Fractal and Fractional

14 pages, 328 KiB

Open AccessArticle

Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra-Fredholm Fuzzy Integral Equations

by Atanaska Georgieva and Snezhana Hristova

Fractal Fract. 2020, 4(1), 9; https://doi.org/10.3390/fractalfract4010009 - 18 Mar 2020

Cited by 17 | Viewed by 2366

The main goal of the paper is to present an approximate method for solving of a two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integral [...] Read more.

The main goal of the paper is to present an approximate method for solving of a two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integral equations in a crisp case. Approximate solutions of this system are obtained by the help with HAM and hence an approximation for the fuzzy solution of the nonlinear Volterra-Fredholm fuzzy integral equation is presented. The convergence of the proposed method is proved and the error estimate between the exact and the approximate solution is obtained. The validity and applicability of the proposed method is illustrated on a numerical example. Full article

11 pages, 403 KiB

Open AccessArticle

Admissibility of Fractional Order Descriptor Systems Based on Complex Variables: An LMI Approach

by Xuefeng Zhang and Yuqing Yan

Fractal Fract. 2020, 4(1), 8; https://doi.org/10.3390/fractalfract4010008 - 16 Mar 2020

Cited by 8 | Viewed by 2162

Abstract

This paper is devoted to the admissibility issue of singular fractional order systems with order

α \in (0, 1)

based on complex variables. Firstly, with regard to admissibility, necessary and sufficient conditions are obtained by strict LMI in complex plane. [...] Read more.

This paper is devoted to the admissibility issue of singular fractional order systems with order

α \in (0, 1)

based on complex variables. Firstly, with regard to admissibility, necessary and sufficient conditions are obtained by strict LMI in complex plane. Then, an observer-based controller is designed to ensure system admissible. Finally, numerical examples are given to reveal the validity of the theoretical conclusions. Full article

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10 pages, 768 KiB

Open AccessArticle

Kamenev-Type Asymptotic Criterion of Fourth-Order Delay Differential Equation

by Omar Bazighifan

Fractal Fract. 2020, 4(1), 7; https://doi.org/10.3390/fractalfract4010007 - 10 Mar 2020

Cited by 3 | Viewed by 1691

Abstract

In this paper, we obtain necessary and sufficient conditions for a Kamenev-type oscillation criterion of a fourth order differential equation of the form [...] Read more.

In this paper, we obtain necessary and sufficient conditions for a Kamenev-type oscillation criterion of a fourth order differential equation of the form

{(r_{3} (t) {(r_{2} (t) {(r_{1} (t) y^{'} (t))}^{'})}^{'})}^{'} + q (t) f (y (σ (t))) = 0,

where

t \geq t_{0} .

The results presented here complement some of the known results reported in the literature. Moreover, the importance of the obtained conditions is illustrated via some examples. Full article

5 pages, 581 KiB

Open AccessArticle

Fractal Dimensions of Cell Wall in Growing Cotton Fibers

by Michael Ioelovich

Fractal Fract. 2020, 4(1), 6; https://doi.org/10.3390/fractalfract4010006 - 09 Mar 2020

Cited by 2 | Viewed by 1804

Abstract

In this research, fractal properties of a cell wall in growing cotton fibers were studied. It was found that dependences of specific pore volume (P) and apparent density (ρ) on the scale factor, F = H/h, can be expressed by power-law equations: P [...] Read more.

In this research, fractal properties of a cell wall in growing cotton fibers were studied. It was found that dependences of specific pore volume (P) and apparent density (ρ) on the scale factor, F = H/h, can be expressed by power-law equations: P = P_o F^(D_v^−E) and ρ = ρ_o F^(E−Dρ), where h is minimum thickness of the microfibrilar network in the primary cell wall, H is total thickness of cell wall in growing cotton, D_v = 2.556 and D_ρ = 2.988 are fractal dimensions. From the obtained results it follows that microfibrilar network of the primary cell wall in immature fibers is loose and disordered, and therefore it has an increased pore volume (P_o = 0.037 cm³/g) and low density (ρ_o = 1.47 g/cm³). With enhance days post anthesis of growing cotton fibers, the wall thickness and density increase, while the pore volume decreases, until dense structure of completely mature fibers is formed with maximum density (1.54 g/cm³) and minimum pore volume (0.006 cm³/g). The fractal dimension for specific pore volume, D_v = 2.556, evidences the mixed surface-volume sorption mechanism of sorbate vapor in the pores. On the other hand, the fractal dimension for apparent density, D_ρ = 2.988, is very close to Euclidean volume dimension, E = 3, for the three-dimensional space. Full article

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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 1947

Abstract

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)

2 pages, 179 KiB

Open AccessEditorial

Acknowledgement to Reviewers of Fractal Fract in 2019

by Fractal and Fractional Editorial Office

Fractal Fract. 2020, 4(1), 4; https://doi.org/10.3390/fractalfract4010004 - 23 Jan 2020

Viewed by 1319

Abstract

The editorial team greatly appreciates the reviewers who have dedicated their considerable time and expertise to the journal’s rigorous editorial process over the past 12 months, regardless of whether the papers are finally published or not [...] Full article

26 pages, 60950 KiB

Open AccessEditor’s ChoiceReview

Fractal Antennas: An Historical Perspective

by Jaume Anguera, Aurora Andújar, Jeevani Jayasinghe, V. V. S. S. Sameer Chakravarthy, P. S. R. Chowdary, Joan L. Pijoan, Tanweer Ali and Carlo Cattani

Fractal Fract. 2020, 4(1), 3; https://doi.org/10.3390/fractalfract4010003 - 19 Jan 2020

Cited by 59 | Viewed by 11609

Abstract

Fractal geometry has been proven to be useful in several disciplines. In the field of antenna engineering, fractal geometry is useful to design small and multiband antenna and arrays, and high-directive elements. A historic overview of the most significant fractal mathematic pioneers is [...] Read more.

Fractal geometry has been proven to be useful in several disciplines. In the field of antenna engineering, fractal geometry is useful to design small and multiband antenna and arrays, and high-directive elements. A historic overview of the most significant fractal mathematic pioneers is presented, at the same time showing how the fractal patterns inspired engineers to design antennas. Full article

(This article belongs to the Special Issue Fractals in Antenna and Microwave Engineering 2019)

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12 pages, 484 KiB

Open AccessArticle

Parametric Identification of Nonlinear Fractional Hammerstein Models

by Vineet Prasad, Kajal Kothari and Utkal Mehta

Fractal Fract. 2020, 4(1), 2; https://doi.org/10.3390/fractalfract4010002 - 30 Dec 2019

Cited by 9 | Viewed by 2574

Abstract

In this paper, a system identification method for continuous fractional-order Hammerstein models is proposed. A block structured nonlinear system constituting a static nonlinear block followed by a fractional-order linear dynamic system is considered. The fractional differential operator is represented through the generalized operational [...] Read more.

In this paper, a system identification method for continuous fractional-order Hammerstein models is proposed. A block structured nonlinear system constituting a static nonlinear block followed by a fractional-order linear dynamic system is considered. The fractional differential operator is represented through the generalized operational matrix of block pulse functions to reduce computational complexity. A special test signal is developed to isolate the identification of the nonlinear static function from that of the fractional-order linear dynamic system. The merit of the proposed technique is indicated by concurrent identification of the fractional order with linear system coefficients, algebraic representation of the immeasurable nonlinear static function output, and permitting use of non-iterative procedures for identification of the nonlinearity. The efficacy of the proposed method is exhibited through simulation at various signal-to-noise ratios. Full article

(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering III)

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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 2798

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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Fractal Fract., Volume 4, Issue 1 (March 2020) – 9 articles

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