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Fractal Fract, Volume 2, Issue 4 (December 2018)

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Open AccessArticle Fractal Calculus of Functions on Cantor Tartan Spaces
Fractal Fract 2018, 2(4), 30; https://doi.org/10.3390/fractalfract2040030
Received: 5 December 2018 / Revised: 16 December 2018 / Accepted: 17 December 2018 / Published: 18 December 2018
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Abstract
In this manuscript, integrals and derivatives of functions on Cantor tartan spaces are defined. The generalisation of standard calculus, which is called Fη-calculus, is utilised to obtain definitions of the integral and derivative of functions on Cantor tartan spaces of different [...] Read more.
In this manuscript, integrals and derivatives of functions on Cantor tartan spaces are defined. The generalisation of standard calculus, which is called F η -calculus, is utilised to obtain definitions of the integral and derivative of functions on Cantor tartan spaces of different dimensions. Differential equations involving the new derivatives are solved. Illustrative examples are presented to check the details. Full article
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Open AccessArticle Approximate Controllability of Semilinear Stochastic Integrodifferential System with Nonlocal Conditions
Fractal Fract 2018, 2(4), 29; https://doi.org/10.3390/fractalfract2040029
Received: 8 October 2018 / Revised: 12 November 2018 / Accepted: 16 November 2018 / Published: 20 November 2018
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Abstract
The objective of this paper is to analyze the approximate controllability of a semilinear stochastic integrodifferential system with nonlocal conditions in Hilbert spaces. The nonlocal initial condition is a generalization of the classical initial condition and is motivated by physical phenomena. The results [...] Read more.
The objective of this paper is to analyze the approximate controllability of a semilinear stochastic integrodifferential system with nonlocal conditions in Hilbert spaces. The nonlocal initial condition is a generalization of the classical initial condition and is motivated by physical phenomena. The results are obtained by using Sadovskii’s fixed point theorem. At the end, an example is given to show the effectiveness of the result. Full article
Open AccessArticle Mathematical Modeling of Solutes Migration under the Conditions of Groundwater Filtration by the Model with the k-Caputo Fractional Derivative
Fractal Fract 2018, 2(4), 28; https://doi.org/10.3390/fractalfract2040028
Received: 1 October 2018 / Revised: 20 October 2018 / Accepted: 23 October 2018 / Published: 24 October 2018
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Abstract
Within the framework of a new mathematical model of convective diffusion with the k-Caputo derivative, we simulate the dynamics of anomalous soluble substances migration under the conditions of two-dimensional steady-state plane-vertical filtration with a free surface. As a corresponding filtration scheme, we [...] Read more.
Within the framework of a new mathematical model of convective diffusion with the k-Caputo derivative, we simulate the dynamics of anomalous soluble substances migration under the conditions of two-dimensional steady-state plane-vertical filtration with a free surface. As a corresponding filtration scheme, we consider the scheme for the spread of pollution from rivers, canals, or storages of industrial wastes. On the base of a locally one-dimensional finite-difference scheme, we develop a numerical method for obtaining solutions of boundary value problem for fractional differential equation with k-Caputo derivative with respect to the time variable that describes the convective diffusion of salt solution. The results of numerical experiments on modeling the dynamics of the considered process are presented. The results that show an existence of a time lag in the process of diffusion field formation are presented. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering 2018)
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Open AccessArticle Fatigue-Induced Cole Electrical Impedance Model Changes of Biceps Tissue Bioimpedance
Fractal Fract 2018, 2(4), 27; https://doi.org/10.3390/fractalfract2040027
Received: 13 September 2018 / Revised: 17 October 2018 / Accepted: 22 October 2018 / Published: 24 October 2018
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Abstract
Bioimpedance, or the electrical impedance of biological tissues, describes the passive electrical properties of these materials. To simplify bioimpedance datasets, fractional-order equivalent circuit presentations are often used, with the Cole-impedance model being one of the most widely used fractional-order circuits for this purpose. [...] Read more.
Bioimpedance, or the electrical impedance of biological tissues, describes the passive electrical properties of these materials. To simplify bioimpedance datasets, fractional-order equivalent circuit presentations are often used, with the Cole-impedance model being one of the most widely used fractional-order circuits for this purpose. In this work, bioimpedance measurements from 10 kHz to 100 kHz were collected from participants biceps tissues immediately prior and immediately post completion of a fatiguing exercise protocol. The Cole-impedance parameters that best fit these datasets were determined using numerical optimization procedures, with relative errors of within approximately ± 0.5 % and ± 2 % for the simulated resistance and reactance compared to the experimental data. Comparison between the pre and post fatigue Cole-impedance parameters shows that the R , R 1 , and f p components exhibited statistically significant mean differences as a result of the fatigue induced changes in the study participants. Full article
(This article belongs to the Special Issue Fractional Behavior in Nature)
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Open AccessArticle Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings
Fractal Fract 2018, 2(4), 26; https://doi.org/10.3390/fractalfract2040026
Received: 23 August 2018 / Accepted: 5 September 2018 / Published: 4 October 2018
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Abstract
The theory of complex dimensions describes the oscillations in the geometry (spectra and dynamics) of fractal strings. Such geometric oscillations can be seen most clearly in the explicit volume formula for the tubular neighborhoods of a p-adic fractal string Lp, [...] Read more.
The theory of complex dimensions describes the oscillations in the geometry (spectra and dynamics) of fractal strings. Such geometric oscillations can be seen most clearly in the explicit volume formula for the tubular neighborhoods of a p-adic fractal string L p , expressed in terms of the underlying complex dimensions. The general fractal tube formula obtained in this paper is illustrated by several examples, including the nonarchimedean Cantor and Euler strings. Moreover, we show that the Minkowski dimension of a p-adic fractal string coincides with the abscissa of convergence of the geometric zeta function associated with the string, as well as with the asymptotic growth rate of the corresponding geometric counting function. The proof of this new result can be applied to both real and p-adic fractal strings and hence, yields a unifying explanation of a key result in the theory of complex dimensions for fractal strings, even in the archimedean (or real) case. Full article
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Open AccessArticle Dynamic Fractional Inequalities Amplified on Time Scale Calculus Revealing Coalition of Discreteness and Continuity
Fractal Fract 2018, 2(4), 25; https://doi.org/10.3390/fractalfract2040025
Received: 10 September 2018 / Revised: 21 September 2018 / Accepted: 21 September 2018 / Published: 29 September 2018
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Abstract
In this paper, we present a generalization of Radon’s inequality on dynamic time scale calculus, which is widely studied by many authors and an intrinsic inequality. Further, we present the classical Bergström’s inequality and refinement of Nesbitt’s inequality unified on dynamic time scale [...] Read more.
In this paper, we present a generalization of Radon’s inequality on dynamic time scale calculus, which is widely studied by many authors and an intrinsic inequality. Further, we present the classical Bergström’s inequality and refinement of Nesbitt’s inequality unified on dynamic time scale calculus in extended form. Full article
Open AccessArticle Power Laws in Fractionally Electronic Elements
Fractal Fract 2018, 2(4), 24; https://doi.org/10.3390/fractalfract2040024
Received: 28 August 2018 / Revised: 10 September 2018 / Accepted: 21 September 2018 / Published: 26 September 2018
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Abstract
The highlight presented in this short article is about the power laws with respect to fractional capacitance and fractional inductance in terms of frequency. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering 2018)
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Open AccessArticle Generalized Memory: Fractional Calculus Approach
Fractal Fract 2018, 2(4), 23; https://doi.org/10.3390/fractalfract2040023
Received: 23 August 2018 / Revised: 20 September 2018 / Accepted: 21 September 2018 / Published: 24 September 2018
Cited by 2 | Viewed by 444 | PDF Full-text (334 KB) | HTML Full-text | XML Full-text
Abstract
The memory means an existence of output (response, endogenous variable) at the present time that depends on the history of the change of the input (impact, exogenous variable) on a finite (or infinite) time interval. The memory can be described by the function [...] Read more.
The memory means an existence of output (response, endogenous variable) at the present time that depends on the history of the change of the input (impact, exogenous variable) on a finite (or infinite) time interval. The memory can be described by the function that is called the memory function, which is a kernel of the integro-differential operator. The main purpose of the paper is to answer the question of the possibility of using the fractional calculus, when the memory function does not have a power-law form. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla (TRB) form for the memory function, we represent the integro-differential equations with memory functions by fractional integral and differential equations with derivatives and integrals of non-integer orders. This allows us to describe general economic dynamics with memory by the methods of fractional calculus. We prove that equation of the generalized accelerator with the TRB memory function can be represented by as a composition of actions of the accelerator with simplest power-law memory and the multi-parametric power-law multiplier. As an example of application of the suggested approach, we consider a generalization of the Harrod-Domar growth model with continuous time. Full article
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