Fractal Fract., Volume 4, Issue 4 (December 2020) – 13 articles

To study the peculiarities of anomalous consolidation processes in saturated porous (soil) media in the conditions of salt transfer, we present a new mathematical model developed on the base of the fractional-fractal approach that allows considering temporal non-locality of transfer processes in media of fractal structure. For the case of the finite thickness domain with permeable boundaries, a finite-difference technique for numerical solution of the corresponding one-dimensional non-linear boundary value problem is developed. The paper also presents a fractional-fractal model of a filtration-consolidation process in clay soils of fractal structure saturated with salt solutions. An analytical solution is found for the corresponding one-dimensional boundary value problem in the domain of finite thickness with permeable upper and impermeable lower boundaries. Full article

The issue of robust admissibility and control for singular fractional-order systems (FOSs) with polytopic uncertainties is investigated in this paper. Firstly, a new method based on linear matrix inequalities (LMIs) is presented to solve the admissibility problems of uncertain linear systems. Then, a solid criterion of robust admissibility and a corresponding state feedback controller are derived, which overcome the conservatism of the existing results. Finally, for the sake of demonstrating the validity of proposed results, some relevant examples are provided. Full article

A numerical method for solving fractional partial differential equations (fPDEs) of the diffusion and reaction–diffusion type, subject to Dirichlet boundary data, in three dimensions is developed. Such fPDEs may describe fluid flows through porous media better than classical diffusion equations. This is a new, fractional version of the Alternating Direction Implicit (ADI) method, where the source term is balanced, in that its effect is split in the three space directions, and it may be relevant, especially in the case of anisotropy. The method is unconditionally stable, second-order in space, and third-order in time. A strategy is devised in order to improve its speed of convergence by means of an extrapolation method that is coupled to the PageRank algorithm. Some numerical examples are given. Full article

Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated. Full article

This paper presents the optimal rational approximation of (1+α) order Butterworth filter, where α ∊ (0,1) under the continued fraction expansion framework, by employing a new cost function. Two simple techniques based on the constrained optimization and the optimal pole-zero placements are proposed to model the magnitude-frequency response of the fractional-order lowpass Butterworth filter (FOLBF). The third-order FOLBF approximants achieve good agreement to the ideal characteristic for six decades of design bandwidth. Circuit realization using the current feedback operational amplifier is presented, and the modelling efficacy is validated in the OrCAD PSPICE platform. Full article

The Cole–Davidson function is an efficient tool for describing the tissue behavior, but the conventional methods of approximation are not applicable due the form of this function. In order to overcome this problem, a novel scheme for approximating the Cole–Davidson function, based on the utilization of a curve fitting procedure offered by the MATLAB software, is introduced in this work. The derived rational transfer function is implemented using the conventional Cauer and Foster RC networks. As an application example, the impedance model of the membrane of mesophyll cells is realized, with simulation results verifying the validity of the introduced procedure. Full article

In this article, a new reproducing kernel approach is developed for obtaining a numerical solution of multi-order fractional nonlinear three-point boundary value problems. This approach is based on a reproducing kernel, which is constructed by shifted Legendre polynomials (L-RKM). In the considered problem, fractional derivatives with respect to $\alpha$ and $\beta$ are defined in the Caputo sense. This method has been applied to some examples that have exact solutions. In order to show the robustness of the proposed method, some examples are solved and numerical results are given in tabulated forms. Full article

A random walk on a two dimensional square in $R^2$ space with a hidden absorbing fractal set $F_{\mu }$ is considered. This search-like problem is treated in the framework of a diffusion–reaction equation, when an absorbing term is included inside a Fokker–Planck equation as a reaction term. This macroscopic approach for the 2D transport in the $R^2$ space corresponds to the comb geometry, when the random walk consists of 1D movements in the x and y directions, respectively, as a direct-Cartesian product of the 1D movements. The main value in task is the first arrival time distribution (FATD) to sink points of the fractal set, where travelling particles are absorbed. Analytical expression for the FATD is obtained in the subdiffusive regime for both the fractal set of sinks and for a single sink. Full article

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs. Full article

This paper focuses on the sliding mode control (SMC) problem for a class of uncertain singular fractional order systems (SFOSs). The uncertainties occur in both state and derivative matrices. A radial basis function (RBF) neural network strategy was utilized to estimate the nonlinear terms of SFOSs. Firstly, by expanding the dimension of the SFOS, a novel sliding surface was constructed. A necessary and sufficient condition was given to ensure the admissibility of the SFOS while the system state moves on the sliding surface. The obtained results are linear matrix inequalities (LMIs), which are more general than the existing research. Then, the adaptive control law based on the RBF neural network was organized to guarantee that the SFOS reaches the sliding surface in a finite time. Finally, a simulation example is proposed to verify the validity of the designed procedures. Full article

In this work we present three new models of the fractal-fractional Ebola virus. We investigate the numerical solutions of the fractal-fractional Ebola virus in the sense of three different kernels based on the power law, the exponential decay and the generalized Mittag-Leffler function by using the concepts of the fractal differentiation and fractional differentiation. These operators have two parameters: The first parameter $\rho$ is considered as the fractal dimension and the second parameter k is the fractional order. We evaluate the numerical solutions of the fractal-fractional Ebola virus for these operators with the theory of fractional calculus and the help of the Lagrange polynomial functions. In the case of $\rho =k=1$, all of the numerical solutions based on the power kernel, the exponential kernel and the generalized Mittag-Leffler kernel are found to be close to each other and, therefore, one of the kernels is compared with such numerical methods as the finite difference methods. This has led to an excellent agreement. For the effect of fractal-fractional on the behavior, we study the numerical solutions for different values of $\rho$ and $k.$ All calculations in this work are accomplished by using the Mathematica package. Full article

Our present investigation is mainly based on the k-hypergeometric functions which are constructed by making use of the Pochhammer k-symbol in Diaz et al. 2007, which are one of the vital generalizations of hypergeometric functions. In this study, we focus on the k-analogues of $F_1$Appell function introduced by Mubeen et al. 2015 and the k-generalizations of F₂ and F₃ Appell functions indicated in Kıymaz et al. 2017. we present some important transformation formulas and some reduction formulas which show close relation not only with k-Appell functions but also with k-hypergeometric functions. Employing the theory of Riemann–Liouville k-fractional derivative from Rahman et al. 2020, and using the relations which we consider in this paper, we acquire linear and bilinear generating relations for k-analogue of hypergeometric functions and Appell functions. Full article