Fractional Derivatives, Fractional Integrals, And Fractional ....pdf

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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] Around the same time, Leibniz wrote to one of the Bernoulli brothers describing the similarity between the binomial theorem and the Leibniz rule for the fractional derivative of a product of two functions.[citation needed] 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][6][7]The autodidact Oliver Heaviside introduced the practical use of fractional differential operators in electrical transmission line analysis circa 1890.[8] 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.[9]

The a {\\displaystyle a} -th derivative of a function f {\\displaystyle f} at a point x {\\displaystyle x} is a local property only when a {\\displaystyle a} is an integer; this is not the case for non-integer power derivatives. In other words, a non-integer fractional derivative of f {\\displaystyle f} at x = a {\\displaystyle x=a} depends on all values of f {\\displaystyle f} , even those far away from a {\\displaystyle a} . Therefore, it is expected that the fractional derivative operation involves some sort of boundary conditions, involving information on the function further out.[10]

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.[11]

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.

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

In a paper of 2015, M. Caputo and M. Fabrizio presented a definition of fractional derivative with a non singular kernel, for a function f ( t ) {\\displaystyle f(t)} of C 1 {\\displaystyle C^{1}} given by:

As described by Wheatcraft and Meerschaert (2008),[31] 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:

Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models.[40][41] 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

A simple extension of the 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 the reference.[39][42][43]

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

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).[45]

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 (2011)[46] 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)[47] and in the survey paper,[48] as well as the Acoustic attenuation article. See Holm & Nasholm (2013)[49] 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.[50]

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.[51] Interestingly, Pandey and Holm derived Lomnitz's law in seismology and Nutting's law in non-Newtonian rheology using the framework of fractional calculus.[52] Nutting's law was used to model the wave propagation in marine sediments using fractional derivatives.[51]

In this paper we prove that local fractional derivatives of differentiable functions are integer-order derivative or zero operator. We demonstrate that the local fractional derivatives are limits of the left-sided Caputo fractional derivatives. The Caputo derivative of fractional order \\(\\alpha \\) of function \\(f(x)\\) is defined as a fractional integration of order \\(n-\\alpha \\) of the derivative \\(f^{(n)}(x)\\) of integer order \\(n\\). The requirement of the existence of integer-order derivatives allows us to conclude that the local fractional derivative cannot be considered as the best method to describe nowhere differentiable functions and fractal objects. We also prove that unviolated Leibniz rule cannot hold for derivatives of orders \\(\\alpha \\ne 1\\).

In this paper, we demonstrate that the definition of local fractional derivative can be presented as a limit of the Caputo fractional derivatives. We note that the Caputo fractional derivative of order \\(\\alpha \\) (\\(n-1

The Caputo fractional derivatives \\(\\ _a^CD^{\\alpha }_{x}\\) can be defined for functions belonging to the space \\(AC^n[a,b]\\) of absolutely continuous functions [2]. If \\(f(x) \\in AC^n[a,b]\\), then the Caputo fractional derivatives exist almost everywhere on \\([a,b]\\).

It is important to emphasize that the Caputo fractional derivative is represented by Eq. (8) as a sequence of two operations, namely, first taking the ordinary derivative of integer order \\(n\\) and then the integration of fractional order \\(n-\\alpha \\), (see Eq. 2.4.17 of [2]):

Local fractional derivative of order \\(\\alpha \\), where \\(n-1 < \\alpha \\le n\\), \\(n\\in {\\mathbb {N}}\\), on a set of \\(n\\)-differentiable functions is a derivative of order \\(n\\in {\\mathbb {N}}\\) or zero operator.

As a corollary of the proved statment we can say that generalizations of the Leibniz rule for local fractional derivatives \\({\\mathcal {D}}^{\\alpha }\\) of order \\(\\alpha >0\\) for \\(n\\)-differentiable functions \\(f(x)\\) and \\(g(x)\\) should give the relation

In paper [7] it was proved that violation of relation (17) is a characteristic property of all types of fractional-order derivatives with \\(\\alpha > 0\\) (and it is obvious for derivatives of integer-orders \\(n \\in {\\mathbb {N}}\\) greater than one).

Let us give some remarks about Leibniz rule (17) for local fractional derivatives on a set of non-differentiable functions. It should be noted that statement [26, 27] that the Leibniz rule in the unviolated form (17) holds for non-differentiable functions is incorrect [8]. Let us give some explanations below.

It should be noted that the Leibniz rule (17) cannot be used for non-differentiable functions that are not fractional-differentiable since the derivatives \\({\\mathcal {D}}^{\\alpha } f(x)\\), \\({\\mathcal {D}}^{\\alpha } g(x)\\) and \\({\\mathcal {D}}^{\\alpha } \\big (f(x) \\, g(x)\\big )\\) should exist.

It is easy to see that nowhere in the proofs proposed in [26, 27], the requirement that the functions \\(f(x)\\) and \\(g(x)\\) are not classically differentiable is not used. Therefore, using the same proofs, we can get the statement that the Leibniz rule (17) holds for fractional-differentiable functions \\(f(x)\\), \\(g(x)\\) without the useless assumption that these functions are not classically differentiable.

The Caputo fractional derivatives \\(\\ _a^CD^{\\alpha }_{x}\\) can be defined for functions belonging to the space \\(AC^n[a,b]\\) of absolutely continuous functions [2]. If \\(f(x) \\in AC^n[a,b]\\), then the Caputo fractional derivatives exist almost everywhere on \\([a,b]\\). If \\(f(x) \\in AC^n[a,b]\\), then this functions is \\(n\\)-differentiable almost everywhere on \\([a,b]\\). The Leibniz rule (17) holds on a set of differentiable functions only for \\(\\alpha =1\\).

As a result, we get that the Leibniz rule (17) with local fractional derivative for \\(f(x), g(x) \\in AC^n[a,b]\\) holds only for \\(\\alpha =1\\) almost everywhere on \\([a,b]\\). In the countable number of points the Leibniz rule is not performed since the derivatives \\({\\mathcal {D}}^{\\alpha } f(x)\\), \\({\\mathcal {D}}^{\\alpha } g(x)\\) and \\({\\mathcal {D}}^{\\alpha } (f(x) \\, g(x))\\) does not exist. Note that the Leibniz rule for fractional derivatives of orders \\(\\alpha =n>1\\) (\\(n\\in {\\mathbb {N}}\\)) cannot have the form (17). It should