### 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 1, pp. 84 – 94 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 1, 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.

### Some Opial-type inequalities involving fractional integral operators

JPRM-Vol. 17 (2021), Issue 1, pp. 48 – 58 Open Access Full-Text PDF
Abstract: The core idea of this paper is to provide the Opial-type inequalities for Hadamard fractional integral operator and fractional integral of a function with respect to an increasing function $$g$$. Moreover, related extreme cases and counter part of our main results are also given in the paper.

### Reversed degree-based topological indices for Benzenoid systems

JPRM-Vol. 17 (2021), Issue 1, pp. 40 – 47 Open Access Full-Text PDF
Abdul Jalil M. Khalaf, Abaid ur Rehman Virk, Ashaq Ali, Murat Cancan
Abstract: Topological indices are numerical values that correlate the chemical structures with physical properties. In this article, we compute some reverse topological indices namely reverse Atom-bond connectivity index and reverse Geometric-arithmetic index for four different types of Benzenoid systems.

### Determinant Spectrum of Diagonal Block Matrix

JPRM-Vol. 17 (2021), Issue 1, pp. 35 – 39 Open Access Full-Text PDF
Elif OTKUN CEVIK, Zameddin I. ISMAILOV
Abstract: It is known that in mathematical literature one of important questions of spectral theory of operators is to describe spectrum of diagonal block matrices in the direct sum of Banach spaces with the spectrums of their coordinate operators. This problem has been investigated in works [1] and [2]. Also for the singular numbers similar investigation has been made in [3]. In this paper the analogous question is researched. Namely, the relationships between $$\epsilon$$-determinat spectrums of the diagonal block matrices and their block matrices are investigated. Later on, some applications are given.

### Inequalities of Hardy-type for Multiple Integrals on Time Scales

JPRM-Vol. 17 (2021), Issue 1, pp. 21 – 34 Open Access Full-Text PDF
Dawood Ahmad, Khuram Ali Khan, Ammara Nosheen
Abstract: We extend some inequalities of Hardy-type on time scales for functions depending on more than one parameter. The results are proved by using induction principle, properties of integrals on time scales, chain rules for composition of two functions, Hölder’s inequality and Fubini’s theorem in time scales settings.

### On algebraic aspects of SSC associated to the subdivided prism graph

JPRM-Vol. 17 (2021), Issue 1, pp. 7 – 20 Open Access Full-Text PDF
Mehwish Javed, Agha Kashif, Muhammad Javaid
Abstract: In this article, some important combinatorial and algebraic properties of spanning simplicial complex associated to the subdivided prism graph $$P(n,m)$$ are presented. The $${f}-$$vector of the spanning simplicial complex $$\Delta_s(P(n,m))$$ and the Hilbert series for the face ring $$K\big[\Delta_s(P(n,m))\big]$$ are computed. Further, the associated primes of the facet ideal $$I_{\mathcal{F}}(\Delta_s(P(n,m)))$$ are determined. Finally, the Cohen-Macaulay characterization of the SR-ring of $$\Delta_s(P(n,m))$$ is discussed.