# 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

### On A Mathematical Contradiction to Rethink Associativity and Commutativity for Infinite Series and Infinite Products

JPRM-Vol. 18 (2022), Issue 1, pp. 1 – 6 Open Access Full-Text PDF
Md. Shafiqul Islam, Sumit Bhowmick
Abstract: This paper deals with two contradictory values of π, focusing on the invalidity of associative and commutative laws for infinite series. The argument shows that operating with some infinite products leads to some dangerous contradictions such as the π value turns out to be 4 or 8/3. The study and the findings embedded with the article’s methodology points out that these classical operations like associativity and commutativity on infinite series or infinite products must be handled carefully.

### 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
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.

### Partition dimension of generalized Peterson and Harary graphs

JPRM-Vol. 17 (2021), Issue 2, pp. 84 – 94 Open Access Full-Text PDF
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.

### Degree-based topological indices and polynomials of cellulose

JPRM-Vol. 17 (2021), Issue 1, pp. 70 – 83 Open Access Full-Text PDF
Abdul Jalil M. Khalaf, M.C. Shanmukha, A. Usha, K.C. Shilpa, Murat Cancan
Abstract: This work attempts to compute cellulose’s chemical structure using topological indices based on the degree and its neighbourhood. The study of graphs using chemistry attracts a lot of researchers globally because of its enormous applications. One such application is discussed in this work, where the structure of cellulose is considered for which the computation of topological indices and analysis of the same are carried out. A polymer is a repeated chain of the same molecule stuck together. Glucose is a natural polymer also called, Polysaccharide. The diet of the humans include fibre which contains cellulose but direct consumption of the same may not be digestible by them.

### On a third-order fuzzy difference equation

JPRM-Vol. 17 (2021), Issue 1, pp. 59 – 69 Open Access Full-Text PDF
Ibrahim Yalcinkaya, Nur Atak, Durhasan Turgut Tollu
Abstract: In this paper, we investigate the qualitative behavior of the fuzzy
difference equation
\begin{equation*}
z_{n+1}=\frac{z_{n-2}}{C+z_{n-2}z_{n-1}z_{n}}\
\end{equation*}
where $$n\in \mathbb{N}_{0}=\mathbb{N}\cup \left\{ 0\right\}$$, $$(z_{n})$$ is a sequence of positive fuzzy numbers, $$C$$ and initial conditions $$z_{-2},z_{-1},z_{0}$$ are positive fuzzy numbers.