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Latest Published Articles

The smallest sum-connectivity index on trees with \(n\) vertices and \(k\) pendant vertices

ODAM-Vol. 2 (2019), Issue 2, pp. 23-30 Open Access Full-Text PDF
Yuedan Yao
Abstract: For a given connected graph \(G\) and a real number \(\alpha\), denote by \(d(u)\) the degree of vertex \(u\) of \(G\), and denote by \(\chi_{\alpha}(G)=\sum_{uv\in E(G)} \big(d(u)+d(v)\big)^{\alpha}\) the general sum-connectivity index of \(G\). In the present note, we determine the smallest general sum-connectivity index of trees (resp., chemical trees) together with corresponding extremal trees among all trees (resp., chemical trees) with \(n\) vertices and \(k\) pendant vertices for \(0<\alpha<1.\)
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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 \(d\).
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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.
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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 \({MK}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n}).\)
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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.
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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.
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BOOK-foundations-of-mathematical-analysis-and-semigroups-theory
BOOK - NULL CONTROLLABILITY OF DEGENERATE AND NON-DEGENERATE SINGULAR PARABOLIC