A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind*Mathematics* **2017**, *5*(2), 28; doi:10.3390/math5020028 - 22 May 2017**Abstract **

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A numerical method is proposed for estimating piecewise-constant solutions for Fredholm integral equations of the first kind. Two functionals, namely the weighted total variation (WTV) functional and the simplified Modica-Mortola (MM) functional, are introduced. The solution procedure consists of two stages. In the

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A numerical method is proposed for estimating piecewise-constant solutions for Fredholm integral equations of the first kind. Two functionals, namely the weighted total variation (WTV) functional and the simplified Modica-Mortola (MM) functional, are introduced. The solution procedure consists of two stages. In the first stage, the WTV functional is minimized to obtain an approximate solution ${\mathbf{f}}_{\mathrm{TV}}^{*}$ . In the second stage, the simplified MM functional is minimized to obtain the final result by using the damped Newton (DN) method with ${\mathbf{f}}_{\mathrm{TV}}^{*}$ as the initial guess. The numerical implementation is given in detail, and numerical results of two examples are presented to illustrate the efficiency of the proposed approach.
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Analysis of Magneto-hydrodynamics Flow and Heat Transfer of a Viscoelastic Fluid through Porous Medium in Wire Coating Analysis*Mathematics* **2017**, *5*(2), 27; doi:10.3390/math5020027 - 16 May 2017**Abstract **

Wire coating process is a continuous extrusion process for primary insulation of conducting wires with molten polymers for mechanical strength and protection in aggressive environments. Nylon, polysulfide, low/high density polyethylene (LDPE/HDPE) and plastic polyvinyl chloride (PVC) are the common and important plastic resin

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Wire coating process is a continuous extrusion process for primary insulation of conducting wires with molten polymers for mechanical strength and protection in aggressive environments. Nylon, polysulfide, low/high density polyethylene (LDPE/HDPE) and plastic polyvinyl chloride (PVC) are the common and important plastic resin used for wire coating. In the current study, wire coating is performed using viscoelastic third grade fluid in the presence of applied magnetic field and porous medium. The governing equations are first modeled and then solved analytically by utilizing the homotopy analysis method (HAM). The convergence of the series solution is established. A numerical technique called ND-solve method is used for comparison and found good agreement. The effect of pertinent parameters on the velocity field and temperature profile is shown with the help of graphs. It is observed that the velocity profiles increase as the value of viscoelastic third grade parameter $\beta $ increase and decrease as the magnetic parameter $M$ and permeability parameter $K$ increase. It is also observed that the temperature profiles increases as the Brinkman number $Br$ , permeability parameter $K$ , magnetic parameter $M$ and viscoelastic third grade parameter (non-Newtonian parameter) $\beta $ increase.
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A New Variational Iteration Method for a Class of Fractional Convection-Diffusion Equations in Large Domains*Mathematics* **2017**, *5*(2), 26; doi:10.3390/math5020026 - 11 May 2017**Abstract **

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In this paper, we introduced a new generalization method to solve fractional convection–diffusion equations based on the well-known variational iteration method (VIM) improved by an auxiliary parameter. The suggested method was highly effective in controlling the convergence region of the approximate solution. By

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In this paper, we introduced a new generalization method to solve fractional convection–diffusion equations based on the well-known variational iteration method (VIM) improved by an auxiliary parameter. The suggested method was highly effective in controlling the convergence region of the approximate solution. By solving some fractional convection–diffusion equations with a propounded method and comparing it with standard VIM, it was concluded that complete reliability, efficiency, and accuracy of this method are guaranteed. Additionally, we studied and investigated the convergence of the proposed method, namely the VIM with an auxiliary parameter. We also offered the optimal choice of the auxiliary parameter in the proposed method. It was noticed that the approach could be applied to other models of physics.
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Fixed Points of Set Valued Mappings in Terms of Start Point on a Metric Space Endowed with a Directed Graph*Mathematics* **2017**, *5*(2), 24; doi:10.3390/math5020024 - 19 April 2017**Abstract **

In the present article, we introduce the new concept of start point in a directed graph and provide the characterizations required for a directed graph to have a start point. We also define the notion of a self path set valued map and

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In the present article, we introduce the new concept of start point in a directed graph and provide the characterizations required for a directed graph to have a start point. We also define the notion of a self path set valued map and establish its relation with start point in the setting of a metric space endowed with a directed graph. Further, some fixed point theorems for set valued maps have been proven in this context. A version of the Knaster–Tarski theorem has also been established using our results.
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Discrete-Time Fractional Optimal Control*Mathematics* **2017**, *5*(2), 25; doi:10.3390/math5020025 - 19 April 2017**Abstract **

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A formulation and solution of the discrete-time fractional optimal control problem in terms of the Caputo fractional derivative is presented in this paper. The performance index (PI) is considered in a quadratic form. The necessary and transversality conditions are obtained using a Hamiltonian

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A formulation and solution of the discrete-time fractional optimal control problem in terms of the Caputo fractional derivative is presented in this paper. The performance index (PI) is considered in a quadratic form. The necessary and transversality conditions are obtained using a Hamiltonian approach. Both the free and fixed final state cases have been considered. Numerical examples are taken up and their solution technique is presented. Results are produced for different values of $\alpha $ .
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Best Proximity Point Results in Non-Archimedean Modular Metric Space*Mathematics* **2017**, *5*(2), 23; doi:10.3390/math5020023 - 5 April 2017**Abstract **

In this paper, we introduce the new notion of Suzuki-type $(\alpha ,\beta ,\theta ,\gamma )$ -contractive mapping and investigate the existence and uniqueness of the best proximity point for such mappings in non-Archimedean modular metric space using the weak

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In this paper, we introduce the new notion of Suzuki-type $(\alpha ,\beta ,\theta ,\gamma )$ -contractive mapping and investigate the existence and uniqueness of the best proximity point for such mappings in non-Archimedean modular metric space using the weak ${P}_{\lambda}$ -property. Meanwhile, we present an illustrative example to emphasize the realized improvements. These obtained results extend and improve certain well-known results in the literature.
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On Optimal Fuzzy Best Proximity Coincidence Points of Proximal Contractions Involving Cyclic Mappings in Non-Archimedean Fuzzy Metric Spaces*Mathematics* **2017**, *5*(2), 22; doi:10.3390/math5020022 - 1 April 2017**Abstract **

The main objective of this paper is to deal with some properties of interest in two types of fuzzy ordered proximal contractions of cyclic self-mappings $T$ integrated in a pair $\left(g\text{},\text{}T\right)$ of mappings. In particular, $g$ is a non-contractive

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The main objective of this paper is to deal with some properties of interest in two types of fuzzy ordered proximal contractions of cyclic self-mappings $T$ integrated in a pair $\left(g\text{},\text{}T\right)$ of mappings. In particular, $g$ is a non-contractive fuzzy self-mapping, in the framework of non-Archimedean ordered fuzzy complete metric spaces and $T$ is a $p$ -cyclic proximal contraction. Two types of such contractions (so called of type I and of type II) are dealt with. In particular, the existence, uniqueness and limit properties for sequences to optimal fuzzy best proximity coincidence points are investigated for such pairs of mappings.
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On Some Extended Block Krylov Based Methods for Large Scale Nonsymmetric Stein Matrix Equations*Mathematics* **2017**, *5*(2), 21; doi:10.3390/math5020021 - 27 March 2017**Abstract **

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In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term $AXB-X+E{F}^{T}=0$ . These matrix equations appear in many applications in discrete-time control problems, filtering and image

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In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term $AXB-X+E{F}^{T}=0$ . These matrix equations appear in many applications in discrete-time control problems, filtering and image restoration and others. The proposed methods are based on projection onto the extended block Krylov subspace with a Galerkin approach (GA) or with the minimization of the norm of the residual. We give some results on the residual and error norms and report some numerical experiments.
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In this paper, some properties of $\mathcal{F}$ -harmonic and conformal $\mathcal{F}$ -harmonic maps between doubly warped product manifolds are studied and new examples of non-harmonic $\mathcal{F}$ -harmonic maps are constructed.

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In this paper, some properties of $\mathcal{F}$ -harmonic and conformal $\mathcal{F}$ -harmonic maps between doubly warped product manifolds are studied and new examples of non-harmonic $\mathcal{F}$ -harmonic maps are constructed.
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A Generalization of *b*-Metric Space and Some Fixed Point Theorems*Mathematics* **2017**, *5*(2), 19; doi:10.3390/math5020019 - 23 March 2017**Abstract **
In this paper, inspired by the concept of *b*-metric space, we introduce the concept of extended *b*-metric space. We also establish some fixed point theorems for self-mappings defined on such spaces. Our results extend/generalize many pre-existing results in literature.
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Characterization of the Minimizing Graph of the Connected Graphs Whose Complements Are Bicyclic*Mathematics* **2017**, *5*(1), 18; doi:10.3390/math5010018 - 11 March 2017**Abstract **

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In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to

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In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. Let ${\mathcal{G}}_{1,n}^{c}$ and ${\mathcal{G}}_{2,n}^{c}$ be the classes of the connected graphs of order *n* whose complements are bicyclic with exactly two and three cycles, respectively. In this paper, we characterize the unique minimizing graph among all the graphs which belong to ${\mathcal{G}}_{n}^{c}={\mathcal{G}}_{1,n}^{c}\cup {\mathcal{G}}_{2,n}^{c}$ , a class of the connected graphs of order *n* whose complements are bicyclic.
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On the Additively Weighted Harary Index of Some Composite Graphs*Mathematics* **2017**, *5*(1), 16; doi:10.3390/math5010016 - 7 March 2017**Abstract **

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The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The additively weighted Harary index ${H}_{A}\left(G\right)$ is a modification of the Harary index in which the contributions of

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The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The additively weighted Harary index ${H}_{A}\left(G\right)$ is a modification of the Harary index in which the contributions of vertex pairs are weighted by the sum of their degrees. This new invariant was introduced in (Alizadeh, Iranmanesh and Došlić. *Additively weighted Harary index of some composite graphs*, Discrete Math, 2013) and they posed the following question: *What is the behavior of *${H}_{A}\left(G\right)$ when G is a composite graph resulting for example by: splice, link, corona and rooted product? We investigate the additively weighted Harary index for these standard graph products. Then we obtain lower and upper bounds for some of them.
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Certain Concepts of Bipolar Fuzzy Directed Hypergraphs*Mathematics* **2017**, *5*(1), 17; doi:10.3390/math5010017 - 4 March 2017**Abstract **

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A hypergraph is the most developed tool for modeling various practical problems in different fields, including computer sciences, biological sciences, social networks and psychology. Sometimes, given data in a network model are based on bipolar information rather than one sided. To deal with

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A hypergraph is the most developed tool for modeling various practical problems in different fields, including computer sciences, biological sciences, social networks and psychology. Sometimes, given data in a network model are based on bipolar information rather than one sided. To deal with such types of problems, we use mathematical models that are based on bipolar fuzzy (BF) sets. In this research paper, we introduce the concept of BF directed hypergraphs. We describe certain operations on BF directed hypergraphs, including addition, multiplication, vertex-wise multiplication and structural subtraction. We introduce the concept of $B=({m}^{+},{m}^{-})$ -tempered BF directed hypergraphs and investigate some of their properties. We also present an algorithm to compute the minimum arc length of a BF directed hyperpath.
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Dialectical Multivalued Logic and Probabilistic Theory*Mathematics* **2017**, *5*(1), 15; doi:10.3390/math5010015 - 23 February 2017**Abstract **

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There are two probabilistic algebras: one for classical probability and the other for quantum mechanics. Naturally, it is the relation to the object that decides, as in the case of logic, which algebra is to be used. From a paraconsistent multivalued logic therefore,

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There are two probabilistic algebras: one for classical probability and the other for quantum mechanics. Naturally, it is the relation to the object that decides, as in the case of logic, which algebra is to be used. From a paraconsistent multivalued logic therefore, one can derive a probability theory, adding the correspondence between truth value and fortuity.
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A Novel Iterative Algorithm Applied to Totally Asymptotically Nonexpansive Mappings in CAT(0) Spaces*Mathematics* **2017**, *5*(1), 14; doi:10.3390/math5010014 - 22 February 2017**Abstract **

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In this paper we introduce a new iterative algorithm for approximating fixed points of totally asymptotically quasi-nonexpansive mappings on CAT(0) spaces. We prove a strong convergence theorem under suitable conditions. The result we obtain improves and extends several recent results stated by many

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In this paper we introduce a new iterative algorithm for approximating fixed points of totally asymptotically quasi-nonexpansive mappings on CAT(0) spaces. We prove a strong convergence theorem under suitable conditions. The result we obtain improves and extends several recent results stated by many others; they also complement many known recent results in the literature. We then provide some numerical examples to illustrate our main result and to display the efficiency of the proposed algorithm.
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A Few Finite Trigonometric Sums*Mathematics* **2017**, *5*(1), 13; doi:10.3390/math5010013 - 18 February 2017**Abstract **

Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known; however, sums with products of trigonometric functions can become complicated, and

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Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known; however, sums with products of trigonometric functions can become complicated, and may not have a simple expression in a number of cases. Some of these sums have interesting properties, and can have amazingly simple values. However, only some of them are available in the literature. We obtain a number of such sums using the method of residues.
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The Split Common Fixed Point Problem for a Family of Multivalued Quasinonexpansive Mappings and Totally Asymptotically Strictly Pseudocontractive Mappings in Banach Spaces*Mathematics* **2017**, *5*(1), 11; doi:10.3390/math5010011 - 11 February 2017**Abstract **

In this paper, we introduce an iterative algorithm for solving the split common fixed point problem for a family of multi-valued quasinonexpansive mappings and totally asymptotically strictly pseudocontractive mappings, as well as for a family of totally quasi-*ϕ*-asymptotically nonexpansive mappings and

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In this paper, we introduce an iterative algorithm for solving the split common fixed point problem for a family of multi-valued quasinonexpansive mappings and totally asymptotically strictly pseudocontractive mappings, as well as for a family of totally quasi-*ϕ*-asymptotically nonexpansive mappings and *k*-quasi-strictly pseudocontractive mappings in the setting of Banach spaces. Our results improve and extend the results of Tang et al., Takahashi, Moudafi, Censor et al., and Byrne et al.
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Fractional Fokker-Planck Equation*Mathematics* **2017**, *5*(1), 12; doi:10.3390/math5010012 - 11 February 2017**Abstract **

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We shall discuss the numerical solution of the Cauchy problem for the fully fractional Fokker-Planck (fFP) equation in connection with Sinc convolution methods. The numerical approximation is based on *Caputo* and *Riesz-Feller* fractional derivatives. The use of the transfer function in Laplace and

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We shall discuss the numerical solution of the Cauchy problem for the fully fractional Fokker-Planck (fFP) equation in connection with Sinc convolution methods. The numerical approximation is based on *Caputo* and *Riesz-Feller* fractional derivatives. The use of the transfer function in Laplace and Fourier spaces in connection with Sinc convolutions allow to find exponentially converging computing schemes. Examples using different initial conditions demonstrate the effective computations with a small number of grid points on an infinite spatial domain.
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Existence of Mild Solutions for Impulsive Fractional Integro-Differential Inclusions with State-Dependent Delay*Mathematics* **2017**, *5*(1), 9; doi:10.3390/math5010009 - 25 January 2017**Abstract **
In this manuscript, we implement Bohnenblust–Karlin’s fixed point theorem to demonstrate the existence of mild solutions for a class of impulsive fractional integro-differential inclusions (IFIDI) with state-dependent delay (SDD) in Banach spaces. An example is provided to illustrate the obtained abstract results.
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Approximation in Müntz Spaces M_{Λ,p} of L_{p} Functions for 1 < *p* < ∞ and Bases*Mathematics* **2017**, *5*(1), 10; doi:10.3390/math5010010 - 25 January 2017**Abstract **

Müntz spaces satisfying the Müntz and gap conditions are considered. A Fourier approximation of functions in the Müntz spaces M_{Λ,p} of L_{p} functions is studied, where 1 < *p* < ∞. It is proven that up to an isomorphism and

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Müntz spaces satisfying the Müntz and gap conditions are considered. A Fourier approximation of functions in the Müntz spaces M_{Λ,p} of L_{p} functions is studied, where 1 < *p* < ∞. It is proven that up to an isomorphism and a change of variables, these spaces are contained in Weil–Nagy’s class. Moreover, the existence of Schauder bases in the Müntz spaces M_{Λ,p} is investigated.
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