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 Viterbo, Italy
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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
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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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Keywords

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

Published Papers (12 papers)

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Editorial

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Open AccessEditorial Fractional Dynamics
Fractal Fract 2018, 2(2), 19; https://doi.org/10.3390/fractalfract2020019
Received: 5 June 2018 / Accepted: 11 June 2018 / Published: 17 June 2018
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(This article belongs to the Special Issue Fractional Dynamics)

Research

Jump to: Editorial

Open AccessArticle Identifying the Fractional Orders in Anomalous Diffusion Models from Real Data
Fractal Fract 2018, 2(1), 14; https://doi.org/10.3390/fractalfract2010014
Received: 10 October 2017 / Revised: 8 February 2018 / Accepted: 15 February 2018 / Published: 24 February 2018
Cited by 1 | PDF Full-text (410 KB) | HTML Full-text | XML Full-text
Abstract
An attempt is made to identify the orders of the fractional derivatives in a simple anomalous diffusion model, starting from real data. We consider experimental data taken at the Columbus Air Force Base in Mississippi. Using as a model a one-dimensional fractional diffusion
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An attempt is made to identify the orders of the fractional derivatives in a simple anomalous diffusion model, starting from real data. We consider experimental data taken at the Columbus Air Force Base in Mississippi. Using as a model a one-dimensional fractional diffusion equation in both space and time, we fit the data by choosing several values of the fractional orders and computing the infinite-norm “errors”, representing the discrepancy between the numerical solution to the model equation and the experimental data. Data were also filtered before being used, to see possible improvements. The minimal discrepancy is attained correspondingly to a fractional order in time around 0 . 6 and a fractional order in space near 2. These results may describe well the memory properties of the porous medium that can be observed. Full article
(This article belongs to the Special Issue Fractional Dynamics)
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Open AccessArticle A Fractional B-spline Collocation Method for the Numerical Solution of Fractional Predator-Prey Models
Fractal Fract 2018, 2(1), 13; https://doi.org/10.3390/fractalfract2010013
Received: 22 December 2017 / Revised: 12 February 2018 / Accepted: 12 February 2018 / Published: 17 February 2018
Cited by 2 | PDF Full-text (1289 KB) | HTML Full-text | XML Full-text
Abstract
We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the
[...] Read more.
We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost. Full article
(This article belongs to the Special Issue Fractional Dynamics)
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Open AccessArticle Poiseuille Flow of a Non-Local Non-Newtonian Fluid with Wall Slip: A First Step in Modeling Cerebral Microaneurysms
Fractal Fract 2018, 2(1), 9; https://doi.org/10.3390/fractalfract2010009
Received: 16 October 2017 / Revised: 1 February 2018 / Accepted: 2 February 2018 / Published: 6 February 2018
Cited by 1 | PDF Full-text (628 KB) | HTML Full-text | XML Full-text
Abstract
Cerebral aneurysms and microaneurysms are abnormal vascular dilatations with high risk of rupture. An aneurysmal rupture could cause permanent disability and even death. Finding and treating aneurysms before their rupture is very difficult since symptoms can be easily attributed mistakenly to other common
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Cerebral aneurysms and microaneurysms are abnormal vascular dilatations with high risk of rupture. An aneurysmal rupture could cause permanent disability and even death. Finding and treating aneurysms before their rupture is very difficult since symptoms can be easily attributed mistakenly to other common brain diseases. Mathematical models could highlight possible mechanisms of aneurysmal development and suggest specialized biomarkers for aneurysms. Existing mathematical models of intracranial aneurysms focus on mechanical interactions between blood flow and arteries. However, these models cannot be applied to microaneurysms since the anatomy and physiology at the length scale of cerebral microcirculation are different. In this paper, we propose a mechanism for the formation of microaneurysms that involves the chemo-mechanical coupling of blood and endothelial and neuroglial cells. We model the blood as a non-local non-Newtonian incompressible fluid and solve analytically the Poiseuille flow of such a fluid through an axi-symmetric circular rigid and impermeable pipe in the presence of wall slip. The spatial derivatives of the proposed generalization of the rate of deformation tensor are expressed using Caputo fractional derivatives. The wall slip is represented by the classic Navier law and a generalization of this law involving fractional derivatives. Numerical simulations suggest that hypertension could contribute to microaneurysmal formation. Full article
(This article belongs to the Special Issue Fractional Dynamics)
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Open AccessArticle Towards a Generalized Beer-Lambert Law
Fractal Fract 2018, 2(1), 8; https://doi.org/10.3390/fractalfract2010008
Received: 31 December 2017 / Revised: 26 January 2018 / Accepted: 27 January 2018 / Published: 31 January 2018
Cited by 1 | PDF Full-text (679 KB) | HTML Full-text | XML Full-text
Abstract
Anomalous deviations from the Beer-Lambert law have been observed for a long time in a wide range of application. Despite all the attempts, a reliable and accepted model has not been provided so far. In addition, in some cases the attenuation of radiation
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Anomalous deviations from the Beer-Lambert law have been observed for a long time in a wide range of application. Despite all the attempts, a reliable and accepted model has not been provided so far. In addition, in some cases the attenuation of radiation seems to follow a hyperbolic more than an exponential extinction law. Starting from a probabilistic interpretation of the Beer-Lambert law based on Poissonian distribution of extinction events, in this paper we consider deviations from the classical exponential extinction introducing a weighted version of the classical law. The generalized law is able to account for both sub or super-exponential extinction of radiation, and can be extended to the case of inhomogeneous media. Focusing on this case, we consider a generalized Beer-Lambert law based on an inhomogeneous weighted Poisson distribution involving a Mittag-Leffler function, and show how it can be directly related to hyperbolic decay laws observed in some applications particularly relevant to microbiology and pharmacology. Full article
(This article belongs to the Special Issue Fractional Dynamics)
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Open AccessArticle Emergence of Fractional Kinetics in Spiny Dendrites
Fractal Fract 2018, 2(1), 6; https://doi.org/10.3390/fractalfract2010006
Received: 21 December 2017 / Revised: 22 January 2018 / Accepted: 22 January 2018 / Published: 25 January 2018
Cited by 1 | PDF Full-text (233 KB) | HTML Full-text | XML Full-text
Abstract
Fractional extensions of the cable equation have been proposed in the literature to describe transmembrane potential in spiny dendrites. The anomalous behavior has been related in the literature to the geometrical properties of the system, in particular, the density of spines, by experiments,
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Fractional extensions of the cable equation have been proposed in the literature to describe transmembrane potential in spiny dendrites. The anomalous behavior has been related in the literature to the geometrical properties of the system, in particular, the density of spines, by experiments, computer simulations, and in comb-like models. The same PDE can be related to more than one stochastic process leading to anomalous diffusion behavior. The time-fractional diffusion equation can be associated to a continuous time random walk (CTRW) with power-law waiting time probability or to a special case of the Erdély-Kober fractional diffusion, described by the ggBm. In this work, we show that time fractional generalization of the cable equation arises naturally in the CTRW by considering a superposition of Markovian processes and in a ggBm-like construction of the random variable. Full article
(This article belongs to the Special Issue Fractional Dynamics)
Open AccessArticle Fractional Diffusion Models for the Atmosphere of Mars
Fractal Fract 2018, 2(1), 1; https://doi.org/10.3390/fractalfract2010001
Received: 5 December 2017 / Revised: 23 December 2017 / Accepted: 24 December 2017 / Published: 28 December 2017
Cited by 1 | PDF Full-text (258 KB) | HTML Full-text | XML Full-text
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
[...] Read more.
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; https://doi.org/10.3390/fractalfract1010015
Received: 1 November 2017 / Revised: 22 November 2017 / Accepted: 23 November 2017 / Published: 27 November 2017
Cited by 2 | 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
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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; https://doi.org/10.3390/fractalfract1010013
Received: 24 October 2017 / Revised: 13 November 2017 / Accepted: 13 November 2017 / Published: 21 November 2017
Cited by 1 | PDF Full-text (3205 KB) | HTML Full-text | XML Full-text
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
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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; https://doi.org/10.3390/fractalfract1010011
Received: 31 October 2017 / Revised: 14 November 2017 / Accepted: 15 November 2017 / Published: 17 November 2017
Cited by 1 | PDF Full-text (238 KB) | HTML Full-text | XML Full-text
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.
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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; https://doi.org/10.3390/fractalfract1010009
Received: 9 October 2017 / Revised: 3 November 2017 / Accepted: 3 November 2017 / Published: 8 November 2017
Cited by 1 | PDF Full-text (289 KB) | HTML Full-text | XML Full-text
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; https://doi.org/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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