Fractal and Fractional

13 pages, 925 KiB

Open AccessArticle

Fractal Calculus of Functions on Cantor Tartan Spaces

by Alireza Khalili Golmankhaneh and Arran Fernandez

Fractal Fract. 2018, 2(4), 30; https://doi.org/10.3390/fractalfract2040030 - 18 Dec 2018

Cited by 30 | Viewed by 3700

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 [...] 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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13 pages, 290 KiB

Open AccessArticle

Approximate Controllability of Semilinear Stochastic Integrodifferential System with Nonlocal Conditions

by Annamalai Anguraj and K. Ramkumar

Fractal Fract. 2018, 2(4), 29; https://doi.org/10.3390/fractalfract2040029 - 20 Nov 2018

Cited by 14 | Viewed by 2308

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

10 pages, 327 KiB

Open AccessArticle

Mathematical Modeling of Solutes Migration under the Conditions of Groundwater Filtration by the Model with the k-Caputo Fractional Derivative

by Vsevolod Bohaienko and Volodymyr Bulavatsky

Fractal Fract. 2018, 2(4), 28; https://doi.org/10.3390/fractalfract2040028 - 24 Oct 2018

Cited by 3 | Viewed by 2373

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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13 pages, 1193 KiB

Open AccessArticle

Fatigue-Induced Cole Electrical Impedance Model Changes of Biceps Tissue Bioimpedance

by Todd J. Freeborn and Bo Fu

Fractal Fract. 2018, 2(4), 27; https://doi.org/10.3390/fractalfract2040027 - 24 Oct 2018

Cited by 25 | Viewed by 3828

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

\pm 0.5 %

and

\pm 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_{\infty}

,

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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30 pages, 480 KiB

Open AccessArticle

Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings

by Michel L. Lapidus, Hùng Lũ’ and Machiel Van Frankenhuijsen

Fractal Fract. 2018, 2(4), 26; https://doi.org/10.3390/fractalfract2040026 - 04 Oct 2018

Cited by 5 | Viewed by 3525

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

L_{p}

, [...] 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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9 pages, 277 KiB

Open AccessArticle

Dynamic Fractional Inequalities Amplified on Time Scale Calculus Revealing Coalition of Discreteness and Continuity

by Muhammad Jibril Shahab Sahir

Fractal Fract. 2018, 2(4), 25; https://doi.org/10.3390/fractalfract2040025 - 29 Sep 2018

Viewed by 2269

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

4 pages, 414 KiB

Open AccessArticle

Power Laws in Fractionally Electronic Elements

by Ming Li

Fractal Fract. 2018, 2(4), 24; https://doi.org/10.3390/fractalfract2040024 - 26 Sep 2018

Cited by 1 | Viewed by 2178

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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17 pages, 334 KiB

Open AccessArticle

Generalized Memory: Fractional Calculus Approach

by Vasily E. Tarasov

Fractal Fract. 2018, 2(4), 23; https://doi.org/10.3390/fractalfract2040023 - 24 Sep 2018

Cited by 60 | Viewed by 4110

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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Fractal Fract., Volume 2, Issue 4 (December 2018) – 8 articles

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