Fractal Fract., Volume 2, Issue 1 (March 2018) – 15 articles

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. Full article

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

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

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. Full article

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. Full article

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. Full article

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

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

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. Full article

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

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. Full article

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. Full article

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. Full article

We study the convergence of the parameter family of series: $V_{\alpha ,\beta }(t)={\sum }_p{p}^{-\alpha }\exp (2\pi i{p}^{\beta }t)$ , $\alpha ,\beta \in {\mathbb{R}}_{>0},t\in [0,1)$ defined over prime numbers p and, subsequently, their differentiability properties. The visible fractal nature of the graphs as a function of $\alpha ,\beta$ 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. Full article

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