach to a class of fractional differential-difference equations of rational type via symbolic computation |journal=Mathematical Methods in the Applied Sciences |language=en |volume=38 |issue=1 |pages=27–36 |doi=10.1002/mma.3047 |bibcode=2015MMAS...38...27A |hdl=11147/5562 |s2cid=120881978 |hdl-access=free}}</ref>
* Sonin–Letnikov derivative<ref name=Aslan2015/>
* Liouville derivative<ref name=deOliveira2014/>
* [[Differintegral|Caputo derivative]]<ref name=deOliveira2014/>
* Hadamard derivative<ref name=deOliveira2014/><ref>{{cite journal |last1=Ma |first1=Li |last2=Li |first2=Changpin |date=2017-05-11 |title=On hadamard fractional calculus |journal=Fractals |volume=25 |issue=3 |pages=1750033–2980 |doi=10.1142/S0218348X17500335 |bibcode=2017Fract..2550033M |issn=0218-348X}}</ref>
* Marchaud derivative<ref name=deOliveira2014/>
* Riesz derivative<ref name=Aslan2015/>
* Miller–Ross derivative<ref name=deOliveira2014/>
* [[Weyl integral|Weyl derivative]]<ref>{{cite book |last=Miller |first=Kenneth S.  |chapter=The Weyl fractional calculus |date=1975 |title=Fractional Calculus and Its Applications: Proceedings of the International Conference Held at the University of New Haven, June 1974 |pages=80–89 |editor-last=Ross |editor-first=Bertram |series=Lecture Notes in Mathematics |volume=457 |publisher=Springer |language=en |doi=10.1007/bfb0067098 |isbn=978-3-540-69975-0}}</ref><ref>{{cite journal |last=Ferrari |first=Fausto |date=January 2018 |title=Weyl and Marchaud Derivatives: A Forgotten History |journal=Mathematics |language=en |volume=6 |issue=1 |pages=6 |doi=10.3390/math6010006 |doi-access=free|arxiv=1711.08070 }}</ref><ref name=deOliveira2014/>
* [[Erdelyi–Kober operator|Erdélyi–Kober derivative]]<ref name=deOliveira2014/>
* [[Fractal calculus|<math>F^{\alpha}</math>-derivative]]<ref name="Ali">{{cite book |last= Khalili Golmankhaneh|first= Alireza |date=2022 |title=Fractal Calculus and its Applications |url=https://worldscientific.com/worldscibooks/10.1142/12988#t=aboutBook|location=Singapore |publisher= World Scientific Pub Co Inc|page=328 |doi= 10.1142/12988 |isbn=978-981-126-110-7 |s2cid= 248575991 }}</ref>
New fractional derivatives include:
* Coimbra derivative<ref name=deOliveira2014/>
* [[Katugampola fractional operators|Katugampola derivative]]<ref>{{cite journal |last1=Anderson |first1=Douglas R. |last2=Ulness |first2=Darin J. |date=2015-06-01 |title=Properties of the Katugampola fractional derivative with potential application in quantum mechanics |journal=Journal of Mathematical Physics |volume=56 |issue=6 |pages=063502 |doi=10.1063/1.4922018 |bibcode=2015JMP....56f3502A |issn=0022-2488}}</ref>
* Hilfer derivative<ref name=deOliveira2014/>
* Davidson derivative<ref name=deOliveira2014/>
* Chen derivative<ref name=deOliveira2014/>
* [[Caputo Fabrizio derivative]]<ref name=Algahtani2016>{{cite journal |last=Algahtani |first=Obaid Jefain Julaighim |date=2016-08-01 |title=Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model |url=https://www.sciencedirect.com/science/article/abs/pii/S0960077916301059 |journal=Chaos, Solitons & Fractals |series=Nonlinear Dynamics and Complexity |language=en |volume=89 |pages=552–559 |doi=10.1016/j.chaos.2016.03.026 |bibcode=2016CSF....89..552A |issn=0960-0779}}</ref><ref>{{cite journal |last1=Caputo |first1=Michele |last2=Fabrizio |first2=Mauro |date=2016-01-01 |title=Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels |journal=Progress in Fractional Differentiation and Applications |volume=2 |issue=1 |pages=1–11 |doi=10.18576/pfda/020101 |issn=2356-9336}}</ref>
* Atangana–Baleanu derivative<ref name=Algahtani2016/><ref name="doiserbia.nb.rs"/>

====Coimbra derivative====
The '''Coimbra derivative''' is used for physical modeling:<ref> C. F. M. Coimbra (2003) "Mechanics with Variable Order Differential Equations," Annalen der Physik (12), No. 11-12, pp. 692-703.</ref> A number of applications in both mechanics and optics can be found in the works by Coimbra and collaborators,<ref>L. E. S. Ramirez, and C. F. M. Coimbra (2007) "A Variable Order Constitutive Relation for Viscoelasticity"– Annalen der Physik (16) 7-8, pp. 543-552.</ref><ref>H. T. C. Pedro, M. H. Kobayashi, J. M. C. Pereira, and C. F. M. Coimbra (2008) "Variable Order Modeling of Diffusive-Convective Effects on the Oscillatory Flow Past a Sphere" – Journal of Vibration and Control, (14) 9-10, pp. 1569-1672.</ref><ref>G. Diaz, and C. F. M. Coimbra (2009) "Nonlinear Dynamics and Control of a Variable Order Oscillator with Application to the van der Pol Equation" – Nonlinear Dynamics, 56, pp. 145—157.</ref><ref>L. E. S. Ramirez, and C. F. M. Coimbra (2010) "On the Selection and Meaning of Variable Order Operators for Dynamic Modeling"– International Journal of Differential Equations Vol. 2010, Article ID 846107.</ref><ref> L. E. S. Ramirez and C. F. M. Coimbra (2011) "On the Variable Order Dynamics of the Nonlinear Wake Caused by a Sedimenting Particle," Physica D (240) 13, pp. 1111-1118.</ref><ref>E. A. Lim, M. H. Kobayashi and C. F. M. Coimbra (2014) "Fractional Dynamics of Tethered Particles in Oscillatory Stokes Flows," Journal of Fluid Mechanics (746) pp. 606-625.</ref><ref>J. Orosco and C. F. M. Coimbra (2016) "On the Control and Stability of Variable Order Mechanical Systems" Nonlinear Dynamics, (86:1), pp. 695–710.</ref> as well as additional applications to physical problems and numerical implementations studied in a number of works by other authors<ref>E. C. de Oliveira, J. A. Tenreiro Machado (2014), "A Review of Definitions for Fractional Derivatives and Integral", Mathematical Problems in Engineering, vol. 2014, Article ID 238459.</ref><ref>S. Shen, F. Liu, J. Chen, I. Turner, and V. Anh (2012) "Numerical techniques for the variable order time fractional diffusion equation" Applied Mathematics and Computation
Volume 218, Issue 22, pp. 10861-10870.</ref><ref>H. Zhang and S. Shen, "The Numerical Simulation of Space-Time Variable Fractional Order Diffusion Equation," Numer. Math. Theor. Meth. Appl. Vol. 6, No. 4, pp. 571-585.</ref><ref>H. Zhang, F. Liu, M. S. Phanikumar, and M. M. Meerschaert (2013) "A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model," Computers & Mathematics with Applications, 66, issue 5, pp. 693–701.</ref>

 For <math>q(t) < 1 </math>
 <math display="block"> \begin{align}
^{\mathbb{C}}_{ a}\mathbb{D}^{q(t)} f(t)=\frac{1}{\Gamma[1-q(t)]}  \int_{0^+}^t (t-\tau)^{-q(t)}\frac{d\,f(\tau)}{d\tau}d\tau\,+\,\frac{(f(0^+)-f(0^-))\,t^{-q(t)}}{\Gamma(1-q(t))},
\end{align}</math>
where the lower limit <math>a</math> can be taken as either <math>0^-</math> or <math>-\infty</math> as long as <math>f(t)</math> is identically zero from or <math>-\infty</math> to <math>0^-</math>. Note that this operator returns the correct fractional derivatives for all values of <math>t</math> and can be applied to either the dependent function itself <math> f(t)</math> with a variable order of the form <math>q(f(t))</math> or to the independent variable with a variable order of the form <math>q(t)</math>.<math>^{[1]}</math>  

The Coimbra derivative can be generalized to any order,<ref> C. F. M. Coimbra "Methods of using generalized order
differentiation and integration of input variables to forecast
trends," U.S. Patent Application 13,641,083 (2013). </ref> leading to the Coimbra Generalized Order Differintegration Operator (GODO)<ref>J. Orosco and C. F. M. Coimbra (2018) "Variable-order Modeling of Nonlocal Emergence in Many-body Systems: Application to Radiative Dispersion," Physical Review E (98), 032208.</ref>
 For <math>q(t) < m </math>
 <math display="block"> \begin{align}
^{\mathbb{\quad C}}_{\,\,-\infty}\mathbb{D}^{q(t)} f(t)=\frac{1}{\Gamma[
m-q(t)]}  \int_{0^+}^t (t-\tau)^{m-1-q(t)}\frac{d^m f(\tau)}{d\tau^m}d\tau\,+\,\sum^{m-1}_{n = 0} \frac{(\frac{d^n f(t)}{dt^n }|_{0^+}-\frac{d^n f(t)}{dt^n}|_{0^-})\,t^{n -q(t)}}{\Gamma[n+1-q(t)]},
\end{align}</math>
where <math>m</math> is an integer larger than the larger value of <math>q(t)</math> for all values of <math>t</math>. Note that the second (summation) term on the right side of the definition above can be expressed as 

 <math display="block"> \begin{align}
\frac{1}{\Gamma[m-q(t)]}\sum^{m-1}_{n = 0} \{[\frac{d^n\!f(t)}{dt^n}|_{0^+}-\frac{d^n\!f(t)}{dt^n }|_{0^-}]\,t^{n -q(t)} \prod^{m-1}_{j=n+1} [j- q(t)]\}
\end{align}</math>
so to keep the denominator on the positive branch of the Gamma (<math>\Gamma</math>) function and for ease of numerical calculation.

=== Nature of the fractional derivative ===
The {{nowrap|<math>a</math>-th}} derivative of a function <math>f</math> at a point <math>x</math> is a ''local property'' only when <math>a</math> is an integer; this is not the case for non-integer power derivatives. In other words, a non-integer fractional derivative of <math>f</math> at <math>x=c</math> depends on all values of {{nowrap|<math>f</math>,}} even those far away from {{nowrap|<math>c</math>.}} Therefore, it is expected that the fractional derivative operation involves some sort of [[boundary condition]]s, involving information on the function further out.<ref>{{MathPages|id=home/kmath616/kmath616.htm|title=Fractional Calculus}}</ref>

The fractional derivative of a function of order <math>a</math> is nowadays often defined by means of the [[Fourier transform|Fourier]] or [[Mellin transform|Mellin]] integral transforms.{{Citation needed|date=November 2022|reason=Examination of recent papers does not mention this}}

==Generalizations==

===Erdélyi–Kober operator===
The [[Erdélyi–Kober operator]] is an integral operator introduced by [[Arthur Erdélyi]] (1940).<ref>{{cite journal |last=Erdélyi |first=Arthur |author-link=Arthur Erdélyi |title=On some functional transformations |journal=Rendiconti del Seminario Matematico dell'Università e del Politecnico di Torino |volume=10 |pages=217–234 |year=1950–1951 |mr=0047818}}</ref> and [[Hermann Kober]] (1940)<ref>{{cite journal |last=Kober |first=Hermann |title=On fractional integrals and derivatives |journal=The Quarterly Journal of Mathematics |volume=os-11 |issue=1 |pages=193–211 |year=1940 |doi=10.1093/qmath/os-11.1.193 |bibcode= 1940QJMat..11..193K}}</ref> and is given by
<math display="block">\frac{x^{-\nu-\alpha+1}}{\Gamma(\alpha)}\int_0^x \left(t-x\right)^{\alpha-1}t^{-\alpha-\nu}f(t) \,dt\,, </math>

which generalizes the [[#Fractional integrals|Riemann–Liouville fractional integral]] and the Weyl integral.

==Functional calculus==
In the context of [[functional analysis]], functions {{math|''f''(''D'')}} more general than powers are studied in the [[functional calculus]] of [[spectral theorem|spectral theory]]. The theory of [[pseudo-differential operator]]s also allows one to consider powers of {{mvar|D}}. The operators arising are examples of [[singular integral operator]]s; and the generalisation of the classical theory to higher dimensions is called the theory of [[Riesz potential]]s. So there are a number of contemporary theories available, within which ''fractional calculus'' can be discussed. See also [[Erdélyi–Kober operator]], important in [[special function]] theory {{harv|Kober|1940}}, {{harv|Erdélyi|1950–1951}}.

==Applications==

===Fractional conservation of mass===
As described by Wheatcraft and Meerschaert (2008),<ref>{{cite journal |last1=Wheatcraft |first1=Stephen W. |last2=Meerschaert |first2=Mark M. |date=October 2008 |title=Fractional conservation of mass |url=https://www.stt.msu.edu/users/mcubed/fCOM.pdf |journal=Advances in Water Resources |language=en |volume=31 |issue=10 |pages=1377–1381 |doi=10.1016/j.advwatres.2008.07.004 |issn=0309-1708 |bibcode=2008AdWR...31.1377W}}</ref> a fractional conservation of mass equation is needed to model fluid flow when the [[control volume]] is not large enough compared to the scale of [[heterogeneity]] and when the flux within the control volume is non-linear. In the referenced paper, the fractional conservation of mass equation for fluid flow is:
<math display="block">-\rho \left(\nabla^\alpha \cdot \vec{u} \right) = \Gamma(\alpha +1)\Delta x^{1-\alpha} \rho \left (\beta_s+\phi \beta_w \right ) \frac{\partial p}{\partial t} </math>

===Electrochemical analysis===
{{See also|Neopolarogram}}
When studying the redox behavior of a substrate in solution, a voltage is applied at an electrode surface to force electron transfer between electrode and substrate. The resulting electron transfer is measured as a current. The current depends upon the concentration of substrate at the electrode surface. As substrate is consumed, fresh substrate diffuses to the electrode as described by [[Fick's laws of diffusion]]. Taking the Laplace transform of Fick's second law yields an ordinary second-order differential equation (here in dimensionless form):
<math display="block">\frac{d^2}{d x^2} C(x,s) = sC(x,s) </math>

whose solution {{math|''C''(''x'',''s'')}} contains a one-half power dependence on {{mvar|s}}. Taking the derivative of {{math|''C''(''x'',''s'')}} and then the inverse Laplace transform yields the following relationship:
<math display="block">\frac{d}{d x} C(x,t) = \frac{d^{\scriptstyle{\frac{1}{2}}}}{d t^{\scriptstyle{\frac{1}{2}}}}C(x,t) </math>

which relates the concentration of substrate at the electrode surface to the current.<ref>Oldham, K. B. ''Analytical Chemistry'' 44(1) '''1972''' 196-198.</ref> This relationship is applied in electrochemical kinetics to elucidate mechanistic behavior. For example, it has been used to study the rate of dimerization of substrates upon electrochemical reduction.<ref>Pospíšil, L. et al. ''Electrochimica Acta'' 300 '''2019''' 284-289.</ref>

===Groundwater flow problem===
In 2013–2014 Atangana et al. described some groundwater flow problems using the concept of a derivative with fractional order.<ref>{{cite journal
|last1=Atangana |first1=Abdon
|last2=Bildik |first2=Necdet
|title=The Use of Fractional Order Derivative to Predict the Groundwater Flow
|year=2013
|journal=Mathematical Problems in Engineering
|volume=2013
|pages=1–9
|doi=10.1155/2013/543026
|doi-access=free
}}</ref><ref>{{cite journal
|last1=Atangana |first1=Abdon
|last2=Vermeulen |first2=P. D.
|title=Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation
|year=2014
|journal=Abstract and Applied Analysis
|volume=2014
|pages=1–11
|doi=10.1155/2014/381753
|doi-access=free
}}</ref> In these works, the classical [[Darcy law]] is generalized by regarding the water flow as a function of a non-integer order derivative of the piezometric head. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow.

===Fractional advection dispersion equation===
This equation{{clarify|date=January 2017}} has been shown useful for modeling contaminant flow in heterogenous porous media.<ref>{{cite journal |last1=Benson |first1=D. |last2=Wheatcraft |first2=S. |last3=Meerschaert |first3=M. |year=2000 |title=Application of a fractional advection-dispersion equation |journal=Water Resources Research |volume=36 |issue=6 |pages=1403–1412 |bibcode=2000WRR....36.1403B |citeseerx=10.1.1.1.4838 |doi=10.1029/2000wr900031|s2cid=7669161 }}</ref><ref>{{cite journal |last1=Benson |first1=D. |last2=Wheatcraft |first2=S. |last3=Meerschaert |first3=M. |s2cid=16579630 |year=2000 |title=The fractional-order governing equation of Lévy motion |journal= Water Resources Research |volume=36 |issue=6 |pages=1413–1423 |bibcode=2000WRR....36.1413B |doi=10.1029/2000wr900032 |doi-access=free}}</ref><ref>{{cite journal |last1=Wheatcraft |first1=Stephen W. |last2=Meerschaert |first2=Mark M. |last3=Schumer |first3=Rina |last4=Benson |first4=David A. |date=2001-01-01 |title=Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests |journal=[[Transport in Porous Media]] |language=en |volume=42 |issue=1–2 |pages=211–240 |citeseerx=10.1.1.58.2062 |doi=10.1023/A:1006733002131 |bibcode=2001TPMed..42..211B |s2cid=189899853 |issn=1573-1634}}</ref>

Atangana and Kilicman extended the fractional advection dispersion equation to a variable order equation. In their work, the hydrodynamic dispersion equation was generalized using the concept of a [[variational order derivative]]. The modified equation was numerically solved via the [[Crank–Nicolson method]]. The stability and convergence in numerical simulations showed that the modified equation is more reliable in predicting the movement of pollution in deformable aquifers than equations with constant fractional and integer derivatives<ref name=Atangana2014a>{{cite journal
|last1=Atangana |first1=Abdon
|last2=Kilicman |first2=Adem
|title=On the Generalized Mass Transport Equation to the Concept of Variable Fractional Derivative
|journal=Mathematical Problems in Engineering
|volume=2014
|year=2014
|page=9
|doi=10.1155/2014/542809
|doi-access=free
}}</ref>

===Time-space fractional diffusion equation models===
Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models.<ref>{{cite journal |last1=Metzler |first1=R. |last2=Klafter |first2=J. |year=2000 |title=The random walk's guide to anomalous diffusion: a fractional dynamics approach |journal=Phys. Rep. |volume=339 |issue=1 |pages=1–77 |doi=10.1016/s0370-1573(00)00070-3 |bibcode=2000PhR...339....1M}}</ref><ref>{{cite journal |last1=Mainardi |first1=F. |author-link2=Yuri Luchko |last2=Luchko |first2=Y. |last3=Pagnini |first3=G. |year=2001 |title=The fundamental solution of the space-time fractional diffusion equation |arxiv=cond-mat/0702419 |journal=Fractional Calculus and Applied Analysis |volume=4 |issue=2 |pages=153–192 |bibcode=2007cond.mat..2419M}}</ref> The time derivative term corresponds to long-time heavy tail decay and the spatial derivative for diffusion nonlocality. The time-space fractional diffusion governing equation can be written as
<math display="block"> \frac{\partial^\alpha u}{\partial t^\alpha}=-K (-\Delta)^\beta u.</math>

A simple extension of the fractional derivative is the variable-order fractional derivative, {{mvar|α}} and {{mvar|β}} are changed into {{math|''α''(''x'', ''t'')}} and {{math|''β''(''x'', ''t'')}}. Its applications in anomalous diffusion modeling can be found in the reference.<ref name=Atangana2014a/><ref>{{cite book |last1=Gorenflo |first1=Rudolf |last2=Mainardi |first2=Francesco |title=Processes with Long-Range Correlations |date=2007 |editor-last=Rangarajan |editor-first=G. |series=Lecture Notes in Physics |volume=621 |pages=148–166 |chapter=Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk |doi=10.1007/3-540-44832-2_8 |arxiv=0709.3990 |editor-last2=Ding |editor-first2=M. |bibcode=2003LNP...621..148G |isbn=978-3-540-40129-2 |s2cid=14946568}}</ref><ref>{{cite journal |last1=Colbrook |first1=Matthew J. |last2=Ma |first2=Xiangcheng |last3=Hopkins |first3=Philip F. |last4=Squire |first4=Jonathan |year=2017 |title=Scaling laws of passive-scalar diffusion in the interstellar medium |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=467 |issue=2 |pages=2421–2429 |arxiv=1610.06590 |doi=10.1093/mnras/stx261 |doi-access=free |bibcode=2017MNRAS.467.2421C |s2cid=20203131}}</ref>

===Structural damping models===
Fractional derivatives are used to model [[viscoelastic]] [[damping]] in certain types of materials like polymers.<ref name=Mainardi>{{cite book |title=Fractional Calculus and Waves in Linear Viscoelasticity |last=Mainardi |first=Francesco |s2cid=118719247 |date=May 2010 |publisher=[[Imperial College Press]] |isbn=978-1-84816-329-4 |language=en |doi=10.1142/p614}}</ref>

===PID controllers===
Generalizing [[PID controller]]s to use fractional orders can increase their degree of freedom. The new equation relating the ''control variable'' {{math|''u''(''t'')}} in terms of a measured ''error value'' {{math|''e''(''t'')}} can be written as
<math display="block">u(t) = K_\mathrm{p} e(t) + K_\mathrm{i} D_t^{-\alpha} e(t) + K_\mathrm{d} D_t^{\beta} e(t)</math>

where {{mvar|α}} and {{math|β}} are positive fractional orders and {{math|''K''<sub>p</sub>}}, {{math|''K''<sub>i</sub>}}, and {{math|''K''<sub>d</sub>}}, all non-negative, denote the coefficients for the [[proportional control|proportional]], [[integral]], and [[derivative]] terms, respectively (sometimes denoted {{mvar|P}}, {{mvar|I}}, and {{mvar|D}}).<ref>{{cite journal |last1=Tenreiro Machado |first1=J. A. |last2=Silva |first2=Manuel F. |last3=Barbosa |first3=Ramiro S. |last4=Jesus |first4=Isabel S. |last5=Reis |first5=Cecília M. |last6=Marcos |first6=Maria G. |last7=Galhano |first7=Alexandra F. |date=2010 |title=Some Applications of Fractional Calculus in Engineering |journal=[[Mathematical Problems in Engineering]] |language=en |volume=2010 |pages=1–34 |doi=10.1155/2010/639801 |doi-access=free|hdl=10400.22/13143 |hdl-access=free }}</ref>

===Acoustic wave equations for complex media===
The propagation of acoustical waves in complex media, such as in biological tissue, commonly implies attenuation obeying a frequency power-law. This kind of phenomenon may be described using a causal wave equation which incorporates fractional time derivatives:
<math display="block">\nabla^2 u -\dfrac 1{c_0^2} \frac{\partial^2 u}{\partial t^2} + \tau_\sigma^\alpha \dfrac{\partial^\alpha}{\partial t^\alpha}\nabla^2 u - \dfrac {\tau_\epsilon^\beta}{c_0^2} \dfrac{\partial^{\beta+2} u}{\partial t^{\beta+2}} = 0\,.</math>

See also Holm & Näsholm (2011)<ref>{{cite journal |last1=Holm |first1=S. |last2=Näsholm |first2=S. P. |s2cid=7804006 |year=2011 |title=A causal and fractional all-frequency wave equation for lossy media |journal=Journal of the Acoustical Society of America |volume=130 |issue=4 |pages=2195–2201 |bibcode=2011ASAJ..130.2195H |doi=10.1121/1.3631626 |pmid=21973374|hdl=10852/103311 |hdl-access=free }}</ref> and the references therein. Such models are linked to the commonly recognized hypothesis that multiple relaxation phenomena give rise to the attenuation measured in complex media. This link is further described in Näsholm & Holm (2011b)<ref>{{cite journal |last1=Näsholm |first1=S. P. |last2=Holm |first2=S. |s2cid=10376751 |year=2011 |title=Linking multiple relaxation, power-law attenuation, and fractional wave equations |journal=Journal of the Acoustical Society of America |volume=130 |issue=5 |pages=3038–3045 |bibcode=2011ASAJ..130.3038N |doi=10.1121/1.3641457 |pmid=22087931|hdl=10852/103312 |hdl-access=free }}</ref> and in the survey paper,<ref name=Nasholm2>{{cite journal |last1=Näsholm |first1=S. P. |last2=Holm |first2=S. |year=2012 |title=On a Fractional Zener Elastic Wave Equation |journal=Fract. Calc. Appl. Anal. |volume=16 |pages=26–50 |arxiv=1212.4024 |doi=10.2478/s13540-013-0003-1 |s2cid=120348311}}</ref> as well as the ''[[Acoustic attenuation]]'' article. See Holm & Nasholm (2013)<ref name=HolmNasholm2014>{{cite journal |last1=Holm |first1=S. |last2=Näsholm |first2=S. P. |year=2013 |title=Comparison of fractional wave equations for power law attenuation in ultrasound and elastography |journal=Ultrasound in Medicine & Biology |volume=40 |issue=4 |pages=695–703 |arxiv=1306.6507 |citeseerx=10.1.1.765.120 |doi=10.1016/j.ultrasmedbio.2013.09.033 |pmid=24433745  |s2cid=11983716}}</ref> for a paper which compares fractional wave equations which model power-law attenuation. This book on power-law attenuation also covers the topic in more detail.<ref name=Holm2019>{{cite book |last=Holm |first=S. |url=https://link.springer.com/book/10.1007/978-3-030-14927-7 |title=Waves with Power-Law Attenuation |publisher=Springer and Acoustical Society of America Press |year=2019 |doi=10.1007/978-3-030-14927-7 |bibcode=2019wpla.book.....H |isbn=978-3-030-14926-0|s2cid=145880744 }}</ref>

Pandey and Holm gave a physical meaning to fractional differential equations by deriving them from physical principles and interpreting the fractional-order in terms of the parameters of the acoustical media, example in fluid-saturated granular unconsolidated marine sediments.<ref name=Pandey2016>{{cite journal |last1=Pandey |first1=Vikash |last2=Holm |first2=Sverre |date=2016-12-01 |title=Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations |journal=The Journal of the Acoustical Society of America |volume=140 |issue=6 |pages=4225–4236 |doi=10.1121/1.4971289 |pmid=28039990 |issn=0001-4966 |arxiv=1612.05557 |bibcode=2016ASAJ..140.4225P |s2cid=29552742}}</ref> Interestingly, Pandey and Holm derived [[Cinna Lomnitz|Lomnitz's law]] in [[seismology]] and Nutting's law in [[non-Newtonian fluid|non-Newtonian rheology]] using the framework of fractional calculus.<ref>{{cite journal |last1=Pandey |first1=Vikash |last2=Holm |first2=Sverre |date=2016-09-23 |title=Linking the fractional derivative and the Lomnitz creep law to non-Newtonian time-varying viscosity |journal=Physical Review E |volume=94 |issue=3 |pages=032606 |doi=10.1103/PhysRevE.94.032606 |pmid=27739858 |bibcode=2016PhRvE..94c2606P |doi-access=free|hdl=10852/53091 |hdl-access=free }}</ref> Nutting's law was used to model the wave propagation in marine sediments using fractional derivatives.<ref name=Pandey2016/>

===Fractional Schrödinger equation in quantum theory===
The [[fractional Schrödinger equation]], a fundamental equation of [[fractional quantum mechanics]], has the following form:<ref>{{cite journal |last=Laskin |first=N. |year=2002 |title=Fractional Schrodinger equation |journal=Phys. Rev. E |volume=66 |issue=5 |pages=056108 |arxiv=quant-ph/0206098 |citeseerx=10.1.1.252.6732 |doi=10.1103/PhysRevE.66.056108 |pmid=12513557 |bibcode=2002PhRvE..66e6108L |s2cid=7520956}}</ref><ref>{{cite book |doi=10.1142/10541 |title=Fractional Quantum Mechanics |year=2018 |last1=Laskin |first1=Nick |isbn=978-981-322-379-0 |citeseerx=10.1.1.247.5449}}</ref>
<math display="block">i\hbar \frac{\partial \psi (\mathbf{r},t)}{\partial t}=D_{\alpha } \left(-\hbar^2\Delta \right)^{\frac{\alpha}{2}}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t)\,.</math>

where the solution of the equation is the [[wavefunction]] {{math|''ψ''('''r''', ''t'')}} – the quantum mechanical [[probability amplitude]] for the particle to have a given [[position vector]] {{math|'''r'''}} at any given time {{mvar|t}}, and {{mvar|ħ}} is the [[reduced Planck constant]]. The [[potential energy]] function {{math|''V''('''r''', ''t'')}} depends on the system.

Further, <math display="inline">\Delta = \frac{\partial^2}{\partial\mathbf{r}^2}</math> is the [[Laplace operator]], and {{mvar|D<sub>α</sub>}} is a scale constant with physical [[dimensional analysis|dimension]] {{math|1=[''D<sub>α</sub>''] = J<sup>1 − ''α''</sup>·m<sup>''α''</sup>·s<sup>−''α''</sup> = kg<sup>1 − ''α''</sup>·m<sup>2 − ''α''</sup>·s<sup>''α'' − 2</sup>}}, (at {{math|1=''α'' = 2}}, <math display="inline">D_2 = \frac{1}{2m}</math> for a particle of mass {{mvar|m}}), and the operator {{math|(−''ħ''<sup>2</sup>Δ)<sup>''α''/2</sup>}} is the 3-dimensional fractional quantum Riesz derivative defined by
<math display="block">(-\hbar^2\Delta)^\frac{\alpha}{2}\psi (\mathbf{r},t) = \frac 1 {(2\pi \hbar)^3} \int d^3 p e^{\frac{i}{\hbar} \mathbf{p}\cdot\mathbf{r}}|\mathbf{p}|^\alpha \varphi (\mathbf{p},t) \,.</math>

The index {{mvar|α}} in the fractional Schrödinger equation is the Lévy index, {{math|1 < ''α'' ≤ 2}}.

====Variable-order fractional Schrödinger equation====
As a natural generalization of the [[fractional Schrödinger equation]], the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena:<ref>{{cite journal |last1=Bhrawy |first1=A.H. |last2=Zaky |first2=M.A. |year=2017 |title=An improved collocation method for multi-dimensional space–time variable-order fractional Schrödinger equations |journal=Applied Numerical Mathematics |volume=111 |pages=197–218 |doi=10.1016/j.apnum.2016.09.009}}</ref>
<math display="block">i\hbar \frac{\partial \psi^{\alpha(\mathbf{r})} (\mathbf{r},t)}{\partial t^{\alpha(\mathbf{r})} } = \left(-\hbar^2\Delta \right)^{\frac{\beta(t)}{2}}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t),</math>

where <math display="inline">\Delta = \frac{\partial^2}{\partial\mathbf{r}^2}</math> is the [[Laplace operator]] and the operator {{math|(−''ħ''<sup>2</sup>Δ)<sup>''β''(''t'')/2</sup>}} is the variable-order fractional quantum Riesz derivative.

==See also==
* [[Acoustic attenuation]]
* [[Autoregressive fractionally integrated moving average]]
* [[Initialized fractional calculus]]
* [[Nonlocal operator]]

===Other fractional theories===
* [[Fractional-order system]]
* [[Fractional Fourier transform]]
* [[Prabhakar function]]

==Notes==
{{Reflist|group=Note}}

==References==
{{Reflist|30em}}

==Further reading==

===Articles regarding the history of fractional calculus===
<!-- Order list by year/author -->
* {{cite journal
|last=Debnath |first=L.
|title=A brief historical introduction to fractional calculus
|journal=International Journal of Mathematical Education in Science and Technology
|volume=35 |issue=4
|year=2004
|pages=487–501
|doi=10.1080/00207390410001686571
|s2cid=122198977
}}

===Books===
<!-- Order list by year/first author) -->
* {{cite book
|title=An Introduction to the Fractional Calculus and Fractional Differential Equations
|editor1-last=Miller |editor1-first=Kenneth S.
|editor2-last=Ross |editor2-first=Bertram
|publisher=John Wiley & Sons
|year=1993
|isbn=978-0-471-58884-9
}}
* {{cite book
|title=Fractional Integrals and Derivatives: Theory and Applications
|last1=Samko |first1=S.
|last2=Kilbas |first2=A.A.
|last3=Marichev |first3=O.
|publisher=Taylor & Francis Books
|isbn=978-2-88124-864-1
|year=1993
}}
* {{cite book
|title=Fractals and Fractional Calculus in Continuum Mechanics
|editor1-last=Carpinteri |editor1-first=A.
|editor2-last=Mainardi |editor2-first=F.
|publisher=Springer-Verlag Telos
|year=1998
|isbn=978-3-211-82913-4
}}
* {{cite book |author=Igor Podlubny |title=Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications |url=https://books.google.com/books?id=K5FdXohLto0C |date=27 October 1998 |publisher=Elsevier |isbn=978-0-08-053198-4}}
* {{cite book
|title=Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media
|last=Tarasov |first=V.E.
|publisher=Springer
|year=2010
|isbn=978-3-642-14003-7
|series=Nonlinear Physical Science
|doi=10.1007/978-3-642-14003-7 |url=https://link.springer.com/book/10.1007/978-3-642-14003-7
}}
* {{cite book
|title=Theory and Numerical Approximations of Fractional Integrals and Derivatives
|last1=Li |first1=Changpin
|last2=Cai |first2=Min
|publisher=SIAM
|isbn=978-1-61197-587-1
|year=2019
|doi=10.1137/1.9781611975888 |url=https://doi.org/10.1137/1.9781611975888
}}

==External links==
{{Refbegin}}
* {{MathWorld |id=FractionalCalculus |title=Fractional calculus}}
* {{MathPages | id=home/kmath616/kmath616.htm |title=Fractional Calculus}}
* [https://www.degruyter.com/journal/key/fca/html Journal of Fractional Calculus and Applied Analysis] {{ISSN|1314-2224}} 2015—
* {{cite web |last1=Lorenzo |first1=Carl F. |first2=Tom T. |last2=Hartley |title=Initialized Fractional Calculus |date=2002 |publisher=NASA John H. Glenn Research Center |work=Tech Briefs |url=https://www.techbriefs.com/component/content/article/tb/pub/briefs/information-sciences/2264}}
* {{cite web |url=https://fractionalcalculus.org/ |title=GigaHedron |last=Herrmann |first=Richard |date=2018}} collection of books, articles, preprints, etc.
* {{cite web |last=Loverro |first=Adam |title=History, Definitions, and Applications for the Engineer |date=2005 |publisher=[[University of Notre Dame]] |url=http://www.nd.edu/~msen/Teaching/UnderRes/FracCalc.pdf|archive-url=https://web.archive.org/web/20051029113800/http://www.nd.edu/~msen/Teaching/UnderRes/FracCalc.pdf |archive-date=2005-10-29 }}

{{Refend}}

{{Differential equations topics}}
{{Authority control}}

[[Category:Fractional calculus| ]]
[[Category:Generalizations]]