Axioms

15 pages, 232 KiB

Open AccessArticle

Characterizations of the Total Space (Indefinite Trans-Sasakian Manifolds) Admitting a Semi-Symmetric Metric Connection

by Dae Ho Jin and Jae Won Lee

Axioms 2018, 7(3), 68; https://doi.org/10.3390/axioms7030068 - 10 Sep 2018

Viewed by 3179

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 [...] Read more.

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)

8 pages, 240 KiB

Open AccessArticle

Neutrosophic Triplet v-Generalized Metric Space

by Memet Şahin and Abdullah Kargın

Axioms 2018, 7(3), 67; https://doi.org/10.3390/axioms7030067 - 6 Sep 2018

Cited by 17 | Viewed by 3255

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 [...] Read more.

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)

17 pages, 281 KiB

Open AccessArticle

Solutions to Abel’s Integral Equations in Distributions

by Chenkuan Li, Thomas Humphries and Hunter Plowman

Axioms 2018, 7(3), 66; https://doi.org/10.3390/axioms7030066 - 2 Sep 2018

Cited by 6 | Viewed by 4172

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 [...] Read more.

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

\int_{θ}^{π / 2} \frac{y (φ)}{\cos^{β} φ {(\cos θ - \cos φ)}^{α}} d φ = f (θ)

, and

\int_{0}^{\infty} 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)

16 pages, 856 KiB

Open AccessArticle

Interior Regularity Estimates for a Degenerate Elliptic Equation with Mixed Boundary Conditions

by Jean-Daniel Djida and Arran Fernandez

Axioms 2018, 7(3), 65; https://doi.org/10.3390/axioms7030065 - 1 Sep 2018

Cited by 3 | Viewed by 3892

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 [...] Read more.

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)

15 pages, 272 KiB

Open AccessArticle

Relations between the Complex Neutrosophic Sets with Their Applications in Decision Making

by Ashraf Al-Quran and Shawkat Alkhazaleh

Axioms 2018, 7(3), 64; https://doi.org/10.3390/axioms7030064 - 1 Sep 2018

Cited by 17 | Viewed by 3288

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 [...] Read more.

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)

26 pages, 5646 KiB

Open AccessArticle

Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine

by Michael Dumbser, Francesco Fambri, Maurizio Tavelli, Michael Bader and Tobias Weinzierl

Axioms 2018, 7(3), 63; https://doi.org/10.3390/axioms7030063 - 1 Sep 2018

Cited by 41 | Viewed by 5159

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 [...] Read more.

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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9 pages, 243 KiB

Open AccessArticle

Umbral Methods and Harmonic Numbers

by Giuseppe Dattoli, Bruna Germano, Silvia Licciardi and Maria Renata Martinelli

Axioms 2018, 7(3), 62; https://doi.org/10.3390/axioms7030062 - 1 Sep 2018

Cited by 7 | Viewed by 3422

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)

14 pages, 487 KiB

Open AccessArticle

Sub-Optimal Control in the Zika Virus Epidemic Model Using Differential Evolution

by Nonthamon Chaikham and Wannika Sawangtong

Axioms 2018, 7(3), 61; https://doi.org/10.3390/axioms7030061 - 23 Aug 2018

Cited by 2 | Viewed by 3608

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 [...] Read more.

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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6 pages, 223 KiB

Open AccessArticle

George-Veeramani Fuzzy Metrics Revised

by Alexander Šostak

Axioms 2018, 7(3), 60; https://doi.org/10.3390/axioms7030060 - 23 Aug 2018

Cited by 7 | Viewed by 3329

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 [...] Read more.

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

6 pages, 2917 KiB

Open AccessArticle

An Alternative to Real Number Axioms

by Igor Líška, Beloslav Riečan and Anna Tirpáková

Axioms 2018, 7(3), 59; https://doi.org/10.3390/axioms7030059 - 21 Aug 2018

Viewed by 2998

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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20 pages, 993 KiB

Open AccessArticle

On a Class of Hermite-Obreshkov One-Step Methods with Continuous Spline Extension

by Francesca Mazzia and Alessandra Sestini

Axioms 2018, 7(3), 58; https://doi.org/10.3390/axioms7030058 - 20 Aug 2018

Cited by 12 | Viewed by 3379 | Correction

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. [...] Read more.

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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12 pages, 882 KiB

Open AccessArticle

Single-Valued Neutrosophic Clustering Algorithm Based on Tsallis Entropy Maximization

by Qiaoyan Li, Yingcang Ma, Florentin Smarandache and Shuangwu Zhu

Axioms 2018, 7(3), 57; https://doi.org/10.3390/axioms7030057 - 17 Aug 2018

Cited by 12 | Viewed by 3559

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 [...] Read more.

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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19 pages, 894 KiB

Open AccessArticle

Some Identities for Euler and Bernoulli Polynomials and Their Zeros

by Taekyun Kim and Cheon Seoung Ryoo

Axioms 2018, 7(3), 56; https://doi.org/10.3390/axioms7030056 - 14 Aug 2018

Cited by 45 | Viewed by 4798 | Correction

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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12 pages, 289 KiB

Open AccessArticle

Conformable Laplace Transform of Fractional Differential Equations

by Fernando S. Silva, Davidson M. Moreira and Marcelo A. Moret

Axioms 2018, 7(3), 55; https://doi.org/10.3390/axioms7030055 - 7 Aug 2018

Cited by 72 | Viewed by 7211

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 [...] Read more.

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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11 pages, 273 KiB

Open AccessArticle

A New Type of Generalization on W—Asymptotically J _λ—Statistical Equivalence with the Number of α

by Hafize Gümüş and Nihal Demir

Axioms 2018, 7(3), 54; https://doi.org/10.3390/axioms7030054 - 2 Aug 2018

Cited by 1 | Viewed by 3307

Abstract

In our paper, by using the concept of

W -

asymptotically

J -

statistical equivalence of order

α

which has been previously defined, we present the definitions of

W -

asymptotically

J_{λ} -

statistical equivalence of order

α

,

W -

strongly [...] Read more.

In our paper, by using the concept of

W -

asymptotically

J -

statistical equivalence of order

α

which has been previously defined, we present the definitions of

W -

asymptotically

J_{λ} -

statistical equivalence of order

α

,

W -

strongly asymptotically

J_{λ} -

statistical equivalence of order

α

, and

W -

strongly Cesáro asymptotically

J -

statistical equivalence of order

α

where

0 < α \leq 1

. We also extend these notions with a sequence of positive real numbers,

p = (p_{k})

, and we investigate how our results change if

p

is constant. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications)

13 pages, 4277 KiB

Open AccessArticle

A Convex Model for Edge-Histogram Specification with Applications to Edge-Preserving Smoothing

by Kelvin C. K. Chan, Raymond H. Chan and Mila Nikolova

Axioms 2018, 7(3), 53; https://doi.org/10.3390/axioms7030053 - 2 Aug 2018

Cited by 2 | Viewed by 3584

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 [...] Read more.

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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13 pages, 264 KiB

Open AccessArticle

Trees, Stumps, and Applications

by John C. Butcher

Axioms 2018, 7(3), 52; https://doi.org/10.3390/axioms7030052 - 1 Aug 2018

Cited by 4 | Viewed by 2908

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 [...] Read more.

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)

14 pages, 323 KiB

Open AccessArticle

A Gradient System for Low Rank Matrix Completion

by Carmela Scalone and Nicola Guglielmi

Axioms 2018, 7(3), 51; https://doi.org/10.3390/axioms7030051 - 24 Jul 2018

Cited by 4 | Viewed by 3824

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 [...] Read more.

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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30 pages, 831 KiB

Open AccessArticle

Certain Notions of Energy in Single-Valued Neutrosophic Graphs

by Sumera Naz, Muhammad Akram and Florentin Smarandache

Axioms 2018, 7(3), 50; https://doi.org/10.3390/axioms7030050 - 20 Jul 2018

Cited by 32 | Viewed by 4015

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 [...] Read more.

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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29 pages, 887 KiB

Open AccessArticle

Block Generalized Locally Toeplitz Sequences: From the Theory to the Applications

by Carlo Garoni, Mariarosa Mazza and Stefano Serra-Capizzano

Axioms 2018, 7(3), 49; https://doi.org/10.3390/axioms7030049 - 19 Jul 2018

Cited by 28 | Viewed by 3622

Abstract

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices

A_{n}

arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to [...] Read more.

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices

A_{n}

arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to infinity, these matrices

A_{n}

give rise to a sequence

{A_{n}}_{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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18 pages, 2902 KiB

Open AccessArticle

Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative

by Konstantin Zhukovsky, Dmitrii Oskolkov and Nadezhda Gubina

Axioms 2018, 7(3), 48; https://doi.org/10.3390/axioms7030048 - 18 Jul 2018

Cited by 11 | Viewed by 4135

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 [...] Read more.

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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14 pages, 736 KiB

Open AccessArticle

Neutrosophic Incidence Graphs With Application

by Muhammad Akram, Sidra Sayed and Florentin Smarandache

Axioms 2018, 7(3), 47; https://doi.org/10.3390/axioms7030047 - 18 Jul 2018

Cited by 13 | Viewed by 3105

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 [...] Read more.

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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19 pages, 629 KiB

Open AccessArticle

Optimal B-Spline Bases for the Numerical Solution of Fractional Differential Problems

by Francesca Pitolli

Axioms 2018, 7(3), 46; https://doi.org/10.3390/axioms7030046 - 2 Jul 2018

Cited by 15 | Viewed by 4001

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 [...] Read more.

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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19 pages, 335 KiB

Open AccessReview

Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

by Angelamaria Cardone, Dajana Conte, Raffaele D’Ambrosio and Beatrice Paternoster

Axioms 2018, 7(3), 45; https://doi.org/10.3390/axioms7030045 - 1 Jul 2018

Cited by 25 | Viewed by 4893

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 [...] Read more.

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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14 pages, 302 KiB

Open AccessArticle

On Partial Cholesky Factorization and a Variant of Quasi-Newton Preconditioners for Symmetric Positive Definite Matrices

by Benedetta Morini

Axioms 2018, 7(3), 44; https://doi.org/10.3390/axioms7030044 - 1 Jul 2018

Cited by 1 | Viewed by 3606

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 [...] Read more.

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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Graphical abstract

25 pages, 6401 KiB

Open AccessArticle

Refinement Algorithms for Adaptive Isogeometric Methods with Hierarchical Splines

by Cesare Bracco, Carlotta Giannelli and Rafael Vázquez

Axioms 2018, 7(3), 43; https://doi.org/10.3390/axioms7030043 - 21 Jun 2018

Cited by 26 | Viewed by 5390

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 [...] Read more.

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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Axioms, Volume 7, Issue 3 (September 2018) – 26 articles

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