Fractal and Fractional — Open Access Journal

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. Full article

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 $\pm 0.5\%$ and $\pm 2\%$ for the simulated resistance and reactance compared to the experimental data. Comparison between the pre and post fatigue Cole-impedance parameters shows that the $R_{\infty }$, $R_1$, and $f_p$ components exhibited statistically significant mean differences as a result of the fatigue induced changes in the study participants. Full article

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 ${\mathcal{L}}_p$, 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. Full article

In this paper, we present a generalization of Radon’s inequality on dynamic time scale calculus, which is widely studied by many authors and an intrinsic inequality. Further, we present the classical Bergström’s inequality and refinement of Nesbitt’s inequality unified on dynamic time scale calculus in extended form. Full article

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. Full article

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. Full article

In the present study, the nonlinear vibration of a nanobeam resting on the fractional order viscoelastic Winkler–Pasternak foundation is studied using nonlocal elasticity theory. The D’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. Full article

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. Full article

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. Full article

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. Full article

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