Fractal and Fractional

Editorial

Jump to: Research

2 pages, 168 KiB

Open AccessEditorial

Editorial for Special Issue “Fractional Behavior in Nature 2019”

by Manuel Duarte Ortigueira

Fractal Fract. 2021, 5(4), 186; https://doi.org/10.3390/fractalfract5040186 - 26 Oct 2021

Cited by 1 | Viewed by 1197

Abstract

The presence of fractional behavior in nature is unquestionable [...] Full article

(This article belongs to the Special Issue Fractional Behavior in Nature 2019)

Research

Jump to: Editorial

17 pages, 917 KiB

Open AccessArticle

Signal Propagation in Electromagnetic Media Modelled by the Two-Sided Fractional Derivative

by Jacek Gulgowski, Dariusz Kwiatkowski and Tomasz P. Stefański

Fractal Fract. 2021, 5(1), 10; https://doi.org/10.3390/fractalfract5010010 - 18 Jan 2021

Cited by 8 | Viewed by 2118

Abstract

In this paper, wave propagation is considered in a medium described by a fractional-order model, which is formulated with the use of the two-sided fractional derivative of Ortigueira and Machado. Although the relation of the derivative to causality is clearly specified in its [...] Read more.

In this paper, wave propagation is considered in a medium described by a fractional-order model, which is formulated with the use of the two-sided fractional derivative of Ortigueira and Machado. Although the relation of the derivative to causality is clearly specified in its definition, there is no obvious relation between causality of the derivative and causality of the transfer function induced by this derivative. Hence, causality of the system is investigated; its output is an electromagnetic signal propagating in media described by the time-domain two-sided fractional derivative. It is demonstrated that, for the derivative order in the range

[1, + \infty)

, the transfer function describing attenuated signal propagation is not causal for any value of the asymmetry parameter of the derivative. On the other hand, it is shown that, for derivative orders in the range

(0, 1)

, the transfer function is causal if and only if the asymmetry parameter is equal to certain specific values corresponding to the left-sided Grünwald–Letnikov derivative. The results are illustrated by numerical simulations and analyses. Some comments on the Kramers–Krönig relations for logarithm of the transfer function are presented as well. Full article

(This article belongs to the Special Issue Fractional Behavior in Nature 2019)

► Show Figures

Figure 1

29 pages, 554 KiB

Open AccessArticle

Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

by Thomas M. Michelitsch, Federico Polito and Alejandro P. Riascos

Fractal Fract. 2020, 4(4), 51; https://doi.org/10.3390/fractalfract4040051 - 31 Oct 2020

Cited by 13 | Viewed by 2458

Abstract

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 [...] Read more.

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 article belongs to the Special Issue Fractional Behavior in Nature 2019)

► Show Figures

Figure 1

15 pages, 6290 KiB

Open AccessArticle

Fractal and Fractional Derivative Modelling of Material Phase Change

by Harry Esmonde

Fractal Fract. 2020, 4(3), 46; https://doi.org/10.3390/fractalfract4030046 - 14 Sep 2020

Cited by 8 | Viewed by 2373

Abstract

An iterative approach is taken to develop a fractal topology that can describe the material structure of phase changing materials. Transfer functions and frequency response functions based on fractional calculus are used to describe this topology and then applied to model phase transformations [...] Read more.

An iterative approach is taken to develop a fractal topology that can describe the material structure of phase changing materials. Transfer functions and frequency response functions based on fractional calculus are used to describe this topology and then applied to model phase transformations in liquid/solid transitions in physical processes. Three types of transformation are tested experimentally, whipping of cream (rheopexy), solidification of gelatine and melting of ethyl vinyl acetate (EVA). A liquid-type model is used throughout the cream whipping process while liquid and solid models are required for gelatine and EVA to capture the yield characteristic of these materials. Full article

(This article belongs to the Special Issue Fractional Behavior in Nature 2019)

► Show Figures

Figure 1

18 pages, 554 KiB

Open AccessArticle

Fractional SIS Epidemic Models

by Caterina Balzotti, Mirko D’Ovidio and Paola Loreti

Fractal Fract. 2020, 4(3), 44; https://doi.org/10.3390/fractalfract4030044 - 31 Aug 2020

Cited by 20 | Viewed by 3608

Abstract

In this paper, we consider the fractional SIS (susceptible-infectious-susceptible) epidemic model (α-SIS model) in the case of constant population size. We provide a representation of the explicit solution to the fractional model and we illustrate the results by numerical schemes. A comparison with [...] Read more.

In this paper, we consider the fractional SIS (susceptible-infectious-susceptible) epidemic model (α-SIS model) in the case of constant population size. We provide a representation of the explicit solution to the fractional model and we illustrate the results by numerical schemes. A comparison with the limit case when the fractional order α converges to 1 (the SIS model) is also given. We analyze the effects of the fractional derivatives by comparing the SIS and the α-SIS models. Full article

(This article belongs to the Special Issue Fractional Behavior in Nature 2019)

► Show Figures

Figure 1

Figure 1
Comparison between the solutions to the susceptible-infectious-susceptible (SIS) model and the explicit and numerical fractional solutions to (<a href="#FD1-fractalfract-04-00044" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.99</mn> </mrow> </semantics></math>. The analysis shows correspondence between SIS model and the case <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> of our model. This result was expected, and it confirms the continuity wit respect to <math display="inline"><semantics> <mi>α</mi> </semantics></math> (see (P4)). Full article ">Figure 2
Comparison between the explicit and numerical fractional solutions to (<a href="#FD1-fractalfract-04-00044" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>. Full article ">Figure 3
Comparison between the explicit and numerical fractional solutions to (<a href="#FD1-fractalfract-04-00044" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>. Full article ">Figure 4
Comparison between the solutions to the SIS model and the fractional solutions to (<a href="#FD1-fractalfract-04-00044" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.99</mn> </mrow> </semantics></math> (continuity w.r. to <math display="inline"><semantics> <mi>α</mi> </semantics></math>). Full article ">Figure 5
Comparison between the fractional solutions to (<a href="#FD1-fractalfract-04-00044" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>. Full article ">Figure 6
Comparison between the fractional solutions to (<a href="#FD1-fractalfract-04-00044" class="html-disp-formula">1</a>) with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>. Full article ">

15 pages, 474 KiB

Open AccessArticle

Mathematical Description of the Groundwater Flow and that of the Impurity Spread, which Use Temporal Caputo or Riemann–Liouville Fractional Partial Derivatives, Is Non-Objective

by Agneta M. Balint and Stefan Balint

Fractal Fract. 2020, 4(3), 36; https://doi.org/10.3390/fractalfract4030036 - 21 Jul 2020

Cited by 13 | Viewed by 2969

Abstract

In this paper, it is shown that the mathematical description of the bulk fluid flow and that of content impurity spread, which uses temporal Caputo or temporal Riemann–Liouville fractional order partial derivatives, having integral representation on a finite interval, in the case of [...] Read more.

In this paper, it is shown that the mathematical description of the bulk fluid flow and that of content impurity spread, which uses temporal Caputo or temporal Riemann–Liouville fractional order partial derivatives, having integral representation on a finite interval, in the case of a horizontal unconfined aquifer is non-objective. The basic idea is that different observers using this type of description obtain different results which cannot be reconciled, in other words, transformed into each other using only formulas that link the numbers representing a moment in time for two different choices from the origin of time measurement. This is not an academic curiosity; it is rather a problem to find which one of the obtained results is correct. Full article

(This article belongs to the Special Issue Fractional Behavior in Nature 2019)

► Show Figures

Figure 1

13 pages, 285 KiB

Open AccessArticle

Fractional Derivatives and Dynamical Systems in Material Instability

by Peter B. Béda

Fractal Fract. 2020, 4(2), 14; https://doi.org/10.3390/fractalfract4020014 - 16 Apr 2020

Cited by 2 | Viewed by 2238

Abstract

Loss of stability is studied extensively in nonlinear investigations, and classified as generic bifurcations. It requires regularity, being connected with non-locality. Such behavior comes from gradient terms in constitutive equations. Most fractional derivatives are non-local, thus by using them in defining strain, a [...] Read more.

Loss of stability is studied extensively in nonlinear investigations, and classified as generic bifurcations. It requires regularity, being connected with non-locality. Such behavior comes from gradient terms in constitutive equations. Most fractional derivatives are non-local, thus by using them in defining strain, a non-local strain appears. In such a way, various versions of non-localities are obtained by using various types of fractional derivatives. The study aims for constitutive modeling via instability phenomena, that is, by observing the way of loss of stability of material, we can be informed about the proper form of its mathematical model. Full article

(This article belongs to the Special Issue Fractional Behavior in Nature 2019)

► Show Figures

Figure 1

12 pages, 3713 KiB

Open AccessArticle

Fractional-Order Models for Biochemical Processes

by Eva-H. Dulf, Dan C. Vodnar, Alex Danku, Cristina-I. Muresan and Ovidiu Crisan

Fractal Fract. 2020, 4(2), 12; https://doi.org/10.3390/fractalfract4020012 - 10 Apr 2020

Cited by 12 | Viewed by 2939

Abstract

Biochemical processes present complex mechanisms and can be described by various computational models. Complex systems present a variety of problems, especially the loss of intuitive understanding. The present work uses fractional-order calculus to obtain mathematical models for erythritol and mannitol synthesis. The obtained [...] Read more.

Biochemical processes present complex mechanisms and can be described by various computational models. Complex systems present a variety of problems, especially the loss of intuitive understanding. The present work uses fractional-order calculus to obtain mathematical models for erythritol and mannitol synthesis. The obtained models are useful for both prediction and process optimization. The models present the complex behavior of the process due to the fractional order, without losing the physical meaning of gain and time constants. To validate each obtained model, the simulation results were compared with experimental data. In order to highlight the advantages of fractional-order models, comparisons with the corresponding integer-order models are presented. Full article

(This article belongs to the Special Issue Fractional Behavior in Nature 2019)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Fractional Behavior in Nature 2019

Share This Special Issue

Special Issue Editor

Special Issue Information

Keywords

Benefits of Publishing in a Special Issue

Published Papers (8 papers)

Editorial

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI