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Axioms, Volume 7, Issue 3 (September 2018)

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Open AccessArticle Characterizations of the Total Space (Indefinite Trans-Sasakian Manifolds) Admitting a Semi-Symmetric Metric Connection
Received: 19 August 2018 / Revised: 4 September 2018 / Accepted: 7 September 2018 / Published: 10 September 2018
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Abstract
We investigate recurrent, Lie-recurrent, and Hopf lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. In these hypersurfaces, we obtain several new results. Moreover, we characterize that the total space (an indefinite generalized Sasakian space form) with a semi-symmetric metric
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We investigate recurrent, Lie-recurrent, and Hopf lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. In these hypersurfaces, we obtain several new results. Moreover, we characterize that the total space (an indefinite generalized Sasakian space form) with a semi-symmetric metric connection is an indefinite Kenmotsu space form under various lightlike hypersurfaces. Full article
(This article belongs to the Special Issue Applications of Differential Geometry)
Open AccessArticle Neutrosophic Triplet v-Generalized Metric Space
Received: 2 August 2018 / Revised: 26 August 2018 / Accepted: 2 September 2018 / Published: 6 September 2018
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Abstract
The notion of Neutrosophic triplet (NT) is a new theory in Neutrosophy. Also, the v-generalized metric is a specific form of the classical metrics. In this study, we introduced the notion of neutrosophic triplet v-generalized metric space (NTVGM), and we obtained
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The notion of Neutrosophic triplet (NT) is a new theory in Neutrosophy. Also, the v-generalized metric is a specific form of the classical metrics. In this study, we introduced the notion of neutrosophic triplet v-generalized metric space (NTVGM), and we obtained properties of NTVGM. Also, we showed that NTVGM is different from the classical metric and neutrosophic triplet metric (NTM). Furthermore, we introduced completeness of NTVGM. Full article
(This article belongs to the Special Issue Neutrosophic Topology)
Open AccessArticle Solutions to Abel’s Integral Equations in Distributions
Received: 10 August 2018 / Revised: 27 August 2018 / Accepted: 31 August 2018 / Published: 2 September 2018
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Abstract
The goal of this paper is to study fractional calculus of distributions, the generalized Abel’s integral equations, as well as fractional differential equations in the distributional space D(R+) based on inverse convolutional operators and Babenko’s approach. Furthermore, we
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The goal of this paper is to study fractional calculus of distributions, the generalized Abel’s integral equations, as well as fractional differential equations in the distributional space D ( R + ) based on inverse convolutional operators and Babenko’s approach. Furthermore, we provide interesting applications of Abel’s integral equations in viscoelastic systems, as well as solving other integral equations, such as θ π / 2 y ( φ ) cos β φ ( cos θ cos φ ) α d φ = f ( θ ) , and 0 x 1 / 2 g ( x ) y ( x + t ) d x = f ( t ) . Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications)
Open AccessArticle Interior Regularity Estimates for a Degenerate Elliptic Equation with Mixed Boundary Conditions
Received: 9 August 2018 / Revised: 24 August 2018 / Accepted: 27 August 2018 / Published: 1 September 2018
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Abstract
The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation
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The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation arises from the Bernardis–Reyes–Stinga–Torrea extension of the Dirichlet problem for the Marchaud fractional derivative. Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications)
Open AccessArticle Relations between the Complex Neutrosophic Sets with Their Applications in Decision Making
Received: 23 July 2018 / Revised: 20 August 2018 / Accepted: 24 August 2018 / Published: 1 September 2018
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Abstract
The basic aim of soft computing is to trade precision for a tractableness and reduction in solution cost by pushing the limits of tolerance for imprecision and uncertainty. This paper introduces a novel soft computing technique called complex neutrosophic relation (CNR) to evaluate
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The basic aim of soft computing is to trade precision for a tractableness and reduction in solution cost by pushing the limits of tolerance for imprecision and uncertainty. This paper introduces a novel soft computing technique called complex neutrosophic relation (CNR) to evaluate the degree of interaction between two complex neutrosophic sets (CNSs). CNSs are used to represent two-dimensional information that are imprecise, uncertain, incomplete and indeterminate. The Cartesian product of CNSs and subsequently the complex neutrosophic relation is formally defined. This relation is generalised from a conventional single valued neutrosophic relation (SVNR), based on CNSs, where the ranges of values of CNR are extended to the unit circle in complex plane for its membership functions instead of [0, 1] as in the conventional SVNR. A new algorithm is created using a comparison matrix of the SVNR after mapping the complex membership functions from complex space to the real space. This algorithm is then applied to scrutinise the impact of some teaching strategies on the student performance and the time frame(phase) of the interaction between these two variables. The notion of inverse, complement and composition of CNRs along with some related theorems and properties are introduced. The performance and utility of the composition concept in real-life situations is also demonstrated. Then, we define the concepts of projection and cylindric extension for CNRs along with illustrative examples. Some interesting properties are also obtained. Finally, a comparison between different existing relations and CNR to show the ascendancy of our proposed CNR is provided. Full article
(This article belongs to the Special Issue Neutrosophic Topology)
Open AccessArticle Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine
Received: 21 May 2018 / Revised: 21 August 2018 / Accepted: 22 August 2018 / Published: 1 September 2018
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Abstract
In this paper we discuss a new and very efficient implementation of high order accurate arbitrary high order schemes using derivatives discontinuous Galerkin (ADER-DG) finite element schemes on modern massively parallel supercomputers. The numerical methods apply to a very broad class of nonlinear
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In this paper we discuss a new and very efficient implementation of high order accurate arbitrary high order schemes using derivatives discontinuous Galerkin (ADER-DG) finite element schemes on modern massively parallel supercomputers. The numerical methods apply to a very broad class of nonlinear systems of hyperbolic partial differential equations. ADER-DG schemes are by construction communication-avoiding and cache-blocking, and are furthermore very well-suited for vectorization, and so they appear to be a good candidate for the future generation of exascale supercomputers. We introduce the numerical algorithm and show some applications to a set of hyperbolic equations with increasing levels of complexity, ranging from the compressible Euler equations over the equations of linear elasticity and the unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics to general relativistic magnetohydrodynamics (GRMHD) and the Einstein field equations of general relativity. We present strong scaling results of the new ADER-DG schemes up to 180,000 CPU cores. To our knowledge, these are the largest runs ever carried out with high order ADER-DG schemes for nonlinear hyperbolic PDE systems. We also provide a detailed performance comparison with traditional Runge-Kutta DG schemes. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle Umbral Methods and Harmonic Numbers
Received: 4 June 2018 / Revised: 22 August 2018 / Accepted: 24 August 2018 / Published: 1 September 2018
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Abstract
The theory of harmonic-based functions is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals. Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications)
Open AccessArticle Sub-Optimal Control in the Zika Virus Epidemic Model Using Differential Evolution
Received: 18 June 2018 / Revised: 17 August 2018 / Accepted: 18 August 2018 / Published: 23 August 2018
Cited by 1 | Viewed by 413 | PDF Full-text (487 KB) | HTML Full-text | XML Full-text
Abstract
A dynamical model of Zika virus (ZIKV) epidemic with direct transmission, sexual transmission, and vertical transmission is developed. A sub-optimal control problem to counter against the disease is proposed including three controls: vector elimination, vector-to-human contact reduction, and sexual contact reduction. Each control
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A dynamical model of Zika virus (ZIKV) epidemic with direct transmission, sexual transmission, and vertical transmission is developed. A sub-optimal control problem to counter against the disease is proposed including three controls: vector elimination, vector-to-human contact reduction, and sexual contact reduction. Each control variable is discretized into piece-wise constant intervals. The problem is solved by Differential Evolution (DE), which is one of the evolutionary algorithm developed for optimization. Two scenarios, namely four time horizons and eight time horizons, are compared and discussed. The simulations show that models with controls lead to decreasing the number of patients as well as epidemic period length. From the optimal solution, vector elimination is the prioritized strategy for disease control. Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications)
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Open AccessArticle George-Veeramani Fuzzy Metrics Revised
Received: 23 June 2018 / Revised: 14 August 2018 / Accepted: 17 August 2018 / Published: 23 August 2018
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Abstract
In this note, we present an alternative approach to the concept of a fuzzy metric, calling it a revised fuzzy metric. In contrast to the traditional approach to the theory of fuzzy metric spaces which is based on the use of a t
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In this note, we present an alternative approach to the concept of a fuzzy metric, calling it a revised fuzzy metric. In contrast to the traditional approach to the theory of fuzzy metric spaces which is based on the use of a t-norm, we proceed from a t-conorm in the definition of a revised fuzzy metric. Here, we restrict our study to the case of fuzzy metrics as they are defined by George-Veeramani, however, similar revision can be done also for some other approaches to the concept of a fuzzy metric. Full article
Open AccessArticle An Alternative to Real Number Axioms
Received: 26 June 2018 / Revised: 9 August 2018 / Accepted: 18 August 2018 / Published: 21 August 2018
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Abstract
In the present paper we consider one of the basic theorems of probability theory on real numbers. We prove that it is equivalent with the supremum axiom of real numbers. Full article
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Open AccessArticle On a Class of Conjugate Symplectic Hermite-Obreshkov One-Step Methods with Continuous Spline Extension
Received: 23 May 2018 / Revised: 13 August 2018 / Accepted: 15 August 2018 / Published: 20 August 2018
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Abstract
The class of A-stable symmetric one-step Hermite–Obreshkov (HO) methods introduced by F. Loscalzo in 1968 for dealing with initial value problems is analyzed. Such schemes have the peculiarity of admitting a multiple knot spline extension collocating the differential equation at the mesh points.
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The class of A-stable symmetric one-step Hermite–Obreshkov (HO) methods introduced by F. Loscalzo in 1968 for dealing with initial value problems is analyzed. Such schemes have the peculiarity of admitting a multiple knot spline extension collocating the differential equation at the mesh points. As a new result, it is shown that these maximal order schemes are conjugate symplectic, which is a benefit when the methods have to be applied to Hamiltonian problems. Furthermore, a new efficient approach for the computation of the spline extension is introduced, adopting the same strategy developed for the BS linear multistep methods. The performances of the schemes are tested in particular on some Hamiltonian benchmarks and compared with those of the Gauss–Runge–Kutta schemes and Euler–Maclaurin formulas of the same order. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle Single-Valued Neutrosophic Clustering Algorithm Based on Tsallis Entropy Maximization
Received: 10 May 2018 / Revised: 19 July 2018 / Accepted: 20 July 2018 / Published: 17 August 2018
Cited by 1 | Viewed by 401 | PDF Full-text (882 KB) | HTML Full-text | XML Full-text
Abstract
Data clustering is an important field in pattern recognition and machine learning. Fuzzy c-means is considered as a useful tool in data clustering. The neutrosophic set, which is an extension of the fuzzy set, has received extensive attention in solving many real-life
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Data clustering is an important field in pattern recognition and machine learning. Fuzzy c-means is considered as a useful tool in data clustering. The neutrosophic set, which is an extension of the fuzzy set, has received extensive attention in solving many real-life problems of inaccuracy, incompleteness, inconsistency and uncertainty. In this paper, we propose a new clustering algorithm, the single-valued neutrosophic clustering algorithm, which is inspired by fuzzy c-means, picture fuzzy clustering and the single-valued neutrosophic set. A novel suitable objective function, which is depicted as a constrained minimization problem based on a single-valued neutrosophic set, is built, and the Lagrange multiplier method is used to solve the objective function. We do several experiments with some benchmark datasets, and we also apply the method to image segmentation using the Lena image. The experimental results show that the given algorithm can be considered as a promising tool for data clustering and image processing. Full article
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Open AccessArticle Some Identities for Euler and Bernoulli Polynomials and Their Zeros
Received: 30 June 2018 / Revised: 27 July 2018 / Accepted: 11 August 2018 / Published: 14 August 2018
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Abstract
In this paper, we study some special polynomials which are related to Euler and Bernoulli polynomials. In addition, we give some identities for these polynomials. Finally, we investigate the zeros of these polynomials by using the computer. Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications)
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Open AccessArticle Conformable Laplace Transform of Fractional Differential Equations
Received: 30 June 2018 / Revised: 31 July 2018 / Accepted: 4 August 2018 / Published: 7 August 2018
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Abstract
In this paper, we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant
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In this paper, we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, the analytical solution for a class of fractional models associated with the logistic model, the von Foerster model and the Bertalanffy model is presented graphically for various fractional orders. The solution of the corresponding classical model is recovered as a particular case. Full article
(This article belongs to the Special Issue Fractional Differential Equations)
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Open AccessArticle A New Type of Generalization on W—Asymptotically J λ—Statistical Equivalence with the Number of α
Received: 19 June 2018 / Revised: 30 July 2018 / Accepted: 31 July 2018 / Published: 2 August 2018
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Abstract
In our paper, by using the concept of Wasymptotically J statistical equivalence of order α which has been previously defined, we present the definitions of Wasymptotically Jλstatistical equivalence of order α, Wstrongly
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In our paper, by using the concept of Wasymptotically J statistical equivalence of order α which has been previously defined, we present the definitions of Wasymptotically Jλstatistical equivalence of order α, Wstrongly asymptotically Jλstatistical equivalence of order α, and Wstrongly Cesáro asymptotically Jstatistical equivalence of order α where 0<α1. We also extend these notions with a sequence of positive real numbers, p=(pk), and we investigate how our results change if p is constant. Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications)
Open AccessArticle A Convex Model for Edge-Histogram Specification with Applications to Edge-Preserving Smoothing
Received: 18 June 2018 / Revised: 1 August 2018 / Accepted: 1 August 2018 / Published: 2 August 2018
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Abstract
The goal of edge-histogram specification is to find an image whose edge image has a histogram that matches a given edge-histogram as much as possible. Mignotte has proposed a non-convex model for the problem in 2012. In his work, edge magnitudes of an
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The goal of edge-histogram specification is to find an image whose edge image has a histogram that matches a given edge-histogram as much as possible. Mignotte has proposed a non-convex model for the problem in 2012. In his work, edge magnitudes of an input image are first modified by histogram specification to match the given edge-histogram. Then, a non-convex model is minimized to find an output image whose edge-histogram matches the modified edge-histogram. The non-convexity of the model hinders the computations and the inclusion of useful constraints such as the dynamic range constraint. In this paper, instead of considering edge magnitudes, we directly consider the image gradients and propose a convex model based on them. Furthermore, we include additional constraints in our model based on different applications. The convexity of our model allows us to compute the output image efficiently using either Alternating Direction Method of Multipliers or Fast Iterative Shrinkage-Thresholding Algorithm. We consider several applications in edge-preserving smoothing including image abstraction, edge extraction, details exaggeration, and documents scan-through removal. Numerical results are given to illustrate that our method successfully produces decent results efficiently. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle Trees, Stumps, and Applications
Received: 23 May 2018 / Revised: 16 July 2018 / Accepted: 16 July 2018 / Published: 1 August 2018
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Abstract
The traditional derivation of Runge–Kutta methods is based on the use of the scalar test problem y(x)=f(x,y(x)). However, above order 4, this gives less restrictive order conditions than
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The traditional derivation of Runge–Kutta methods is based on the use of the scalar test problem y(x)=f(x,y(x)). However, above order 4, this gives less restrictive order conditions than those obtained from a vector test problem using a tree-based theory. In this paper, stumps, or incomplete trees, are introduced to explain the discrepancy between the two alternative theories. Atomic stumps can be combined multiplicatively to generate all trees. For the scalar test problem, these quantities commute, and certain sets of trees form isomeric classes. There is a single order condition for each class, whereas for the general vector-based problem, for which commutation of atomic stumps does not occur, there is exactly one order condition for each tree. In the case of order 5, the only nontrivial isomeric class contains two trees, and the number of order conditions reduces from 17 to 16 for scalar problems. A method is derived that satisfies the 16 conditions for scalar problems but not the complete set based on 17 trees. Hence, as a practical numerical method, it has order 4 for a general initial value problem, but this increases to order 5 for a scalar problem. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
Open AccessArticle A Gradient System for Low Rank Matrix Completion
Received: 7 May 2018 / Revised: 18 July 2018 / Accepted: 18 July 2018 / Published: 24 July 2018
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Abstract
In this article we present and discuss a two step methodology to find the closest low rank completion of a sparse large matrix. Given a large sparse matrix M, the method consists of fixing the rank to r and then looking for
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In this article we present and discuss a two step methodology to find the closest low rank completion of a sparse large matrix. Given a large sparse matrix M, the method consists of fixing the rank to r and then looking for the closest rank-r matrix X to M, where the distance is measured in the Frobenius norm. A key element in the solution of this matrix nearness problem consists of the use of a constrained gradient system of matrix differential equations. The obtained results, compared to those obtained by different approaches show that the method has a correct behaviour and is competitive with the ones available in the literature. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle Certain Notions of Energy in Single-Valued Neutrosophic Graphs
Received: 12 June 2018 / Revised: 14 July 2018 / Accepted: 16 July 2018 / Published: 20 July 2018
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Abstract
A single-valued neutrosophic set is an instance of a neutrosophic set, which provides us an additional possibility to represent uncertainty, imprecise, incomplete and inconsistent information existing in real situations. In this research study, we present concepts of energy, Laplacian energy and signless Laplacian
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A single-valued neutrosophic set is an instance of a neutrosophic set, which provides us an additional possibility to represent uncertainty, imprecise, incomplete and inconsistent information existing in real situations. In this research study, we present concepts of energy, Laplacian energy and signless Laplacian energy in single-valued neutrosophic graphs (SVNGs), describe some of their properties and develop relationship among them. We also consider practical examples to illustrate the applicability of the our proposed concepts. Full article
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Open AccessArticle Block Generalized Locally Toeplitz Sequences: From the Theory to the Applications
Received: 9 May 2018 / Revised: 6 July 2018 / Accepted: 16 July 2018 / Published: 19 July 2018
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Abstract
The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices An arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to
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The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices An arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to infinity, these matrices An give rise to a sequence {An}n, which often turns out to be a GLT sequence or one of its “relatives”, i.e., a block GLT sequence or a reduced GLT sequence. In particular, block GLT sequences are encountered in the discretization of systems of DEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar DEs. Despite the applicative interest, a solid theory of block GLT sequences has been developed only recently, in 2018. The purpose of the present paper is to illustrate the potential of this theory by presenting a few noteworthy examples of applications in the context of DE discretizations. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative
Received: 10 April 2018 / Revised: 3 July 2018 / Accepted: 13 July 2018 / Published: 18 July 2018
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Abstract
One-dimensional equations of telegrapher’s-type (TE) and Guyer–Krumhansl-type (GK-type) with substantial derivative considered and operational solutions to them are given. The role of the exponential differential operators is discussed. The examples of their action on some initial functions are explored. Proper solutions are constructed
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One-dimensional equations of telegrapher’s-type (TE) and Guyer–Krumhansl-type (GK-type) with substantial derivative considered and operational solutions to them are given. The role of the exponential differential operators is discussed. The examples of their action on some initial functions are explored. Proper solutions are constructed in the integral form and some examples are studied with solutions in elementary functions. A system of hyperbolic-type inhomogeneous differential equations (DE), describing non-Fourier heat transfer with substantial derivative thin films, is considered. Exact harmonic solutions to these equations are obtained for the Cauchy and the Dirichlet conditions. The application to the ballistic heat transport in thin films is studied; the ballistic properties are accounted for by the Knudsen number. Two-speed heat propagation process is demonstrated—fast evolution of the ballistic quasi-temperature component in low-dimensional systems is elucidated and compared with slow diffusive heat-exchange process. The comparative analysis of the obtained solutions is performed. Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications)
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Open AccessArticle Neutrosophic Incidence Graphs With Application
Received: 12 June 2018 / Revised: 7 July 2018 / Accepted: 13 July 2018 / Published: 18 July 2018
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Abstract
In this research study, we introduce the notion of single-valued neutrosophic incidence graphs. We describe certain concepts, including bridges, cut vertex and blocks in single-valued neutrosophic incidence graphs. We present some properties of single-valued neutrosophic incidence graphs. We discuss the edge-connectivity, vertex-connectivity and
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In this research study, we introduce the notion of single-valued neutrosophic incidence graphs. We describe certain concepts, including bridges, cut vertex and blocks in single-valued neutrosophic incidence graphs. We present some properties of single-valued neutrosophic incidence graphs. We discuss the edge-connectivity, vertex-connectivity and pair-connectivity in neutrosophic incidence graphs. We also deal with a mathematical model of the situation of illegal migration from Pakistan to Europe. Full article
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Open AccessArticle Optimal B-Spline Bases for the Numerical Solution of Fractional Differential Problems
Received: 14 May 2018 / Revised: 17 June 2018 / Accepted: 22 June 2018 / Published: 2 July 2018
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Abstract
Efficient numerical methods to solve fractional differential problems are particularly required in order to approximate accurately the nonlocal behavior of the fractional derivative. The aim of the paper is to show how optimal B-spline bases allow us to construct accurate numerical methods that
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Efficient numerical methods to solve fractional differential problems are particularly required in order to approximate accurately the nonlocal behavior of the fractional derivative. The aim of the paper is to show how optimal B-spline bases allow us to construct accurate numerical methods that have a low computational cost. First of all, we describe in detail how to construct optimal B-spline bases on bounded intervals and recall their main properties. Then, we give the analytical expression of their derivatives of fractional order and use these bases in the numerical solution of fractional differential problems. Some numerical tests showing the good performances of the bases in solving a time-fractional diffusion problem by a collocation–Galerkin method are also displayed. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessReview Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review
Received: 27 April 2018 / Revised: 18 June 2018 / Accepted: 19 June 2018 / Published: 1 July 2018
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Abstract
We present a collection of recent results on the numerical approximation of Volterra integral equations and integro-differential equations by means of collocation type methods, which are able to provide better balances between accuracy and stability demanding. We consider both exact and discretized one-step
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We present a collection of recent results on the numerical approximation of Volterra integral equations and integro-differential equations by means of collocation type methods, which are able to provide better balances between accuracy and stability demanding. We consider both exact and discretized one-step and multistep collocation methods, and illustrate main convergence results, making some comparisons in terms of accuracy and efficiency. Some numerical experiments complete the paper. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle On Partial Cholesky Factorization and a Variant of Quasi-Newton Preconditioners for Symmetric Positive Definite Matrices
Received: 23 April 2018 / Revised: 19 June 2018 / Accepted: 20 June 2018 / Published: 1 July 2018
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Abstract
This work studies limited memory preconditioners for linear symmetric positive definite systems of equations. Connections are established between a partial Cholesky factorization from the literature and a variant of Quasi-Newton type preconditioners. Then, a strategy for enhancing the Quasi-Newton preconditioner via available information
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This work studies limited memory preconditioners for linear symmetric positive definite systems of equations. Connections are established between a partial Cholesky factorization from the literature and a variant of Quasi-Newton type preconditioners. Then, a strategy for enhancing the Quasi-Newton preconditioner via available information is proposed. Numerical experiments show the behaviour of the resulting preconditioner. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle Refinement Algorithms for Adaptive Isogeometric Methods with Hierarchical Splines
Received: 11 May 2018 / Revised: 16 June 2018 / Accepted: 18 June 2018 / Published: 21 June 2018
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Abstract
The construction of suitable mesh configurations for spline models that provide local refinement capabilities is one of the fundamental components for the analysis and development of adaptive isogeometric methods. We investigate the design and implementation of refinement algorithms for hierarchical B-spline spaces that
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The construction of suitable mesh configurations for spline models that provide local refinement capabilities is one of the fundamental components for the analysis and development of adaptive isogeometric methods. We investigate the design and implementation of refinement algorithms for hierarchical B-spline spaces that enable the construction of locally graded meshes. The refinement rules properly control the interaction of basis functions at different refinement levels. This guarantees a bounded number of nonvanishing (truncated) hierarchical B-splines on any mesh element. The performances of the algorithms are validated with standard benchmark problems. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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