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

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (30 November 2022) | Viewed by 21217

Special Issue Editor

Prof. Dr. Yuri Luchko

E-Mail Website
Guest Editor
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
Special Issues, Collections and Topics in MDPI journals

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.

This issue is a continuation of the previous successful Special Issue “Fractional Integrals and Derivatives: “True” versus “False””.

Prof. Dr. Yuri Luchko
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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

Published Papers (9 papers)

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Editorial

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2 pages, 145 KiB  
Editorial
Fractional Integrals and Derivatives: “True” versus “False”
by Yuri Luchko
Mathematics 2023, 11(13), 3003; https://0-doi-org.brum.beds.ac.uk/10.3390/math11133003 - 05 Jul 2023
Viewed by 610
Abstract
Within the last few years, many of the efforts of the fractional calculus (FC) community have been directed towards clarifying the nature and essential properties of the operators known as fractional integrals and derivatives [...] Full article
(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False” II)

Research

Jump to: Editorial

17 pages, 313 KiB  
Article
An Inverse Problem for a Non-Homogeneous Time-Space Fractional Equation
by Abdallah El Hamidi, Mokhtar Kirane and Ali Tfayli
Mathematics 2022, 10(15), 2586; https://0-doi-org.brum.beds.ac.uk/10.3390/math10152586 - 25 Jul 2022
Cited by 4 | Viewed by 1428
Abstract
We consider the inverse problem of finding the solution of a generalized time-space fractional equation and the source term knowing the spatial mean of the solution at any times t∈(0,T], as well as the initial and [...] Read more.
We consider the inverse problem of finding the solution of a generalized time-space fractional equation and the source term knowing the spatial mean of the solution at any times t∈(0,T], as well as the initial and the boundary conditions. The existence and the continuity with respect to the data of the solution for the direct and the inverse problem are proven by Fourier’s method and the Schauder fixed-point theorem in an adequate convex bounded subset. In the published articles on this topic, the incorrect use of the estimates in the generalized Mittag–Leffler functions is commonly performed. This leads to false proofs of the Fourier series’ convergence to recover the equation satisfied by the solution, the initial data or the boundary conditions. In the present work, the correct framework to recover the decay of fractional Fourier coefficients is established; this allows one to recover correctly the initial data, the boundary conditions and the partial differential equations within the space-time domain. Full article
(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False” II)
17 pages, 328 KiB  
Article
A New Look at the Initial Condition Problem
by Manuel D. Ortigueira
Mathematics 2022, 10(10), 1771; https://0-doi-org.brum.beds.ac.uk/10.3390/math10101771 - 23 May 2022
Cited by 15 | Viewed by 1778
Abstract
In this paper, some myths associated to the initial condition problem are studied and demystified. It is shown that the initial conditions provided by the one-sided Laplace transform are not those required for Riemann-Liouville and Caputo derivatives. The problem is studied and solved [...] Read more.
In this paper, some myths associated to the initial condition problem are studied and demystified. It is shown that the initial conditions provided by the one-sided Laplace transform are not those required for Riemann-Liouville and Caputo derivatives. The problem is studied and solved with generality as well as applied to continuous-time fractional autoregressive-moving average systems. Full article
(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False” II)
13 pages, 344 KiB  
Article
Internal Variable Theory in Viscoelasticity: Fractional Generalizations and Thermodynamical Restrictions
by Teodor M. Atanackovic, Cemal Dolicanin and Enes Kacapor
Mathematics 2022, 10(10), 1708; https://0-doi-org.brum.beds.ac.uk/10.3390/math10101708 - 16 May 2022
Cited by 2 | Viewed by 1381
Abstract
Here, we study the internal variable approach to viscoelasticity. First, we generalize the classical approach by introducing a fractional derivative into the equation for time evolution of the internal variables. Next, we derive restrictions on the coefficients that follow from the dissipation inequality [...] Read more.
Here, we study the internal variable approach to viscoelasticity. First, we generalize the classical approach by introducing a fractional derivative into the equation for time evolution of the internal variables. Next, we derive restrictions on the coefficients that follow from the dissipation inequality (entropy inequality under isothermal conditions). In the example of wave propagation, we show that the restrictions that follow from entropy inequality are sufficient to guarantee the existence of the solution. We present a numerical solution to the wave equation for several values of the parameters. Full article
(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False” II)
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34 pages, 435 KiB  
Article
Fractional Dynamics with Depreciation and Obsolescence: Equations with Prabhakar Fractional Derivatives
by Vasily E. Tarasov
Mathematics 2022, 10(9), 1540; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091540 - 04 May 2022
Cited by 7 | Viewed by 1111
Abstract
In economics, depreciation functions (operator kernels) are certain decreasing functions, which are assumed to be equal to unity at zero. Usually, an exponential function is used as a depreciation function. However, exponential functions in operator kernels do not allow simultaneous consideration of memory [...] Read more.
In economics, depreciation functions (operator kernels) are certain decreasing functions, which are assumed to be equal to unity at zero. Usually, an exponential function is used as a depreciation function. However, exponential functions in operator kernels do not allow simultaneous consideration of memory effects and depreciation effects. In this paper, it is proposed to consider depreciation of a non-exponential type, and simultaneously take into account memory effects by using the Prabhakar fractional derivatives and integrals. Integro-differential operators with the Prabhakar (generalized Mittag-Leffler) function in the kernels are considered. The important distinguishing features of the Prabhakar function in operator kernels, which allow us to take into account non-exponential depreciation and fading memory in economics, are described. In this paper, equations with the following operators are considered: (a) the Prabhakar fractional integral, which contains the Prabhakar function as the kernels; (b) the Prabhakar fractional derivative of Riemann–Liouville type proposed by Kilbas, Saigo, and Saxena in 2004, which is left inverse for the Prabhakar fractional integral; and (c) the Prabhakar operator of Caputo type proposed by D’Ovidio and Polito, which is also called the regularized Prabhakar fractional derivative. The solutions of fractional differential equations with the Prabhakar operator and its special cases are suggested. The asymptotic behavior of these solutions is discussed. Full article
(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False” II)
24 pages, 390 KiB  
Article
Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense
by Yuri Luchko
Mathematics 2022, 10(6), 849; https://0-doi-org.brum.beds.ac.uk/10.3390/math10060849 - 08 Mar 2022
Cited by 28 | Viewed by 3076
Abstract
In this paper, we first consider the general fractional derivatives of arbitrary order defined in the Riemann–Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of fractional calculus that leads to a closed [...] Read more.
In this paper, we first consider the general fractional derivatives of arbitrary order defined in the Riemann–Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of fractional calculus that leads to a closed form formula for their projector operator. These results allow us to formulate the natural initial conditions for the fractional differential equations with the general fractional derivatives of arbitrary order in the Riemann–Liouville sense. In the second part of the paper, we develop an operational calculus of the Mikusiński type for the general fractional derivatives of arbitrary order in the Riemann–Liouville sense and apply it for derivation of an explicit form of solutions to the Cauchy problems for the single- and multi-term linear fractional differential equations with these derivatives. The solutions are provided in form of the convolution series generated by the kernels of the corresponding general fractional integrals. Full article
(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False” II)
55 pages, 510 KiB  
Article
Fractional Calculus and Time-Fractional Differential Equations: Revisit and Construction of a Theory
by Masahiro Yamamoto
Mathematics 2022, 10(5), 698; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050698 - 23 Feb 2022
Cited by 12 | Viewed by 2895
Abstract
For fractional derivatives and time-fractional differential equations, we construct a framework on the basis of operator theory in fractional Sobolev spaces. Our framework provides a feasible extension of the classical Caputo and the Riemann–Liouville derivatives within Sobolev spaces of fractional orders, including negative [...] Read more.
For fractional derivatives and time-fractional differential equations, we construct a framework on the basis of operator theory in fractional Sobolev spaces. Our framework provides a feasible extension of the classical Caputo and the Riemann–Liouville derivatives within Sobolev spaces of fractional orders, including negative ones. Our approach enables a unified treatment for fractional calculus and time-fractional differential equations. We formulate initial value problems for fractional ordinary differential equations and initial boundary value problems for fractional partial differential equations to prove well-posedness and other properties. Full article
(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False” II)
24 pages, 422 KiB  
Article
On the Quantitative Properties of Some Market Models Involving Fractional Derivatives
by Jean-Philippe Aguilar, Jan Korbel and Nicolas Pesci
Mathematics 2021, 9(24), 3198; https://0-doi-org.brum.beds.ac.uk/10.3390/math9243198 - 11 Dec 2021
Cited by 2 | Viewed by 2068
Abstract
We review and discuss the properties of various models that are used to describe the behavior of stock returns and are related in a way or another to fractional pseudo-differential operators in the space variable; we compare their main features and discuss what [...] Read more.
We review and discuss the properties of various models that are used to describe the behavior of stock returns and are related in a way or another to fractional pseudo-differential operators in the space variable; we compare their main features and discuss what behaviors they are able to capture. Then, we extend the discussion by showing how the pricing of contingent claims can be integrated into the framework of a model featuring a fractional derivative in both time and space, recall some recently obtained formulas in this context, and derive new ones for some commonly traded instruments and a model involving a Riesz temporal derivative and a particular case of Riesz–Feller space derivative. Finally, we provide formulas for implied volatility and first- and second-order market sensitivities in this model, discuss hedging and profit and loss policies, and compare with other fractional (Caputo) or non-fractional models. Full article
(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False” II)
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17 pages, 311 KiB  
Article
General Fractional Integrals and Derivatives with the Sonine Kernels
by Yuri Luchko
Mathematics 2021, 9(6), 594; https://0-doi-org.brum.beds.ac.uk/10.3390/math9060594 - 10 Mar 2021
Cited by 76 | Viewed by 4759
Abstract
In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. First, the Sonine kernels and their important special classes and particular cases are discussed. In [...] Read more.
In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. First, the Sonine kernels and their important special classes and particular cases are discussed. In particular, we introduce a class of the Sonine kernels that possess an integrable singularity of power function type at the point zero. For the general fractional integrals and derivatives with the Sonine kernels from this class, two fundamental theorems of fractional calculus are proved. Then, we construct the n-fold general fractional integrals and derivatives and study their properties. Full article
(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False” II)
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