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Fractional Integrals and Derivatives: “True” versus “False”

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (30 September 2020) | Viewed by 54780

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Prof. Dr. Yuri Luchko

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Department of Mathematics, Physics, and Chemistry, Beuth University of Applied Sciences Berlin, Luxemburger Str. 10, 13353 Berlin, Germany
Interests: fractional calculus; ordinary and partial fractional differential equations; mathematical modelling with fractional calculus models; fractional anomalous diffusion and wave propagation; integral transforms and special functions in fractional calculus
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Special Issue Information

Dear Colleagues,

This Special Issue is devoted to some serious problems regarding the current development of Fractional Calculus (FC).

Even if FC is nearly as old as the conventional calculus, for long time it was addressed and used just sporadically and only by few scientists. Within the last few decades, the situation changed dramatically and nowadays we observe an exponential growth of FC publications, conferences, and scientists involved in this topic. One of the explanations for this phenomenon is in active attempts to introduce a new kind of mathematical models containing fractional order operators into physics, chemistry, engineering, biology, medicine, and other sciences. This speeds up the development of the mathematical theory of FC, including fractional ordinary and partial differential equations, fractional calculus of variations, inverse problems for fractional differential equations, fractional stochastic models, etc. Unfortunately, some of these new models and results are just formal “fractionalisations” of the known conventional theories, often without any justification and motivation.

Additionally, a recent trend in FC is in introducing “new fractional derivatives and integrals” and considering classical equations and models with these fractional order operators instead of conventional integrals and derivatives. This development led to an uncontrolled flood of FC publications both in mathematical and physical journals. Some of these publications contain trivial, well-known, and sometimes even wrong results that threaten the image of FC in scientific community. Thus, we have to think about and to answer questions like “What are the fractional integrals and derivatives?”, “What are their decisive mathematical properties?”, “What fractional operators make sense in applications and why?’’, etc. These and similar questions have remained mostly unanswered until now.

The aim of this Special Issue is to provide an independent platform for discussing these essential problems in the current development of FC. Contributions devoted both to the “new fractional integrals and derivatives” and their justification and those containing constructive criticism of these concepts are welcome.

Prof. Dr. Yuri Luchko

Guest Editor

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Keywords

fractional integrals and derivatives
fractional differential equations
mathematical properties of fractional operators
applications of fractional integrals and derivatives
physical justification of fractional operators
modeling with fractional operators

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Published Papers (13 papers)

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Research

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16 pages, 301 KiB

Open AccessArticle

Intrinsic Discontinuities in Solutions of Evolution Equations Involving Fractional Caputo–Fabrizio and Atangana–Baleanu Operators

by Christopher Nicholas Angstmann, Byron Alexander Jacobs, Bruce Ian Henry and Zhuang Xu

Mathematics 2020, 8(11), 2023; https://doi.org/10.3390/math8112023 - 13 Nov 2020

Cited by 14 | Viewed by 2125

Abstract

There has been considerable recent interest in certain integral transform operators with non-singular kernels and their ability to be considered as fractional derivatives. Two such operators are the Caputo–Fabrizio operator and the Atangana–Baleanu operator. Here we present solutions to simple initial value problems involving these two operators and show that, apart from some special cases, the solutions have an intrinsic discontinuity at the origin. The intrinsic nature of the discontinuity in the solution raises concerns about using such operators in modelling. Solutions to initial value problems involving the traditional Caputo operator, which has a singularity inits kernel, do not have these intrinsic discontinuities. Full article

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)

27 pages, 593 KiB

Open AccessArticle

Finite Difference Method for Two-Sided Two Dimensional Space Fractional Convection-Diffusion Problem with Source Term

by Eyaya Fekadie Anley and Zhoushun Zheng

Mathematics 2020, 8(11), 1878; https://doi.org/10.3390/math8111878 - 29 Oct 2020

Cited by 7 | Viewed by 3230

Abstract

In this paper, we have considered a numerical difference approximation for solving two-dimensional Riesz space fractional convection-diffusion problem with source term over a finite domain. The convection and diffusion equation can depend on both spatial and temporal variables. Crank-Nicolson scheme for time combined with weighted and shifted Grünwald-Letnikov difference operator for space are implemented to get second order convergence both in space and time. Unconditional stability and convergence order analysis of the scheme are explained theoretically and experimentally. The numerical tests are indicated that the Crank-Nicolson scheme with weighted shifted Grünwald-Letnikov approximations are effective numerical methods for two dimensional two-sided space fractional convection-diffusion equation. Full article

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)

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4 pages, 228 KiB

Open AccessCommunication

Maximal Domains for Fractional Derivatives and Integrals

by R. Hilfer and T. Kleiner

Mathematics 2020, 8(7), 1107; https://doi.org/10.3390/math8071107 - 6 Jul 2020

Cited by 3 | Viewed by 1714

Abstract

The purpose of this short communication is to announce the existence of fractional calculi on precisely specified domains of distributions. The calculi satisfy desiderata proposed above in Mathematics 7, 149 (2019). For the desiderata (a)–(c) the examples are optimal in the sense of having maximal domains with respect to convolvability of distributions. The examples suggest to modify desideratum (f) in the original list. Full article

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)

17 pages, 308 KiB

Open AccessArticle

Fractional Integral Equations Tell Us How to Impose Initial Values in Fractional Differential Equations

by Daniel Cao Labora

Mathematics 2020, 8(7), 1093; https://doi.org/10.3390/math8071093 - 4 Jul 2020

Cited by 2 | Viewed by 2653

Abstract

One major question in Fractional Calculus is to better understand the role of the initial values in fractional differential equations. In this sense, there is no consensus about what is the reasonable fractional abstraction of the idea of “initial value problem”. This work provides an answer to this question. The techniques that are used involve known results concerning Volterra integral equations, and the spaces of summable fractional differentiability introduced by Samko et al. In a few words, we study the natural consequences in fractional differential equations of the already existing results involving existence and uniqueness for their integral analogues, in terms of the Riemann–Liouville fractional integral. In particular, we show that a fractional differential equation of a certain order with Riemann–Liouville derivatives demands, in principle, less initial values than the ceiling of the order to have a uniquely determined solution, in contrast to a widely extended opinion. We compute explicitly the amount of necessary initial values and the orders of differentiability where these conditions need to be imposed. Full article

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)

21 pages, 403 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Good (and Not So Good) Practices in Computational Methods for Fractional Calculus

by Kai Diethelm, Roberto Garrappa and Martin Stynes

Mathematics 2020, 8(3), 324; https://doi.org/10.3390/math8030324 - 2 Mar 2020

Cited by 43 | Viewed by 5381

Abstract

The solution of fractional-order differential problems requires in the majority of cases the use of some computational approach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties. As a consequence, numerical methods are often applied in an incorrect way or unreliable methods are devised and proposed in the literature. In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results. Full article

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)

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16 pages, 2810 KiB

Open AccessArticle

Some Alternative Solutions to Fractional Models for Modelling Power Law Type Long Memory Behaviours

by Jocelyn Sabatier, Christophe Farges and Vincent Tartaglione

Mathematics 2020, 8(2), 196; https://doi.org/10.3390/math8020196 - 5 Feb 2020

Cited by 20 | Viewed by 2616

Abstract

The paper first describes a process that exhibits a power law-type long memory behaviour: the dynamical behaviour of the heap top of falling granular matter such as sand. Fractional modelling is proposed for this process, and some drawbacks and difficulties associated to fractional models are reviewed and illustrated with the sand pile process. Alternative models that solve the drawbacks and difficulties mentioned while producing power law-type long memory behaviours are presented. Full article

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)

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22 pages, 354 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Fractional Derivatives and Integrals: What Are They Needed For?

by Vasily E. Tarasov and Svetlana S. Tarasova

Mathematics 2020, 8(2), 164; https://doi.org/10.3390/math8020164 - 25 Jan 2020

Cited by 34 | Viewed by 5020

Abstract

The question raised in the title of the article is not philosophical. We do not expect general answers of the form “to describe the reality surrounding us”. The question should actually be formulated as a mathematical problem of applied mathematics, a task for new research. This question should be answered in mathematically rigorous statements about the interrelations between the properties of the operator’s kernels and the types of phenomena. This article is devoted to a discussion of the question of what is fractional operator from the point of view of not pure mathematics, but applied mathematics. The imposed restrictions on the kernel of the fractional operator should actually be divided by types of phenomena, in addition to the principles of self-consistency of mathematical theory. In applications of fractional calculus, we have a fundamental question about conditions of kernels of fractional operator of non-integer orders that allow us to describe a particular type of phenomenon. It is necessary to obtain exact correspondences between sets of properties of kernel and type of phenomena. In this paper, we discuss the properties of kernels of fractional operators to distinguish the following types of phenomena: fading memory (forgetting) and power-law frequency dispersion, spatial non-locality and power-law spatial dispersion, distributed lag (time delay), distributed scaling (dilation), depreciation, and aging. Full article

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)

10 pages, 283 KiB

Open AccessArticle

On Fractional Operators and Their Classifications

by Dumitru Baleanu and Arran Fernandez

Mathematics 2019, 7(9), 830; https://doi.org/10.3390/math7090830 - 8 Sep 2019

Cited by 186 | Viewed by 7243

Abstract

Fractional calculus dates its inception to a correspondence between Leibniz and L’Hopital in 1695, when Leibniz described “paradoxes” and predicted that “one day useful consequences will be drawn” from them. In today’s world, the study of non-integer orders of differentiation has become a thriving field of research, not only in mathematics but also in other parts of science such as physics, biology, and engineering: many of the “useful consequences” predicted by Leibniz have been discovered. However, the field has grown so far that researchers cannot yet agree on what a “fractional derivative” can be. In this manuscript, we suggest and justify the idea of classification of fractional calculus into distinct classes of operators. Full article

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)

14 pages, 358 KiB

Open AccessEditor’s ChoiceArticle

Fractional Derivatives: The Perspective of System Theory

by Manuel Duarte Ortigueira and José Tenreiro Machado

Mathematics 2019, 7(2), 150; https://doi.org/10.3390/math7020150 - 5 Feb 2019

Cited by 48 | Viewed by 4179

Abstract

This paper addresses the present day problem of multiple proposals for operators under the umbrella of “fractional derivatives”. Several papers demonstrated that various of those “novel” definitions are incorrect. Here the classical system theory is applied to develop a unified framework to clarify [...] Read more.

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)

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5 pages, 250 KiB

Open AccessEditor’s ChoiceArticle

Desiderata for Fractional Derivatives and Integrals

by Rudolf Hilfer and Yuri Luchko

Mathematics 2019, 7(2), 149; https://doi.org/10.3390/math7020149 - 4 Feb 2019

Cited by 94 | Viewed by 5763

Abstract

The purpose of this brief article is to initiate discussions in this special issue by proposing desiderata for calling an operator a fractional derivative or a fractional integral. Our desiderata are neither axioms nor do they define fractional derivatives or integrals uniquely. Instead [...] Read more.

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)

Review

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35 pages, 551 KiB

Open AccessReview

Unified Approach to Fractional Calculus Images of Special Functions—A Survey

by Virginia Kiryakova

Mathematics 2020, 8(12), 2260; https://doi.org/10.3390/math8122260 - 21 Dec 2020

Cited by 28 | Viewed by 2910

Abstract

Evaluation of images of special functions under operators of fractional calculus has become a hot topic with hundreds of recently published papers. These are growing daily and we are able to comment here only on a few of them, including also some of the latest of 2019–2020, just for the purpose of illustrating our unified approach. Many authors are producing a flood of results for various operators of fractional order integration and differentiation and their generalizations of different special (and elementary) functions. This effect is natural because there are great varieties of special functions, respectively, of operators of (classical and generalized) fractional calculus, and thus, their combinations amount to a large number. As examples, we mentioned only two such operators from thousands of results found by a Google search. Most of the mentioned works use the same formal and standard procedures. Furthermore, in such results, often the originals and the images are special functions of different kinds, or the images are not recognized as known special functions, and thus are not easy to use. In this survey we present a unified approach to fulfill the mentioned task at once in a general setting and in a well visible form: for the operators of generalized fractional calculus (including also the classical operators of fractional calculus); and for all generalized hypergeometric functions such as

_{p} Ψ_{q}

and

_{p} F_{q}

, Fox H- and Meijer G-functions, thus incorporating wide classes of special functions. In this way, a great part of the results in the mentioned publications are well predicted and appear as very special cases of ours. The proposed general scheme is based on a few basic classical results (from the Bateman Project and works by Askey, Lavoie–Osler–Tremblay, etc.) combined with ideas and developments from more than 30 years of author’s research, and reflected in the cited recent works. The main idea is as follows: From one side, the operators considered by other authors are cases of generalized fractional calculus and so, are shown to be (m-times) compositions of weighted Riemann–Lioville, i.e., Erdélyi–Kober operators. On the other side, from each generalized hypergeometric function

_{p} Ψ_{q}

_{p} F_{q}

(

p \leq q

p = q + 1

) we can reach, from the final number of applications of such operators, one of the simplest cases where the classical results are known, for example: to

_{0} F_{q - p}

(hyper-Bessel functions, in particular trigonometric functions of order

(q - p)

_{0} F_{0}

(exponential function), or

_{1} F_{0}

(beta-distribution of form

{(1 - z)}^{α} z^{β})

. The final result, written explicitly, is that any GFC operator (of multiplicity

m \geq 1

) transforms a generalized hypergeometric function into the same kind of special function with indices p and q increased by m. Full article

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)

20 pages, 300 KiB

Open AccessEditor’s ChoiceReview

The General Fractional Derivative and Related Fractional Differential Equations

by Yuri Luchko and Masahiro Yamamoto

Mathematics 2020, 8(12), 2115; https://doi.org/10.3390/math8122115 - 26 Nov 2020

Cited by 55 | Viewed by 3793

Abstract

In this survey paper, we start with a discussion of the general fractional derivative (GFD) introduced by A. Kochubei in his recent publications. In particular, a connection of this derivative to the corresponding fractional integral and the Sonine relation for their kernels are presented. Then we consider some fractional ordinary differential equations (ODEs) with the GFD including the relaxation equation and the growth equation. The main part of the paper is devoted to the fractional partial differential equations (PDEs) with the GFD. We discuss both the Cauchy problems and the initial-boundary-value problems for the time-fractional diffusion equations with the GFD. In the final part of the paper, some results regarding the inverse problems for the differential equations with the GFD are presented. Full article

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)

53 pages, 1057 KiB

Open AccessReview

Numerical Approaches to Fractional Integrals and Derivatives: A Review

by Min Cai and Changpin Li

Mathematics 2020, 8(1), 43; https://doi.org/10.3390/math8010043 - 1 Jan 2020

Cited by 39 | Viewed by 6573

Abstract

Fractional calculus, albeit a synonym of fractional integrals and derivatives which have two main characteristics—singularity and nonlocality—has attracted increasing interest due to its potential applications in the real world. This mathematical concept reveals underlying principles that govern the behavior of nature. The present paper focuses on numerical approximations to fractional integrals and derivatives. Almost all the results in this respect are included. Existing results, along with some remarks are summarized for the applied scientists and engineering community of fractional calculus. Full article

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)

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