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Fractal Fract, Volume 1, Issue 1 (December 2017)

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Editorial

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Open AccessEditorial Fractal and Fractional
Fractal Fract 2017, 1(1), 1; doi:10.3390/fractalfract1010001
Received: 20 March 2017 / Revised: 20 March 2017 / Accepted: 20 March 2017 / Published: 26 March 2017
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Abstract
Fractal and Fractional are two words referring to some characteristics and fundamental problems which arise in all fields of science and technology. [...]
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Research

Jump to: Editorial

Open AccessArticle Fractional Definite Integral
Fractal Fract 2017, 1(1), 2; doi:10.3390/fractalfract1010002
Received: 14 June 2017 / Revised: 30 June 2017 / Accepted: 30 June 2017 / Published: 2 July 2017
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Abstract
This paper proposes the definition of fractional definite integral and analyses the corresponding fundamental theorem of fractional calculus. In this context, we studied the relevant properties of the fractional derivatives that lead to such a definition. Finally, integrals on R2R2
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This paper proposes the definition of fractional definite integral and analyses the corresponding fundamental theorem of fractional calculus. In this context, we studied the relevant properties of the fractional derivatives that lead to such a definition. Finally, integrals on R2 R 2 and R3 R 3 are also proposed. Full article
Open AccessArticle Which Derivative?
Fractal Fract 2017, 1(1), 3; doi:10.3390/fractalfract1010003
Received: 6 July 2017 / Revised: 22 July 2017 / Accepted: 23 July 2017 / Published: 25 July 2017
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Abstract
The actual state of interplay between Fractional Calculus, Signal Processing, and Applied Sciences is discussed in this paper. A framework for compatible integer and fractional derivatives/integrals in signals and systems context is described. It is shown how suitable fractional formulations are really extensions
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The actual state of interplay between Fractional Calculus, Signal Processing, and Applied Sciences is discussed in this paper. A framework for compatible integer and fractional derivatives/integrals in signals and systems context is described. It is shown how suitable fractional formulations are really extensions of the integer order definitions currently used in Signal Processing. The particular case of fractional linear systems is considered and the problem of initial conditions is tackled. Full article
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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Open AccessArticle Dynamics and Stability Results for Hilfer Fractional Type Thermistor Problem
Fractal Fract 2017, 1(1), 5; doi:10.3390/fractalfract1010005
Received: 22 August 2017 / Revised: 5 September 2017 / Accepted: 6 September 2017 / Published: 9 September 2017
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Abstract
In this paper, we study the dynamics and stability of thermistor problem for Hilfer fractional type. Classical fixed point theorems are utilized in deriving the results. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering)
Open AccessArticle Exact Discretization of an Economic Accelerator and Multiplier with Memory
Fractal Fract 2017, 1(1), 6; doi:10.3390/fractalfract1010006
Received: 31 July 2017 / Revised: 7 September 2017 / Accepted: 7 September 2017 / Published: 11 September 2017
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Abstract
Fractional differential equations of macroeconomics, which allow us to take into account power-law memory effects, are considered. We describe an economic accelerator and multiplier with fading memory in the framework of discrete-time and continuous-time approaches. A relationship of the continuous- and discrete-time fractional-order
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Fractional differential equations of macroeconomics, which allow us to take into account power-law memory effects, are considered. We describe an economic accelerator and multiplier with fading memory in the framework of discrete-time and continuous-time approaches. A relationship of the continuous- and discrete-time fractional-order equations is considered. We propose equations of the accelerator and multiplier for economic processes with power-law memory. Exact discrete analogs of these equations are suggested by using the exact fractional differences of integer and non-integer orders. Exact correspondence between the equations with finite differences and differential equations lies not so much in the limiting condition, when the step of discretization tends to zero, as in the fact that mathematical operations, which are used in these equations, satisfy in many cases the same mathematical laws. Full article
Open AccessArticle Stokes’ First Problem for Viscoelastic Fluids with a Fractional Maxwell Model
Fractal Fract 2017, 1(1), 7; doi:10.3390/fractalfract1010007
Received: 21 September 2017 / Revised: 20 October 2017 / Accepted: 23 October 2017 / Published: 24 October 2017
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Abstract
Stokes’ first problem for a class of viscoelastic fluids with the generalized fractional Maxwell constitutive model is considered. The constitutive equation is obtained from the classical Maxwell stress–strain relation by substituting the first-order derivatives of stress and strain by derivatives of non-integer orders
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Stokes’ first problem for a class of viscoelastic fluids with the generalized fractional Maxwell constitutive model is considered. The constitutive equation is obtained from the classical Maxwell stress–strain relation by substituting the first-order derivatives of stress and strain by derivatives of non-integer orders in the interval ( 0 , 1 ] . Explicit integral representation of the solution is derived and some of its characteristics are discussed: non-negativity and monotonicity, asymptotic behavior, analyticity, finite/infinite propagation speed, and absence of wave front. To illustrate analytical findings, numerical results for different values of the parameters are presented. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering)
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Open AccessArticle Fractional Divergence of Probability Densities
Fractal Fract 2017, 1(1), 8; doi:10.3390/fractalfract1010008
Received: 12 September 2017 / Revised: 18 October 2017 / Accepted: 19 October 2017 / Published: 25 October 2017
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Abstract
The divergence or relative entropy between probability densities is examined. Solutions that minimise the divergence between two distributions are usually “trivial” or unique. By using a fractional-order formulation for the divergence with respect to the parameters, the distance between probability densities can be
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The divergence or relative entropy between probability densities is examined. Solutions that minimise the divergence between two distributions are usually “trivial” or unique. By using a fractional-order formulation for the divergence with respect to the parameters, the distance between probability densities can be minimised so that multiple non-trivial solutions can be obtained. As a result, the fractional divergence approach reduces the divergence to zero even when this is not possible via the conventional method. This allows replacement of a more complicated probability density with one that has a simpler mathematical form for more general cases. Full article
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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 The Fractal Nature of an Approximate Prime Counting Function
Fractal Fract 2017, 1(1), 10; doi:10.3390/fractalfract1010010
Received: 20 October 2017 / Revised: 31 October 2017 / Accepted: 3 November 2017 / Published: 8 November 2017
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Abstract
Prime number related fractal polygons and curves are derived by combining two different aspects. One is an approximation of the prime counting function build on an additive function. The other is prime number indexed basis entities taken from the discrete or continuous Fourier
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Prime number related fractal polygons and curves are derived by combining two different aspects. One is an approximation of the prime counting function build on an additive function. The other is prime number indexed basis entities taken from the discrete or continuous Fourier basis. Full article
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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.
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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 Fractal Simulation of Flocculation Processes Using a Diffusion-Limited Aggregation Model
Fractal Fract 2017, 1(1), 12; doi:10.3390/fractalfract1010012
Received: 25 October 2017 / Revised: 15 November 2017 / Accepted: 15 November 2017 / Published: 18 November 2017
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Abstract
In flocculation processes, particulates randomly collide and coagulate with each other, leading to the formation and sedimention of aggregates exhibiting fractal characteristics. The diffusion-limited aggregation (DLA) model is extensively employed to describe and study flocculation processes. To more accurately simulate flocculation processes with
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In flocculation processes, particulates randomly collide and coagulate with each other, leading to the formation and sedimention of aggregates exhibiting fractal characteristics. The diffusion-limited aggregation (DLA) model is extensively employed to describe and study flocculation processes. To more accurately simulate flocculation processes with the DLA model, the effects of particle number (denoting flocculation time), motion step length (denoting water temperature), launch radius (representing initial particulate concentration), and finite motion step (representing the motion energy of the particles) on the morphology and structure of the two-dimensional (2D) as well as three-dimensional (3D) DLA aggregates are studied. The results show that the 2D DLA aggregates possess conspicuous fractal features when the particle number is above 1000, motion step length is 1.5–3.5, launch radius is 1–10, and finite motion step is more than 3000; the 3D DLA aggregates present clear fractal characteristics when the particle number is above 500, the motion step length is 1.5–3.5, the launch radius is 1–10, and the finite motion step exceeds 200. The fractal dimensions of 3D DLA aggregates are appreciably higher than those of 2D DLA aggregates. Full article
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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
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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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