# Journal of Prime Research in Mathematics

Journal of Prime Research in Mathematics (JPRM) ISSN: 1817-3462 (Online) 1818-5495 (Print) is an HEC recognized, Scopus indexed, open access journal which provides a plate forum to the international community all over the world to publish their work in mathematical sciences. JPRM is very much focused on timely processed publications keeping in view the high frequency of upcoming new ideas and make those new ideas readily available to our readers from all over the world for free of cost. Starting from 2020, we publish one Volume each year containing two issues in June and December. The accepted papers will be published online immediate in the running issue. All issues will be gathered in one volume which will be published in December of every year.

Latest Published Articles

### Combined effect of slip and radiation on MHD flow past a constantly moving vertical plate with variable temperature

JPRM-Vol. 1 (2016), Issue 1, pp. 130 – 144 Open Access Full-Text PDF
M. A. Imran, Shakila Sarwar, M. Imran, Maryam Aleem.
Abstract: The unsteady free convection of an MHD flow of a viscous fluid passing a vertical plate which is constantly moving with variable temperature is analyzed by taking slip and radiation into consideration. The dimensionless governing equations for temperature and velocity fields are solved using Laplace transform technique. The radiative and slip effects are taken into consideration and the whole system is rotating as a rigid body with a constant angular velocity about the z-axis. Exact solutions are obtained for the two components of velocity. Some known solutions from the literature are obtained as a limiting case. The obtained solutions satisfy the initial and boundary conditions. Some physical aspects of flow parameters on the fluid motion are graphically presented.

### Cartesian product of interval-valued fuzzy ideals in ordered semigroup

JPRM-Vol. 1 (2016), Issue 1, pp. 120 – 129 Open Access Full-Text PDF
Hidayat Ullah Khan, Asghar Khan, Nor Haniza Sarmin
Abstract: Interval-valued fuzzy set theory is a more generalized theory that can deal with real world problems more precisely than ordinary fuzzy set theory. In this paper, the concepts of interval-valued fuzzy (prime, semiprime) ideal and the Cartesian product of interval-valued fuzzy subsets have been introduced. Some interesting results about Cartesian product of interval-valued fuzzy ideals, interval-valued fuzzy prime ideals, intervalvalued fuzzy semiprime ideals, interval-valued fuzzy bi-ideals and intervalvalued fuzzy interior ideals in ordered semigroups are obtained. The purport of this paper is to link ordinary ideals with interval-valued fuzzy ideals by means of level subset of Cartesian product of interval-valued fuzzy subsets.

### A novel approach to approximate unsteady squeezing flow through porous medium

JPRM-Vol. 1 (2016), Issue 1, pp. 91 – 109 Open Access Full-Text PDF
Mubashir Qayyum, Hamid Khan, M.T. Rahim
Abstract: In this article, a new alteration of the Homotopy Perturbation Method (HPM) is proposed to approximate the solution of unsteady axisymmetric flow of Newtonian fluid. The flow is squeezed between two circular plates and passes through a porous medium channel. The alteration extends the Homotopy Perturbation with a Laplace transform, which is referred to as the Laplace Transform Homotopy Perturbation Method (LTHPM) in this manuscript. A single fourth order non-linear ordinary differential equation is obtained using similarity transformations. The resulting boundary value problem is then solved through LTHPM, HPM and fourth order Implicit Runge Kutta Method (IRK4). Convergence of the proposed scheme is checked by finding absolute residual errors of various order solutions. Also, the validity is confirmed by comparing numerical and analytical (LTHPM) solutions. The comparison of obtained residual errors shows that LTHPM is an effective scheme that can be applied to various initial and boundary value problems in science and engineering

### Reciprocal product degree distance of strong product of graphs

JPRM-Vol. 1 (2016), Issue 1, pp. 79 – 90 Open Access Full-Text PDF
K. Pattabiraman, A. Arivalagan, V.S.A. Subramanian
Abstract: In this paper, the exact formula for the reciprocal product degree distance of strong product of a connected graph and the complete multipartite graph with partite sets of sizes $$m_0, m_1, . . . , m_{r−1}$$ is obtained. Using the results obtained here, the formula for the reciprocal degree distance of the closed fence graph is computed.

### Hardy-type inequalities involving generalized fractional integrals via superquadratic functions

JPRM-Vol. 1 (2016), Issue 1, pp. 60 – 78 Open Access Full-Text PDF
Sajid Iqbal, Josip Pecaric, Muhammad Samraiz

### Vertex-to-clique detour distance in graphs

JPRM-Vol. 1 (2016), Issue 1, pp. 45 – 59 Open Access Full-Text PDF
I. Keerthi Asir, S. Athisayanathan
Abstract: Let $$v$$ be a vertex and $$C$$ a clique in a connected graph $$G$$. A vertex-to-clique $$u − C$$ path P is a $$u − v$$ path, where v is a vertex in $$C$$ such that $$P$$ contains no vertices of $$C$$ other than $$v$$. The vertex-to-clique distance, $$d(u, C)$$ is the length of a smallest $$u−C$$ path in $$G$$. A $$u−C$$ path of length $$d(u, C)$$ is called a $$u − C$$ geodesic. The vertex-to-clique eccentricity $$e_1(u)$$ of a vertex $$u$$ in $$G$$ is the maximum vertex-to-clique distance from $$u$$ to a clique $$C ∈ ζ$$, where $$ζ$$ is the set of all cliques in $$G$$. The vertex-to-clique radius $$r_1$$ of $$G$$ is the minimum vertex-to-clique eccentricity among the vertices of $$G$$, while the vertex-to-clique diameter $$d_1$$ of $$G$$ is the maximum vertex-to-clique eccentricity among the vertices of $$G$$. Also the vertex toclique detour distance, $$D(u, C)$$ is the length of a longest $$u−C$$ path in $$G$$. A $$u−C$$ path of length $$D(u, C)$$ is called a $$u−C$$ detour. The vertex-to-clique detour eccentricity $$e_{D1}(u)$$ of a vertex $$u$$ in $$G$$ is the maximum vertex-toclique detour distance from u to a clique $$C ∈ ζ$$ in $$G$$. The vertex-to-clique detour radius $$R_1$$ of $$G$$ is the minimum vertex-to-clique detour eccentricity among the vertices of $$G$$, while the vertex-to-clique detour diameter $$D_1$$ of $$G$$ is the maximum vertex-to-clique detour eccentricity among the vertices of $$G$$. It is shown that $$R_1 ≤ D_1$$ for every connected graph $$G$$ and that every two positive integers a and b with $$2 ≤ a ≤ b$$ are realizable as the vertex-to-clique detour radius and the vertex-to-clique detour diameter, respectively, of some connected graph. Also it is shown that for any three positive integers $$a$$, $$b$$, $$c$$ with $$2 ≤ a ≤ b < c$$, there exists a connected graph G such that $$r_1 = a$$, $$R_1 = b$$, $$R = c$$ and for any three positive integers $$a$$, $$b$$, $$c$$ with $$2 ≤ a ≤ b < c$$ and $$a + c ≤ 2b$$, there exists a connected graph $$G$$ such that $$d_1 = a$$, $$D_1 = b$$, $$D = c$$.