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Fractional calculus

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}} 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 differentiation operator <math>D</math> <math display="block">D f(x) = \frac{d}{dx} f(x)\,,</math>

and of the 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 Template:Nowrap glyphs, such as 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 Template:Nowrap that is, repeatedly 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 Template:Em to any function, will have the same effect as 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 Template:Nowrap it coincides with the usual Template:Nowrap differentiation <math>D</math> if Template:Nowrap and with the Template:Nowrap power of <math>J</math> when Template:Nowrap

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

Fractional differential equations, also known as extraordinary differential equations,<ref name=Zwillinger2014>Template:Cite book</ref> are a generalization of differential equations through the application of fractional calculus.

Historical notesEdit

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>Template:Cite journal</ref> Around the same time, Leibniz wrote to Johann Bernoulli about derivatives of "general order".<ref name=":1">Template:Cite book</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 Template:Sfrac.<ref name=":1" />

Fractional calculus was introduced in one of Niels Henrik Abel's early papers<ref>Template:Cite journal</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>Template:Cite journal</ref> Independently, the foundations of the subject were laid by Liouville in a paper from 1832.<ref>Template:Citation.</ref><ref>Template:Citation.</ref><ref>For the history of the subject, see the thesis (in French): Stéphane Dugowson, 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 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: Template:Cite journal</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>Template:Cite journal</ref>

Computing the fractional integralEdit

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 Template:Mvar: 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 Template:Mvar 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>

Template:Collapse top <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 Template:Math factor from the Template:Mvar integration.

Changing variables to Template:Mvar defined by Template:Math, <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 Template:Mvar and Template:Mvar shows that the order in which the Template:Mvar operator is applied is irrelevant and completes the proof. Template:Collapse bottom

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

Riemann–Liouville fractional integralEdit

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 functions (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 Template:Closed-closed, 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 Template:Math and the latter is valid for Template:Math.<ref>Template:Cite book</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 integralEdit

The Hadamard fractional integral was introduced by Jacques Hadamard<ref>Template:Cite journal</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)Edit

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 derivativesEdit

Template:Distinguish

Unfortunately, the comparable process for the derivative operator Template:Mvar is significantly more complex, but it can be shown that Template:Mvar is neither commutative nor additive in general.<ref>Template:Cite book</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
Fractional derivatives of a Gaussian, interpolating continuously between the function and its first derivative

Riemann–Liouville fractional derivativeEdit

The corresponding derivative is calculated using Lagrange's rule for differential operators. To find the Template:Mvarth order derivative, the Template:Mvarth order derivative of the integral of order Template:Math is computed, where Template:Mvar is the smallest integer greater than Template:Mvar (that is, Template:Math). 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">Template:Cite journal</ref><ref name="Al-Raeei">Template:Cite journal</ref> Similar to the definitions for the Riemann–Liouville integral, the derivative has upper and lower variants.<ref>Template:Cite book</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 derivativeEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Another option for computing fractional derivatives is the Caputo fractional derivative. It was introduced by Michele Caputo in his 1967 paper.<ref>Template:Cite journal.</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 Template:Math: <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 Template:Math 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 Template:Math is a weight function and which is used to represent mathematically the presence of multiple memory formalisms.

Caputo–Fabrizio fractional derivativeEdit

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 Template:Nowrap<ref>Template:Cite journal</ref>

Atangana–Baleanu fractional derivativeEdit

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">Template:Cite journal</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 Template:Nowrap the function <math>E_\alpha</math> is increasing on the real line, converges to <math>0</math> in Template:Nowrap and Template:Nowrap 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 Template:Nowrap It is also very well-known that, all these probability distributions are absolutely continuous. In particular, the function Mittag-Leffler has a particular case Template:Nowrap which is the exponential function, the Mittag-Leffler distribution of order <math>1</math> is therefore an exponential distribution. However, for Template:Nowrap the Mittag-Leffler distributions are 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 Template:Nowrap the expectation is infinite. In addition, these distributions are geometric stable distributions.

Riesz derivativeEdit

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>Template:Cite journal</ref><ref>Template:Cite journal</ref>

Conformable fractional derivativeEdit

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 and quotient rule has analogs to Rolle's theorem and the mean value theorem.<ref>Template:Cite journal</ref><ref name=":0">Template:Cite journal</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>Template:Cite journal</ref>

Other typesEdit

Classical fractional derivatives include:

New fractional derivatives include:

Coimbra derivativeEdit

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 derivativeEdit

The Template:Nowrap 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 Template:Nowrap even those far away from Template:Nowrap Therefore, it is expected that the fractional derivative operation involves some sort of boundary conditions, involving information on the function further out.<ref>Template:MathPages</ref>

The fractional derivative of a function of order <math>a</math> is nowadays often defined by means of the Fourier or Mellin integral transforms.Template:Citation needed

GeneralizationsEdit

Erdélyi–Kober operatorEdit

The Erdélyi–Kober operator is an integral operator introduced by Arthur Erdélyi (1940).<ref>Template:Cite journal</ref> and Hermann Kober (1940)<ref>Template:Cite journal</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 Riemann–Liouville fractional integral and the Weyl integral.

Functional calculusEdit

In the context of functional analysis, functions Template:Math more general than powers are studied in the functional calculus of spectral theory. The theory of pseudo-differential operators also allows one to consider powers of Template:Mvar. The operators arising are examples of singular integral operators; and the generalisation of the classical theory to higher dimensions is called the theory of Riesz potentials. 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 Template:Harv, Template:Harv.

ApplicationsEdit

Fractional conservation of massEdit

As described by Wheatcraft and Meerschaert (2008),<ref>Template:Cite journal</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 analysisEdit

Template:See also 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 Template:Math contains a one-half power dependence on Template:Mvar. Taking the derivative of Template:Math 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 problemEdit

In 2013–2014 Atangana et al. described some groundwater flow problems using the concept of a derivative with fractional order.<ref>Template:Cite journal</ref><ref>Template:Cite journal</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 equationEdit

This equationTemplate:Clarify has been shown useful for modeling contaminant flow in heterogenous porous media.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</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>Template:Cite journal</ref>

Time-space fractional diffusion equation modelsEdit

Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models.<ref>Template:Cite journal</ref><ref>Template:Cite journal</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, Template:Mvar and Template:Mvar are changed into Template:Math and Template:Math. Its applications in anomalous diffusion modeling can be found in the reference.<ref name=Atangana2014a/><ref>Template:Cite book</ref><ref>Template:Cite journal</ref>

Structural damping modelsEdit

Fractional derivatives are used to model viscoelastic damping in certain types of materials like polymers.<ref name=Mainardi>Template:Cite book</ref>

PID controllersEdit

Generalizing PID controllers to use fractional orders can increase their degree of freedom. The new equation relating the control variable Template:Math in terms of a measured error value Template:Math 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 Template:Mvar and Template:Math are positive fractional orders and Template:Math, Template:Math, and Template:Math, all non-negative, denote the coefficients for the proportional, integral, and derivative terms, respectively (sometimes denoted Template:Mvar, Template:Mvar, and Template:Mvar).<ref>Template:Cite journal</ref>

Acoustic wave equations for complex mediaEdit

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>Template:Cite journal</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>Template:Cite journal</ref> and in the survey paper,<ref name=Nasholm2>Template:Cite journal</ref> as well as the Acoustic attenuation article. See Holm & Nasholm (2013)<ref name=HolmNasholm2014>Template:Cite journal</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>Template:Cite book</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>Template:Cite journal</ref> Interestingly, Pandey and Holm derived Lomnitz's law in seismology and Nutting's law in non-Newtonian rheology using the framework of fractional calculus.<ref>Template:Cite journal</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 theoryEdit

The fractional Schrödinger equation, a fundamental equation of fractional quantum mechanics, has the following form:<ref>Template:Cite journal</ref><ref>Template:Cite book</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 Template:Math – the quantum mechanical probability amplitude for the particle to have a given position vector Template:Math at any given time Template:Mvar, and Template:Mvar is the reduced Planck constant. The potential energy function Template:Math depends on the system.

Further, <math display="inline">\Delta = \frac{\partial^2}{\partial\mathbf{r}^2}</math> is the Laplace operator, and Template:Mvar is a scale constant with physical dimension Template:Math, (at Template:Math, <math display="inline">D_2 = \frac{1}{2m}</math> for a particle of mass Template:Mvar), and the operator Template:Math 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 Template:Mvar in the fractional Schrödinger equation is the Lévy index, Template:Math.

Variable-order fractional Schrödinger equationEdit

As a natural generalization of the fractional Schrödinger equation, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena:<ref>Template:Cite journal</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 Template:Math is the variable-order fractional quantum Riesz derivative.

See alsoEdit

Other fractional theoriesEdit

NotesEdit

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ReferencesEdit

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Further readingEdit

Articles regarding the history of fractional calculusEdit

BooksEdit

External linksEdit

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