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
and of the integration operator
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.
For example, one may ask for a meaningful interpretation of:
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
for every real-number in such a way that, when takes an integer value, it coincides with the usual -fold differentiation if, and with the -th power of when.
One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator is that the sets of operator powers defined in this way are continuous semigroups with parameter, of which the original discrete semigroup of for integer 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, 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.. Fractional calculus was introduced in one of Niels Henrik Abel’s early papers 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.
Independently, the foundations of the subject were laid by Liouville in a paper from 1832.
The autodidact Oliver Heaviside introduced the practical use of fractional differential operators in electrical transmission line analysis circa 1890. The theory and applications of fractional calculus expanded greatly over the 19th and 20th centuries, and numerous contributors have given definitions for fractional derivatives and integrals.

Nature of the fractional derivative

The th derivative of a function at a point is a local property only when 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 of a function depends only on values of very near, 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.
The fractional derivative of a function to order 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, or half-derivative, such that
It turns out that there is such an operator, and indeed for any, there exists an operator such that
or to put it another way, the definition of can be extended to all real values of.
Let be a function defined for. Form the definite integral from 0 to. Call this
Repeating this process gives
and this can be extended arbitrarily.
The Cauchy formula for repeated integration, namely
leads in a straightforward way to a generalization for real.
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.
This is in fact a well-defined operator.
It is straightforward to show that the operator satisfies
where in the last step we exchanged the order of integration and pulled out the factor from the integration. Changing variables to defined by,
The inner integral is the beta function which satisfies the following property:
Substituting back into the equation
Interchanging and shows that the order in which the operator is applied is irrelevant and completes the proof.
This relationship is called the semigroup property of fractional differintegral operators. Unfortunately the comparable process for the derivative operator is significantly more complex, but it can be shown that is neither commutative nor additive in general.

Fractional derivative of a basic power function

Let us assume that is a monomial of the form
The first derivative is as usual
Repeating this gives the more general result that
Which, after replacing the factorials with the gamma function, leads us to
For and, we obtain the half-derivative of the function as
To demonstrate that this is, in fact, the "half derivative", we repeat the process to get:
which is indeed the expected result of
For negative integer power k, the gamma function is undefined and we have to use the following relation:
This extension of the above differential operator need not be constrained only to real powers. For example, the th derivative of the th derivative yields the 2nd derivative. Also setting negative values for yields integrals.
For a general function and, the complete fractional derivative is
For arbitrary, 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,

Laplace transform

We can also come at the question via the Laplace transform. Knowing that
and
and so on, we assert
For example,
as expected. Indeed, given the convolution rule
and shorthanding for clarity, we find that
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 is the Weyl integral. It is defined on Fourier series, and requires the constant Fourier coefficient to vanish. The Riemann-Liouville integral exists in two forms, upper and lower. Considering the interval, the integrals are defined as
Where the former is valid for and the latter is valid for.
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 and is given by the following formula,

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:
where is a normalization function such that.

Fractional derivatives

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

Riemann–Liouville fractional derivative

The corresponding derivative is calculated using Lagrange's rule for differential operators. Computing th order derivative over the integral of order, the order derivative is obtained. It is important to remark that is the smallest integer greater than . Similar to the definitions for the Riemann-Liouville integral, the derivative has upper and lower variants.

Caputo fractional derivative

Another option for computing fractional derivatives is the Caputo fractional derivative. It was introduced by Michele Caputo in his 1967 paper. 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.
There is the Caputo fractional derivative defined as:
which has the advantage that is zero when 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
where 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 of given by:
where

Atangana–Baleanu derivative

Like the integral, there is also a fractional derivative using the general Mittag-Leffler function as a kernel. The authors introduced two versions, the Atangana–Baleanu in Caputo sense 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 derivative, which is the derivative of a convolution of a given function that is not differentiable with the generalized Mittag-Leffler function. The Atangana-Baleanu fractional derivative in Caputo sense is defined as:
And the Atangana–Baleanu fractional derivative in Riemann–Liouville is defined as:

Riesz derivative

where denotes the Fourier transform.

Other types

Classical fractional derivatives include:
New fractional derivatives include:

Erdélyi–Kober operator

The Erdélyi–Kober operator is an integral operator introduced by Arthur Erdélyi. and Hermann Kober and is given by
which generalizes the Riemann–Liouville fractional integral and the Weyl integral.

Functional calculus

In the context of functional analysis, functions 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. 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,.

Applications

Fractional conservation of mass

As described by Wheatcraft and Meerschaert, 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:

Groundwater flow problem

In 2013–2014 Atangana et al. described some groundwater flow problems using the concept of derivative with fractional order. 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 has been shown useful for modeling contaminant flow in heterogenous porous media.
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

Time-space fractional diffusion equation models

Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models. 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
A simple extension of fractional derivative is the variable-order fractional derivative, and are changed into and. Its applications in anomalous diffusion modeling can be found in reference.

Structural damping models

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

PID controllers

Generalizing PID controllers to use fractional orders can increase their degree of freedom. The new equation relating the control variable in terms of a measured error value can be written as
where and are positive fractional orders and,, and, all non-negative, denote the coefficients for the proportional, integral, and derivative terms, respectively.

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:
See also Holm & Näsholm 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 and in the survey paper, as well as the acoustic attenuation article. See Holm & Nasholm 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.
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. Interestingly, Pandey and Holm derived Lomnitz's law in seismology and Nutting's law in non-Newotnian rheology using the framework of fractional calculus. Nutting's law was used to model the wave propagation in marine sediments using fractional derivatives.

Fractional Schrödinger equation in quantum theory

The fractional Schrödinger equation, a fundamental equation of fractional quantum mechanics, has the following form:
where the solution of the equation is the wavefunction – the quantum mechanical probability amplitude for the particle to have a given position vector at any given time, and is the reduced Planck constant. The potential energy function depends on the system.
Further, is the Laplace operator, and is a scale constant with physical dimension,, and the operator is the 3-dimensional fractional quantum Riesz derivative defined by
The index in the fractional Schrödinger equation is the Lévy index,.

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:
where is the Laplace operator and the operator is the variable-order fractional quantum Riesz derivative.

Other fractional theories

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