Fractal and Fractional

13 pages, 352 KiB

Open AccessArticle

Comb Model: Non-Markovian versus Markovian

by Alexander Iomin, Vicenç Méndez and Werner Horsthemke

Fractal Fract. 2019, 3(4), 54; https://doi.org/10.3390/fractalfract3040054 - 10 Dec 2019

Cited by 10 | Viewed by 2587

Combs are a simple caricature of various types of natural branched structures, which belong to the category of loopless graphs and consist of a backbone and branches. We study two generalizations of comb models and present a generic method to obtain their transport [...] Read more.

Combs are a simple caricature of various types of natural branched structures, which belong to the category of loopless graphs and consist of a backbone and branches. We study two generalizations of comb models and present a generic method to obtain their transport properties. The first is a continuous time random walk on a many dimensional

m + n

comb, where m and n are the dimensions of the backbone and branches, respectively. We observe subdiffusion, ultra-slow diffusion and random localization as a function of n. The second deals with a quantum particle in the

1 + 1

comb. It turns out that the comb geometry leads to a power-law relaxation, described by a wave function in the framework of the Schrödinger equation. Full article

(This article belongs to the Special Issue 2019 Selected Papers from Fractal Fract’s Editorial Board Members)

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7 pages, 274 KiB

Open AccessArticle

A Fractional Measles Model Having Monotonic Real Statistical Data for Constant Transmission Rate of the Disease

by Ricardo Almeida and Sania Qureshi

Fractal Fract. 2019, 3(4), 53; https://doi.org/10.3390/fractalfract3040053 - 21 Nov 2019

Cited by 26 | Viewed by 2914

Abstract

Non-Markovian effects have a vital role in modeling the processes related with natural phenomena such as epidemiology. Various infectious diseases have long-range memory characteristics and, thus, non-local operators are one of the best choices to be used to understand the transmission dynamics of [...] Read more.

Non-Markovian effects have a vital role in modeling the processes related with natural phenomena such as epidemiology. Various infectious diseases have long-range memory characteristics and, thus, non-local operators are one of the best choices to be used to understand the transmission dynamics of such diseases and epidemics. In this paper, we study a fractional order epidemiological model of measles. Some relevant features, such as well-posedness and stability of the underlying Cauchy problem, are considered accompanying the proofs for a locally asymptotically stable equilibrium point for basic reproduction number

R_{0} < 1

, which is most sensitive to the fractional order parameter and to the percentage of vaccination. We show the efficiency of the model through a real life application of the spread of the epidemic in Pakistan, comparing the fractional and classical models, while assuming constant transmission rate of the epidemic with monotonically increasing and decreasing behavior of the infected population. Secondly, the fractional Caputo type model, based upon nonlinear least squares curve fitting technique, is found to have smaller residuals when compared with the classical model. Full article

(This article belongs to the Special Issue 2019 Selected Papers from Fractal Fract’s Editorial Board Members)

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13 pages, 7560 KiB

Open AccessArticle

Dynamic Analysis of the Viscoelastic Pipeline Conveying Fluid with an Improved Variable Fractional Order Model Based on Shifted Legendre Polynomials

by Yuanhui Wang and Yiming Chen

Fractal Fract. 2019, 3(4), 52; https://doi.org/10.3390/fractalfract3040052 - 17 Nov 2019

Cited by 15 | Viewed by 2835

Abstract

Viscoelastic pipeline conveying fluid is analyzed with an improved variable fractional order model for researching its dynamic properties accurately in this study. After introducing the improved model, an involuted variable fractional order, which is an unknown piecewise nonlinear function for analytical solution, an [...] Read more.

Viscoelastic pipeline conveying fluid is analyzed with an improved variable fractional order model for researching its dynamic properties accurately in this study. After introducing the improved model, an involuted variable fractional order, which is an unknown piecewise nonlinear function for analytical solution, an equation is established as the governing equation for the dynamic displacement of the viscoelastic pipeline. In order to solve this class of equations, a numerical method based on shifted Legendre polynomials is presented for the first time. The method is effective and accurate after the numerical example verifying. Numerical results show that how dynamic properties are influenced by internal fluid velocity, force excitation, and variable fractional order through the proposed method. More importantly, the numerical method has shown great potentials for dynamic problems with the high precision model. Full article

(This article belongs to the Special Issue 2019 Selected Papers from Fractal Fract’s Editorial Board Members)

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14 pages, 283 KiB

Open AccessArticle

Multi-Point and Anti-Periodic Conditions for Generalized Langevin Equation with Two Fractional Orders

by Ahmed Salem and Balqees Alghamdi

Fractal Fract. 2019, 3(4), 51; https://doi.org/10.3390/fractalfract3040051 - 8 Nov 2019

Cited by 29 | Viewed by 2670

Abstract

With anti-periodic and a new class of multi-point boundary conditions, we investigate, in this paper, the existence and uniqueness of solutions for the Langevin equation that has Caputo fractional derivatives of two different orders. Existence of solutions is obtained by applying Krasnoselskii–Zabreiko’s and [...] Read more.

With anti-periodic and a new class of multi-point boundary conditions, we investigate, in this paper, the existence and uniqueness of solutions for the Langevin equation that has Caputo fractional derivatives of two different orders. Existence of solutions is obtained by applying Krasnoselskii–Zabreiko’s and the Leray–Schauder fixed point theorems. The Banach contraction mapping principle is used to investigate the uniqueness. Illustrative examples are provided to apply of the fundamental investigations. Full article

16 pages, 322 KiB

Open AccessArticle

Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to h-Manifolds

by Gani Stamov, Anatoliy Martynyuk and Ivanka Stamova

Fractal Fract. 2019, 3(4), 50; https://doi.org/10.3390/fractalfract3040050 - 7 Nov 2019

Cited by 11 | Viewed by 2622

Abstract

In this paper, an impulsive fractional-like system of differential equations is introduced. The notions of practical stability and boundedness with respect to h-manifolds for fractional-like differential equations are generalized to the impulsive case. For the first time in the literature, Lyapunov-like functions [...] Read more.

In this paper, an impulsive fractional-like system of differential equations is introduced. The notions of practical stability and boundedness with respect to h-manifolds for fractional-like differential equations are generalized to the impulsive case. For the first time in the literature, Lyapunov-like functions and their derivatives with respect to impulsive fractional-like systems are defined. As an application, an impulsive fractional-like system of Lotka–Volterra equations is considered and new criteria for practical exponential stability are proposed. In addition, the uncertain case is also investigated. Full article

(This article belongs to the Special Issue 2019 Selected Papers from Fractal Fract’s Editorial Board Members)

9 pages, 5738 KiB

Open AccessArticle

The Unexpected Fractal Signatures in Fibonacci Chains

by Fang Fang, Raymond Aschheim and Klee Irwin

Fractal Fract. 2019, 3(4), 49; https://doi.org/10.3390/fractalfract3040049 - 6 Nov 2019

Cited by 2 | Viewed by 5881

Abstract

In this paper, a new fractal signature possessing the cardioid shape in the Mandelbrot set is presented in the Fourier space of a Fibonacci chain with two lengths, L and S, where

L / S = ϕ

. The corresponding pointwise dimension [...] Read more.

In this paper, a new fractal signature possessing the cardioid shape in the Mandelbrot set is presented in the Fourier space of a Fibonacci chain with two lengths, L and S, where

L / S = ϕ

. The corresponding pointwise dimension is 1.7. Various modifications, such as truncation from the head or tail, scrambling the orders of the sequence and changing the ratio of the L and S, are done on the Fibonacci chain. The resulting patterns in the Fourier space show that that the fractal signature is very sensitive to changes in the Fibonacci order but not to the

L / S

ratio. Full article

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7 pages, 800 KiB

Open AccessArticle

Cabinet of Curiosities: The Interesting Geometry of the Angle β = arccos ((3ϕ − 1)/4)

by Fang Fang, Klee Irwin, Julio Kovacs and Garrett Sadler

Fractal Fract. 2019, 3(4), 48; https://doi.org/10.3390/fractalfract3040048 - 30 Oct 2019

Cited by 4 | Viewed by 3874

Abstract

In this paper, we present the construction of several aggregates of tetrahedra. Each construction is obtained by performing rotations on an initial set of tetrahedra that either (1) contains gaps between adjacent tetrahedra, or (2) exhibits an aperiodic nature. Following this rotation, gaps [...] Read more.

In this paper, we present the construction of several aggregates of tetrahedra. Each construction is obtained by performing rotations on an initial set of tetrahedra that either (1) contains gaps between adjacent tetrahedra, or (2) exhibits an aperiodic nature. Following this rotation, gaps of the former case are “closed” (in the sense that faces of adjacent tetrahedra are brought into contact to form a “face junction”), while translational and rotational symmetries are obtained in the latter case. In all cases, an angular displacement of

β = arccos (3 ϕ - 1) / 4

(or a closely related angle), where

ϕ = (1 + \sqrt{5}) / 2

is the golden ratio, is observed between faces of a junction. Additionally, the overall number of plane classes, defined as the number of distinct facial orientations in the collection of tetrahedra, is reduced following the transformation. Finally, we present several “curiosities” involving the structures discussed here with the goal of inspiring the reader’s interest in constructions of this nature and their attending, interesting properties. Full article

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11 pages, 1172 KiB

Open AccessArticle

Tempered Fractional Equations for Quantum Transport in Mesoscopic One-Dimensional Systems with Fractal Disorder

by Renat T. Sibatov and HongGuang Sun

Fractal Fract. 2019, 3(4), 47; https://doi.org/10.3390/fractalfract3040047 - 19 Oct 2019

Cited by 12 | Viewed by 2742

Abstract

New aspects of electron transport in quantum wires with Lévy-type disorder are described. We study the weak scattering and the incoherent sequential tunneling in one-dimensional quantum systems characterized by a tempered Lévy stable distribution of spacing between scatterers or tunneling barriers. The generalized [...] Read more.

New aspects of electron transport in quantum wires with Lévy-type disorder are described. We study the weak scattering and the incoherent sequential tunneling in one-dimensional quantum systems characterized by a tempered Lévy stable distribution of spacing between scatterers or tunneling barriers. The generalized Dorokhov–Mello–Pereyra–Kumar equation contains the tempered fractional derivative on wire length. The solution describes the evolution from the anomalous conductance distribution to the Dorokhov function for a long wire. For sequential tunneling, average values and relative fluctuations of conductance and resistance are calculated for different parameters of spatial distributions. A tempered Lévy stable distribution of spacing between barriers leads to a transition in conductance scaling. Full article

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Fractal Fract., Volume 3, Issue 4 (December 2019) – 8 articles

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