Correction: Thabet, H.; Kendre, S.; Chalishajar, D. New Analytical Technique for Solving a System of Nonlinear Fractional Partial Differential Equations *Mathematics* 2017, *5*, 47*Mathematics* **2018**, *6*(2), 26; doi:10.3390/math6020026 (registering DOI) - 14 February 2018**Abstract **

We have found some errors in the caption of Figure 1 and Figure 2 in our paper [1], and thus would like to make the following corrections:[...]
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Statics of Shallow Inclined Elastic Cables under General Vertical Loads: A Perturbation Approach*Mathematics* **2018**, *6*(2), 24; doi:10.3390/math6020024 - 13 February 2018**Abstract **

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The static problem for elastic shallow cables suspended at points at different levels under general vertical loads is addressed. The cases of both suspended and taut cables are considered. The funicular equation and the compatibility condition, well known in literature, are here shortly

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The static problem for elastic shallow cables suspended at points at different levels under general vertical loads is addressed. The cases of both suspended and taut cables are considered. The funicular equation and the compatibility condition, well known in literature, are here shortly re-derived, and the commonly accepted simplified hypotheses are recalled. Furthermore, with the aim of obtaining simple asymptotic expressions with a desired degree of accuracy, a perturbation method is designed, using the taut string solution as the generator system. The method is able to solve the static problem for any distributions of vertical loads and shows that the usual, simplified analysis is just the first step of the perturbation procedure proposed here.
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On Generalized Pata Type Contractions*Mathematics* **2018**, *6*(2), 25; doi:10.3390/math6020025 - 13 February 2018**Abstract **

In this paper, the existence of fixed point for Pata type Zamfirescu mapping in a complete metric space is proved. Our result give existence of fixed point for a wider class of functions and also prove the existence of best proximity point to

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In this paper, the existence of fixed point for Pata type Zamfirescu mapping in a complete metric space is proved. Our result give existence of fixed point for a wider class of functions and also prove the existence of best proximity point to the result on “A fixed point theorem in metric spaces” by vittorino Pata.
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Singularity Penetration with Unit Delay (SPUD)*Mathematics* **2018**, *6*(2), 23; doi:10.3390/math6020023 - 11 February 2018**Abstract **

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This manuscript reveals both the full experimental and methodical details of a most-recent patent that demonstrates a much-desired goal of rotational maneuvers via angular exchange momentum, namely extremely high torque without mathematical singularity and accompanying loss of attitude control while the angular momentum

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This manuscript reveals both the full experimental and methodical details of a most-recent patent that demonstrates a much-desired goal of rotational maneuvers via angular exchange momentum, namely extremely high torque without mathematical singularity and accompanying loss of attitude control while the angular momentum trajectory resides in the mathematical singularity. The paper briefly reviews the most recent literature, and then gives theoretical development for implementing the new control methods described in the patent to compute a non-singular steering command to the angular momentum actuators. The theoretical developments are followed by computer simulations used to verify the theoretical computation methodology, and then laboratory experiments are used for validation on a free-floating hardware simulator. A typical 3/4 CMG array skewed at 54.73° yields 0.15H. Utilizing the proposed singularity penetration techniques, 3H momentum is achieved about yaw, 2H about roll, and 1H about pitch representing performance increases of 1900%, 1233%, and 566% respectfully.
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Nonlinear Stability of *ρ*-Functional Equations in Latticetic Random Banach Lattice Spaces*Mathematics* **2018**, *6*(2), 22; doi:10.3390/math6020022 - 9 February 2018**Abstract **

In this paper, we prove the generalized nonlinear stability of the first and second of the following $\rho $ -functional equations, $G\left(\right|a|{\Delta}_{\mathcal{A}}^{*}\left|b\right|){\Delta}_{\mathcal{B}}^{*}G\left(\right|a|{\Delta}_{\mathcal{A}}^{*}$

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In this paper, we prove the generalized nonlinear stability of the first and second of the following $\rho $ -functional equations, $G\left(\right|a|{\Delta}_{\mathcal{A}}^{*}\left|b\right|){\Delta}_{\mathcal{B}}^{*}G\left(\right|a|{\Delta}_{\mathcal{A}}^{**}\left|b\right|)-G\left(\right|a\left|\right){\Delta}_{\mathcal{B}}^{**}G\left(\right|b\left|\right)=\rho (2\left[G,\left({\displaystyle \frac{\left|a\right|{\Delta}_{\mathcal{A}}^{*}\left|b\right|}{2}}\right),{\Delta}_{\mathcal{B}}^{*},G,\left({\displaystyle \frac{\left|a\right|{\Delta}_{\mathcal{A}}^{**}\left|b\right|}{2}}\right)\right]-G\left(\right|a\left|\right){\Delta}_{\mathcal{B}}^{**}G\left(\right|b\left|\right))$ , and $2\left[G,\left({\displaystyle \frac{\left|a\right|{\Delta}_{\mathcal{A}}^{*}\left|b\right|}{2}}\right),{\Delta}_{\mathcal{B}}^{*},G,\left({\displaystyle \frac{\left|a\right|{\Delta}_{\mathcal{A}}^{**}\left|b\right|}{2}}\right)\right]-G\left(\right|a\left|\right){\Delta}_{\mathcal{B}}^{**}G\left(\right|b\left|\right)=\rho \left(G\left(\right|a|,{\Delta}_{\mathcal{A}}^{*},\left|b\right|),{\Delta}_{\mathcal{B}}^{*},G\left(\right|a|,{\Delta}_{\mathcal{A}}^{**},\left|b\right|),-,G,\left(\right|a\left|\right),{\Delta}_{\mathcal{B}}^{**},G,\left(\right|b\left|\right)\right)$ in latticetic random Banach lattice spaces, where $\rho $ is a fixed real or complex number with $\rho \ne 1$ .
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A New Proof of a Conjecture on Nonpositive Ricci Curved Compact Kähler–Einstein Surfaces*Mathematics* **2018**, *6*(2), 21; doi:10.3390/math6020021 - 7 February 2018**Abstract **

In an earlier paper, we gave a proof of the conjecture of the pinching of the bisectional curvature mentioned in those two papers of Hong et al. of 1988 and 2011. Moreover, we proved that any compact Kähler–Einstein surface *M* is a quotient

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In an earlier paper, we gave a proof of the conjecture of the pinching of the bisectional curvature mentioned in those two papers of Hong et al. of 1988 and 2011. Moreover, we proved that any compact Kähler–Einstein surface *M* is a quotient of the complex two-dimensional unit ball or the complex two-dimensional plane if (1) *M* has a nonpositive Einstein constant, and (2) at each point, the average holomorphic sectional curvature is closer to the minimal than to the maximal. Following Siu and Yang, we used a minimal holomorphic sectional curvature direction argument, which made it easier for the experts in this direction to understand our proof. On this note, we use a maximal holomorphic sectional curvature direction argument, which is shorter and easier for the readers who are new in this direction.
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Numerical Methods for Solving Fuzzy Linear Systems*Mathematics* **2018**, *6*(2), 19; doi:10.3390/math6020019 - 1 February 2018**Abstract **

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In this article, three numerical iterative schemes, namely: Jacobi, Gauss–Seidel and Successive over-relaxation (SOR) have been proposed to solve a fuzzy system of linear equations (FSLEs). The convergence properties of these iterative schemes have been discussed. To display the validity of these iterative

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In this article, three numerical iterative schemes, namely: Jacobi, Gauss–Seidel and Successive over-relaxation (SOR) have been proposed to solve a fuzzy system of linear equations (FSLEs). The convergence properties of these iterative schemes have been discussed. To display the validity of these iterative schemes, an illustrative example with known exact solution is considered. Numerical results show that the SOR iterative method with $\omega =1.3$ provides more efficient results in comparison with other iterative techniques.
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Finite Automata Capturing Winning Sequences for All Possible Variants of the *PQ* Penny Flip Game*Mathematics* **2018**, *6*(2), 20; doi:10.3390/math6020020 - 1 February 2018**Abstract **

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The meticulous study of finite automata has produced many important and useful results. Automata are simple yet efficient finite state machines that can be utilized in a plethora of situations. It comes, therefore, as no surprise that they have been used in classic

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The meticulous study of finite automata has produced many important and useful results. Automata are simple yet efficient finite state machines that can be utilized in a plethora of situations. It comes, therefore, as no surprise that they have been used in classic game theory in order to model players and their actions. Game theory has recently been influenced by ideas from the field of quantum computation. As a result, quantum versions of classic games have already been introduced and studied. The $PQ$ penny flip game is a famous quantum game introduced by Meyer in 1999. In this paper, we investigate *all* possible finite games that can be played between the two players Q and Picard of the original $PQ$ game. For this purpose, we establish a rigorous connection between finite automata and the $PQ$ game along with all its possible variations. Starting from the automaton that corresponds to the original game, we construct more elaborate automata for certain extensions of the game, before finally presenting a semiautomaton that captures the intrinsic behavior of all possible variants of the $PQ$ game. What this means is that, from the semiautomaton in question, by setting appropriate initial and accepting states, one can construct deterministic automata able to capture every possible finite game that can be played between the two players Q and Picard. Moreover, we introduce the new concepts of a winning automaton and complete automaton for either player.
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Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches*Mathematics* **2018**, *6*(2), 17; doi:10.3390/math6020017 - 29 January 2018**Abstract **

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Pulsed-field gradient (PFG) diffusion experiments can be used to measure anomalous diffusion in many polymer or biological systems. However, it is still complicated to analyze PFG anomalous diffusion, particularly the finite gradient pulse width (FGPW) effect. In practical applications, the FGPW effect may

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Pulsed-field gradient (PFG) diffusion experiments can be used to measure anomalous diffusion in many polymer or biological systems. However, it is still complicated to analyze PFG anomalous diffusion, particularly the finite gradient pulse width (FGPW) effect. In practical applications, the FGPW effect may not be neglected, such as in clinical diffusion magnetic resonance imaging (MRI). Here, two significantly different methods are proposed to analyze PFG anomalous diffusion: the effective phase-shift diffusion equation (EPSDE) method and a method based on observing the signal intensity at the origin. The EPSDE method describes the phase evolution in virtual phase space, while the method to observe the signal intensity at the origin describes the magnetization evolution in real space. However, these two approaches give the same general PFG signal attenuation including the FGPW effect, which can be numerically evaluated by a direct integration method. The direct integration method is fast and without overflow. It is a convenient numerical evaluation method for Mittag-Leffler function-type PFG signal attenuation. The methods here provide a clear view of spin evolution under a field gradient, and their results will help the analysis of PFG anomalous diffusion.
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Notions of Rough Neutrosophic Digraphs*Mathematics* **2018**, *6*(2), 18; doi:10.3390/math6020018 - 29 January 2018**Abstract **

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Graph theory has numerous applications in various disciplines, including computer networks, neural networks, expert systems, cluster analysis, and image capturing. Rough neutrosophic set (NS) theory is a hybrid tool for handling uncertain information that exists in real life. In this research paper, we

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Graph theory has numerous applications in various disciplines, including computer networks, neural networks, expert systems, cluster analysis, and image capturing. Rough neutrosophic set (NS) theory is a hybrid tool for handling uncertain information that exists in real life. In this research paper, we apply the concept of rough NS theory to graphs and present a new kind of graph structure, rough neutrosophic digraphs. We present certain operations, including lexicographic products, strong products, rejection and tensor products on rough neutrosophic digraphs. We investigate some of their properties. We also present an application of a rough neutrosophic digraph in decision-making.
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Storage and Dissipation of Energy in Prabhakar Viscoelasticity*Mathematics* **2018**, *6*(2), 15; doi:10.3390/math6020015 - 23 January 2018**Abstract **

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In this paper, after a brief review of the physical notion of quality factor in viscoelasticity, we present a complete discussion of the attenuation processes emerging in the Maxwell–Prabhakar model, recently developed by Giusti and Colombaro. Then, taking profit of some illuminating plots,

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In this paper, after a brief review of the physical notion of quality factor in viscoelasticity, we present a complete discussion of the attenuation processes emerging in the Maxwell–Prabhakar model, recently developed by Giusti and Colombaro. Then, taking profit of some illuminating plots, we discuss some potential connections between the presented model and the modern mathematical modelling of seismic processes.
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Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial*Mathematics* **2018**, *6*(2), 16; doi:10.3390/math6020016 - 23 January 2018**Abstract **

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Solving differential equations of fractional (i.e., non-integer) order in an accurate, reliable and efficient way is much more difficult than in the standard integer-order case; moreover, the majority of the computational tools do not provide built-in functions for this kind of problem. In

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Solving differential equations of fractional (i.e., non-integer) order in an accurate, reliable and efficient way is much more difficult than in the standard integer-order case; moreover, the majority of the computational tools do not provide built-in functions for this kind of problem. In this paper, we review two of the most effective families of numerical methods for fractional-order problems, and we discuss some of the major computational issues such as the efficient treatment of the persistent memory term and the solution of the nonlinear systems involved in implicit methods. We present therefore a set of MATLAB routines specifically devised for solving three families of fractional-order problems: fractional differential equations (FDEs) (also for the non-scalar case), multi-order systems (MOSs) of FDEs and multi-term FDEs (also for the non-scalar case); some examples are provided to illustrate the use of the routines.
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Acknowledgement to Reviewers of *Mathematics* in 2017*Mathematics* **2018**, *6*(1), 14; doi:10.3390/math6010014 - 18 January 2018**Abstract **

Peer review is an essential part in the publication process, ensuring that Mathematics maintains high quality standards for its published papers.[...]
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Iterative Methods for Computing Vibrational Spectra*Mathematics* **2018**, *6*(1), 13; doi:10.3390/math6010013 - 16 January 2018**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.
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Best Approximation of the Fractional Semi-Derivative Operator by Exponential Series*Mathematics* **2018**, *6*(1), 12; doi:10.3390/math6010012 - 16 January 2018**Abstract **

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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

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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.
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Length-Fuzzy Subalgebras in *BCK*/*BCI*-Algebras*Mathematics* **2018**, *6*(1), 11; doi:10.3390/math6010011 - 12 January 2018**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 $BCK/BCI$ -algebras. The notion of length-fuzzy subalgebras in $BCK/BCI$ -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.
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The Collapse of Ecosystem Engineer Populations*Mathematics* **2018**, *6*(1), 9; doi:10.3390/math6010009 - 12 January 2018**Abstract **

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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

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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.
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Global Dynamics of Certain Mix Monotone Difference Equation*Mathematics* **2018**, *6*(1), 10; doi:10.3390/math6010010 - 12 January 2018**Abstract **

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We investigate global dynamics of the following second order rational difference equation ${x}_{n+1}=\frac{{x}_{n}{x}_{n-1}+\alpha {x}_{n}+\beta {x}_{n-1}}{a{x}_{n}{x}_{n-1}+b{x}_{}}$

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We investigate global dynamics of the following second order rational difference equation ${x}_{n+1}=\frac{{x}_{n}{x}_{n-1}+\alpha {x}_{n}+\beta {x}_{n-1}}{a{x}_{n}{x}_{n-1}+b{x}_{n-1}},$ where the parameters $\alpha ,\beta ,\phantom{\rule{0.166667em}{0ex}}a,\phantom{\rule{0.166667em}{0ex}}b$ 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.
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Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions*Mathematics* **2018**, *6*(1), 7; doi:10.3390/math6010007 - 9 January 2018**Abstract **

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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

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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.
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A Note on the Equivalence of Fractional Relaxation Equations to Differential Equations with Varying Coefficients*Mathematics* **2018**, *6*(1), 8; doi:10.3390/math6010008 - 9 January 2018**Abstract **

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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

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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.
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