Axioms

15 pages, 806 KiB

Open AccessArticle

A Metric for Finite Power Multisets of Positive Real Numbers Based on Minimal Matching

by Ray-Ming Chen

Axioms 2018, 7(4), 94; https://doi.org/10.3390/axioms7040094 - 14 Dec 2018

Cited by 6 | Viewed by 2697

In this article, we show how to define a metric on the finite power multisets of positive real numbers. The metric, based on the minimal matching, consists of two parts: the matched part and the mismatched part. We also give some concrete applications [...] Read more.

In this article, we show how to define a metric on the finite power multisets of positive real numbers. The metric, based on the minimal matching, consists of two parts: the matched part and the mismatched part. We also give some concrete applications and examples to demonstrate the validity of this metric. Full article

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10 pages, 239 KiB

Open AccessArticle

Diffeomorphisms with Shadowable Measures

by Kazumine Moriyasu, Kazuhiro Sakai and Naoya Sumi

Axioms 2018, 7(4), 93; https://doi.org/10.3390/axioms7040093 - 7 Dec 2018

Cited by 5 | Viewed by 3244

Abstract

In this paper, the notion of shadowable measures is introduced as a generalization of the shadowing property from the measure theoretical view point, and the set of diffeomorphisms satisfying the notion is considered. The dynamics of the

C^{1}

-interior of the set [...] Read more.

In this paper, the notion of shadowable measures is introduced as a generalization of the shadowing property from the measure theoretical view point, and the set of diffeomorphisms satisfying the notion is considered. The dynamics of the

C^{1}

-interior of the set of diffeomorphisms possessing the shadowable measures is characterized as the uniform hyperbolicity. Full article

(This article belongs to the Special Issue Shadowing in Dynamical Systems)

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13 pages, 307 KiB

Open AccessArticle

Regional Enlarged Observability of Fractional Differential Equations with Riemann—Liouville Time Derivatives

by Hayat Zouiten, Ali Boutoulout and Delfim F. M. Torres

Axioms 2018, 7(4), 92; https://doi.org/10.3390/axioms7040092 - 1 Dec 2018

Cited by 2 | Viewed by 2923

Abstract

We introduce the concept of regional enlarged observability for fractional evolution differential equations involving Riemann–Liouville derivatives. The Hilbert Uniqueness Method (HUM) is used to reconstruct the initial state between two prescribed functions, in an interested subregion of the whole domain, without the knowledge [...] Read more.

We introduce the concept of regional enlarged observability for fractional evolution differential equations involving Riemann–Liouville derivatives. The Hilbert Uniqueness Method (HUM) is used to reconstruct the initial state between two prescribed functions, in an interested subregion of the whole domain, without the knowledge of the state. Full article

(This article belongs to the Special Issue Fractional Differential Equations)

14 pages, 11136 KiB

Open AccessReview

Stability Issues for Selected Stochastic Evolutionary Problems: A Review

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

Axioms 2018, 7(4), 91; https://doi.org/10.3390/axioms7040091 - 1 Dec 2018

Cited by 15 | Viewed by 3015

Abstract

We review some recent contributions of the authors regarding the numerical approximation of stochastic problems, mostly based on stochastic differential equations modeling random damped oscillators and stochastic Volterra integral equations. The paper focuses on the analysis of selected stability issues, i.e., the preservation [...] Read more.

We review some recent contributions of the authors regarding the numerical approximation of stochastic problems, mostly based on stochastic differential equations modeling random damped oscillators and stochastic Volterra integral equations. The paper focuses on the analysis of selected stability issues, i.e., the preservation of the long-term character of stochastic oscillators over discretized dynamics and the analysis of mean-square and asymptotic stability properties of

ϑ

-methods for Volterra integral equations. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)

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14 pages, 328 KiB

Open AccessArticle

Special Types of Locally Conformal Closed G₂-Structures

by Giovanni Bazzoni and Alberto Raffero

Axioms 2018, 7(4), 90; https://doi.org/10.3390/axioms7040090 - 28 Nov 2018

Cited by 1 | Viewed by 3056

Abstract

Motivated by known results in locally conformal symplectic geometry, we study different classes of G

_{2}

-structures defined by a locally conformal closed 3-form. In particular, we provide a complete characterization of invariant exact locally conformal closed G

_{2}

-structures on simply connected [...] Read more.

Motivated by known results in locally conformal symplectic geometry, we study different classes of G

_{2}

-structures defined by a locally conformal closed 3-form. In particular, we provide a complete characterization of invariant exact locally conformal closed G

_{2}

-structures on simply connected Lie groups, and we present examples of compact manifolds with different types of locally conformal closed G

_{2}

-structures. Full article

(This article belongs to the Special Issue Applications of Differential Geometry)

11 pages, 341 KiB

Open AccessArticle

A Two Dimensional Discrete Mollification Operator and the Numerical Solution of an Inverse Source Problem

by Manuel D. Echeverry and Carlos E. Mejía

Axioms 2018, 7(4), 89; https://doi.org/10.3390/axioms7040089 - 23 Nov 2018

Cited by 5 | Viewed by 2916

Abstract

We consider a two-dimensional time fractional diffusion equation and address the important inverse problem consisting of the identification of an ingredient in the source term. The fractional derivative is in the sense of Caputo. The necessary regularization procedure is provided by a two-dimensional [...] Read more.

We consider a two-dimensional time fractional diffusion equation and address the important inverse problem consisting of the identification of an ingredient in the source term. The fractional derivative is in the sense of Caputo. The necessary regularization procedure is provided by a two-dimensional discrete mollification operator. Convergence results and illustrative numerical examples are included. Full article

(This article belongs to the Special Issue Applications of Differential Equations and Dynamical Systems)

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7 pages, 311 KiB

Open AccessArticle

Exponentially Harmonic Maps into Spheres

by Sorin Dragomir and Francesco Esposito

Axioms 2018, 7(4), 88; https://doi.org/10.3390/axioms7040088 - 22 Nov 2018

Cited by 1 | Viewed by 2569

Abstract

We study smooth exponentially harmonic maps from a compact, connected, orientable Riemannian manifold M into a sphere

S^{m} \subset R^{m + 1}

. Given a codimension two totally geodesic submanifold

Σ \subset S^{m}

, we show that every nonconstant exponentially [...] Read more.

We study smooth exponentially harmonic maps from a compact, connected, orientable Riemannian manifold M into a sphere

S^{m} \subset R^{m + 1}

. Given a codimension two totally geodesic submanifold

Σ \subset S^{m}

, we show that every nonconstant exponentially harmonic map

ϕ : M \to S^{m}

either meets or links

Σ

. If

H^{1} (M, Z) = 0

then

ϕ (M) \cap Σ \neq \emptyset

. Full article

(This article belongs to the Special Issue Applications of Differential Geometry)

9 pages, 221 KiB

Open AccessArticle

Extended Partial S_b-Metric Spaces

by Aiman Mukheimer

Axioms 2018, 7(4), 87; https://doi.org/10.3390/axioms7040087 - 21 Nov 2018

Cited by 5 | Viewed by 3416

Abstract

In this paper, we introduce the concept of extended partial

S_{b}

-metric spaces, which is a generalization of the extended

S_{b}

-metric spaces. Basically, in the triangle inequality, we add a control function with some very interesting properties. These new metric [...] Read more.

In this paper, we introduce the concept of extended partial

S_{b}

-metric spaces, which is a generalization of the extended

S_{b}

-metric spaces. Basically, in the triangle inequality, we add a control function with some very interesting properties. These new metric spaces generalize many results in the literature. Moreover, we prove some fixed point theorems under some different contractions, and some examples are given to illustrate our results. Full article

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

23 pages, 416 KiB

Open AccessArticle

Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets

by Dmitri Shakhmatov and Víctor Hugo Yañez

Axioms 2018, 7(4), 86; https://doi.org/10.3390/axioms7040086 - 16 Nov 2018

Cited by 3 | Viewed by 3429

Abstract

We give a “naive” (i.e., using no additional set-theoretic assumptions beyond ZFC, the Zermelo-Fraenkel axioms of set theory augmented by the Axiom of Choice) example of a Boolean topological group G without infinite separable pseudocompact subsets having the following “selective” compactness property: For [...] Read more.

We give a “naive” (i.e., using no additional set-theoretic assumptions beyond ZFC, the Zermelo-Fraenkel axioms of set theory augmented by the Axiom of Choice) example of a Boolean topological group G without infinite separable pseudocompact subsets having the following “selective” compactness property: For each free ultrafilter p on the set

N

of natural numbers and every sequence

(U_{n})

of non-empty open subsets of G, one can choose a point

x_{n} \in U_{n}

for all

n \in N

in such a way that the resulting sequence

(x_{n})

has a p-limit in G; that is,

{n \in N : x_{n} \in V} \in p

for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first listed author. The group G above is not pseudo-

ω

-bounded either. Furthermore, we show that the free precompact Boolean group of a topological sum

⨁_{i \in I} X_{i}

, where each space

X_{i}

is either maximal or discrete, contains no infinite separable pseudocompact subsets. Full article

(This article belongs to the Collection Topological Groups)

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5 pages, 224 KiB

Open AccessArticle

Unification Theories: Examples and Applications

by Florin F. Nichita

Axioms 2018, 7(4), 85; https://doi.org/10.3390/axioms7040085 - 16 Nov 2018

Cited by 6 | Viewed by 3189

Abstract

We consider several unification problems in mathematics. We refer to transcendental numbers. Furthermore, we present some ways to unify the main non-associative algebras (Lie algebras and Jordan algebras) and associative algebras. Full article

(This article belongs to the Special Issue Non-associative Structures and Other Related Structures)

5 pages, 219 KiB

Open AccessArticle

Equicontinuity, Expansivity, and Shadowing for Linear Operators

by Keonhee Lee and C. A. Morales

Axioms 2018, 7(4), 84; https://doi.org/10.3390/axioms7040084 - 15 Nov 2018

Cited by 1 | Viewed by 3230

Abstract

We prove that a linear operator of a complex Banach space has a shadowable point if and only if it has the shadowing property. In addition, every equicontinuous linear operator does not have the shadowing property and its spectrum is contained in the [...] Read more.

We prove that a linear operator of a complex Banach space has a shadowable point if and only if it has the shadowing property. In addition, every equicontinuous linear operator does not have the shadowing property and its spectrum is contained in the unit circle. Finally, we prove that if a linear operator is expansive and has the shadowing property, then the origin is the only nonwandering point. Full article

(This article belongs to the Special Issue Shadowing in Dynamical Systems)

17 pages, 349 KiB

Open AccessArticle

Application of Neutrosophic Soft Sets to K-Algebras

by Muhammad Akram, Hina Gulzar, Florentin Smarandache and Said Broumi

Axioms 2018, 7(4), 83; https://doi.org/10.3390/axioms7040083 - 12 Nov 2018

Cited by 3 | Viewed by 3110

Abstract

Neutrosophic sets and soft sets are two different mathematical tools for representing vagueness and uncertainty. We apply these models in combination to study vagueness and uncertainty in K-algebras. We introduce the notion of single-valued neutrosophic soft (SNS) K-algebras and investigate some [...] Read more.

Neutrosophic sets and soft sets are two different mathematical tools for representing vagueness and uncertainty. We apply these models in combination to study vagueness and uncertainty in K-algebras. We introduce the notion of single-valued neutrosophic soft (SNS) K-algebras and investigate some of their properties. We establish the notion of

(\in, \in \lor q)

-single-valued neutrosophic soft K-algebras and describe some of their related properties. We also illustrate the concepts with numerical examples. Full article

(This article belongs to the Special Issue Neutrosophic Topology)

2 pages, 158 KiB

Open AccessEditorial

Mathematical Analysis and Applications

by Hari M. Srivastava

Axioms 2018, 7(4), 82; https://doi.org/10.3390/axioms7040082 - 12 Nov 2018

Viewed by 2882

Abstract

Website: http://www.math.uvic.ca/faculty/harimsri/ [...] Full article

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

18 pages, 361 KiB

Open AccessArticle

The Generalized Schur Algorithm and Some Applications

by Teresa Laudadio, Nicola Mastronardi and Paul Van Dooren

Axioms 2018, 7(4), 81; https://doi.org/10.3390/axioms7040081 - 9 Nov 2018

Cited by 5 | Viewed by 4489

Abstract

The generalized Schur algorithm is a powerful tool allowing to compute classical decompositions of matrices, such as the

Q R

and

L U

factorizations. When applied to matrices with particular structures, the generalized Schur algorithm computes these factorizations with a complexity of one [...] Read more.

The generalized Schur algorithm is a powerful tool allowing to compute classical decompositions of matrices, such as the

Q R

and

L U

factorizations. When applied to matrices with particular structures, the generalized Schur algorithm computes these factorizations with a complexity of one order of magnitude less than that of classical algorithms based on Householder or elementary transformations. In this manuscript, we describe the main features of the generalized Schur algorithm. We show that it helps to prove some theoretical properties of the R factor of the

Q R

factorization of some structured matrices, such as symmetric positive definite Toeplitz and Sylvester matrices, that can hardly be proven using classical linear algebra tools. Moreover, we propose a fast implementation of the generalized Schur algorithm for computing the rank of Sylvester matrices, arising in a number of applications. Finally, we propose a generalized Schur based algorithm for computing the null-space of polynomial matrices. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)

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10 pages, 240 KiB

Open AccessArticle

On the Fixed-Circle Problem and Khan Type Contractions

by Nabil Mlaiki, Nihal Taş and Nihal Yılmaz Özgür

Axioms 2018, 7(4), 80; https://doi.org/10.3390/axioms7040080 - 8 Nov 2018

Cited by 34 | Viewed by 3770

Abstract

In this paper, we consider the fixed-circle problem on metric spaces and give new results on this problem. To do this, we present three types of

F_{C}

-Khan type contractions. Furthermore, we obtain some solutions to an open problem related to the [...] Read more.

In this paper, we consider the fixed-circle problem on metric spaces and give new results on this problem. To do this, we present three types of

F_{C}

-Khan type contractions. Furthermore, we obtain some solutions to an open problem related to the common fixed-circle problem. Full article

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

4 pages, 207 KiB

Open AccessArticle

Extending Characters of Fixed Point Algebras

by Stefan Wagner

Axioms 2018, 7(4), 79; https://doi.org/10.3390/axioms7040079 - 7 Nov 2018

Cited by 3 | Viewed by 2692

Abstract

A dynamical system is a triple

(A, G, α)

consisting of a unital locally convex algebra A, a topological group G, and a group homomorphism

α : G \to Aut (A)

that induces a continuous [...] Read more.

A dynamical system is a triple

(A, G, α)

consisting of a unital locally convex algebra A, a topological group G, and a group homomorphism

α : G \to Aut (A)

that induces a continuous action of G on A. Furthermore, a unital locally convex algebra A is called a continuous inverse algebra, or CIA for short, if its group of units

A^{\times}

is open in A and the inversion map

ι : A^{\times} \to A^{\times}

,

a \mapsto a^{- 1}

is continuous at

1_{A}

. Given a dynamical system

(A, G, α)

with a complete commutative CIA A and a compact group G, we show that each character of the corresponding fixed point algebra can be extended to a character of A. Full article

(This article belongs to the Collection Topological Groups)

10 pages, 249 KiB

Open AccessArticle

Consistency Properties for Fuzzy Choice Functions: An Analysis with the Łukasiewicz T-Norm

by Susana Díaz, José Carlos R. Alcantud and Susana Montes

Axioms 2018, 7(4), 78; https://doi.org/10.3390/axioms7040078 - 29 Oct 2018

Viewed by 2248

Abstract

We focus on the relationships among some consistency axioms in the framework of fuzzy choice functions. In order to help disclose the role of the t-norm in existing analyses, we start to study the situation that arises when we replace the standard t-norm [...] Read more.

We focus on the relationships among some consistency axioms in the framework of fuzzy choice functions. In order to help disclose the role of the t-norm in existing analyses, we start to study the situation that arises when we replace the standard t-norm with other t-norms. Our results allow us to conclude that unless we impose further structure on the domain of application for the choices, the use of the Łukasiewicz t-norm as a replacement for the standard t-norm does not guarantee a better performance. Full article

6 pages, 242 KiB

Open AccessArticle

A Note on the Topological Group c₀

by Michael Megrelishvili

Axioms 2018, 7(4), 77; https://doi.org/10.3390/axioms7040077 - 29 Oct 2018

Cited by 1 | Viewed by 2758

Abstract

A well-known result of Ferri and Galindo asserts that the topological group

c_{0}

is not reflexively representable and the algebra WAP

(c_{0})

of weakly almost periodic functions does not separate points and closed subsets. However, it is unknown if [...] Read more.

A well-known result of Ferri and Galindo asserts that the topological group

c_{0}

is not reflexively representable and the algebra WAP

(c_{0})

of weakly almost periodic functions does not separate points and closed subsets. However, it is unknown if the same remains true for a larger important algebra Tame

(c_{0})

of tame functions. Respectively, it is an open question if

c_{0}

is representable on a Rosenthal Banach space. In the present work we show that Tame

(c_{0})

is small in a sense that the unit sphere S and

2 S

cannot be separated by a tame function f ∈ Tame

(c_{0})

. As an application we show that the Gromov’s compactification of

c_{0}

is not a semigroup compactification. We discuss some questions. Full article

(This article belongs to the Collection Topological Groups)

15 pages, 293 KiB

Open AccessArticle

On the Shape Differentiability of Objectives: A Lagrangian Approach and the Brinkman Problem

by José Rodrigo González Granada, Joachim Gwinner and Victor A. Kovtunenko

Axioms 2018, 7(4), 76; https://doi.org/10.3390/axioms7040076 - 27 Oct 2018

Cited by 6 | Viewed by 3087

Abstract

This paper establishes the shape derivative of geometry-dependent objective functions for use in constrained variational problems. Using a Lagrangian approach, our differentiablity result is based on the theorem of Delfour–Zolésio on directional derivatives with respect to a parameter of shape perturbation. As the [...] Read more.

This paper establishes the shape derivative of geometry-dependent objective functions for use in constrained variational problems. Using a Lagrangian approach, our differentiablity result is based on the theorem of Delfour–Zolésio on directional derivatives with respect to a parameter of shape perturbation. As the key issue of the paper, we analyze the bijection under the kinematic transport of geometries that is needed for function spaces and feasible sets involved in variational problems. Our abstract theoretical result is applied to the Brinkman flow problem under incompressibility and mixed Dirichlet–Neumann boundary conditions, and provides an analytic formula of the shape derivative based on the velocity method. Full article

(This article belongs to the Special Issue Applications of Differential Equations and Dynamical Systems)

12 pages, 271 KiB

Open AccessReview

Selective Survey on Spaces of Closed Subgroups of Topological Groups

by Igor V. Protasov

Axioms 2018, 7(4), 75; https://doi.org/10.3390/axioms7040075 - 26 Oct 2018

Viewed by 2821

Abstract

We survey different topologizations of the set

S (G)

of closed subgroups of a topological group G and demonstrate some applications using Topological Groups, Model Theory, Geometric Group Theory, and Topological Dynamics. [...] Read more.

We survey different topologizations of the set

S (G)

of closed subgroups of a topological group G and demonstrate some applications using Topological Groups, Model Theory, Geometric Group Theory, and Topological Dynamics. Full article

(This article belongs to the Collection Topological Groups)

17 pages, 789 KiB

Open AccessArticle

Common Fixed Point Theorems for Generalized Geraghty (α,ψ,ϕ)-Quasi Contraction Type Mapping in Partially Ordered Metric-Like Spaces

by Haitham Qawaqneh, Mohd Noorani, Wasfi Shatanawi and Habes Alsamir

Axioms 2018, 7(4), 74; https://doi.org/10.3390/axioms7040074 - 25 Oct 2018

Cited by 17 | Viewed by 3777

Abstract

The aim of this paper is to establish the existence of some common fixed point results for generalized Geraghty

(α, ψ, ϕ)

-quasi contraction self-mapping in partially ordered metric-like spaces. We display an example and an application to show [...] Read more.

The aim of this paper is to establish the existence of some common fixed point results for generalized Geraghty

(α, ψ, ϕ)

-quasi contraction self-mapping in partially ordered metric-like spaces. We display an example and an application to show the superiority of our results. The obtained results progress some well-known fixed (common fixed) point results in the literature. Our main results cannot be specifically attained from the corresponding metric space versions. This paper is scientifically novel because we take Geraghty contraction self-mapping in partially ordered metric-like spaces via

α -

admissible mapping. This opens the door to other possible fixed (common fixed) point results for non-self-mapping and in other generalizing metric spaces. Full article

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

13 pages, 307 KiB

Open AccessArticle

On the Degree-Based Topological Indices of the Tickysim SpiNNaker Model

by Muhammad Imran, Muhammad Kamran Siddiqui, Ali Ahmad, Usman Ali and Nazia Hanif

Axioms 2018, 7(4), 73; https://doi.org/10.3390/axioms7040073 - 19 Oct 2018

Cited by 8 | Viewed by 3271

Abstract

Tickysim is a clock tick-based simulator for the inter-chip interconnection network of the SpiNNaker architecture. Network devices such as arbiters, routers, and packet generators store, read, and write forward data through fixed-length FIFO buffers. At each clock tick, every component executes a “read” [...] Read more.

Tickysim is a clock tick-based simulator for the inter-chip interconnection network of the SpiNNaker architecture. Network devices such as arbiters, routers, and packet generators store, read, and write forward data through fixed-length FIFO buffers. At each clock tick, every component executes a “read” phase followed by a “write” phase. The structures of any finite graph which represents numerical quantities are known as topological indices. In this paper, we compute degree-based topological indices of the Tickysim SpiNNaker Model (

T S M

) sheet. Full article

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32 pages, 461 KiB

Open AccessArticle

The Role of Spin(9) in Octonionic Geometry

by Maurizio Parton and Paolo Piccinni

Axioms 2018, 7(4), 72; https://doi.org/10.3390/axioms7040072 - 12 Oct 2018

Cited by 4 | Viewed by 3674

Abstract

Starting from the 2001 Thomas Friedrich’s work on

Spin (9)

, we review some interactions between

Spin (9)

and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the

Spin (9)

canonical [...] Read more.

Starting from the 2001 Thomas Friedrich’s work on

Spin (9)

, we review some interactions between

Spin (9)

and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the

Spin (9)

canonical 8-form and its analogies with quaternionic geometry as well as the role of

Spin (9)

both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel

Spin (9)

manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley–Rosenfeld planes and to three series of Grassmannians. Full article

(This article belongs to the Special Issue Applications of Differential Geometry)

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10 pages, 233 KiB

Open AccessArticle

New Bell–Sheffer Polynomial Sets

by Pierpaolo Natalini and Paolo Emilio Ricci

Axioms 2018, 7(4), 71; https://doi.org/10.3390/axioms7040071 - 8 Oct 2018

Cited by 6 | Viewed by 3365

Abstract

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to preceding results [...] Read more.

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to preceding results by Dattoli and Ben Cheikh on the monomiality principle, showing the possibility to derive explicitly the main properties of Sheffer polynomial families starting from the basic elements of their generating functions. The introduction of iterated exponential and logarithmic functions allows to construct new sets of Bell–Sheffer polynomials which exhibit an iterative character of the obtained shift operators and differential equations. In this context, it is possible, for every integer r, to define polynomials of higher type, which are linked to the higher order Bell-exponential and logarithmic numbers introduced in preceding papers. Connections with integer sequences appearing in Combinatorial analysis are also mentioned. Naturally, the considered technique can also be used in similar frameworks, where the iteration of exponential and logarithmic functions appear. Full article

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

13 pages, 782 KiB

Open AccessArticle

A New Efficient Method for the Numerical Solution of Linear Time-Dependent Partial Differential Equations

by Mina Torabi and Mohammad-Mehdi Hosseini

Axioms 2018, 7(4), 70; https://doi.org/10.3390/axioms7040070 - 1 Oct 2018

Cited by 4 | Viewed by 3086

Abstract

This paper presents a new efficient method for the numerical solution of a linear time-dependent partial differential equation. The proposed technique includes the collocation method with Legendre wavelets for spatial discretization and the three-step Taylor method for time discretization. This procedure is third-order [...] Read more.

This paper presents a new efficient method for the numerical solution of a linear time-dependent partial differential equation. The proposed technique includes the collocation method with Legendre wavelets for spatial discretization and the three-step Taylor method for time discretization. This procedure is third-order accurate in time. A comparative study between the proposed method and the one-step wavelet collocation method is provided. In order to verify the stability of these methods, asymptotic stability analysis is employed. Numerical illustrations are investigated to show the reliability and efficiency of the proposed method. An important property of the presented method is that unlike the one-step wavelet collocation method, it is not necessary to choose a small time step to achieve stability. Full article

(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)

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13 pages, 809 KiB

Open AccessArticle

Periodically Forced Nonlinear Oscillatory Acoustic Vacuum

by Makrina Agaoglou, Michal Fečkan, Michal Pospíšil, Vassilis M. Rothos and Alexander F. Vakakis

Axioms 2018, 7(4), 69; https://doi.org/10.3390/axioms7040069 - 22 Sep 2018

Cited by 1 | Viewed by 3399

Abstract

In this work, we study the in-plane oscillations of a finite lattice of particles coupled by linear springs under distributed harmonic excitation. Melnikov-type analysis is applied for the persistence of periodic oscillations of a reduced system. Full article

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

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