Journal of Prime Research in Mathematics

Abstract: The major idea of this paper is to establish some new improvements of the Hardy inequality by using \(k\)-fractional integral of Riemann-type, Caputo \(k\)-fractional derivative, Hilfer \(k\)-fractional derivative and Riemann-Liouville \((k,r)\)-fractional integral. We discuss the \(\log\)-convexity of the related linear functionals. We also deduce some known results from our general results.

Abstract: Topological indices helps us to collect information about algebraic graphs and gives us mathematical approach to understand the properties of algebraic structures. Here, we will introduce Multiplicative version of Shingali and Kanabour indices and compute these indices for Bismuth Tri-Iodide chain \(m-Bil_{3}\) and sheet \(Bil_{3}(m\times n)\).

Abstract: Theoretical analysis of unsteady incompressible viscous fluid has been carried with constant proportional Caputo fractional derivative namely constant proportional Caputo type with singular kernel. The modeled considered in this paper is the fundament problem of fluid dynamics. The resulting governing equations are modeled with hybrid fractional operator of singular kernel and its solution obtained by using Laplace transform method and expressed in terms of series. Some graphs are captured for fractional parameter \(\alpha\) for large and small time and found that velocity shows dual trend for small and large values of time for different values of fractional parameter $\alpha$. Further, compared the present results with the results obtained with new fractional operators and found that constant proportional Caputo type operator portrait better velocity decay. Moreover, for increasing time, momentum boundary layer thickness increases while for grater values of fractional parameter it reduces.

Abstract: Various partial attempts to count cycles and components of digraphs from congruences have been made earlier. While the problem is still open till date. In this work, we introduce a new approach to solve the problem over quadratic congruence equations. Define a mapping \(g:Z_{m}\mapsto Z_{m}\) by \(g(t)=t^{2}\), where \(Z_{m}\) is the ring of residue classes modulo \(m\). The digraph \(G(2,m)\) over the set of residue classes assumes an edge between the residue classes \(\overline{x}\) and \(\overline{y}\) if and only if \(g(\overline{x})\equiv \overline{y}~(\text{mod}~m)\) for \(m\in Z^{+}\). Classifications of cyclic and non-cyclic vertices are proposed and proved using basic modular arithmetic. Finally, explicit formulas for the enumeration of non-isomorphic components are proposed followed by simple proofs from number theory.

Abstract: This article introduces a novel technique called modified fractional Taylor series method (MFTSM) to find numerical solutions for the fractional SIS epidemic model. The fractional derivative is considered in the sense of Caputo. The most important feature of the MFTSM is that it is very effective, accurate, simple, and more computational than the methods found in literature. The validity and effectiveness of the proposed technique are investigated and verified through numerical example.

Abstract: In this work, a new efficient method called, Elzaki’s fractional decomposition method (EFDM) has been used to give an approximate series solutions to time-fractional Sine-Gordon equation. The time-fractional derivatives are described in the Caputo and Caputo-Fabrizio sense. The EFDM is based on the combination of two different methods which are: the Elzaki transform method and the Adomian decomposition method. To demonstrate the accuracy and efficiency of the proposed method, a numerical example is provided. The obtained results indicate that the EFDM is simple and practical for solving the fractional partial differential equations which appear in various fields of applied sciences.