# Fractional 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 differentiation operator D

${\displaystyle Df(x)={\frac }f(x)\,,}$

and of the integration operator J[Note 1]

${\displaystyle Jf(x)=\int _^f(s)\,ds\,,}$

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 D to a function f, that is, repeatedly composing D with itself, as in ${\displaystyle D^(f)=(D\circ D)(f)=D(D(f))}$.

For example, one may ask for a meaningful interpretation of:

${\displaystyle {\sqrt }=D^{\frac }}$

as an analogue of the functional square root for the differentiation operator, that is, an expression for some linear operator that when applied twice to any function will have the same effect as differentiation. More generally, one can look at the question of defining a linear functional

${\displaystyle D^}$

for every real-number a in such a way that, when a takes an integer value n ∈ ℤ, it coincides with the usual n-fold differentiation D if n > 0, and with the −nth power of J when n < 0.

One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator D is that the sets of operator powers { Da |a ∈ ℝ } defined in this way are continuous semigroups with parameter a, of which the original discrete semigroup of { Dn | n ∈ ℤ } for integer n 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,[1] 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.[2]. Fractional calculus was introduced in one of Niels Henrik Abel’s early papers[3] 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 even the unified notation for differentiation and integration of arbitrary real order.[4] Independently, the foundations of the subject were laid by Liouville in a paper from 1832.[5] The autodidact Oliver Heaviside introduced the practical use of fractional differential operators in electrical transmission line analysis circa 1890.[6]

## Nature of the fractional derivative

The ath derivative of a function f (x) at a point x is a local property only when a is an integer; this is not the case for non-integer power derivatives. In other words, it is not correct to say that the fractional derivative at x of a function f (x) depends only on values of f very near x, in the way that integer-power derivatives certainly do. Therefore, it is expected that the theory involves some sort of boundary conditions, involving information on the function further out.[7]

The fractional derivative of a function to order a is often now defined by means of the Fourier or Mellin integral transforms.

## Heuristics

A fairly natural question to ask is whether there exists a linear operator H, or half-derivative, such that

${\displaystyle H^f(x)=Df(x)={\dfrac }f(x)=f'(x)\,.}$

It turns out that there is such an operator, and indeed for any a > 0, there exists an operator P such that

${\displaystyle \left(P^f\right)(x)=f'(x),}$

or to put it another way, the definition of dny/dxn can be extended to all real values of n.

Let f (x) be a function defined for x > 0. Form the definite integral from 0 to x. Call this

${\displaystyle (Jf)(x)=\int _^f(t)\,dt\,.}$

Repeating this process gives

${\displaystyle \left(J^f\right)(x)=\int _^(Jf)(t)\,dt=\int _^\left(\int _^f(s)\,ds\right)\,dt\,,}$

and this can be extended arbitrarily.

The Cauchy formula for repeated integration, namely

${\displaystyle \left(J^f\right)(x)={\frac {(n-1)!}}\int _^\left(x-t\right)^f(t)\,dt\,,}$

leads in a straightforward way to a generalization for real n.

Using the gamma function to remove the discrete nature of the factorial function gives us a natural candidate for fractional applications of the integral operator.

${\displaystyle \left(J^{\alpha }f\right)(x)={\frac {\Gamma (\alpha )}}\int _^\left(x-t\right)^{\alpha -1}f(t)\,dt\,.}$

This is in fact a well-defined operator.

It is straightforward to show that the J operator satisfies

${\displaystyle \left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)=\left(J^{\beta }\right)\left(J^{\alpha }f\right)(x)=\left(J^{\alpha +\beta }f\right)(x)={\frac {\Gamma (\alpha +\beta )}}\int _^\left(x-t\right)^{\alpha +\beta -1}f(t)\,dt\,.}$

This relationship is called the semigroup property of fractional differintegral operators. Unfortunately the comparable process for the derivative operator D is significantly more complex, but it can be shown that D is neither commutative nor additive in general.[8]

## Fractional derivative of a basic power function

The half derivative (purple curve) of the function f (x) = x (blue curve) together with the first derivative (red curve).
The animation shows the derivative operator oscillating between the antiderivative (α = −1: y = 1/2x2) and the derivative (α = +1: y = 1) of the simple power function y = x continuously.

Let us assume that f (x) is a monomial of the form

${\displaystyle f(x)=x^\,.}$

The first derivative is as usual

${\displaystyle f'(x)={\frac }f(x)=kx^\,.}$

Repeating this gives the more general result that

${\displaystyle {\frac }}}x^={\dfrac {(k-a)!}}x^\,,}$

Which, after replacing the factorials with the gamma function, leads us to

${\displaystyle {\frac }}}x^={\dfrac {\Gamma (k+1)}{\Gamma (k-a+1)}}x^,\qquad k\geq 0}$

For k = 1 and a = 1/2, we obtain the half-derivative of the function x as

${\displaystyle {\frac }}}}}x={\frac {\Gamma (1+1)}{\Gamma \left(1-{\frac }+1\right)}}x^}}={\frac {\Gamma (2)}{\Gamma \left({\frac }\right)}}x^{\frac }={\frac {\frac {\sqrt {\pi }}}}x^{\frac }.}$

To demonstrate that this is, in fact, the "half derivative" (where H2f (x) = Df (x)), we repeat the process to get:

${\displaystyle {\dfrac }}}}}{\dfrac }}{\sqrt {\pi }}}={\frac {\sqrt {\pi }}}{\dfrac {\Gamma (1+{\frac })}{\Gamma ({\frac }-{\frac }+1)}}x^{{\frac }-{\frac }}={\frac {\sqrt {\pi }}}{\frac {\Gamma \left({\frac }\right)}{\Gamma (1)}}x^={\frac }}x^}{\sqrt {\pi }}}=1\,,}$

(because ${\textstyle \Gamma \left({\frac }\right)={\frac {\sqrt {\pi }}}}$ and Γ(1) = 1) which is indeed the expected result of

${\displaystyle \left({\frac }}}}}{\frac }}}}}\right)x={\frac }x=1\,.}$

For negative integer power k, the gamma function is undefined and we have to use the following relation:[9]

${\displaystyle {\frac }}}x^{-k}=\left(-1\right)^{\dfrac {\Gamma (k+a)}{\Gamma (k)}}x^{-(k+a)}\quad {\text{ for }}k\geq 0}$

This extension of the above differential operator need not be constrained only to real powers. For example, the (1 + i)th derivative of the (1 − i)th derivative yields the 2nd derivative. Also setting negative values for a yields integrals.

For a general function f (x) and 0 < α < 1, the complete fractional derivative is

${\displaystyle D^{\alpha }f(x)={\frac {\Gamma (1-\alpha )}}{\frac }\int _^{\frac {\left(x-t\right)^{\alpha }}}\,dt}$

For arbitrary α, since the gamma function is undefined for arguments whose real part is a negative integer and whose imaginary part is zero, it is necessary to apply the fractional derivative after the integer derivative has been performed. For example,

${\displaystyle D^{\frac }f(x)=D^{\frac }D^f(x)=D^{\frac }{\frac }f(x)}$

## Laplace transform

We can also come at the question via the Laplace transform. Knowing that

${\displaystyle {\mathcal }\left\(s)={\mathcal }\left\{\int _^f(\tau )\,d\tau \right\}(s)={\frac }{\bigl (}{\mathcal }\left\{\bigr )}(s)}$

and

${\displaystyle {\mathcal }\left\f\right\}={\frac }{\bigl (}{\mathcal }\left\{\bigr )}(s)={\frac }}{\bigl (}{\mathcal }\left\{\bigr )}(s)}$

and so on, we assert

${\displaystyle J^{\alpha }f={\mathcal }^{-1}\left\{\bigl (}{\mathcal }\{\bigr )}(s)\right\}}$.

For example,

${\displaystyle J^{\alpha }(t^)={\mathcal }^{-1}\left\{{\frac {\Gamma (k+1)}}}\right\}={\frac {\Gamma (k+1)}{\Gamma (\alpha +k+1)}}t^{\alpha +k}}$

as expected. Indeed, given the convolution rule

${\displaystyle {\mathcal }\={\bigl (}{\mathcal }\{\bigr )}{\bigl (}{\mathcal }\{\bigr )}}$

and shorthanding p(x) = xα − 1 for clarity, we find that

${\displaystyle {\begin\left(J^{\alpha }f\right)(t)&={\frac {\Gamma (\alpha )}}{\mathcal }^{-1}\left\{{\bigl (}{\mathcal }\{\bigr )}{\bigl (}{\mathcal }\{\bigr )}\right\}\\&={\frac {\Gamma (\alpha )}}(p*f)\\&={\frac {\Gamma (\alpha )}}\int _^p(t-\tau )f(\tau )\,d\tau \\&={\frac {\Gamma (\alpha )}}\int _^\left(t-\tau \right)^{\alpha -1}f(\tau )\,d\tau \\\end}}$

which is what Cauchy gave us above.

Laplace transforms "work" on relatively few functions, but they are often useful for solving fractional differential equations.

## Fractional integrals

### 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 for periodic functions (therefore including the 'boundary condition' of repeating after a period) is 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 0). The Riemann-Liouville integral exists in two forms, upper and lower. Considering the interval [a,b], the integrals are defined as

${\displaystyle _D_^{-\alpha }f(t)={}_I_^{\alpha }f(t)={\frac {\Gamma (\alpha )}}\int _^\left(t-\tau \right)^{\alpha -1}f(\tau )\,d\tau }$
${\displaystyle _D_^{-\alpha }f(t)={}_I_^{\alpha }f(t)={\frac {\Gamma (\alpha )}}\int _^\left(\tau -t\right)^{\alpha -1}f(\tau )\,d\tau }$

Where the former is valid for t > a and the latter is valid for t < b.[10]

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

### Hadamard fractional integral

The Hadamard fractional integral is introduced by Jacques Hadamard[11] and is given by the following formula,

${\displaystyle _\mathbf _^{-\alpha }f(t)={\frac {\Gamma (\alpha )}}\int _^\left(\log {\frac {\tau }}\right)^{\alpha -1}f(\tau ){\frac {\tau }},\qquad t>a\,.}$

### Atangana–Baleanu fractional integral

Recently, using the generalized Mittag-Leffler function, Atangana and Baleanu suggested a new formulation of the fractional derivative with a nonlocal and nonsingular kernel. The integral is defined as:

${\displaystyle _^D_^{-\alpha }f(t)=_^I_^{\alpha }f(t)={\frac }f(t)+{\frac {\alpha }}\int _^\left(t-\tau \right)^{\alpha -1}f(\tau )\,d\tau ,}$

where AB(α) is a normalization function such that AB(0) = AB(1) = 1.[12]

## Fractional derivatives

Unlike classical Newtonian derivatives, a fractional derivative is defined via a fractional integral.

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. Computing nth order derivative over the integral of order (nα), the α order derivative is obtained. It is important to remark that n is the smallest integer greater than α ( that is, n = ⌈α). Similar to the definitions for the Riemann-Liouville integral, the derivative has upper and lower variants.[13]

${\displaystyle _D_^{\alpha }f(t)={\frac }}}{}_D_^{-(n-\alpha )}f(t)={\frac }}}{}_I_^f(t)}$
${\displaystyle _D_^{\alpha }f(t)={\frac }}}{}_D_^{-(n-\alpha )}f(t)={\frac }}}{}_I_^f(t)}$

### Caputo fractional derivative

Another option for computing fractional derivatives is the Caputo fractional derivative. It was introduced by Michele Caputo in his 1967 paper.[14] 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.

${\displaystyle {}_^D_^{\alpha }f(t)={\frac {\Gamma (n-\alpha )}}\int _^{\frac (\tau )\,d\tau }{\left(t-\tau \right)^{\alpha +1-n}}}.}$

There is the Caputo fractional derivative defined as:

${\displaystyle {}D^{\nu }f(t)={\frac {\Gamma (n-\nu )}}\int _^(t-u)^{(n-\nu -1)}f^{(n)}(u)du\qquad (n-1)<\nu

which has the advantage that is zero when 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

${\displaystyle {}_^D^{\nu }f(t)=\int _^\phi (\nu )\left[D^{(\nu )}f(t)\right]\,d\nu =\int _^\left[{\frac {\phi (\nu )}{\Gamma (1-\nu )}}\int _^\left(t-u\right)^{-\nu }f'(u)\,du\right]\,d\nu }$

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

### Atangana–Baleanu derivative

Like the integral, there is also a fractional derivative using the general Mittag-Leffler function as a kernel.[12] The authors introduced two versions, the Atangana–Baleanu in Caputo sense (ABC) derivative, which is the convolution of a local derivative of a given function with the generalized Mittag-Leffler function, and the Atangana–Baleanu in Riemann–Liouville sense (ABR) derivative, which is the derivative of a convolution of a given function that is not differentiable with the generalized Mittag-Leffler function.[15] The Atangana-Baleanu fractional derivative in Caputo sense is defined as:

${\displaystyle {}_^D_^{\alpha }f(t)={\frac }\int _^f'(\tau )E_{\alpha }\left(-\alpha {\frac {\left(t-\tau \right)^{\alpha }}}\right)\,d\tau \,.}$

And the Atangana–Baleanu fractional derivative in Riemann–Liouville is defined as:

${\displaystyle {}_^D_^{\alpha }f(t)={\frac }{\frac }\int _^f(\tau )E_{\alpha }\left(-\alpha {\frac {\left(t-\tau \right)^{\alpha }}}\right)\,d\tau \,.}$

### Riesz derivative

${\displaystyle {\mathcal }\left\{{\frac {\partial ^{\alpha }u}{\partial \left|x\right|^{\alpha }}}\right\}(k)=-\left|k\right|^{\alpha }{\mathcal }\(k)}$

where F denotes the Fourier transform.[16][17]

### Other types

Classical fractional derivatives include:

• Grünwald–Letnikov derivative
• Sonin–Letnikov derivative
• Liouville derivative
• Caputo derivative
• Marchaud derivative
• Riesz derivative
• Riesz–Miller derivative
• Miller–Ross derivative
• Weyl derivative
• Erdélyi–Kober derivative

New fractional derivatives include:

• Machado derivative[citation needed] (This derivative does not exist anywhere in the literature)
• Coimbra derivative
• Katugampola derivative
• Caputo–Katugampola derivative
• Hilfer derivative
• Hilfer–Katugampola derivative
• Davidson derivative
• Chen derivative
• Caputo Fabrizio derivative
• Atangana–Baleanu derivative
• Pichaghchi derivative

## Generalizations

### Erdélyi–Kober operator

The Erdélyi–Kober operator is an integral operator introduced by Arthur Erdélyi (1940).[18] and Hermann Kober (1940)[19] and is given by

${\displaystyle {\frac }{\Gamma (\alpha )}}\int _^\left(t-x\right)^{\alpha -1}t^{-\alpha -\nu }f(t)\,dt\,,}$

which generalizes the Riemann–Liouville fractional integral and the Weyl integral.

## Functional calculus

In the context of functional analysis, functions f (D) 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 D. 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 (Kober 1940), (Erdélyi 1950–51).

## Applications

### Fractional conservation of mass

As described by Wheatcraft and Meerschaert (2008),[20] 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:

${\displaystyle -\rho \left(\nabla ^{\alpha }\cdot {\vec }\right)=\Gamma (\alpha +1)\Delta x^\rho \left(\beta _+\phi \beta _\right){\frac {\partial p}{\partial t}}}$

### Groundwater flow problem

In 2013–2014 Atangana et al. described some groundwater flow problems using the concept of derivative with fractional order.[21][22] 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[clarification needed] has been shown useful for modeling contaminant flow in heterogenous porous media.[23][24][25]

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[26]

### Time-space fractional diffusion equation models

Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models.[27][28] The time derivative term is corresponding to long-time heavy tail decay and the spatial derivative for diffusion nonlocality. The time-space fractional diffusion governing equation can be written as

${\displaystyle {\frac {\partial ^{\alpha }u}{\partial t^{\alpha }}}=-K(-\Delta )^{\beta }u.}$

A simple extension of fractional derivative is the variable-order fractional derivative, α and β are changed into α(x, t) and β(x, t). Its applications in anomalous diffusion modeling can be found in reference.[26][29][30]

### Structural damping models

Fractional derivatives are used to model viscoelastic damping in certain types of materials like polymers.[31]

### PID controllers

Generalizing PID controllers to use fractional orders can increase their degree of freedom. The new equation relating the control variable u(t) in terms of a measured error value e(t) can be written as

${\displaystyle u(t)=K_{\mathrm }e(t)+K_{\mathrm }D_^{-\alpha }e(t)+K_{\mathrm }D_^{\beta }e(t)}$

where α and β are positive fractional orders and Kp, Ki, and Kd, all non-negative, denote the coefficients for the proportional, integral, and derivative terms, respectively (sometimes denoted P, I, and D).[32]

### Acoustical 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:

${\displaystyle \nabla ^u-{\dfrac ^}}{\frac {\partial ^u}{\partial t^}}+\tau _{\sigma }^{\alpha }{\dfrac {\partial ^{\alpha }}{\partial t^{\alpha }}}\nabla ^u-{\dfrac {\tau _{\epsilon }^{\beta }}^}}{\dfrac {\partial ^{\beta +2}u}{\partial t^{\beta +2}}}=0\,.}$

See also Holm & Näsholm (2011)[33] 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)[34] and in the survey paper,[35] as well as the acoustic attenuation article. See Holm & Nasholm (2013)[36] 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.[37]

### Fractional Schrödinger equation in quantum theory

The fractional Schrödinger equation, a fundamental equation of fractional quantum mechanics, has the following form:[38][39]

${\displaystyle i\hbar {\frac {\partial \psi (\mathbf ,t)}{\partial t}}=D_{\alpha }\left(-\hbar ^\Delta \right)^{\frac {\alpha }}\psi (\mathbf ,t)+V(\mathbf ,t)\psi (\mathbf ,t)\,.}$

where the solution of the equation is the wavefunction ψ(r, t) – the quantum mechanical probability amplitude for the particle to have a given position vector r at any given time t, and ħ is the reduced Planck constant. The potential energy function V(r, t) depends on the system.

Further, Δ = 2/r2 is the Laplace operator, and Dα is a scale constant with physical dimension [Dα] = J1 − α·mα·sα = kg1 − α·m2 − α·sα − 2, (at α = 2, D2 = 1/2m for a particle of mass m), and the operator (−ħ2Δ)α/2 is the 3-dimensional fractional quantum Riesz derivative defined by

${\displaystyle (-\hbar ^\Delta )^{\frac {\alpha }}\psi (\mathbf ,t)={\frac {(2\pi \hbar )^}}\int d^pe^{{\frac {\hbar }}\mathbf \cdot \mathbf }|\mathbf |^{\alpha }\varphi (\mathbf ,t)\,.}$

The index α in the fractional Schrödinger equation is the Lévy index, 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:[40]

${\displaystyle i\hbar {\frac {\partial \psi ^{\alpha (\mathbf )}(\mathbf ,t)}{\partial t^{\alpha (\mathbf )}}}=\left(-\hbar ^\Delta \right)^{\frac {\beta (t)}}\psi (\mathbf ,t)+V(\mathbf ,t)\psi (\mathbf ,t),}$

where Δ = 2/r2 is the Laplace operator and the operator (−ħ2Δ)β(t)/2 is the variable-order fractional quantum Riesz derivative.

## Notes

1. ^ The symbol J is commonly used instead of the intuitive I in order to avoid confusion with other concepts identified by similar I–like glyphs, such as identities.

## References

1. ^ Daniel Zwillinger (12 May 2014). Handbook of Differential Equations. Elsevier Science. ISBN 978-1-4832-2096-3.
2. ^ Katugampola, Udita N. (15 October 2014). "A New Approach To Generalized Fractional Derivatives" (PDF). Bulletin of Mathematical Analysis and Applications. 6 (4): 1–15. arXiv:1106.0965. Bibcode:2011arXiv1106.0965K.
3. ^ Niels Henrik Abel (1823). "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)" (PDF). Magazin for Naturvidenskaberne. Kristiania (Oslo): 55–68.
4. ^ Igor Podlubny, Richard L. Magin, and Irina Trymorush (2017). "Niels Henrik Abel and the birth of fractional calculus". Fractional Calculus and Applied Analysis. 20: 1068–1075. doi:10.1515/fca-2017-0057.CS1 maint: multiple names: authors list (link)
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### Sources

• Kilbas, Anatolii Aleksandrovich; Srivastava, Hari Mohan; Trujillo, Juan J. (2006). Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier. ISBN 978-0-444-51832-3.