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Fractal Fract
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Fractional Dynamics
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Fractional Dynamics

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

Deadline for manuscript submissions: closed (31 December 2017) | Viewed by 54428

Special Issue Editors

Prof. Dr. Carlo Cattani

E-Mail Website
Guest Editor
Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
Interests: wavelets; fractals; fractional and stochastic equations; numerical and computational methods; mathematical physics; nonlinear systems; artificial intelligence
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Renato Spigler

E-Mail Website
Guest Editor
Department of Mathematics and Physics, Roma Tre University, 00146 Rome, Italy
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 monthly 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 2700 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

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

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Published Papers (12 papers)

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Editorial

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3 pages, 167 KiB  
Editorial
Fractional Dynamics
by Carlo Cattani and Renato Spigler
Fractal Fract. 2018, 2(2), 19; https://doi.org/10.3390/fractalfract2020019 - 17 Jun 2018
Cited by 4 | Viewed by 3071
Abstract
n/a Full article
(This article belongs to the Special Issue Fractional Dynamics)

Research

Jump to: Editorial

12 pages, 410 KiB  
Article
Identifying the Fractional Orders in Anomalous Diffusion Models from Real Data
by Moreno Concezzi and Renato Spigler
Fractal Fract. 2018, 2(1), 14; https://doi.org/10.3390/fractalfract2010014 - 24 Feb 2018
Cited by 5 | Viewed by 4039
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 [...] Read more.
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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16 pages, 1289 KiB  
Article
A Fractional B-spline Collocation Method for the Numerical Solution of Fractional Predator-Prey Models
by Francesca Pitolli
Fractal Fract. 2018, 2(1), 13; https://doi.org/10.3390/fractalfract2010013 - 17 Feb 2018
Cited by 31 | Viewed by 5201
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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20 pages, 628 KiB  
Article
Poiseuille Flow of a Non-Local Non-Newtonian Fluid with Wall Slip: A First Step in Modeling Cerebral Microaneurysms
by Corina S. Drapaca
Fractal Fract. 2018, 2(1), 9; https://doi.org/10.3390/fractalfract2010009 - 6 Feb 2018
Cited by 9 | Viewed by 4618
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 [...] Read more.
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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7 pages, 679 KiB  
Article
Towards a Generalized Beer-Lambert Law
by Giampietro Casasanta and Roberto Garra
Fractal Fract. 2018, 2(1), 8; https://doi.org/10.3390/fractalfract2010008 - 31 Jan 2018
Cited by 20 | Viewed by 7218
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 [...] Read more.
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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8 pages, 233 KiB  
Article
Emergence of Fractional Kinetics in Spiny Dendrites
by Silvia Vitali, Francesco Mainardi and Gastone Castellani
Fractal Fract. 2018, 2(1), 6; https://doi.org/10.3390/fractalfract2010006 - 25 Jan 2018
Cited by 4 | Viewed by 2960
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, [...] Read more.
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)
258 KiB  
Article
Fractional Diffusion Models for the Atmosphere of Mars
by Salvador Jiménez, David Usero, Luis Vázquez and Maria Pilar Velasco
Fractal Fract. 2018, 2(1), 1; https://doi.org/10.3390/fractalfract2010001 - 28 Dec 2017
Cited by 8 | Viewed by 3396
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)
290 KiB  
Article
Some Nonlocal Operators in the First Heisenberg Group
by Fausto Ferrari
Fractal Fract. 2017, 1(1), 15; https://doi.org/10.3390/fractalfract1010015 - 27 Nov 2017
Cited by 7 | Viewed by 2986
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)
3205 KiB  
Article
The Kernel of the Distributed Order Fractional Derivatives with an Application to Complex Materials
by Michele Caputo and Mauro Fabrizio
Fractal Fract. 2017, 1(1), 13; https://doi.org/10.3390/fractalfract1010013 - 21 Nov 2017
Cited by 25 | Viewed by 3966
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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238 KiB  
Article
A Fractional-Order Infectivity and Recovery SIR Model
by Christopher N. Angstmann, Bruce I. Henry and Anna V. McGann
Fractal Fract. 2017, 1(1), 11; https://doi.org/10.3390/fractalfract1010011 - 17 Nov 2017
Cited by 30 | Viewed by 4998
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)
289 KiB  
Article
From Circular to Bessel Functions: A Transition through the Umbral Method
by Giuseppe Dattoli, Emanuele Di Palma, Silvia Licciardi and Elio Sabia
Fractal Fract. 2017, 1(1), 9; https://doi.org/10.3390/fractalfract1010009 - 8 Nov 2017
Cited by 8 | Viewed by 3218
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 [...] Read more.
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)
2717 KiB  
Article
A Fractional Complex Permittivity Model of Media with Dielectric Relaxation
by Armando Ciancio and Bruno Felice Filippo Flora
Fractal Fract. 2017, 1(1), 4; https://doi.org/10.3390/fractalfract1010004 - 29 Aug 2017
Cited by 5 | Viewed by 7314
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 [...] Read more.
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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