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# Fractal Fract., Volume 1, Issue 1 (December 2017) – 17 articles

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929 KiB
Article
Modeling of Heat Distribution in Porous Aluminum Using Fractional Differential Equation
Fractal Fract. 2017, 1(1), 17; https://doi.org/10.3390/fractalfract1010017 - 12 Dec 2017
Cited by 18 | Viewed by 3178
Abstract
The authors present a model of heat conduction using the Caputo fractional derivative with respect to time. The presented model was used to reconstruct the thermal conductivity coefficient, heat transfer coefficient, initial condition and order of fractional derivative in the fractional heat conduction [...] Read more.
The authors present a model of heat conduction using the Caputo fractional derivative with respect to time. The presented model was used to reconstruct the thermal conductivity coefficient, heat transfer coefficient, initial condition and order of fractional derivative in the fractional heat conduction inverse problem. Additional information for the inverse problem was the temperature measurements obtained from porous aluminum. In this paper, the authors used a finite difference method to solve direct problems and the Real Ant Colony Optimization algorithm to find a minimum of certain functional (solve the inverse problem). Finally, the authors present the temperature values computed from the model and compare them with the measured data from real objects. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering)
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283 KiB
Article
Series Solution of the Pantograph Equation and Its Properties
Fractal Fract. 2017, 1(1), 16; https://doi.org/10.3390/fractalfract1010016 - 08 Dec 2017
Cited by 12 | Viewed by 2979
Abstract
In this paper, we discuss the classical pantograph equation and its generalizations to include fractional order and the higher order case. The special functions are obtained from the series solution of these equations. We study different properties of these special functions and establish [...] Read more.
In this paper, we discuss the classical pantograph equation and its generalizations to include fractional order and the higher order case. The special functions are obtained from the series solution of these equations. We study different properties of these special functions and establish the relation with other functions. Further, we discuss some contiguous relations for these special functions. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering)
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290 KiB
Article
Some Nonlocal Operators in the First Heisenberg Group
Fractal Fract. 2017, 1(1), 15; https://doi.org/10.3390/fractalfract1010015 - 27 Nov 2017
Cited by 7 | Viewed by 2563
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
679 KiB
Article
A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility
Fractal Fract. 2017, 1(1), 14; https://doi.org/10.3390/fractalfract1010014 - 25 Nov 2017
Cited by 7 | Viewed by 4646
Abstract
In the option pricing literature, it is well known that (i) the decrease in the smile amplitude is much slower than the standard stochastic volatility models and (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with [...] Read more.
In the option pricing literature, it is well known that (i) the decrease in the smile amplitude is much slower than the standard stochastic volatility models and (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with the exponent close to zero. These stylized facts cannot be captured by standard models, and while (i) has been explained by using a fractional volatility model with Hurst index $H > 1 / 2$ , (ii) is proven to be satisfied by a rough volatility model with $H < 1 / 2$ under a risk-neutral measure. This paper provides a solution to this fractional puzzle in the implied volatility. Namely, we construct a two-factor fractional volatility model and develop an approximation formula for European option prices. It is shown through numerical examples that our model can resolve the fractional puzzle, when the correlations between the underlying asset process and the factors of rough volatility and persistence belong to a certain range. More specifically, depending on the three correlation values, the implied volatility surface is classified into four types: (1) the roughness exists, but the persistence does not; (2) the persistence exists, but the roughness does not; (3) both the roughness and the persistence exist; and (4) neither the roughness nor the persistence exist. Full article
(This article belongs to the Special Issue Fractional Calculus in Economics and Finance)
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3205 KiB
Article
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 - 21 Nov 2017
Cited by 21 | Viewed by 3377
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
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24630 KiB
Article
Fractal Simulation of Flocculation Processes Using a Diffusion-Limited Aggregation Model
Fractal Fract. 2017, 1(1), 12; https://doi.org/10.3390/fractalfract1010012 - 18 Nov 2017
Cited by 15 | Viewed by 5534
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 [...] Read more.
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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238 KiB
Article
A Fractional-Order Infectivity and Recovery SIR Model
Fractal Fract. 2017, 1(1), 11; https://doi.org/10.3390/fractalfract1010011 - 17 Nov 2017
Cited by 26 | Viewed by 4057
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
289 KiB
Article
From Circular to Bessel Functions: A Transition through the Umbral Method
Fractal Fract. 2017, 1(1), 9; https://doi.org/10.3390/fractalfract1010009 - 08 Nov 2017
Cited by 2 | Viewed by 2776
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
1290 KiB
Article
The Fractal Nature of an Approximate Prime Counting Function
Fractal Fract. 2017, 1(1), 10; https://doi.org/10.3390/fractalfract1010010 - 08 Nov 2017
Cited by 3 | Viewed by 5091
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 [...] Read more.
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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19497 KiB
Article
Fractional Divergence of Probability Densities
Fractal Fract. 2017, 1(1), 8; https://doi.org/10.3390/fractalfract1010008 - 25 Oct 2017
Cited by 4 | Viewed by 4875
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 [...] Read more.
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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392 KiB
Article
Stokes’ First Problem for Viscoelastic Fluids with a Fractional Maxwell Model
Fractal Fract. 2017, 1(1), 7; https://doi.org/10.3390/fractalfract1010007 - 24 Oct 2017
Cited by 10 | Viewed by 3174
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 [...] Read more.
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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277 KiB
Article
Exact Discretization of an Economic Accelerator and Multiplier with Memory
Fractal Fract. 2017, 1(1), 6; https://doi.org/10.3390/fractalfract1010006 - 11 Sep 2017
Cited by 16 | Viewed by 2952
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 [...] Read more.
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
284 KiB
Article
Dynamics and Stability Results for Hilfer Fractional Type Thermistor Problem
Fractal Fract. 2017, 1(1), 5; https://doi.org/10.3390/fractalfract1010005 - 09 Sep 2017
Cited by 15 | Viewed by 3158
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)
2717 KiB
Article
A Fractional Complex Permittivity Model of Media with Dielectric Relaxation
Fractal Fract. 2017, 1(1), 4; https://doi.org/10.3390/fractalfract1010004 - 29 Aug 2017
Cited by 5 | Viewed by 6758
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
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285 KiB
Article
Which Derivative?
Fractal Fract. 2017, 1(1), 3; https://doi.org/10.3390/fractalfract1010003 - 25 Jul 2017
Cited by 68 | Viewed by 4647
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 [...] Read more.
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
236 KiB
Article
Fractional Definite Integral
Fractal Fract. 2017, 1(1), 2; https://doi.org/10.3390/fractalfract1010002 - 02 Jul 2017
Cited by 15 | Viewed by 4213
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 R2 [...] Read more.
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
370 KiB
Editorial
Fractal and Fractional
Fractal Fract. 2017, 1(1), 1; https://doi.org/10.3390/fractalfract1010001 - 26 Mar 2017
Cited by 7 | Viewed by 5048
Abstract
Fractal and Fractional are two words referring to some characteristics and fundamental problems which arise in all fields of science and technology. [...]