Special Issue "Fractals, Fractional Calculus and Applied Statistics"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (28 February 2022) | Viewed by 12579

Special Issue Editor

Prof. Dr. Raoul R. Nigmatullin
E-Mail Website
Guest Editor
Kazan National Research Technical University, 420111 Kazan, Russia
Interests: mathematical statistics; electrochemistry; signal processing; dielectric spectroscopy

Special Issue Information

Dear Colleagues,

This Special Issue provides a recent collection of the papers that are related to the following topics:

  1. Fractals and especially of these applications in applied mathematics and statistics
  2. Fractional calculus and its application in the fractal dynamics.
  3. Applied statistics as a “specific” lens to concentrate and direct corresponding mathematical tools in the solution of some applied problems.

These topics are becoming “hotspots” in the further development of fractional calculus, which covers different regions of mathematics. Many conferences have been trying to solve the following problem: is it true that there is a relationship between fractional integral and some types of fractals that can be observed in nature? This important question has not been solved and there are a variety of fractional integrals that fill in gaps between normal integral and derivative.

Prof. Dr. Raoul R. Nigmatullin
Guest Editor

Manuscript Submission Information

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Keywords

  • Regular fractals
  • Random fractals
  • Fractal physics
  • Fractional integrals/derivatives
  • Fractional statistics
  • Fractional distributions
  • Hurst exponent

Published Papers (7 papers)

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Research

Article
Mathematical Modeling of the Electrophysical Properties of a Layered Nanocomposite Based on Silicon with an Ordered Structure
Mathematics 2021, 9(24), 3167; https://doi.org/10.3390/math9243167 - 09 Dec 2021
Cited by 5 | Viewed by 1191
Abstract
The authors carried out modeling of the electrophysical properties of composite media. The frequency dependences of the dielectric constant on the type of inclusions in the composite are investigated. On the basis of the nanocomposite considered in the work, based on Si, B, [...] Read more.
The authors carried out modeling of the electrophysical properties of composite media. The frequency dependences of the dielectric constant on the type of inclusions in the composite are investigated. On the basis of the nanocomposite considered in the work, based on Si, B, and SiO2, the authors model a reflecting screen, the lattice elements of which have a layered hierarchically constructed structure similar to a fractal formation. The influence of the level of fractality on the optical properties of the object was also investigated, and it was found that the proposed structure makes it possible to increase the operating frequency range of the reflecting screen and the efficiency, in comparison with reflecting screens that have a lattice of traditional structure. The results obtained can be of practical interest for broadband and nonlinear radar devices, localization devices and mobile objects, microelectronics, as well as intelligent applications in the field of information security. Full article
(This article belongs to the Special Issue Fractals, Fractional Calculus and Applied Statistics)
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Article
General Fractional Vector Calculus
Mathematics 2021, 9(21), 2816; https://doi.org/10.3390/math9212816 - 05 Nov 2021
Cited by 10 | Viewed by 2572
Abstract
A generalization of fractional vector calculus (FVC) as a self-consistent mathematical theory is proposed to take into account a general form of non-locality in kernels of fractional vector differential and integral operators. Self-consistency involves proving generalizations of all fundamental theorems of vector calculus [...] Read more.
A generalization of fractional vector calculus (FVC) as a self-consistent mathematical theory is proposed to take into account a general form of non-locality in kernels of fractional vector differential and integral operators. Self-consistency involves proving generalizations of all fundamental theorems of vector calculus for generalized kernels of operators. In the generalization of FVC from power-law nonlocality to the general form of nonlocality in space, we use the general fractional calculus (GFC) in the Luchko approach, which was published in 2021. This paper proposed the following: (I) Self-consistent definitions of general fractional differential vector operators: the regional and line general fractional gradients, the regional and surface general fractional curl operators, the general fractional divergence are proposed. (II) Self-consistent definitions of general fractional integral vector operators: the general fractional circulation, general fractional flux and general fractional volume integral are proposed. (III) The general fractional gradient, Green’s, Stokes’ and Gauss’s theorems as fundamental theorems of general fractional vector calculus are proved for simple and complex regions. The fundamental theorems (Gradient, Green, Stokes, Gauss theorems) of the proposed general FVC are proved for a wider class of domains, surfaces and curves. All these three parts allow us to state that we proposed a calculus, which is a general fractional vector calculus (General FVC). The difficulties and problems of defining general fractional integral and differential vector operators are discussed to the nonlocal case, caused by the violation of standard product rule (Leibniz rule), chain rule (rule of differentiation of function composition) and semigroup property. General FVC for orthogonal curvilinear coordinates, which includes general fractional vector operators for the spherical and cylindrical coordinates, is also proposed. Full article
(This article belongs to the Special Issue Fractals, Fractional Calculus and Applied Statistics)
Article
Acoustics of Fractal Porous Material and Fractional Calculus
Mathematics 2021, 9(15), 1774; https://doi.org/10.3390/math9151774 - 27 Jul 2021
Cited by 2 | Viewed by 1119
Abstract
In this paper, we present a fractal (self-similar) model of acoustic propagation in a porous material with a rigid structure. The fractal medium is modeled as a continuous medium of non-integer spatial dimension. The basic equations of acoustics in a fractal porous material [...] Read more.
In this paper, we present a fractal (self-similar) model of acoustic propagation in a porous material with a rigid structure. The fractal medium is modeled as a continuous medium of non-integer spatial dimension. The basic equations of acoustics in a fractal porous material are written. In this model, the fluid space is considered as fractal while the solid matrix is non-fractal. The fluid–structure interactions are described by fractional operators in the time domain. The resulting propagation equation contains fractional derivative terms and space-dependent coefficients. The fractional wave equation is solved analytically in the time domain, and the reflection and transmission operators are calculated for a slab of fractal porous material. Expressions for the responses of the fractal porous medium (reflection and transmission) to an acoustic excitation show that it is possible to deduce these responses from those obtained for a non-fractal porous medium, only by replacing the thickness of the non-fractal material by an effective thickness depending on the fractal dimension of the material. This result shows us that, thanks to the fractal dimension, we can increase (sometimes by a ratio of 50) and decrease the equivalent thickness of the fractal material. The wavefront speed of the fractal porous material depends on the fractal dimension and admits several supersonic values. These results open a scientific challenge for the creation of new acoustic fractal materials, such as metamaterials with very specific acoustic properties. Full article
(This article belongs to the Special Issue Fractals, Fractional Calculus and Applied Statistics)
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Article
Mathematical Modeling of Layered Nanocomposite of Fractal Structure
Mathematics 2021, 9(13), 1541; https://doi.org/10.3390/math9131541 - 01 Jul 2021
Cited by 9 | Viewed by 1341
Abstract
A model of a layered hierarchically constructed composite is presented, the structure of which demonstrates the properties of similarity at different scales. For the proposed model of the composite, fractal analysis was carried out, including an assessment of the permissible range of scales, [...] Read more.
A model of a layered hierarchically constructed composite is presented, the structure of which demonstrates the properties of similarity at different scales. For the proposed model of the composite, fractal analysis was carried out, including an assessment of the permissible range of scales, calculation of fractal capacity, Hausdorff and Minkovsky dimensions, calculation of the Hurst exponent. The maximum and minimum sizes at which fractal properties are observed are investigated, and a quantitative assessment of the complexity of the proposed model is carried out. A software package is developed that allows calculating the fractal characteristics of hierarchically constructed composite media. A qualitative analysis of the calculated fractal characteristics is carried out. Full article
(This article belongs to the Special Issue Fractals, Fractional Calculus and Applied Statistics)
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Article
ML-LME: A Plant Growth Situation Analysis Model Using the Hierarchical Effect of Fractal Dimension
Mathematics 2021, 9(12), 1322; https://doi.org/10.3390/math9121322 - 08 Jun 2021
Cited by 2 | Viewed by 1645
Abstract
Rice plays an essential role in agricultural production as the most significant food crop. Automated supervision in the process of crop growth is the future development direction of agriculture, and it is also a problem that needs to be solved urgently. Productive cultivation, [...] Read more.
Rice plays an essential role in agricultural production as the most significant food crop. Automated supervision in the process of crop growth is the future development direction of agriculture, and it is also a problem that needs to be solved urgently. Productive cultivation, production and research of crops are attributed to increased automation of supervision in the growth. In this article, for the first time, we propose the concept of rice fractal dimension heterogeneity and define it as rice varieties with different fractal dimension values having various correlations between their traits. To make a comprehensive prediction of the rice growth, Machine Learning and Linear Mixed Effect (ML-LME) model is proposed to model and analyze this heterogeneity, which is based on the existing automatic measurement system RAP and introduces statistical characteristics of fractal dimensions as novel features. Machine learning algorithms are applied to distinguish the rice growth stages with a high degree of accuracy and to excavate the heterogeneity of rice fractal dimensions with statistical meaning. According to the information of growth stage and fractal dimension heterogeneity, a precise prediction of key rice phenotype traits can be received by ML-LME using a Linear Mixed Effect model. In this process, the value of the fractal dimension is divided into groups and then rices of different levels are respectively fitted to improve the accuracy of the subsequent prediction, that is, the heterogeneity of the fractal dimension. Afterwards, we apply the model to analyze the rice pot image. The research results show that the ML-LME model, which possesses the hierarchical effect of fractal dimension, performs more excellently in predicting the growth situation of plants than the traditional regression model does. Further comparison confirmed that the model we proposed is the first to consider the hierarchy structure of plant fractal dimension, and that consideration obviously strengthens the model on the ability of variation interpretation and prediction precision. Full article
(This article belongs to the Special Issue Fractals, Fractional Calculus and Applied Statistics)
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Article
Multivariate Multifractal Detrending Moving Average Analysis of Air Pollutants
Mathematics 2021, 9(7), 711; https://doi.org/10.3390/math9070711 - 25 Mar 2021
Cited by 2 | Viewed by 1208
Abstract
One of the most challenging endeavors of contemporary research is to describe and analyze the dynamic behavior of time series arising from real-world systems. To address the need for analyzing long-range correlations and multifractal properties of multivariate time series, we generalize the multifractal [...] Read more.
One of the most challenging endeavors of contemporary research is to describe and analyze the dynamic behavior of time series arising from real-world systems. To address the need for analyzing long-range correlations and multifractal properties of multivariate time series, we generalize the multifractal detrended moving average algorithm (MFDMA) to the multivariate case and propose a multivariate MFDMA algorithm (MV-MFDMA). The validity and performance of the proposed algorithm are tested by conducting numerical simulations on synthetic multivariate monofractal and multifractal time series. The MV-MFDMA algorithm is then utilized to analyze raw, seasonally adjusted, and remainder components of five air pollutant time series. Results from all three cases reveal multifractal properties with persistent long-range correlations. Full article
(This article belongs to the Special Issue Fractals, Fractional Calculus and Applied Statistics)
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Article
Generalized Hurst Hypothesis: Description of Time-Series in Communication Systems
Mathematics 2021, 9(4), 381; https://doi.org/10.3390/math9040381 - 14 Feb 2021
Cited by 1 | Viewed by 2377
Abstract
In this paper, we focus on the generalization of the Hurst empirical law and suggest a set of reduced parameters for quantitative description of long-time series. These series are usually considered as a specific response of a complex system (economic, geophysical, electromagnetic and [...] Read more.
In this paper, we focus on the generalization of the Hurst empirical law and suggest a set of reduced parameters for quantitative description of long-time series. These series are usually considered as a specific response of a complex system (economic, geophysical, electromagnetic and other systems), where successive fixations of external factors become impossible. We consider applying generalized Hurst laws to obtain a new set of reduced parameters in data associated with communication systems. We analyze three hypotheses. The first one contains one power-law exponent. The second one incorporates two power-law exponents, which are in many cases complex-conjugated. The third hypothesis has three power-law exponents, two of which are complex-conjugated as well. These hypotheses describe with acceptable accuracy (relative error does not exceed 2%) a wide set of trendless sequences (TLS) associated with radiometric measurements. Generalized Hurst laws operate with R/S curves not only in the asymptotic region, but in the entire domain. The fitting parameters can be used as the reduced parameters for the description of the given data. The paper demonstrates that this general approach can also be applied to other TLS. Full article
(This article belongs to the Special Issue Fractals, Fractional Calculus and Applied Statistics)
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