Fractal and Fractional

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function $\eta \left(z\right)=z^2+\lambda$. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented. Full article

This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order $q\in (1,2]$, ${}_0^c{D}_{0^{+}}^{q}u(t)=\lambda u(t)\oplus f(t,u(t))\oplus B(t)C(t),t\in [0,T]$ with initial conditions $u(0)=u_0,u^{\prime }(0)=u_1.$ Moreover, by using direct analytic methods, the $E_q$–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations. Full article

This article deals with the random sequential adsorption (RSA) of 2D disks of the same size on fractal surfaces with a Hausdorff dimension $1<d<2$. According to the literature and confirmed by numerical simulations in the paper, the high coverage regime exhibits fractional dynamics, i.e., dynamics in $t^{-1/d}$ where d is the fractal dimension of the surface. The main contribution this paper is that it proposes to capture this behavior with a particular class of nonlinear model: a driftless control affine model. Full article

Numerical methods for spectral space-fractional elliptic equations are studied. The boundary value problem is defined in a bounded domain of general geometry, $\Omega \subset {\mathbb{R}}^d$, $d\in \{1,2,3\}$. Assuming that the finite difference method (FDM) or the finite element method (FEM) is applied for discretization in space, the approximate solution is described by the system of linear algebraic equations ${\mathbb{A}}^{\alpha }\mathbf{u}=\mathbf{f}$, $\alpha \in (0,1)$. Although matrix $\mathbb{A}\in {\mathbb{R}}^{N\times N}$ is sparse, symmetric and positive definite (SPD), matrix ${\mathbb{A}}^{\alpha }$ is dense. The recent achievements in the field are determined by methods that reduce the original non-local problem to solving k auxiliary linear systems with sparse SPD matrices that can be expressed as positive diagonal perturbations of $\mathbb{A}$. The present study is in the spirit of the BURA method, based on the best uniform rational approximation $r_{\alpha ,k}\left(t\right)$ of degree k of $t^{\alpha }$ in the interval $[0,1]$. The introduced additive BURA-AR and multiplicative BURA-MR methods follow the observation that the matrices of part of the auxiliary systems possess very different properties. As a result, solution methods with substantially improved computational complexity are developed. In this paper, we present new theoretical characterizations of the BURA parameters, which gives a theoretical justification for the new methods. The theoretical estimates are supported by a set of representative numerical tests. The new theoretical and experimental results raise the question of whether the almost optimal estimate of the computational complexity of the BURA method in the form $O\left(N{log}^{2}N\right)$ can be improved. Full article

In the three-dimensional open rectangular domain, the problem of the identification of the redefinition function for a partial differential equation with Gerasimov–Caputo-type fractional operator, degeneration, and integral form condition is considered in the case of the $0<\alpha \le 1$ order. A positive parameter is present in the mixed derivatives. The solution of this fractional differential equation is studied in the class of regular functions. The Fourier series method is used, and a countable system of ordinary fractional differential equations with degeneration is obtained. The presentation for the redefinition function is obtained using a given additional condition. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proven. Full article

We present a general series representation formula for the local solution of the Bernoulli equation with Caputo fractional derivatives. We then focus on a generalization of the fractional logistic equation and present some related numerical simulations. Full article