The nonlinear time-fractional differential equations with integral conditions

JPRM-Vol. 17 (2021), Issue 2, pp. 168 – 181 Open Access Full-Text PDF
Abstract: In this paper, we present a nonlinear equation modeling a time-fractional pseudoparabolic problem, involving fractional Caputo derivative where the fractional order is 0 < α < 1. We first started with the associated linear problem, we establish the energy inequalities to obtaine a priori estimate,and demonstrate the density of the operator’s range generated. Accordingly, the existence and uniqueness of the weak solutions are given, then we use the preceding results to handle the nonlinear case via an iterative process.

Some Integral Inequalities in the Framework of Conformable Fractional Integral

JPRM-Vol. 17 (2021), Issue 2, pp. 159 – 167 Open Access Full-Text PDF
Sikander Mehmood, Mariam Shahzad, Kiran Batool, Nawal Fatima, Juan E. N´apoles Vald$$^´$$es
Abstract: In this paper, we use new definition of left and right conformable fractional integral to obtain some new inequalities. The results obtained are refinements of existing results.

Isomorphism Theorems in Generalized d−algebras

JPRM-Vol. 17 (2021), Issue 2, pp. 149 – 158 Open Access Full-Text PDF
Abstract: We introduce the generalized d-algebras, generalized d-ideals (d∗-ideals, d#-ideals, d\$-ideals) and other related notions. We also prove some properties about d-ideal, d#-ideal and results related to quotient generalized d-algebra. Through these constructions, we prove the first, second and the third isomorphism theorems for the generalized d-algebras. These developments contribute to the theory of the BCI/BCK/BCH and the generalized BCH-algebras.

Computation of Topological Indices for Inner Dual Graph of Honeycomb and Graphene Network

JPRM-Vol. 17 (2021), Issue 2, pp. 138 – 148 Open Access Full-Text PDF
FM Bhatti, Iqra Zaman, Sawaira Sikander
Abstract: In QSPR/QSAR study, the molecular structure indices are now standard methods for studying structureproperty relations. Due to the chemical significance of these indices, the number of proposed molecular descriptors is quickly rising in the last few years. A topological index is a transformation of a chemical structure into a real number. In mathematics, honeycomb networks are widely used because of their extreme importance in computer graphics, image processing, cellular phone base stations, and in chemistry to represent benzenoid hydrocarbons. They are formed by recursively using hexagonal tiling in a particular pattern. HC(n) represents the honeycomb network of dimension n, where n is the number of hexagons between boundary and central hexagon. An atomic-scale honeycomb structure composed of carbon atoms is known as graphene. Professor Andre Geim and Professor Kostya Novoselov separated it from graphite in 2004. It is the first 2D material that is one million times thinner than human hair, two hundred times stronger than steel, and the world’s most conductive material. The graph 2D graphene is expressed as G(r, s) where “r” means the number of rows, and “s” is the number of hexagons in a row. This paper uses the inner dual graph of honeycomb networks and 2D graphene network, which are named as HcID(n) and GID(r, s) respectively. We derive some results related to topological indices for these graphs. We compute degree-based indices, first general Zagreb index, general Randi´c connectivity index, general sum-connectivity index, first Zagreb index, Second Zagreb index, Randi´c index, Atom-bond Connectivity (ABC) index, and Geometric-Arithmetic (GA) index of inner dual graphs of honeycomb networks and graphene network.

A Conceptual Framework of Convex and Concave Sets under Refined Intuitionistic Fuzzy Set Environment

JPRM-Vol. 17 (2021), Issue 2, pp. 122 – 137 Open Access Full-Text PDF
Abstract: Intuitionistic fuzzy set deals with membership and non-membership of a certain element of universe of discourse whereas these are further partitioned into their sub-membership degrees in refined intuitionistic fuzzy set. This study aims to introduce the notions of convex and concave refined intuitionistic fuzzy sets. Moreover, some of its important properties e.g. complement, union, intersection etc. and results are discussed.

Two-sided and modified two-sided group chain sampling plan for Pareto distribution of the 2nd kind

JPRM-Vol. 17 (2021), Issue 2, pp. 115 – 121 Open Access Full-Text PDF
Abdur Razzaque Mughal , Zakiyah Zain , Nazrina Aziz
Abstract: In this article two-sided and modified two-sided group chain sampling plans for Pareto distribution of the 2nd kind is proposed. The minimum group size and mean ratios are obtained by satisfying the consumer’s risk at the pre-specified quality level. Several tables are presented for easy selection of comparative study of the proposed plan with established plans.

A cubic trigonometric B-spline collocation method based on Hermite formula for the numerical solution of the heat equation with classical and non-classical boundary conditions

JPRM-Vol. 17 (2021), Issue 2, pp. 95 – 114 Open Access Full-Text PDF
Aatika Yousaf, Muhammad Yaseen
Abstract: In this article, a new trigonometric cubic B-spline collocation method based on the Hermite formula is presented for the numerical solution of the heat equation with classical and non-classical boundary conditions. This scheme depends on the standard finite difference scheme to discretize the time derivative while cubic trigonometric B-splines are utilized to discretize the derivatives in space. The scheme is further refined utilizing the Hermite formula. The stability analysis of the scheme is established by standard Von-Neumann method. The numerical solution is obtained as a piecewise smooth function empowering us to find approximations at any location in the domain. The relevance of the method is checked by some test problems. The suitability and exactness of the proposed method are shown by computing the error norms. Numerical results are compared with some current numerical procedures to show the effectiveness of the proposed scheme.
Abstract: The distance of a connected, simple graph $$\mathbb{P}$$ is denoted by $$d({\alpha}_1,{\alpha}_2),$$ which is the length of a shortest path between the vertices $${\alpha}_1,{\alpha}_2\in V(\mathbb{P}),$$ where $$V(\mathbb{P})$$ is the vertex set of $$\mathbb{P}.$$ The $$l$$-ordered partition of $$V(\mathbb{P})$$ is $$K=\{K_1,K_2,\dots,K_l\}.$$ A vertex $${\alpha}\in V(\mathbb{P}),$$ and $$r({\alpha}|K)=\{d({\alpha},K_1),d({\alpha},K_2),\dots,d({\alpha},K_l)\}$$ be a $$l$$-tuple distances, where $$r({\alpha}|K)$$ is the representation of a vertex $${\alpha}$$ with respect to set $$K.$$ If $$r({\alpha}|K)$$ of $${\alpha}$$ is unique, for every pair of vertices, then $$K$$ is the resolving partition set of $$V(\mathbb{P}).$$ The minimum number $$l$$ in the resolving partition set $$K$$ is known as partition dimension ($$pd(\mathbb{P})$$). In this paper, we studied the generalized families of Peterson graph, $$P_{{\lambda},{\chi}}$$ and proved that these families have bounded partition dimension.