Mathematics

777 KiB

Open AccessArticle

Lie Symmetries, Optimal System and Invariant Reductions to a Nonlinear Timoshenko System

by Shadi Al-Omari, Fiazuddin Zaman and Hassan Azad

Mathematics 2017, 5(2), 34; https://doi.org/10.3390/math5020034 - 17 Jun 2017

Cited by 3 | Viewed by 3904

Lie symmetries and their Lie group transformations for a class of Timoshenko systems are presented. The class considered is the class of nonlinear Timoshenko systems of partial differential equations (PDEs). An optimal system of one-dimensional sub-algebras of the corresponding Lie algebra is derived. [...] Read more.

Lie symmetries and their Lie group transformations for a class of Timoshenko systems are presented. The class considered is the class of nonlinear Timoshenko systems of partial differential equations (PDEs). An optimal system of one-dimensional sub-algebras of the corresponding Lie algebra is derived. All possible invariant variables of the optimal system are obtained. The corresponding reduced systems of ordinary differential equations (ODEs) are also provided. All possible non-similar invariant conditions prescribed on invariant surfaces under symmetry transformations are given. As an application, explicit solutions of the system are given where the beam is hinged at one end and free at the other end. Full article

806 KiB

Open AccessArticle

An Analysis on the Fractional Asset Flow Differential Equations

by Din Prathumwan, Wannika Sawangtong and Panumart Sawangtong

Mathematics 2017, 5(2), 33; https://doi.org/10.3390/math5020033 - 16 Jun 2017

Cited by 4 | Viewed by 4323

Abstract

The asset flow differential equation (AFDE) is the mathematical model that plays an essential role for planning to predict the financial behavior in the market. In this paper, we introduce the fractional asset flow differential equations (FAFDEs) based on the Liouville-Caputo derivative. We [...] Read more.

The asset flow differential equation (AFDE) is the mathematical model that plays an essential role for planning to predict the financial behavior in the market. In this paper, we introduce the fractional asset flow differential equations (FAFDEs) based on the Liouville-Caputo derivative. We prove the existence and uniqueness of a solution for the FAFDEs. Furthermore, the stability analysis of the model is investigated and the numerical simulation is accordingly performed to support the proposed model. Full article

(This article belongs to the Special Issue Recent Developments in the Theory and Applications of Fractional Calculus)

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664 KiB

Open AccessArticle

Metrization Theorem for Uniform Loops with the Invertibility Property

by Dagmar Markechová, Peter Vrábel and Beáta Stehlíková

Mathematics 2017, 5(2), 32; https://doi.org/10.3390/math5020032 - 02 Jun 2017

Viewed by 2836

Abstract

In this paper, we have proved a metrization theorem that gives the sufficient conditions for a uniform IP-loop X to be metrizable by a left-invariant metric. It is shown that by consideration of topological IP-loop dual to X we obtain an analogical theorem [...] Read more.

In this paper, we have proved a metrization theorem that gives the sufficient conditions for a uniform IP-loop X to be metrizable by a left-invariant metric. It is shown that by consideration of topological IP-loop dual to X we obtain an analogical theorem for the case of the right-invariant metric. Full article

722 KiB

Open AccessArticle

Nonlinear Gronwall–Bellman Type Inequalities and Their Applications

by Weimin Wang, Yuqiang Feng and Yuanyuan Wang

Mathematics 2017, 5(2), 31; https://doi.org/10.3390/math5020031 - 31 May 2017

Cited by 7 | Viewed by 4389

Abstract

In this paper, some nonlinear Gronwall–Bellman type inequalities are established. Then, the obtained results are applied to study the Hyers–Ulam stability of a fractional differential equation and the boundedness of solutions to an integral equation, respectively. Full article

(This article belongs to the Special Issue Recent Developments in the Theory and Applications of Fractional Calculus)

248 KiB

Open AccessArticle

Coincidence Points of a Sequence of Multivalued Mappings in Metric Space with a Graph

by Muhammad Nouman Aslam Khan, Akbar Azam and Nayyar Mehmood

Mathematics 2017, 5(2), 30; https://doi.org/10.3390/math5020030 - 26 May 2017

Cited by 2 | Viewed by 3075

Abstract

In this article the coincidence points of a self map and a sequence of multivalued maps are found in the settings of complete metric space endowed with a graph. A novel result of Asrifa and Vetrivel is generalized and as an application we [...] Read more.

In this article the coincidence points of a self map and a sequence of multivalued maps are found in the settings of complete metric space endowed with a graph. A novel result of Asrifa and Vetrivel is generalized and as an application we obtain an existence theorem for a special type of fractional integral equation. Moreover, we establish a result on the convergence of successive approximation of a system of Bernstein operators on a Banach space. Full article

2030 KiB

Open AccessArticle

Emergence of an Aperiodic Dirichlet Space from the Tetrahedral Units of an Icosahedral Internal Space

by Amrik Sen, Raymond Aschheim and Klee Irwin

Mathematics 2017, 5(2), 29; https://doi.org/10.3390/math5020029 - 26 May 2017

Cited by 3 | Viewed by 7422

Abstract

We present the emergence of a root system in six dimensions from the tetrahedra of an icosahedral core known as the 20-group (20G) within the framework of Clifford’s geometric algebra. Consequently, we establish a connection between a three-dimensional icosahedral seed, a six-dimensional (6D) [...] Read more.

We present the emergence of a root system in six dimensions from the tetrahedra of an icosahedral core known as the 20-group (20G) within the framework of Clifford’s geometric algebra. Consequently, we establish a connection between a three-dimensional icosahedral seed, a six-dimensional (6D) Dirichlet quantized host and a higher dimensional lattice structure. The 20G, owing to its icosahedral symmetry, bears the signature of a 6D lattice that manifests in the Dirichlet integer representation. We present an interpretation whereby the three-dimensional 20G can be regarded as the core substratum from which the higher dimensional lattices emerge. This emergent geometry is based on an induction principle supported by the Clifford multi-vector formalism of three-dimensional (3D) Euclidean space. This lays a geometric framework for understanding several physics theories related to

S U (5)

,

E_{6}

,

E_{8}

Lie algebras and their composition with the algebra associated with the even unimodular lattice in

R^{3, 1}

. The construction presented here is inspired by Penrose’s three world model. Full article

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999 KiB

Open AccessArticle

A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind

by Fu-Rong Lin and Shi-Wei Yang

Mathematics 2017, 5(2), 28; https://doi.org/10.3390/math5020028 - 22 May 2017

Cited by 3 | Viewed by 4007

Abstract

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

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

f_{TV}^{*}

. In the second stage, the simplified MM functional is minimized to obtain the final result by using the damped Newton (DN) method with

f_{TV}^{*}

as the initial guess. The numerical implementation is given in detail, and numerical results of two examples are presented to illustrate the efficiency of the proposed approach. Full article

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5015 KiB

Open AccessArticle

Analysis of Magneto-hydrodynamics Flow and Heat Transfer of a Viscoelastic Fluid through Porous Medium in Wire Coating Analysis

by Zeeshan Khan, Muhammad Altaf Khan, Saeed Islam, Bilal Jan, Fawad Hussain, Haroon Ur Rasheed and Waris Khan

Mathematics 2017, 5(2), 27; https://doi.org/10.3390/math5020027 - 16 May 2017

Cited by 21 | Viewed by 4396

Abstract

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

Wire coating process is a continuous extrusion process for primary insulation of conducting wires with molten polymers for mechanical strength and protection in aggressive environments. Nylon, polysulfide, low/high density polyethylene (LDPE/HDPE) and plastic polyvinyl chloride (PVC) are the common and important plastic resin used for wire coating. In the current study, wire coating is performed using viscoelastic third grade fluid in the presence of applied magnetic field and porous medium. The governing equations are first modeled and then solved analytically by utilizing the homotopy analysis method (HAM). The convergence of the series solution is established. A numerical technique called ND-solve method is used for comparison and found good agreement. The effect of pertinent parameters on the velocity field and temperature profile is shown with the help of graphs. It is observed that the velocity profiles increase as the value of viscoelastic third grade parameter

β

increase and decrease as the magnetic parameter

M

and permeability parameter

K

increase. It is also observed that the temperature profiles increases as the Brinkman number

B r

, permeability parameter

K

, magnetic parameter

M

and viscoelastic third grade parameter (non-Newtonian parameter)

β

increase. Full article

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595 KiB

Open AccessArticle

A New Variational Iteration Method for a Class of Fractional Convection-Diffusion Equations in Large Domains

by Mohammad Abolhasani, Saeid Abbasbandy and Tofigh Allahviranloo

Mathematics 2017, 5(2), 26; https://doi.org/10.3390/math5020026 - 11 May 2017

Cited by 8 | Viewed by 4934

Abstract

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

In this paper, we introduced a new generalization method to solve fractional convection–diffusion equations based on the well-known variational iteration method (VIM) improved by an auxiliary parameter. The suggested method was highly effective in controlling the convergence region of the approximate solution. By solving some fractional convection–diffusion equations with a propounded method and comparing it with standard VIM, it was concluded that complete reliability, efficiency, and accuracy of this method are guaranteed. Additionally, we studied and investigated the convergence of the proposed method, namely the VIM with an auxiliary parameter. We also offered the optimal choice of the auxiliary parameter in the proposed method. It was noticed that the approach could be applied to other models of physics. Full article

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2451 KiB

Open AccessArticle

Discrete-Time Fractional Optimal Control

by Tirumalasetty Chiranjeevi and Raj Kumar Biswas

Mathematics 2017, 5(2), 25; https://doi.org/10.3390/math5020025 - 19 Apr 2017

Cited by 18 | Viewed by 5108

Abstract

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

A formulation and solution of the discrete-time fractional optimal control problem in terms of the Caputo fractional derivative is presented in this paper. The performance index (PI) is considered in a quadratic form. The necessary and transversality conditions are obtained using a Hamiltonian approach. Both the free and fixed final state cases have been considered. Numerical examples are taken up and their solution technique is presented. Results are produced for different values of

α

. Full article

(This article belongs to the Special Issue Recent Developments in the Theory and Applications of Fractional Calculus)

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263 KiB

Open AccessArticle

Fixed Points of Set Valued Mappings in Terms of Start Point on a Metric Space Endowed with a Directed Graph

by Murchana Neog and Pradip Debnath

Mathematics 2017, 5(2), 24; https://doi.org/10.3390/math5020024 - 19 Apr 2017

Cited by 6 | Viewed by 3558

Abstract

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

In the present article, we introduce the new concept of start point in a directed graph and provide the characterizations required for a directed graph to have a start point. We also define the notion of a self path set valued map and establish its relation with start point in the setting of a metric space endowed with a directed graph. Further, some fixed point theorems for set valued maps have been proven in this context. A version of the Knaster–Tarski theorem has also been established using our results. Full article

(This article belongs to the Special Issue Fixed Point Theorems and Applications)

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260 KiB

Open AccessArticle

Best Proximity Point Results in Non-Archimedean Modular Metric Space

by Mohadeshe Paknazar and Manuel De la Sen

Mathematics 2017, 5(2), 23; https://doi.org/10.3390/math5020023 - 05 Apr 2017

Cited by 2 | Viewed by 3340

Abstract

In this paper, we introduce the new notion of Suzuki-type

(α, β, θ, γ)

-contractive mapping and investigate the existence and uniqueness of the best proximity point for such mappings in non-Archimedean modular metric space using the weak [...] Read more.

In this paper, we introduce the new notion of Suzuki-type

(α, β, θ, γ)

-contractive mapping and investigate the existence and uniqueness of the best proximity point for such mappings in non-Archimedean modular metric space using the weak

P_{λ}

-property. Meanwhile, we present an illustrative example to emphasize the realized improvements. These obtained results extend and improve certain well-known results in the literature. Full article

330 KiB

Open AccessArticle

On Optimal Fuzzy Best Proximity Coincidence Points of Proximal Contractions Involving Cyclic Mappings in Non-Archimedean Fuzzy Metric Spaces

by Manuel De la Sen, Mujahid Abbas and Naeem Saleem

Mathematics 2017, 5(2), 22; https://doi.org/10.3390/math5020022 - 01 Apr 2017

Cited by 14 | Viewed by 3448

Abstract

The main objective of this paper is to deal with some properties of interest in two types of fuzzy ordered proximal contractions of cyclic self-mappings

T

integrated in a pair

(g, T)

of mappings. In particular,

g

is a non-contractive [...] Read more.

The main objective of this paper is to deal with some properties of interest in two types of fuzzy ordered proximal contractions of cyclic self-mappings

T

integrated in a pair

(g, T)

of mappings. In particular,

g

is a non-contractive fuzzy self-mapping, in the framework of non-Archimedean ordered fuzzy complete metric spaces and

T

is a

p

-cyclic proximal contraction. Two types of such contractions (so called of type I and of type II) are dealt with. In particular, the existence, uniqueness and limit properties for sequences to optimal fuzzy best proximity coincidence points are investigated for such pairs of mappings. Full article

(This article belongs to the Special Issue Fixed Point Theorems and Applications)

314 KiB

Open AccessArticle

On Some Extended Block Krylov Based Methods for Large Scale Nonsymmetric Stein Matrix Equations

by Abdeslem Hafid Bentbib, Khalide Jbilou and EL Mostafa Sadek

Mathematics 2017, 5(2), 21; https://doi.org/10.3390/math5020021 - 27 Mar 2017

Cited by 7 | Viewed by 3654

Abstract

In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term

A X B - X + E F^{T} = 0

. These matrix equations appear in many applications in discrete-time control problems, filtering and image [...] Read more.

In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term

A X B - X + E F^{T} = 0

. These matrix equations appear in many applications in discrete-time control problems, filtering and image restoration and others. The proposed methods are based on projection onto the extended block Krylov subspace with a Galerkin approach (GA) or with the minimization of the norm of the residual. We give some results on the residual and error norms and report some numerical experiments. Full article

(This article belongs to the Special Issue Numerical Linear Algebra with Applications)

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782 KiB

Open AccessArticle

F-Harmonic Maps between Doubly Warped Product Manifolds

by Seyed Mehdi Kazemi Torbaghan and Morteza Mirmohammad Rezaii

Mathematics 2017, 5(2), 20; https://doi.org/10.3390/math5020020 - 23 Mar 2017

Cited by 3 | Viewed by 3029

Abstract

In this paper, some properties of

F

-harmonic and conformal

F

-harmonic maps between doubly warped product manifolds are studied and new examples of non-harmonic

F

-harmonic maps are constructed. [...] Read more.

In this paper, some properties of

F

-harmonic and conformal

F

-harmonic maps between doubly warped product manifolds are studied and new examples of non-harmonic

F

-harmonic maps are constructed. Full article

231 KiB

Open AccessArticle

A Generalization of b-Metric Space and Some Fixed Point Theorems

by Tayyab Kamran, Maria Samreen and Qurat UL Ain

Mathematics 2017, 5(2), 19; https://doi.org/10.3390/math5020019 - 23 Mar 2017

Cited by 218 | Viewed by 15688

Abstract

In this paper, inspired by the concept of b-metric space, we introduce the concept of extended b-metric space. We also establish some fixed point theorems for self-mappings defined on such spaces. Our results extend/generalize many pre-existing results in literature. Full article

(This article belongs to the Special Issue Fixed Point Theorems and Applications)

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Mathematics, Volume 5, Issue 2 (June 2017) – 16 articles

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