Fractal and Fractional doi: 10.3390/fractalfract3010011

Authors: Alireza Khalili Golmankhaneh

In this paper, fractal stochastic Langevin equations are suggested, providing a mathematical model for random walks on the middle- &tau; Cantor set. The fractal mean square displacement of different random walks on the middle- &tau; Cantor set are presented. Fractal under-damped and over-damped Langevin equations, fractal scaled Brownian motion, and ultra-slow fractal scaled Brownian motion are suggested and the corresponding fractal mean square displacements are obtained. The results are plotted to show the details.

]]>Fractal and Fractional doi: 10.3390/fractalfract3010010

Authors: Khalida Inayat Noor

In this paper, we use concepts of q-calculus to introduce a certain type of q-difference operator, and using it define some subclasses of analytic functions. Inclusion relations, coefficient result, and some other interesting properties of these classes are studied.

]]>Fractal and Fractional doi: 10.3390/fractalfract3010009

Authors: D. G. Prakasha P. Veeresha Haci Mehmet Baskonus

In this paper, the approximated analytical solution for the fractional Swift&ndash;Hohenberg (S&ndash;H) equation has been investigated with the help of the residual power series method (RPSM). To ensure the applicability and efficiency of the proposed technique, we consider a non-linear fractional order Swift&ndash;Hohenberg equation in the presence and absence of dispersive terms. The effect of bifurcation and dispersive parameters with physical importance on the probability density function for distinct fractional Brownian and standard motions are studied and presented through plots. The results obtained show that the proposed technique is simple to implement and very effective for analyzing the complex problems that arise in connected areas of science and technology.

]]>Fractal and Fractional doi: 10.3390/fractalfract3010008

Authors: Fakhri Khajvand Jafari Mohammad Sadegh Asgari Amir Pishkoo

The major problem in the process of mixing fluids (for instance liquid-liquid mixers) is turbulence, which is the outcome of the function of the equipment (engine). Fractal mixing is an alternative method that has symmetry and is predictable. Therefore, fractal structures and fractal reactors find importance. Using F &alpha; -fractal calculus, in this paper, we derive exact F &alpha; -differential forms of an ideal gas. Depending on the dimensionality of space, we should first obtain the integral staircase function and mass function of our geometry. When gases expand inside the fractal structure because of changes from the i + 1 iteration to the i iteration, in fact, we are faced with fluid mixing inside our fractal structure, which can be described by physical quantities P, V, and T. Finally, for the ideal gas equation, we calculate volume expansivity and isothermal compressibility.

]]>Fractal and Fractional doi: 10.3390/fractalfract3010007

Authors: Chi Wah Kok Wing Shan Tam

This paper reviews the implementation of fractal based image interpolation, the associated visual artifacts of the interpolated images, and various techniques, including novel contributions, that alleviate these awkward visual artifacts to achieve visually pleasant interpolated image. The fractal interpolation methods considered in this paper are based on the plain Iterative Function System (IFS) in spatial domain without additional transformation, where we believe that the benefits of additional transformation can be added onto the presented study without complication. Simulation results are presented to demonstrate the discussed techniques, together with the pros and cons of each techniques. Finally, a novel spatial domain interleave layer has been proposed to add to the IFS image system for improving the performance of the system from image zooming to interpolation with the preservation of the pixel intensity from the original low resolution image.

]]>Fractal and Fractional doi: 10.3390/fractalfract3010006

Authors: Vance Blankers Tristan Rendfrey Aaron Shukert Patrick D. Shipman

Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form x + &tau; y for x , y &isin; R , and &tau; 2 = 1 but &tau; &ne; &plusmn; 1 , are the natural number system in which to encode geometric properties of the Minkowski space R 1 , 1 . We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets.

]]>Fractal and Fractional doi: 10.3390/fractalfract3010005

Authors: Rizwan S. Badar Khalida Inayat Noor

Let f be analytic in open unit disc E = { z : | z | &lt; 1 } with f ( 0 ) = 0 and f &prime; ( 0 ) = 1 . The q-derivative of f is defined by: D q f ( z ) = f ( z ) &minus; f ( q z ) ( 1 &minus; q ) z , q &isin; ( 0 , 1 ) , z &isin; B &minus; { 0 } , where B is a q-geometric subset of C . Using operator D q , q-analogue class k &minus; U M q ( &alpha; , &beta; ) , k-uniformly Mocanu functions are defined as: For k = 0 and q &rarr; 1 &minus; , k &minus; reduces to M ( &alpha; ) of Mocanu functions. Subordination is used to investigate many important properties of these functions. Several interesting results are derived as special cases.

]]>Fractal and Fractional doi: 10.3390/fractalfract3010004

Authors: Dimiter Prodanov

The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions are given and the conditions under which these function can be represented by simpler functions are demonstrated. The connection with generalized Erd&eacute;lyi-Kober fractional differential and integral operators is demonstrated and discussed.

]]>Fractal and Fractional doi: 10.3390/fractalfract3010003

Authors: Michael Ioelovich

In this research, the fractal structure of beads of different sizes obtained by the spray-drying of aqueous dispersions of microcrystalline cellulose (MCC) was studied. These beads were formed as a result of the aggregation of rod-shaped cellulose nanocrystalline particles (CNP). It was found that increasing the average radius (R) of the formed MCC beads resulted in increased specific pore volume (P) and reduced apparent density (&rho;). The dependences of P and &rho; on the scale factor (R/r) can be expressed by power-law equations: P = Po (R/r)E&minus;Dp and &rho; = d (R/r)Dd&minus;E, where the fractal dimensions Dp = 2.887 and Dd = 2.986 are close to the Euclidean dimension E = 3 for three-dimensional space; r = 3 nm is the radius of the cellulose nanocrystalline particles, Po = 0.03 cm3/g is the specific pore volume, and d = 1.585 g/cm3 is the true density (specific gravity) of the CNP, respectively. With the increase in the size of the formed MCC beads, the order in the packing of the beads was distorted, conforming to theory of the diffusion-limited aggregation process.

]]>Fractal and Fractional doi: 10.3390/fractalfract3010002

Authors: Fractal Fract Editorial Office

Rigorous peer-review is the corner-stone of high-quality academic publishing [...]

]]>Fractal and Fractional doi: 10.3390/fractalfract3010001

Authors: Dimiter Prodanov

This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine&rsquo;s complex representation is demonstrated. For both real and complex integrals, the regularized representation can be expressed in terms of the two-parameter Mittag-Leffler function. Reference numerical implementations in the Computer Algebra System Maxima are provided.

]]>Fractal and Fractional doi: 10.3390/fractalfract2040030

Authors: Alireza Khalili Golmankhaneh Arran Fernandez

In this manuscript, integrals and derivatives of functions on Cantor tartan spaces are defined. The generalisation of standard calculus, which is called F &eta; -calculus, is utilised to obtain definitions of the integral and derivative of functions on Cantor tartan spaces of different dimensions. Differential equations involving the new derivatives are solved. Illustrative examples are presented to check the details.

]]>Fractal and Fractional doi: 10.3390/fractalfract2040029

Authors: Annamalai Anguraj K. Ramkumar

The objective of this paper is to analyze the approximate controllability of a semilinear stochastic integrodifferential system with nonlocal conditions in Hilbert spaces. The nonlocal initial condition is a generalization of the classical initial condition and is motivated by physical phenomena. The results are obtained by using Sadovskii&rsquo;s fixed point theorem. At the end, an example is given to show the effectiveness of the result.

]]>Fractal and Fractional doi: 10.3390/fractalfract2040028

Authors: Vsevolod Bohaienko Volodymyr Bulavatsky

Within the framework of a new mathematical model of convective diffusion with the k-Caputo derivative, we simulate the dynamics of anomalous soluble substances migration under the conditions of two-dimensional steady-state plane-vertical filtration with a free surface. As a corresponding filtration scheme, we consider the scheme for the spread of pollution from rivers, canals, or storages of industrial wastes. On the base of a locally one-dimensional finite-difference scheme, we develop a numerical method for obtaining solutions of boundary value problem for fractional differential equation with k-Caputo derivative with respect to the time variable that describes the convective diffusion of salt solution. The results of numerical experiments on modeling the dynamics of the considered process are presented. The results that show an existence of a time lag in the process of diffusion field formation are presented.

]]>Fractal and Fractional doi: 10.3390/fractalfract2040027

Authors: Todd J. Freeborn Bo Fu

Bioimpedance, or the electrical impedance of biological tissues, describes the passive electrical properties of these materials. To simplify bioimpedance datasets, fractional-order equivalent circuit presentations are often used, with the Cole-impedance model being one of the most widely used fractional-order circuits for this purpose. In this work, bioimpedance measurements from 10 kHz to 100 kHz were collected from participants biceps tissues immediately prior and immediately post completion of a fatiguing exercise protocol. The Cole-impedance parameters that best fit these datasets were determined using numerical optimization procedures, with relative errors of within approximately &plusmn; 0.5 % and &plusmn; 2 % for the simulated resistance and reactance compared to the experimental data. Comparison between the pre and post fatigue Cole-impedance parameters shows that the R &infin; , R 1 , and f p components exhibited statistically significant mean differences as a result of the fatigue induced changes in the study participants.

]]>Fractal and Fractional doi: 10.3390/fractalfract2040026

Authors: Michel L. Lapidus Hùng Lũ’ Machiel Van Frankenhuijsen

The theory of complex dimensions describes the oscillations in the geometry (spectra and dynamics) of fractal strings. Such geometric oscillations can be seen most clearly in the explicit volume formula for the tubular neighborhoods of a p-adic fractal string L p , expressed in terms of the underlying complex dimensions. The general fractal tube formula obtained in this paper is illustrated by several examples, including the nonarchimedean Cantor and Euler strings. Moreover, we show that the Minkowski dimension of a p-adic fractal string coincides with the abscissa of convergence of the geometric zeta function associated with the string, as well as with the asymptotic growth rate of the corresponding geometric counting function. The proof of this new result can be applied to both real and p-adic fractal strings and hence, yields a unifying explanation of a key result in the theory of complex dimensions for fractal strings, even in the archimedean (or real) case.

]]>Fractal and Fractional doi: 10.3390/fractalfract2040025

Authors: Muhammad Sahir

In this paper, we present a generalization of Radon&rsquo;s inequality on dynamic time scale calculus, which is widely studied by many authors and an intrinsic inequality. Further, we present the classical Bergstr&ouml;m&rsquo;s inequality and refinement of Nesbitt&rsquo;s inequality unified on dynamic time scale calculus in extended form.

]]>Fractal and Fractional doi: 10.3390/fractalfract2040024

Authors: Ming Li

The highlight presented in this short article is about the power laws with respect to fractional capacitance and fractional inductance in terms of frequency.

]]>Fractal and Fractional doi: 10.3390/fractalfract2040023

Authors: Vasily E. Tarasov

The memory means an existence of output (response, endogenous variable) at the present time that depends on the history of the change of the input (impact, exogenous variable) on a finite (or infinite) time interval. The memory can be described by the function that is called the memory function, which is a kernel of the integro-differential operator. The main purpose of the paper is to answer the question of the possibility of using the fractional calculus, when the memory function does not have a power-law form. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla (TRB) form for the memory function, we represent the integro-differential equations with memory functions by fractional integral and differential equations with derivatives and integrals of non-integer orders. This allows us to describe general economic dynamics with memory by the methods of fractional calculus. We prove that equation of the generalized accelerator with the TRB memory function can be represented by as a composition of actions of the accelerator with simplest power-law memory and the multi-parametric power-law multiplier. As an example of application of the suggested approach, we consider a generalization of the Harrod-Domar growth model with continuous time.

]]>Fractal and Fractional doi: 10.3390/fractalfract2030022

Authors: Kamal Ait Touchent Zakia Hammouch Toufik Mekkaoui Fethi B. M. Belgacem

In the present paper, the explicit solutions of some local fractional partial differential equations are constructed through the integration of local fractional Sumudu transform and homotopy perturbation such as local fractional dissipative and damped wave equations. The convergence aspect of this technique is also discussed and presented. The obtained results prove that the employed method is very simple and effective for treating analytically various kinds of problems comprising local fractional derivatives.

]]>Fractal and Fractional doi: 10.3390/fractalfract2030021

Authors: Guy Joseph Eyebe Gambo Betchewe Alidou Mohamadou Timoleon Crepin Kofane

In the present study, the nonlinear vibration of a nanobeam resting on the fractional order viscoelastic Winkler&ndash;Pasternak foundation is studied using nonlocal elasticity theory. The D&rsquo;Alembert principle is used to derive the governing equation and the associated boundary conditions. The approximate analytical solution is obtained by applying the multiple scales method. A detailed parametric study is conducted, and the effects of the variation of different parameters belonging to the application problems on the system are calculated numerically and depicted. We remark that the order and the coefficient of the fractional derivative have a significant effect on the natural frequency and the amplitude of vibrations.

]]>Fractal and Fractional doi: 10.3390/fractalfract2030020

Authors: Maike A. F. Dos Santos

The investigation of diffusive process in nature presents a complexity associated with memory effects. Thereby, it is necessary new mathematical models to involve memory concept in diffusion. In the following, I approach the continuous time random walks in the context of generalised diffusion equations. To do this, I investigate the diffusion equation with exponential and Mittag-Leffler memory-kernels in the context of Caputo-Fabrizio and Atangana-Baleanu fractional operators on Caputo sense. Thus, exact expressions for the probability distributions are obtained, in that non-Gaussian distributions emerge. I connect the distribution obtained with a rich class of diffusive behaviour. Moreover, I propose a generalised model to describe the random walk process with resetting on memory kernel context.

]]>Fractal and Fractional doi: 10.3390/fractalfract2020019

Authors: Carlo Cattani Renato Spigler

n/a

]]>Fractal and Fractional doi: 10.3390/fractalfract2020018

Authors: Yusuf Zakariya Yusuf Afolabi Rahmatullah Nuruddeen Ibrahim Sarumi

In this paper, we provide solutions to the general fractional Caputo-type differential equation models for the dynamics of a sphere immersed in an incompressible viscous fluid and oscillatory process with fractional damping using Laplace transform method. We study the effects of fixing one of the fractional indices while varying the other as particular examples. We conclude this article by explaining the dynamics of the solutions of the models.

]]>Fractal and Fractional doi: 10.3390/fractalfract2020017

Authors: Ndolane Sene

This paper deals with a Lyapunov characterization of the conditional Mittag-Leffler stability and conditional asymptotic stability of the fractional nonlinear systems with exogenous input. A particular class of the fractional nonlinear systems is studied. The paper contributes to giving in particular the Lyapunov characterization of fractional linear systems and fractional bilinear systems with exogenous input.

]]>Fractal and Fractional doi: 10.3390/fractalfract2020016

Authors: Jordan Hristov

n/a

]]>Fractal and Fractional doi: 10.3390/fractalfract2010015

Authors: Jean-Philippe Aguilar Jan Korbel

In this paper, we focus on option pricing models based on space-time fractional diffusion. We briefly revise recent results which show that the option price can be represented in the terms of rapidly converging double-series and apply these results to the data from real markets. We focus on estimation of model parameters from the market data and estimation of implied volatility within the space-time fractional option pricing models.

]]>Fractal and Fractional doi: 10.3390/fractalfract2010014

Authors: Moreno Concezzi Renato Spigler

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract2010013

Authors: Francesca Pitolli

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract2010012

Authors: Martin Ostoja-Starzewski Jun Zhang

The question addressed by this paper is tackled through a continuum micromechanics model of a 2D random checkerboard, in which one phase is linear elastic and another linear viscoelastic of integer-order. The spatial homogeneity and ergodicity of the material statistics justify homogenization in the vein of the Hill–Mandel condition for viscoelastic media. Thus, uniform kinematic- or traction-controlled boundary conditions, applied to sufficiently large domains, provide macroscopic (RVE level) responses. With computational mechanics, this strategy is applied over the entire range of the relative content of both phases. Setting the volume fraction of either the elastic phase or the viscoelastic phase at the critical value (≃0.59) results in fractal patterns of site-percolation. Extensive simulations of boundary value problems show that, for a viscoelastic composite having such a fractal structure, the integer (not fractional) calculus model is adequate. In other words, the spatial randomness of the composite material—even in the fractal regime—is not necessarily the cause of the fractional order viscoelasticity.

]]>Fractal and Fractional doi: 10.3390/fractalfract2010011

Authors: Ervin Lenzi Andrea Ryba Marcelo Lenzi

A fractional-calculus-based model is used to analyze the data obtained from the image analysis of mixtures of olive and soybean oil, which were quantified with the RGB color system. The model consists in a linear fractional differential equation, containing one fractional derivative of order α and an additional term multiplied by a parameter k. Using a hybrid parameter estimation scheme (genetic algorithm and a simplex-based algorithm), the model parameters were estimated as k = 3.42 ± 0.12 and α = 1.196 ± 0.027, while a correlation coefficient value of 0.997 was obtained. For the sake of comparison, parameter α was set equal to 1 and an integer order model was also studied, resulting in a one-parameter model with k = 3.11 ± 0.28. Joint confidence regions are calculated for the fractional order model, showing that the derivative order is statistically different from 1. Finally, an independent validation sample of color component B equal to 96 obtained from a sample with olive oil mass fraction equal to 0.25 is used for prediction purposes. The fractional model predicted the color B value equal to 93.1 ± 6.6.

]]>Fractal and Fractional doi: 10.3390/fractalfract2010010

Authors: José Francisco Gómez-Aguilar Abdon Atangana

This paper considers the Freedman model using the Liouville–Caputo fractional-order derivative and the fractional-order derivative with Mittag–Leffler kernel in the Liouville–Caputo sense. Alternative solutions via Laplace transform, Sumudu–Picard and Adams–Moulton rules were obtained. We prove the uniqueness and existence of the solutions for the alternative model. Numerical simulations for the prediction and interaction between a unilingual and a bilingual population were obtained for different values of the fractional order.

]]>Fractal and Fractional doi: 10.3390/fractalfract2010009

Authors: Corina S. Drapaca

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract2010008

Authors: Giampietro Casasanta Roberto Garra

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract2010007

Authors: Fractal and Fractional Editorial Office

Peer review is an essential part in the publication process, ensuring that Fractal and Fractional maintains high quality standards for its published papers. In 2017, a total of 17 papers were published in the journal. Thanks to the cooperation of our reviewers, the median time to first decision was 14 days and the median time to publication was 24 days. The editors would like to express their sincere gratitude to the reviewers for their time and dedication in 2017.

]]>Fractal and Fractional doi: 10.3390/fractalfract2010006

Authors: Silvia Vitali Francesco Mainardi Gastone Castellani

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract2010005

Authors: Xavier Moreau Roy Abi Zeid Daou Fady Christophy

The control of thermal interfaces has gained importance in recent years because of the high cost of heating and cooling materials in many applications. Thus, the main focus in this work is to compare the second and third generations of the CRONE controller (French acronym of Commande Robuste d’Ordre Non Entier), which means a non-integer order robust controller, and to synthesize a robust controller that can fit several types of systems. For this study, the plant consists of a rectangular homogeneous bar of length L, where the heating element in applied on one boundary, and a temperature sensor is placed at distance x from that boundary (x is considered very small with respect to L). The type of material used is the third parameter, which may help in analyzing the robustness of the synthesized controller. The originality of this work resides in controlling a non-integer plant using a fractional order controller, as, so far, almost all of the systems where the CRONE controller has been implemented were of integer order. Three case studies were defined in order to show how and where each CRONE generation controller can be applied. These case studies were chosen in such a way as to influence the asymptotic behavior of the open-loop transfer function in the Black–Nichols diagram in order to point out the importance of respecting the conditions of the applications of the CRONE generations. Results show that the second generation performs well when the parametric uncertainties do not affect the phase of the plant, whereas the third generation is the most robust, even when both the phase and the gain variations are encountered. However, it also has some limitations, especially when the temperature to be controlled is far from the interface when the density of flux is applied.

]]>Fractal and Fractional doi: 10.3390/fractalfract2010004

Authors: Dimiter Prodanov

Singular functions and, in general, Hölder functions represent conceptual models of nonlinear physical phenomena. The purpose of this survey is to demonstrate the applicability of fractional velocities as tools to characterize Hölder and singular functions, in particular. Fractional velocities are defined as limits of the difference quotients of a fractional power and they generalize the local notion of a derivative. On the other hand, their properties contrast some of the usual properties of derivatives. One of the most peculiar properties of these operators is that the set of their non trivial values is disconnected. This can be used for example to model instantaneous interactions, for example Langevin dynamics. Examples are given by the De Rham and Neidinger’s singular functions, represented by limits of iterative function systems. Finally, the conditions for equivalence with the Kolwankar-Gangal local fractional derivative are investigated.

]]>Fractal and Fractional doi: 10.3390/fractalfract2010003

Authors: Mehmet Yavuz Necati Özdemir

Recently, fractional differential equations (FDEs) have attracted much more attention in modeling real-life problems. Since most FDEs do not have exact solutions, numerical solution methods are used commonly. Therefore, in this study, we have demonstrated a novel approximate-analytical solution method, which is called the Laplace homotopy analysis method (LHAM) using the Caputo–Fabrizio (CF) fractional derivative operator. The recommended method is obtained by combining Laplace transform (LT) and the homotopy analysis method (HAM). We have used the fractional operator suggested by Caputo and Fabrizio in 2015 based on the exponential kernel. We have considered the LHAM with this derivative in order to obtain the solutions of the fractional Black–Scholes equations (FBSEs) with the initial conditions. In addition to this, the convergence and stability analysis of the model have been constructed. According to the results of this study, it can be concluded that the LHAM in the sense of the CF fractional derivative is an effective and accurate method, which is computable in the series easily in a short time.

]]>Fractal and Fractional doi: 10.3390/fractalfract2010002

Authors: Dimitris Vartziotis Doris Bohnet

We study the convergence of the parameter family of series: V α , β ( t ) = ∑ p p − α exp ( 2 π i p β t ) , α , β ∈ R &gt; 0 , t ∈ [ 0 , 1 ) defined over prime numbers p and, subsequently, their differentiability properties. The visible fractal nature of the graphs as a function of α , β is analyzed in terms of Hölder continuity, self-similarity and fractal dimension, backed with numerical results. Although this series is not a lacunary series, it has properties in common, such that we also discuss the link of this series with random walks and, consequently, explore its random properties numerically.

]]>Fractal and Fractional doi: 10.3390/fractalfract2010001

Authors: Salvador Jiménez David Usero Luis Vázquez Maria Velasco

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010017

Authors: Rafał Brociek Damian Słota Mariusz Król Grzegorz Matula Waldemar Kwaśny

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010016

Authors: Sachin Bhalekar Jayvant Patade

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010015

Authors: Fausto Ferrari

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010014

Authors: Hideharu Funahashi Masaaki Kijima

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 &gt; 1 / 2 , (ii) is proven to be satisfied by a rough volatility model with H &lt; 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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010013

Authors: Michele Caputo Mauro Fabrizio

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010012

Authors: Dongjing Liu Weiguo Zhou Xu Song Zumin Qiu

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010011

Authors: Christopher N. Angstmann Bruce I. Henry Anna V. McGann

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010010

Authors: Dimitris Vartziotis Joachim Wipper

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010009

Authors: Giuseppe Dattoli Emanuele Di Palma Silvia Licciardi Elio Sabia

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010008

Authors: Aris Alexopoulos

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010007

Authors: Emilia Bazhlekova Ivan Bazhlekov

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010006

Authors: Valentina Tarasova Vasily Tarasov

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010005

Authors: D. Vivek K. Kanagarajan Seenith Sivasundaram

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010004

Authors: Armando Ciancio Bruno Flora

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010003

Authors: Manuel Ortigueira José Machado

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010002

Authors: Manuel Ortigueira José Machado

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.

]]>Fractal and Fractional doi: 10.3390/fractalfract1010001

Authors: Carlo Cattani

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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