Special Issue "Fractional Dynamics"

A special issue of Fractal and Fractional (ISSN 2504-3110).

Deadline for manuscript submissions: closed (31 December 2017)

Special Issue Editors

Guest Editor
Prof. Dr. Carlo Cattani

Engineering School (DEIM) University of Tuscia, 01100 Largo dell'Università, Viterbo, Italy
Website | E-Mail
Interests: wavelets; fractals; fractional calculus; dynamical systems; data analysis; time series analysis; image analysis; computer science; computational methods; composite materials; elasticity; nonlinear waves
Guest Editor
Prof. Dr. Renato Spigler

Department of Mathematics and Physics, Roma Tre University, 00146 Rome, Italy
Website | E-Mail
Phone: +39-06-5733-8211
Fax: +39-06-5733-8211
Interests: fractional calculus (approximations, numerics, and applications); applied and computational mathematics

Special Issue Information

Dear Colleagues,

Modeling, simulation, and applications of Fractional Calculus have recently become an increasingly popular subject, with an impressive growth concerning applications. The founding and limited ideas on fractional derivatives have achieved an incredibly valuable status. The manifold applications in mathematics, physics, engineering, economics, biology, and medicine have opened new challenging fields of research. For instance, in mechanics, a suitable definition of the fractional operator has shed some light on viscoelasticity, by explaining memory effects on materials. Needless to say, these applications require the development of practical mathematical tools to obtain quantitative information from models, newly reformulated in terms of fractional differential equations. Even confining ourselves to the field of ordinary differential equations, the Bagley-Torvik model showed that fractional derivatives may actually arise naturally within certain physical models, and are not mere fancy mathematical generalizations.This Special Issue focuses on the most recent advances in fractional calculus, applied to dynamic problems, linear and nonlinear fractional ordinaries and partial differential equations, integral fractional differential equations and stochastic integral problems arising in all fields of science, engineering applications, and other applied fields.

Prof. Dr. Carlo Cattani
Prof. Dr. Renato Spigler
Guest Editors

Manuscript Submission Information

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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. Fractal and Fractional is an international peer-reviewed open access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) is waived for well-prepared manuscripts submitted to this issue. 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

  • mathematics
  • physics
  • mathematical physics
  • mechanics
  • fractional calculus
  • fractal
  • fractional dynamical systems
  • fractional partial differential equations

Published Papers (6 papers)

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Research

Open AccessArticle Fractional Diffusion Models for the Atmosphere of Mars
Fractal Fract 2018, 2(1), 1; doi:10.3390/fractalfract2010001
Received: 5 December 2017 / Revised: 23 December 2017 / Accepted: 24 December 2017 / Published: 28 December 2017
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Abstract
The dust aerosols floating in the atmosphere of Mars cause an attenuation of the solar radiation traversing the atmosphere that cannot be modeled through the use of classical diffusion processes. However, the definition of a type of fractional diffusion equation offers a more
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The dust aerosols floating in the atmosphere of Mars cause an attenuation of the solar radiation traversing the atmosphere that cannot be modeled through the use of classical diffusion processes. However, the definition of a type of fractional diffusion equation offers a more accurate model for this dynamic and the second order moment of this equation allows one to establish a connection between the fractional equation and the Ångstrom law that models the attenuation of the solar radiation. In this work we consider both one and three dimensional wavelength-fractional diffusion equations, and we obtain the analytical solutions and numerical methods using two different approaches of the fractional derivative. Full article
(This article belongs to the Special Issue Fractional Dynamics)
Open AccessArticle Some Nonlocal Operators in the First Heisenberg Group
Fractal Fract 2017, 1(1), 15; doi:10.3390/fractalfract1010015
Received: 1 November 2017 / Revised: 22 November 2017 / Accepted: 23 November 2017 / Published: 27 November 2017
Cited by 1 | PDF Full-text (290 KB) | HTML Full-text | XML Full-text
Abstract
In this paper we construct some nonlocal operators in the Heisenberg group. Specifically, starting from the Grünwald-Letnikov derivative and Marchaud derivative in the Euclidean setting, we revisit those definitions with respect to the one of the fractional Laplace operator. Then, we define some
[...] Read more.
In this paper we construct some nonlocal operators in the Heisenberg group. Specifically, starting from the Grünwald-Letnikov derivative and Marchaud derivative in the Euclidean setting, we revisit those definitions with respect to the one of the fractional Laplace operator. Then, we define some nonlocal operators in the non-commutative structure of the first Heisenberg group adapting the approach applied in the Euclidean case to the new framework. Full article
(This article belongs to the Special Issue Fractional Dynamics)
Open AccessArticle The Kernel of the Distributed Order Fractional Derivatives with an Application to Complex Materials
Fractal Fract 2017, 1(1), 13; doi:10.3390/fractalfract1010013
Received: 24 October 2017 / Revised: 13 November 2017 / Accepted: 13 November 2017 / Published: 21 November 2017
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Abstract
The extension of the fractional order derivative to the distributed order fractional derivative (DOFD) is somewhat simple from a formal point of view, but it does not yet have a simple, obvious analytic form that allows its fast numerical calculation, which is necessary
[...] Read more.
The extension of the fractional order derivative to the distributed order fractional derivative (DOFD) is somewhat simple from a formal point of view, but it does not yet have a simple, obvious analytic form that allows its fast numerical calculation, which is necessary when solving differential equations with DOFD. In this paper, we supply a simple analytic kernel for the Caputo DOFD and the Caputo-Fabrizio DOFD, which may be used for numerical calculation in cases where the weight function is unity. This, in turn, could potentially allow faster solution of differential equations containing DOFD. Utilizing an analytical formulation of simple physical systems with phenomenological equations that include a DOFD, we show the relevant differences between the Caputo DOFD and the Caputo-Fabrizio DOFD. Finally, we propose a model based on DOFD for modeling composed materials that comprise different constituents, and show its compatibility with thermodynamics. Full article
(This article belongs to the Special Issue Fractional Dynamics)
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Open AccessArticle A Fractional-Order Infectivity and Recovery SIR Model
Fractal Fract 2017, 1(1), 11; doi:10.3390/fractalfract1010011
Received: 31 October 2017 / Revised: 14 November 2017 / Accepted: 15 November 2017 / Published: 17 November 2017
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Abstract
The introduction of fractional-order derivatives to epidemiological compartment models, such as SIR models, has attracted much attention. When this introduction is done in an ad hoc manner, it is difficult to reconcile parameters in the resulting fractional-order equations with the dynamics of individuals.
[...] Read more.
The introduction of fractional-order derivatives to epidemiological compartment models, such as SIR models, has attracted much attention. When this introduction is done in an ad hoc manner, it is difficult to reconcile parameters in the resulting fractional-order equations with the dynamics of individuals. This issue is circumvented by deriving fractional-order models from an underlying stochastic process. Here, we derive a fractional-order infectivity and recovery Susceptible Infectious Recovered (SIR) model from the stochastic process of a continuous-time random walk (CTRW) that incorporates a time-since-infection dependence on both the infectivity and the recovery of the population. By considering a power-law dependence in the infectivity and recovery, fractional-order derivatives appear in the generalised master equations that govern the evolution of the SIR populations. Under the appropriate limits, this fractional-order infectivity and recovery model reduces to both the standard SIR model and the fractional recovery SIR model. Full article
(This article belongs to the Special Issue Fractional Dynamics)
Open AccessArticle From Circular to Bessel Functions: A Transition through the Umbral Method
Fractal Fract 2017, 1(1), 9; doi:10.3390/fractalfract1010009
Received: 9 October 2017 / Revised: 3 November 2017 / Accepted: 3 November 2017 / Published: 8 November 2017
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Abstract
A common environment in which to place Bessel and circular functions is envisaged. We show, by the use of operational methods, that the Gaussian provides the umbral image of these functions. We emphasize the role of the spherical Bessel functions and a family
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A common environment in which to place Bessel and circular functions is envisaged. We show, by the use of operational methods, that the Gaussian provides the umbral image of these functions. We emphasize the role of the spherical Bessel functions and a family of associated auxiliary polynomials, as transition elements between these families of functions. The consequences of this point of view and the relevant impact on the study of the properties of special functions is carefully discussed. Full article
(This article belongs to the Special Issue Fractional Dynamics)
Open AccessArticle A Fractional Complex Permittivity Model of Media with Dielectric Relaxation
Fractal Fract 2017, 1(1), 4; doi:10.3390/fractalfract1010004
Received: 11 August 2017 / Revised: 25 August 2017 / Accepted: 25 August 2017 / Published: 29 August 2017
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Abstract
In this work, we propose a fractional complex permittivity model of dielectric media with memory. Debye’s generalized equation, expressed in terms of the phenomenological coefficients, is replaced with the corresponding differential equation by applying Caputo’s fractional derivative. We observe how fractional order depends
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In this work, we propose a fractional complex permittivity model of dielectric media with memory. Debye’s generalized equation, expressed in terms of the phenomenological coefficients, is replaced with the corresponding differential equation by applying Caputo’s fractional derivative. We observe how fractional order depends on the frequency band of excitation energy in accordance with the 2nd Principle of Thermodynamics. The model obtained is validated with respect to the measurements made on the biological tissues and in particular on the human aorta. Full article
(This article belongs to the Special Issue Fractional Dynamics)
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Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

 

 

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