No violation of the Leibniz rule. No fractional derivative
01 Nov 2013-Communications in Nonlinear Science and Numerical Simulation (Elsevier)-Vol. 18, Iss: 11, pp 2945-2948
TL;DR: It is proved that all fractional derivatives D α, which satisfy the Leibniz rule D α ( fg ) = ( D α f ) g + f ( D β g ) , should have the integer order α = 1.
About: This article is published in Communications in Nonlinear Science and Numerical Simulation.The article was published on 2013-11-01 and is currently open access. It has received 229 citations till now. The article focuses on the topics: General Leibniz rule & Leibniz integral rule.
Citations
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TL;DR: In this paper, the authors demonstrate the failure of the semi-group principle in modeling real-world problems and demonstrate the importance of non-commutative and non-associative operators under which the Caputo-Fabrizio and Atangana-Baleanu fractional operators fall.
Abstract: To answer some issues raised about the concept of fractional differentiation and integration based on the exponential and Mittag-Leffler laws, we present, in this paper, fundamental differences between the power law, exponential decay, Mittag-Leffler law and their possible applications in nature. We demonstrate the failure of the semi-group principle in modeling real-world problems. We use natural phenomena to illustrate the importance of non-commutative and non-associative operators under which the Caputo-Fabrizio and Atangana-Baleanu fractional operators fall. We present statistical properties of generator for each fractional derivative, including Riemann-Liouville, Caputo-Fabrizio and Atangana-Baleanu ones. The Atangana-Baleanu and Caputo-Fabrizio fractional derivatives show crossover properties for the mean-square displacement, while the Riemann-Liouville is scale invariant. Their probability distributions are also a Gaussian to non-Gaussian crossover, with the difference that the Caputo Fabrizio kernel has a steady state between the transition. Only the Atangana-Baleanu kernel is a crossover for the waiting time distribution from stretched exponential to power law. A new criterion was suggested, namely the Atangana-Gomez fractional bracket, that helps describe the energy needed by a fractional derivative to characterize a 2-pletic manifold. Based on these properties, we classified fractional derivatives in three categories: weak, mild and strong fractional differential and integral operators. We presented some applications of fractional differential operators to describe real-world problems and we proved, with numerical simulations, that the Riemann-Liouville power-law derivative provides a description of real-world problems with much additional information, that can be seen as noise or error due to specific memory properties of its power-law kernel. The Caputo-Fabrizio derivative is less noisy while the Atangana-Baleanu fractional derivative provides an excellent description, due to its Mittag-Leffler memory, able to distinguish between dynamical systems taking place at different scales without steady state. The study suggests that the properties of associativity and commutativity or the semi-group principle are just irrelevant in fractional calculus. Properties of classical derivatives were established for the ordinary calculus with no memory effect and it is a failure of mathematical investigation to attempt to describe more complex natural phenomena using the same notions.
368 citations
TL;DR: Two criteria for required by a fractional operator are formulated and the Grunwald-Letnikov, Riemann-Liouville and Caputo fractional derivatives and the Riesz potential are accessed in the light of the proposed criteria.
351 citations
TL;DR: A new formula for the fractional derivative with Mittag-Leffler kernel is established, in the form of a series of Riemann–Liouville fractional integrals, which brings out more clearly the non-locality of fractional derivatives and is easier to handle for certain computational purposes.
246 citations
Cites background from "No violation of the Leibniz rule. N..."
...As was pointed out by Tarasov [41], the Leibniz rule plays a crucial role in fractional calculus and its applications, to the extent that it can be used as a test for the validity of a given model....
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TL;DR: A list of expressions to have a general overview of the concept of fractional (integrals) derivatives and some formulae that do not involve the term fractional, are also included due to their particular interest in the area.
243 citations
TL;DR: A new factorization technique for nonlinear ODEs involving local fractional derivatives for the first time is proposed by making use of the traveling-wave transformation and the results illustrate that the proposed method is efficient and accurate for finding the exact solutions for a class of local fractionals occurring in mathematical physics.
182 citations
References
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Book•
02 Mar 2006
TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.
Abstract: 1. Preliminaries. 2. Fractional Integrals and Fractional Derivatives. 3. Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems. 4. Methods for Explicitly solving Fractional Differential Equations. 5. Integral Transform Methods for Explicit Solutions to Fractional Differential Equations. 6. Partial Fractional Differential Equations. 7. Sequential Linear Differential Equations of Fractional Order. 8. Further Applications of Fractional Models. Bibliography Subject Index
11,492 citations
"No violation of the Leibniz rule. N..." refers background or methods in this paper
...For example, Dx x 1⁄4 1 Cð2 aÞ x 1 a for Riemann–Liouville fractional derivative [2]....
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...Note that all known fractional derivatives are linear [1,2]....
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...Fractional derivatives of non-integer orders [1,2] have wide applications in physics and mechanics [4–13]....
[...]
...For example, we have Dx1 1⁄4 1 Cð1 aÞ x a for Riemann–Liouville fractional derivative [2]....
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...There are different definitions of fractional derivatives such as Riemann–Liouville, Riesz, Caputo, Grünwald-Letnikov, Marchaud, Weyl, Sonin-Letnikov and others [1,2]....
[...]
Book•
08 Dec 1993
TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.
Abstract: Fractional integrals and derivatives on an interval fractional integrals and derivatives on the real axis and half-axis further properties of fractional integrals and derivatives other forms of fractional integrals and derivatives fractional integrodifferentiation of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations fo the first kind with special function kernels applications to differential equations.
7,096 citations
"No violation of the Leibniz rule. N..." refers background in this paper
...1 in [1]), where D is the Riemann-Liouville derivative, D is derivative of integer order k....
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...Fractional derivatives of non-integer orders [1, 2] have wide applications in physics and mechanics [4]-[13]....
[...]
...Note that all known fractional derivatives are linear [1, 2]....
[...]
...The well-known Leibniz rule D(fg) = (Df)g + f(Dg) is not satisfied for differentiation of non-integer orders [1]....
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...There are different definitions of fractional derivatives such as Riemann-Liouville, Riesz, Caputo, Grünwald-Letnikov, Marchaud, Weyl, Sonin-Letnikov and others [1, 2]....
[...]
Book•
02 Mar 2000
TL;DR: An introduction to fractional calculus can be found in this paper, where Butzer et al. present a discussion of fractional fractional derivatives, derivatives and fractal time series.
Abstract: An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time series, B.J. West and P. Grigolini fractional kinetics of Hamiltonian chaotic systems, G.M. Zaslavsky polymer science applications of path integration, integral equations, and fractional calculus, J.F. Douglas applications to problems in polymer physics and rheology, H. Schiessel et al applications of fractional calculus and regular variation in thermodynamics, R. Hilfer.
5,201 citations
"No violation of the Leibniz rule. N..." refers background in this paper
...Fractional derivatives of non-integer orders [1,2] have wide applications in physics and mechanics [4–13]....
[...]
Book•
31 May 2010
TL;DR: The Eulerian Functions The Bessel Functions The Error Functions The Exponential Integral Functions The Mittag-Leffler Functions The Wright Functions as mentioned in this paper The Eulerians Functions
Abstract: Essentials of Fractional Calculus Essentials of Linear Viscoelasticity Fractional Viscoelastic Media Waves in Linear Viscoelastic Media: Dispersion and Dissipation Waves in Linear Viscoelastic Media: Asymptotic Methods Pulse Evolution in Fractional Viscoelastic Media The Eulerian Functions The Bessel Functions The Error Functions The Exponential Integral Functions The Mittag-Leffler Functions The Wright Functions.
1,593 citations
01 Jan 1997
TL;DR: Panagiotopoulos, O.K.Carpinteri, B. Chiaia, R. Gorenflo, F. Mainardi, and R. Lenormand as mentioned in this paper.
Abstract: A. Carpinteri: Self-Similarity and Fractality in Microcrack Coalescence and Solid Rupture.- B. Chiaia: Experimental Determination of the Fractal Dimension of Microcrack Patterns and Fracture Surfaces.- P.D. Panagiotopoulos, O.K. Panagouli: Fractal Geometry in Contact Mechanics and Numerical Applications.- R. Lenormand: Fractals and Porous Media: from Pore to Geological Scales.- R. Gorenflo, F. Mainardi: Fractional Calculus: Integral and Differential Equations of Fractional Order.- R. Gorenflo: Fractional Calculus: some Numerical Methods.- F. Mainardi: Fractional Calculus: some Basic Problems in Continuum and Statistical Mechanics.
1,389 citations
"No violation of the Leibniz rule. N..." refers background in this paper
...Fractional derivatives of non-integer orders [1,2] have wide applications in physics and mechanics [4–13]....
[...]