Research

9 pages, 279 KiB

Open AccessArticle

Non-Differentiable Solution of Nonlinear Biological Population Model on Cantor Sets

by Djelloul Ziane, Mountassir Hamdi Cherif, Dumitru Baleanu and Kacem Belghaba

Fractal Fract. 2020, 4(1), 5; https://doi.org/10.3390/fractalfract4010005 - 09 Feb 2020

Cited by 5 | Viewed by 1921

The main objective of this study is to apply the local fractional homotopy analysis method (LFHAM) to obtain the non-differentiable solution of two nonlinear partial differential equations of the biological population model on Cantor sets. The derivative operator are taken in the local [...] Read more.

The main objective of this study is to apply the local fractional homotopy analysis method (LFHAM) to obtain the non-differentiable solution of two nonlinear partial differential equations of the biological population model on Cantor sets. The derivative operator are taken in the local fractional sense. Two examples have been presented showing the effectiveness of this method in solving this model on Cantor sets. Full article

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

11 pages, 3682 KiB

Open AccessEditor’s ChoiceArticle

Power Law Type Long Memory Behaviors Modeled with Distributed Time Delay Systems

by Jocelyn Sabatier

Fractal Fract. 2020, 4(1), 1; https://doi.org/10.3390/fractalfract4010001 - 27 Dec 2019

Cited by 17 | Viewed by 2788

Abstract

This paper studies a class of distributed time delay systems that exhibit power law type long memory behaviors. Such dynamical behaviors are present in multiple domains and it is therefore essential to have tools to model them. The literature is full of examples [...] Read more.

This paper studies a class of distributed time delay systems that exhibit power law type long memory behaviors. Such dynamical behaviors are present in multiple domains and it is therefore essential to have tools to model them. The literature is full of examples in which these behaviors are modeled by means of fractional models. However, several limitations of fractional models have recently been reported and other solutions must be found. In the literature, the analysis of distributed delay models and integro-differential equations in general is older than that of fractional models. In this paper, it is shown that particular delay distributions and conditions on the model coefficients make it possible to obtain power laws. The class of systems considered is then used to model the input-output behavior of a lithium-ion cell. 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, 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 2269

Abstract

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 22 | Viewed by 2523

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 2520

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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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 - 07 Nov 2019

Cited by 10 | Viewed by 2315

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)

12 pages, 1376 KiB

Open AccessArticle

Approximate Solutions of the Damped Wave Equation and Dissipative Wave Equation in Fractal Strings

by Dumitru Baleanu and Hassan Kamil Jassim

Fractal Fract. 2019, 3(2), 26; https://doi.org/10.3390/fractalfract3020026 - 11 May 2019

Cited by 29 | Viewed by 3291

Abstract

In this paper, we apply the local fractional Laplace variational iteration method (LFLVIM) and the local fractional Laplace decomposition method (LFLDM) to obtain approximate solutions for solving the damped wave equation and dissipative wave equation within local fractional derivative operators (LFDOs). The efficiency [...] Read more.

In this paper, we apply the local fractional Laplace variational iteration method (LFLVIM) and the local fractional Laplace decomposition method (LFLDM) to obtain approximate solutions for solving the damped wave equation and dissipative wave equation within local fractional derivative operators (LFDOs). The efficiency of the considered methods are illustrated by some examples. The results obtained by LFLVIM and LFLDM are compared with the results obtained by LFVIM. The results reveal that the suggested algorithms are very effective and simple, and can be applied for linear and nonlinear problems in sciences and engineering. Full article

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

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16 pages, 4452 KiB

Open AccessArticle

Residual Power Series Method for Fractional Swift–Hohenberg Equation

by D. G. Prakasha, P. Veeresha and Haci Mehmet Baskonus

Fractal Fract. 2019, 3(1), 9; https://doi.org/10.3390/fractalfract3010009 - 07 Mar 2019

Cited by 51 | Viewed by 3932

Abstract

In this paper, the approximated analytical solution for the fractional Swift–Hohenberg (S–H) equation has been investigated with the help of the residual power series method (RPSM). To ensure the applicability and efficiency of the proposed technique, we consider a non-linear fractional order Swift–Hohenberg [...] Read more.

In this paper, the approximated analytical solution for the fractional Swift–Hohenberg (S–H) equation has been investigated with the help of the residual power series method (RPSM). To ensure the applicability and efficiency of the proposed technique, we consider a non-linear fractional order Swift–Hohenberg equation in the presence and absence of dispersive terms. The effect of bifurcation and dispersive parameters with physical importance on the probability density function for distinct fractional Brownian and standard motions are studied and presented through plots. The results obtained show that the proposed technique is simple to implement and very effective for analyzing the complex problems that arise in connected areas of science and technology. Full article

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

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