Next Issue
Previous Issue

Table of Contents

Fractal Fract, Volume 3, Issue 1 (March 2019)

  • Issues are regarded as officially published after their release is announced to the table of contents alert mailing list.
  • You may sign up for e-mail alerts to receive table of contents of newly released issues.
  • PDF is the official format for papers published in both, html and pdf forms. To view the papers in pdf format, click on the "PDF Full-text" link, and use the free Adobe Readerexternal link to open them.
View options order results:
result details:
Displaying articles 1-13
Export citation of selected articles as:
Open AccessArticle
Intrinsic Metric Formulas on Some Self-Similar Sets via the Code Representation
Fractal Fract 2019, 3(1), 13; https://doi.org/10.3390/fractalfract3010013
Received: 4 February 2019 / Revised: 18 March 2019 / Accepted: 22 March 2019 / Published: 25 March 2019
Viewed by 256 | PDF Full-text (2037 KB) | HTML Full-text | XML Full-text
Abstract
In recent years, intrinsic metrics have been described on various fractals with different formulas. The Sierpinski gasket is given as one of the fundamental models which defined the intrinsic metrics on them via the code representations of the points. In this paper, we [...] Read more.
In recent years, intrinsic metrics have been described on various fractals with different formulas. The Sierpinski gasket is given as one of the fundamental models which defined the intrinsic metrics on them via the code representations of the points. In this paper, we obtain the explicit formulas of the intrinsic metrics on some self-similar sets (but not strictly self-similar), which are composed of different combinations of equilateral and right Sierpinski gaskets, respectively, by using the code representations of their points. We then express geometrical properties of these structures on their code sets and also give some illustrative examples. Full article
Figures

Figure 1

Open AccessArticle
Some New Fractional Trapezium-Type Inequalities for Preinvex Functions
Fractal Fract 2019, 3(1), 12; https://doi.org/10.3390/fractalfract3010012
Received: 28 February 2019 / Revised: 20 March 2019 / Accepted: 21 March 2019 / Published: 24 March 2019
Viewed by 218 | PDF Full-text (284 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, authors the present the discovery of an interesting identity regarding trapezium-type integral inequalities. By using the lemma as an auxiliary result, some new estimates with respect to trapezium-type integral inequalities via general fractional integrals are obtained. It is pointed out [...] Read more.
In this paper, authors the present the discovery of an interesting identity regarding trapezium-type integral inequalities. By using the lemma as an auxiliary result, some new estimates with respect to trapezium-type integral inequalities via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from the main results. Some applications regarding special means for different real numbers are provided as well. The ideas and techniques described in this paper may stimulate further research. Full article
Open AccessArticle
On the Fractal Langevin Equation
Fractal Fract 2019, 3(1), 11; https://doi.org/10.3390/fractalfract3010011
Received: 22 February 2019 / Revised: 7 March 2019 / Accepted: 12 March 2019 / Published: 13 March 2019
Cited by 2 | Viewed by 341 | PDF Full-text (364 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, fractal stochastic Langevin equations are suggested, providing a mathematical model for random walks on the middle-τ Cantor set. The fractal mean square displacement of different random walks on the middle-τ Cantor set are presented. Fractal under-damped and over-damped [...] Read more.
In this paper, fractal stochastic Langevin equations are suggested, providing a mathematical model for random walks on the middle- τ Cantor set. The fractal mean square displacement of different random walks on the middle- τ Cantor set are presented. Fractal under-damped and over-damped Langevin equations, fractal scaled Brownian motion, and ultra-slow fractal scaled Brownian motion are suggested and the corresponding fractal mean square displacements are obtained. The results are plotted to show the details. Full article
Figures

Figure 1

Open AccessArticle
On Analytic Functions Involving the q-Ruscheweyeh Derivative
Fractal Fract 2019, 3(1), 10; https://doi.org/10.3390/fractalfract3010010
Received: 20 February 2019 / Revised: 8 March 2019 / Accepted: 8 March 2019 / Published: 10 March 2019
Viewed by 281 | PDF Full-text (239 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we use concepts of q-calculus to introduce a certain type of q-difference operator, and using it define some subclasses of analytic functions. Inclusion relations, coefficient result, and some other interesting properties of these classes are studied. Full article
Open AccessArticle
Residual Power Series Method for Fractional Swift–Hohenberg Equation
Fractal Fract 2019, 3(1), 9; https://doi.org/10.3390/fractalfract3010009
Received: 20 February 2019 / Revised: 5 March 2019 / Accepted: 6 March 2019 / Published: 7 March 2019
Viewed by 299 | PDF Full-text (4452 KB) | HTML Full-text | XML Full-text
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)
Figures

Figure 1

Open AccessArticle
The Fractal Calculus for Fractal Materials
Fractal Fract 2019, 3(1), 8; https://doi.org/10.3390/fractalfract3010008
Received: 10 February 2019 / Revised: 2 March 2019 / Accepted: 5 March 2019 / Published: 6 March 2019
Cited by 2 | Viewed by 282 | PDF Full-text (743 KB) | HTML Full-text | XML Full-text
Abstract
The major problem in the process of mixing fluids (for instance liquid-liquid mixers) is turbulence, which is the outcome of the function of the equipment (engine). Fractal mixing is an alternative method that has symmetry and is predictable. Therefore, fractal structures and fractal [...] Read more.
The major problem in the process of mixing fluids (for instance liquid-liquid mixers) is turbulence, which is the outcome of the function of the equipment (engine). Fractal mixing is an alternative method that has symmetry and is predictable. Therefore, fractal structures and fractal reactors find importance. Using F α -fractal calculus, in this paper, we derive exact F α -differential forms of an ideal gas. Depending on the dimensionality of space, we should first obtain the integral staircase function and mass function of our geometry. When gases expand inside the fractal structure because of changes from the i + 1 iteration to the i iteration, in fact, we are faced with fluid mixing inside our fractal structure, which can be described by physical quantities P, V, and T. Finally, for the ideal gas equation, we calculate volume expansivity and isothermal compressibility. Full article
Figures

Figure 1

Open AccessArticle
Fractal Image Interpolation: A Tutorial and New Result
Fractal Fract 2019, 3(1), 7; https://doi.org/10.3390/fractalfract3010007
Received: 16 December 2018 / Revised: 13 February 2019 / Accepted: 20 February 2019 / Published: 23 February 2019
Viewed by 306 | PDF Full-text (1079 KB) | HTML Full-text | XML Full-text
Abstract
This paper reviews the implementation of fractal based image interpolation, the associated visual artifacts of the interpolated images, and various techniques, including novel contributions, that alleviate these awkward visual artifacts to achieve visually pleasant interpolated image. The fractal interpolation methods considered in this [...] Read more.
This paper reviews the implementation of fractal based image interpolation, the associated visual artifacts of the interpolated images, and various techniques, including novel contributions, that alleviate these awkward visual artifacts to achieve visually pleasant interpolated image. The fractal interpolation methods considered in this paper are based on the plain Iterative Function System (IFS) in spatial domain without additional transformation, where we believe that the benefits of additional transformation can be added onto the presented study without complication. Simulation results are presented to demonstrate the discussed techniques, together with the pros and cons of each techniques. Finally, a novel spatial domain interleave layer has been proposed to add to the IFS image system for improving the performance of the system from image zooming to interpolation with the preservation of the pixel intensity from the original low resolution image. Full article
Figures

Figure 1

Open AccessBrief Report
Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers
Fractal Fract 2019, 3(1), 6; https://doi.org/10.3390/fractalfract3010006
Received: 31 December 2018 / Revised: 18 February 2019 / Accepted: 18 February 2019 / Published: 20 February 2019
Viewed by 266 | PDF Full-text (844 KB) | HTML Full-text | XML Full-text
Abstract
Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form x+ [...] Read more.
Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form x + τ y for x , y R , and τ 2 = 1 but τ ± 1 , are the natural number system in which to encode geometric properties of the Minkowski space R 1 , 1 . We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets. Full article
Figures

Figure 1

Open AccessArticle
On q-Uniformly Mocanu Functions
Fractal Fract 2019, 3(1), 5; https://doi.org/10.3390/fractalfract3010005
Received: 28 January 2019 / Revised: 6 February 2019 / Accepted: 10 February 2019 / Published: 11 February 2019
Viewed by 273 | PDF Full-text (270 KB) | HTML Full-text | XML Full-text
Abstract
Let f be analytic in open unit disc E={z:|z|<1} with f(0)=0 and f(0)=1. The q-derivative of f is defined by: [...] Read more.
Let f be analytic in open unit disc E = { z : | z | < 1 } with f ( 0 ) = 0 and f ( 0 ) = 1 . The q-derivative of f is defined by: D q f ( z ) = f ( z ) f ( q z ) ( 1 q ) z , q ( 0 , 1 ) , z B { 0 } , where B is a q-geometric subset of C . Using operator D q , q-analogue class k U M q ( α , β ) , k-uniformly Mocanu functions are defined as: For k = 0 and q 1 , k reduces to M ( α ) of Mocanu functions. Subordination is used to investigate many important properties of these functions. Several interesting results are derived as special cases. Full article
Open AccessArticle
Integral Representations and Algebraic Decompositions of the Fox-Wright Type of Special Functions
Fractal Fract 2019, 3(1), 4; https://doi.org/10.3390/fractalfract3010004
Received: 23 December 2018 / Revised: 22 January 2019 / Accepted: 22 January 2019 / Published: 25 January 2019
Viewed by 285 | PDF Full-text (489 KB) | HTML Full-text | XML Full-text
Abstract
The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions are given and the conditions under [...] Read more.
The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions are given and the conditions under which these function can be represented by simpler functions are demonstrated. The connection with generalized Erdélyi-Kober fractional differential and integral operators is demonstrated and discussed. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering 2018)
Open AccessArticle
Study of Fractal Dimensions of Microcrystalline Cellulose Obtained by the Spray-Drying Method
Fractal Fract 2019, 3(1), 3; https://doi.org/10.3390/fractalfract3010003
Received: 8 December 2018 / Revised: 11 January 2019 / Accepted: 22 January 2019 / Published: 24 January 2019
Viewed by 244 | PDF Full-text (911 KB) | HTML Full-text | XML Full-text
Abstract
In this research, the fractal structure of beads of different sizes obtained by the spray-drying of aqueous dispersions of microcrystalline cellulose (MCC) was studied. These beads were formed as a result of the aggregation of rod-shaped cellulose nanocrystalline particles (CNP). It was found [...] Read more.
In this research, the fractal structure of beads of different sizes obtained by the spray-drying of aqueous dispersions of microcrystalline cellulose (MCC) was studied. These beads were formed as a result of the aggregation of rod-shaped cellulose nanocrystalline particles (CNP). It was found that increasing the average radius (R) of the formed MCC beads resulted in increased specific pore volume (P) and reduced apparent density (ρ). The dependences of P and ρ on the scale factor (R/r) can be expressed by power-law equations: P = Po (R/r)E−Dp and ρ = d (R/r)Dd−E, where the fractal dimensions Dp = 2.887 and Dd = 2.986 are close to the Euclidean dimension E = 3 for three-dimensional space; r = 3 nm is the radius of the cellulose nanocrystalline particles, Po = 0.03 cm3/g is the specific pore volume, and d = 1.585 g/cm3 is the true density (specific gravity) of the CNP, respectively. With the increase in the size of the formed MCC beads, the order in the packing of the beads was distorted, conforming to theory of the diffusion-limited aggregation process. Full article
Figures

Figure 1

Open AccessEditorial
Acknowledgement to Reviewers of Fractal Fract in 2018
Fractal Fract 2019, 3(1), 2; https://doi.org/10.3390/fractalfract3010002
Published: 16 January 2019
Viewed by 245 | PDF Full-text (127 KB) | HTML Full-text | XML Full-text
Abstract
Rigorous peer-review is the corner-stone of high-quality academic publishing [...] Full article
Open AccessArticle
Regularized Integral Representations of the Reciprocal Gamma Function
Fractal Fract 2019, 3(1), 1; https://doi.org/10.3390/fractalfract3010001
Received: 16 November 2018 / Revised: 29 December 2018 / Accepted: 8 January 2019 / Published: 12 January 2019
Cited by 1 | Viewed by 362 | PDF Full-text (340 KB) | HTML Full-text | XML Full-text
Abstract
This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine’s complex representation is demonstrated. For [...] Read more.
This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine’s complex representation is demonstrated. For both real and complex integrals, the regularized representation can be expressed in terms of the two-parameter Mittag-Leffler function. Reference numerical implementations in the Computer Algebra System Maxima are provided. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering 2018)
Figures

Figure 1

Fractal Fract EISSN 2504-3110 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top