PSR Press – Page 67

Cyclic-antimagic construction of ladders

EASL-Vol. 2 (2019), Issue 2, pp. 43-47 Open Access Full-Text PDF

Muhammad Awais Umar

Abstract: A simple graph $G=(V,E)$ admits an $H$-covering if every edge in the edge set $E(G)$ belongs to at least one subgraph of $G$ isomorphic to a given graph $H$. A graph $G$ having an $H$-covering is called $(a,d)-H$-antimagic if the function $h:V(G)\cup E(G) \to \{1,2,\dots, |V(G)|+|E(G)| \}$ defines a bijective map such that, for all subgraphs $H’$ of $G$ isomorphic to $H$, the sums of labels of all vertices and edges belonging to $H’$ constitute an arithmetic progression with the initial term $a$ and the common difference $d.$ Such a graph is named as super $(a,d)-H$-antimagic if $h(V(G))= \{ 1,2,3,\dots,|V(G)|\}$. For $d=0$, the super $(a,d)-H$-antimagic graph is called $H$-supermagic. In the present paper, we study the existence of super $(a,d)$-cycle-antimagic labelings of ladder graphs for certain differences $d$.

Effect of filler on solid insulator surface and tracking length in high voltage systems

EASL-Vol. 2 (2019), Issue 2, pp. 37-42 Open Access Full-Text PDF

Irfanullah Khan, Muhammad Ahtasham Abid, Zahid Ullah, Arooj Rashid

Abstract: This paper presents the effect of different fillers on tracking length of electrical insulators. Insulator samples were prepared using polyester resin-c and were tested according to ASTM D2302. A standard test known as “Inclined plane test” is used to test the insulators after the application of high stresses. Track length of each sample is measured using a Polari scope. Track length of filler added insulators is compared to the insulator without filler and a significant change was noted among them.

Multilinear fractional integral with rough kernel on variable exponent Morrey-Herz spaces

OMS-Vol. 3 (2019), Issue 1, pp. 167-183 Open Access Full-Text PDF

Afif Abdalmonem, Omer Abdalrhman, Shuangping Tao

Abstract: In this article, we study a class of the multilinear fractional integral with rough kernel on Morrey-Herz space with $p(\cdot), q(\cdot), \alpha(\cdot).$ By using the properties of the variable exponent spaces, the boundedness of the multilinear fractional integral operator is obtained on variable nonhomogeneous Morrey-Herz spaces ${MK}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n}).$

First analytic solution for the oscillatory flow of a Maxwell’s fluid with annulus

OMS-Vol. 3 (2019), Issue 1, pp. 159-166 Open Access Full-Text PDF

Chaudry Masood Khalique, Rabia Safdar, Madeeha Tahir

Abstract: The major purpose of this article is to discuss the oscillatory flow of an incompressible viscous Maxwell fluids (IVMF) between two infinite coaxial of circular pipes. In the case when time $t=0$ the inner pipe is lying at rest where as at $t>0$ the inner pipe of the annulus starts to oscillate along the common axis of the pipes. The analytical solutions of the problem are obtained via integral transformation technique which is beneficial for time dependent problems. Moreover, the derived solutions are given under the series form of the generalized $G$ functions satisfying all the imposed auxiliary conditions whereas, the solutions for ordinary Maxwell and Newtonian fluids appear as the limiting case of the present obtained results. We include graphical comparison between Maxwell and Newtonian fluid, and we also explored the effects of different physical parameters on the fluid motion.

Positive solutions for nonlinear Caputo-Hadamard fractional differential equations with integral boundary conditions

OMA-Vol. 3 (2019), Issue 1, pp. 61-69 Open Access Full-Text PDF

Abdelouaheb Ardjouni, Ahcene Djoudi

Abstract: We study the existence and uniqueness of positive solutions of the nonlinear fractional differential equation with integral boundary conditions $\mathfrak{D}_{1}^{\alpha }x\left( t\right) =f\left( t,x\left( t\right) \right) ,\;\;\; 1<t\leq e, x\left( 1\right) =\lambda \int_{1}^{e}x\left( s\right) ds+d,$ where $\mathfrak{D}_{1}^{\alpha }$ is the Caputo-Hadamard fractional derivative of order $0<\alpha \leq 1$. In the process we convert the given fractional differential equation into an equivalent integral equation. Then we construct an appropriate mapping and employ the Schauder fixed point theorem and the method of upper and lower solutions to show the existence of a positive solution of this equation. We also use the Banach fixed point theorem to show the existence of a unique positive solution. Finally, an example is given to illustrate our results.

Solutions structures for some systems of fractional difference equations

OMA-Vol. 3 (2019), Issue 1, pp. 51-61 Open Access Full-Text PDF

M. B. Almatrafi

Abstract: It is a well-known fact that the majority of rational difference equations cannot be solved theoretically. As a result, some scientific experts use manual iterations to obtain the exact solutions of some of these equations. In this paper, we obtain the fractional solutions of the following systems of difference equations:
$$
x_{n+1}=\frac{x_{n-1}y_{n-3}}{y_{n-1}\left( -1-x_{n-1}y_{n-3}\right) },\ \ \
y_{n+1}=\frac{y_{n-1}x_{n-3}}{x_{n-1}\left( \pm 1\pm y_{n-1}x_{n-3}\right) }
,\ \ \ n=0,1,…,
$$
where the initial data $x_{-3},\ x_{-2},\ x_{-1},\ $\ $
x_{0},\ y_{-3},\ y_{-2},\ y_{-1}$ and $\ \ y_{0}\;$ are arbitrary non-zero real numbers. All solutions will be depicted under specific initial conditions.

Magnetohydrodynamic flow of dissipative casson-carreau nanofluid over a stretching sheet embedded in a porous medium under the influence of thermal radiation and variable internal heat generation

EASL-Vol. 2 (2019), Issue 2, pp. 18-36 Open Access Full-Text PDF

Gbeminiyi Sobamowo, O. A. Adesina, Lawrence Jayesimi

Abstract: In this paper, finite difference method is used to study the combined effects of thermal radiation, inclined magnetic field and temperature-dependent internal heat generation on unsteady two-dimensional flow and heat transfer analysis of dissipative Casson-Carreau nanofluid over a stretching sheet embedded in a porous medium. In the study, kerosene is used as the base fluid which is embedded with the silver (Ag) and copper (Cu) nanoparticles. Also, effects of other pertinent parameters on the flow and heat transfer characteristics of the Casson-Carreau nanofluids are investigated and discussed. From the results, it is established that the temperature field and the thermal boundary layers of Ag-Kerosene nanofluid are highly effective when compared with the Cu-Kerosene nanofluid. Heat transfer rate is enhanced by increasing power-law index and unsteadiness parameter. Skin friction coefficient and local Nusselt number can be reduced by magnetic field parameter and they can be enhanced by increasing the aligned angle. Friction factor is depreciated and the rate of heat transfer increases by increasing the Weissenberg number. A very good agreement is established between the results of the present study and the previous results. The present analysis can help in expanding the understanding of the thermo-fluidic behaviour of the Casson-Carreau nanofluid over a stretching sheet.

Dominator and total dominator colorings in vague graphs

EASL-Vol. 2 (2019), Issue 2, pp. 10-17 Open Access Full-Text PDF

Lian Chen, Huiqin Jiang, Zehui Shao, Marija Ivanović

Abstract: The concept of vague graph was introduced early by Ramakrishna and substantial graph parameters on vague graphs were proposed such graph coloring, connectivity, dominating set, independent set, total dominating number and independent dominating number. In this paper, we introduce the concept of the dominator coloring and total dominator coloring of a vague graph and establish mathematical modelling for these problems.

On the two-variable generalized Laguerre polynomials

OMS-Vol. 3 (2019), Issue 1, pp. 152-158 Open Access Full-Text PDF

Ahmed Ali Atash

Abstract: In this paper, we introduce the two variable generalized Laguerre polynomials (2VGLP) ${}_GL^{(\alpha,\beta)}_n(x,y)$. Some properties of these polynomials such as generating functions, summation formulae and expansions are also discussed.

Aphids, Ants and Ladybirds: a mathematical model predicting their population dynamics

OMS-Vol. 3 (2019), Issue 1, pp. 139–151 Open Access Full-Text PDF

Gianluca Gabbriellini

Abstract: The interaction between aphids, ants and ladybirds has been investigated from an ecological point of view since many decades, while there are no attempts to describe it from a mathematical point of view. This paper introduces a new mathematical model to describe the within-season population dynamics in an ecological patch of a system composed by aphids, ants and ladybirds, through a set of four differential equations. The proposed model is based on the Kindlmann and Dixon set of differential equations [1], focused on the prediction of the aphids-ladybirds population densities, that share a prey-predator relationship. The population of ants, in mutualistic relationship with aphids and in interspecific competition with ladybirds, is described according to the Holland and De Angelis mathematical model [2], in which the authors faced the problem of mutualistic interactions in general terms. The set of differential equations proposed here is discretized by means the Nonstandard Finite Difference scheme, successfully applied by Gabbriellini to the mutualistic model [3]. The constructed finite-difference scheme is positivity-preserving and characterized by four nonhyperbolic steady-states, as highlighted by the phase-space and time-series analyses. Particular attention is dedicated to the steady-state most interesting from an ecological point of view, whose asymptotic stability is demonstrated via the Centre Manifold Theory. The model allows to numerically confirm that mutualistic relationship effectively influences the population dynamic, by increasing the peaks of the aphids and ants population densities. Nonetheless, it is showed that the asymptotical populations of aphids and ladybirds collapse for any initial condition, unlike that of ants that, after the peak, settle on a constant asymptotic value.

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