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Mathematics, Volume 6, Issue 1 (January 2018)

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Cover Story (view full-size image) We applied a Monte Carlo approach to investigate the impact of medium properties on pollutant [...] Read more.
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Open AccessEditorial Acknowledgement to Reviewers of Mathematics in 2017
Mathematics 2018, 6(1), 14; https://doi.org/10.3390/math6010014
Received: 18 January 2018 / Revised: 18 January 2018 / Accepted: 18 January 2018 / Published: 18 January 2018
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Abstract
Peer review is an essential part in the publication process, ensuring that Mathematics maintains high quality standards for its published papers.[...] Full article
Open AccessFeature PaperArticle Iterative Methods for Computing Vibrational Spectra
Mathematics 2018, 6(1), 13; https://doi.org/10.3390/math6010013
Received: 21 December 2017 / Revised: 10 January 2018 / Accepted: 11 January 2018 / Published: 16 January 2018
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Abstract
I review some computational methods for calculating vibrational spectra. They all use iterative eigensolvers to compute eigenvalues of a Hamiltonian matrix by evaluating matrix-vector products (MVPs). A direct-product basis can be used for molecules with five or fewer atoms. This is done by
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I review some computational methods for calculating vibrational spectra. They all use iterative eigensolvers to compute eigenvalues of a Hamiltonian matrix by evaluating matrix-vector products (MVPs). A direct-product basis can be used for molecules with five or fewer atoms. This is done by exploiting the structure of the basis and the structure of a direct product quadrature grid. I outline three methods that can be used for molecules with more than five atoms. The first uses contracted basis functions and an intermediate (F) matrix. The second uses Smolyak quadrature and a pruned basis. The third uses a tensor rank reduction scheme. Full article
(This article belongs to the Special Issue Computational Spectroscopy)
Open AccessArticle Best Approximation of the Fractional Semi-Derivative Operator by Exponential Series
Mathematics 2018, 6(1), 12; https://doi.org/10.3390/math6010012
Received: 30 November 2017 / Revised: 11 January 2018 / Accepted: 12 January 2018 / Published: 16 January 2018
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Abstract
A significant reduction in the time required to obtain an estimate of the mean frequency of the spectrum of Doppler signals when seeking to measure the instantaneous velocity of dangerous near-Earth cosmic objects (NEO) is an important task being developed to counter the
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A significant reduction in the time required to obtain an estimate of the mean frequency of the spectrum of Doppler signals when seeking to measure the instantaneous velocity of dangerous near-Earth cosmic objects (NEO) is an important task being developed to counter the threat from asteroids. Spectral analysis methods have shown that the coordinate of the centroid of the Doppler signal spectrum can be found by using operations in the time domain without spectral processing. At the same time, an increase in the speed of resolving the algorithm for estimating the mean frequency of the spectrum is achieved by using fractional differentiation without spectral processing. Thus, an accurate estimate of location of the centroid for the spectrum of Doppler signals can be obtained in the time domain as the signal arrives. This paper considers the implementation of a fractional-differentiating filter of the order of ½ by a set of automation astatic transfer elements, which greatly simplifies practical implementation. Real technical devices have the ultimate time delay, albeit small in comparison with the duration of the signal. As a result, the real filter will process the signal with some error. In accordance with this, this paper introduces and uses the concept of a “pre-derivative” of ½ of magnitude. An optimal algorithm for realizing the structure of the filter is proposed based on the criterion of minimum mean square error. Relations are obtained for the quadrature coefficients that determine the structure of the filter. Full article
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
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Open AccessArticle The Collapse of Ecosystem Engineer Populations
Mathematics 2018, 6(1), 9; https://doi.org/10.3390/math6010009
Received: 26 November 2017 / Revised: 18 December 2017 / Accepted: 8 January 2018 / Published: 12 January 2018
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Abstract
Humans are the ultimate ecosystem engineers who have profoundly transformed the world’s landscapes in order to enhance their survival. Somewhat paradoxically, however, sometimes the unforeseen effect of this ecosystem engineering is the very collapse of the population it intended to protect. Here we
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Humans are the ultimate ecosystem engineers who have profoundly transformed the world’s landscapes in order to enhance their survival. Somewhat paradoxically, however, sometimes the unforeseen effect of this ecosystem engineering is the very collapse of the population it intended to protect. Here we use a spatial version of a standard population dynamics model of ecosystem engineers to study the colonization of unexplored virgin territories by a small settlement of engineers. We find that during the expansion phase the population density reaches values much higher than those the environment can support in the equilibrium situation. When the colonization front reaches the boundary of the available space, the population density plunges sharply and attains its equilibrium value. The collapse takes place without warning and happens just after the population reaches its peak number. We conclude that overpopulation and the consequent collapse of an expanding population of ecosystem engineers is a natural consequence of the nonlinear feedback between the population and environment variables. Full article
(This article belongs to the Special Issue Progress in Mathematical Ecology)
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Open AccessFeature PaperArticle Length-Fuzzy Subalgebras in BCK/BCI-Algebras
Mathematics 2018, 6(1), 11; https://doi.org/10.3390/math6010011
Received: 1 December 2017 / Revised: 3 January 2018 / Accepted: 5 January 2018 / Published: 12 January 2018
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Abstract
As a generalization of interval-valued fuzzy sets and fuzzy sets, the concept of hyperfuzzy sets was introduced by Ghosh and Samanta in the paper [J. Ghosh and T.K. Samanta, Hyperfuzzy sets and hyperfuzzy group, Int. J. Advanced Sci Tech. 41 (2012), 27–37]. The
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As a generalization of interval-valued fuzzy sets and fuzzy sets, the concept of hyperfuzzy sets was introduced by Ghosh and Samanta in the paper [J. Ghosh and T.K. Samanta, Hyperfuzzy sets and hyperfuzzy group, Int. J. Advanced Sci Tech. 41 (2012), 27–37]. The aim of this manuscript is to introduce the length-fuzzy set and apply it to B C K / B C I -algebras. The notion of length-fuzzy subalgebras in B C K / B C I -algebras is introduced, and related properties are investigated. Characterizations of a length-fuzzy subalgebra are discussed. Relations between length-fuzzy subalgebras and hyperfuzzy subalgebras are established. Full article
(This article belongs to the Special Issue Fuzzy Mathematics)
Open AccessFeature PaperArticle Global Dynamics of Certain Mix Monotone Difference Equation
Mathematics 2018, 6(1), 10; https://doi.org/10.3390/math6010010
Received: 7 December 2017 / Revised: 3 January 2018 / Accepted: 7 January 2018 / Published: 12 January 2018
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Abstract
We investigate global dynamics of the following second order rational difference equation xn+1=xnxn1+αxn+βxn1axnxn1+bx
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We investigate global dynamics of the following second order rational difference equation x n + 1 = x n x n 1 + α x n + β x n 1 a x n x n 1 + b x n 1 , where the parameters α , β , a , b are positive real numbers and initial conditions x 1 and x 0 are arbitrary positive real numbers. The map associated to the right-hand side of this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the corresponding parametric space. In most cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)
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Open AccessArticle A Note on the Equivalence of Fractional Relaxation Equations to Differential Equations with Varying Coefficients
Mathematics 2018, 6(1), 8; https://doi.org/10.3390/math6010008
Received: 14 December 2017 / Revised: 3 January 2018 / Accepted: 5 January 2018 / Published: 9 January 2018
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Abstract
In this note, we show how an initial value problem for a relaxation process governed by a differential equation of a non-integer order with a constant coefficient may be equivalent to that of a differential equation of the first order with a varying
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In this note, we show how an initial value problem for a relaxation process governed by a differential equation of a non-integer order with a constant coefficient may be equivalent to that of a differential equation of the first order with a varying coefficient. This equivalence is shown for the simple fractional relaxation equation that points out the relevance of the Mittag–Leffler function in fractional calculus. This simple argument may lead to the equivalence of more general processes governed by evolution equations of fractional order with constant coefficients to processes governed by differential equations of integer order but with varying coefficients. Our main motivation is to solicit the researchers to extend this approach to other areas of applied science in order to have a deeper knowledge of certain phenomena, both deterministic and stochastic ones, investigated nowadays with the techniques of the fractional calculus. Full article
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
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Open AccessFeature PaperArticle Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions
Mathematics 2018, 6(1), 7; https://doi.org/10.3390/math6010007
Received: 14 December 2017 / Revised: 29 December 2017 / Accepted: 1 January 2018 / Published: 9 January 2018
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Abstract
Multiterm fractional differential equations (MTFDEs) nowadays represent a widely used tool to model many important processes, particularly for multirate systems. Their numerical solution is then a compelling subject that deserves great attention, not least because of the difficulties to apply general purpose methods
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Multiterm fractional differential equations (MTFDEs) nowadays represent a widely used tool to model many important processes, particularly for multirate systems. Their numerical solution is then a compelling subject that deserves great attention, not least because of the difficulties to apply general purpose methods for fractional differential equations (FDEs) to this case. In this paper, we first transform the MTFDEs into equivalent systems of FDEs, as done by Diethelm and Ford; in this way, the solution can be expressed in terms of Mittag–Leffler (ML) functions evaluated at matrix arguments. We then propose to compute it by resorting to the matrix approach proposed by Garrappa and Popolizio. Several numerical tests are presented that clearly show that this matrix approach is very accurate and fast, also in comparison with other numerical methods. Full article
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
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Open AccessArticle Weyl and Marchaud Derivatives: A Forgotten History
Mathematics 2018, 6(1), 6; https://doi.org/10.3390/math6010006
Received: 22 November 2017 / Revised: 23 December 2017 / Accepted: 29 December 2017 / Published: 3 January 2018
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Abstract
In this paper, we recall the contribution given by Hermann Weyl and André Marchaud to the notion of fractional derivative. In addition, we discuss some relationships between the fractional Laplace operator and Marchaud derivative in the perspective to generalize these objects to different
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In this paper, we recall the contribution given by Hermann Weyl and André Marchaud to the notion of fractional derivative. In addition, we discuss some relationships between the fractional Laplace operator and Marchaud derivative in the perspective to generalize these objects to different fields of the mathematics. Full article
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
Open AccessArticle Application of Tempered-Stable Time Fractional-Derivative Model to Upscale Subdiffusion for Pollutant Transport in Field-Scale Discrete Fracture Networks
Mathematics 2018, 6(1), 5; https://doi.org/10.3390/math6010005
Received: 10 December 2017 / Revised: 29 December 2017 / Accepted: 29 December 2017 / Published: 3 January 2018
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Abstract
Fractional calculus provides efficient physical models to quantify non-Fickian dynamics broadly observed within the Earth system. The potential advantages of using fractional partial differential equations (fPDEs) for real-world problems are often limited by the current lack of understanding of how earth system properties
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Fractional calculus provides efficient physical models to quantify non-Fickian dynamics broadly observed within the Earth system. The potential advantages of using fractional partial differential equations (fPDEs) for real-world problems are often limited by the current lack of understanding of how earth system properties influence observed non-Fickian dynamics. This study explores non-Fickian dynamics for pollutant transport in field-scale discrete fracture networks (DFNs), by investigating how fracture and rock matrix properties influence the leading and tailing edges of pollutant breakthrough curves (BTCs). Fractured reservoirs exhibit erratic internal structures and multi-scale heterogeneity, resulting in complex non-Fickian dynamics. A Monte Carlo approach is used to simulate pollutant transport through DFNs with a systematic variation of system properties, and the resultant non-Fickian transport is upscaled using a tempered-stable fractional in time advection–dispersion equation. Numerical results serve as a basis for determining both qualitative and quantitative relationships between BTC characteristics and model parameters, in addition to the impacts of fracture density, orientation, and rock matrix permeability on non-Fickian dynamics. The observed impacts of medium heterogeneity on tracer transport at late times tend to enhance the applicability of fPDEs that may be parameterized using measurable fracture–matrix characteristics. Full article
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
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Open AccessArticle A Note on Hadamard Fractional Differential Equations with Varying Coefficients and Their Applications in Probability
Mathematics 2018, 6(1), 4; https://doi.org/10.3390/math6010004
Received: 18 November 2017 / Revised: 24 December 2017 / Accepted: 26 December 2017 / Published: 1 January 2018
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Abstract
In this paper, we show several connections between special functions arising from generalized Conway-Maxwell-Poisson (COM-Poisson) type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New analytical results are obtained, showing the particular role of Hadamard-type derivatives in connection with a
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In this paper, we show several connections between special functions arising from generalized Conway-Maxwell-Poisson (COM-Poisson) type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New analytical results are obtained, showing the particular role of Hadamard-type derivatives in connection with a recently introduced generalization of the Le Roy function. We are also able to prove a general connection between fractional hyper-Bessel-type equations involving Hadamard operators and Le Roy functions. Full article
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
Open AccessFeature PaperArticle Letnikov vs. Marchaud: A Survey on Two Prominent Constructions of Fractional Derivatives
Mathematics 2018, 6(1), 3; https://doi.org/10.3390/math6010003
Received: 22 November 2017 / Revised: 14 December 2017 / Accepted: 20 December 2017 / Published: 25 December 2017
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Abstract
In this survey paper, we analyze two constructions of fractional derivatives proposed by Aleksey Letnikov (1837–1888) and by André Marchaud (1887–1973), respectively. These derivatives play very important roles in Fractional Calculus and its applications. Full article
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
Open AccessArticle An Iterative Method for Solving a Class of Fractional Functional Differential Equations with “Maxima”
Mathematics 2018, 6(1), 2; https://doi.org/10.3390/math6010002
Received: 14 November 2017 / Revised: 16 December 2017 / Accepted: 19 December 2017 / Published: 22 December 2017
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Abstract
In the present work, we deal with nonlinear fractional differential equations with “maxima” and deviating arguments. The nonlinear part of the problem under consideration depends on the maximum values of the unknown function taken in time-dependent intervals. Proceeding by an iterative approach, we
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In the present work, we deal with nonlinear fractional differential equations with “maxima” and deviating arguments. The nonlinear part of the problem under consideration depends on the maximum values of the unknown function taken in time-dependent intervals. Proceeding by an iterative approach, we obtain the existence and uniqueness of the solution, in a context that does not fit within the framework of fixed point theory methods for the self-mappings, frequently used in the study of such problems. An example illustrating our main result is also given. Full article
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
Open AccessArticle A Numerical Solution of Fractional Lienard’s Equation by Using the Residual Power Series Method
Mathematics 2018, 6(1), 1; https://doi.org/10.3390/math6010001
Received: 1 October 2017 / Revised: 19 November 2017 / Accepted: 21 November 2017 / Published: 22 December 2017
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Abstract
In this paper, we investigate a numerical solution of Lienard’s equation. The residual power series (RPS) method is implemented to find an approximate solution to this problem. The proposed method is a combination of the fractional Taylor series and the residual functions. Numerical
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In this paper, we investigate a numerical solution of Lienard’s equation. The residual power series (RPS) method is implemented to find an approximate solution to this problem. The proposed method is a combination of the fractional Taylor series and the residual functions. Numerical and theoretical results are presented. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)
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