Journal of Prime Research in Mathematics

Abstract: Let \(R\) be a ring with center \(C(R)\). A ring \(R\) is called a ξring if, for any element \(x ∈ R\), there exists an element \(y ∈ R\) such that \(x − x^2y ∈ C(R)\). In Proc. Japan Acad. Sci., Ser. A – Math. (1957), Utumi describes the structure of these rings as a natural generalization of the classical strongly regular rings, that are rings for which \(x = x^2 y\). In order to make up a natural connection of \(ξ\)-rings with the more general class of von Neumann regular rings, that are rings for which \(x =xyx\), we introduce here the so-called generalized \(ξ\)-rings as those rings in which \(x − xyx ∈ C(R)\). Several characteristic properties of this newly defined class are proved, which extend the corresponding ones established by Utumi in these Proceedings (1957).

Abstract: The fixing number of a graph G is the smallest cardinality of a set of vertices \(F ⊆ V (G)\) such that only the trivial automorphism of \(G\) fixes every vertex in \(F\). In this paper, we introduce and study three new fixing parameters: fixing share, fixing polynomial and fixing value.

Abstract: We propose a meshless adaptive solution of the time-dependent partial differential equations (PDE) using radial basis functions (RBFs). The approximate solution to the PDE is obtained using multiquadrics (MQ). We choose MQ because of its exponential convergence for sufficiently smooth functions. The solution of partial differential equations arising in science and engineering frequently have large variations occurring over small portion of the physical domain. The challenge then is to resolve the solution behaviour there. For the sake of efficiency we require a finer grid in those parts of the physical domain whereas a much coarser grid can be used otherwise. Local scattered data reconstruction is used to compute an error indicator to decide where nodes should be placed. We use polyharmonic spline approximation in this step. The performance of the method is shown for numerical examples of one dimensional Kortwegde-Vries equation, Burger’s equation and Allen-Cahn equation.

Abstract: Steady analytical solution of non-isothermal, second grade fluid through an artery having constriction of cosine shape in two dimension is presented. The governing equations are transformed into stream function formulation which are solved analytically with the help of regular perturbation technique. The solutions thus obtained are presented graphically in terms of streamlines, wall shear stress, separation points, pressure gradient and temperature distribution. It is observed that an increase in height of constriction \((\in)\) gives rise in wall shear stress, pressure gradient and temperature, whereas critical Reynolds  number \((R_e)\) decreases. Further an increase in second grade parameter \((α)\) increases the temperature, pressure gradient, velocity and wall shear stress while critical Re decreases. Its worthy to mention that the present results are compared with the already published results which ensures good agreement.

Abstract: The unsteady time fractional free convection flow of an incompressible Newtonian fluid over an infinite vertical plate due to an impulsive motion of the plate and constant temperature at the boundary is analyzed. The old (Caputo) and new (Caputo-Fabrizio) fractional derivative approaches have been used to develop a physical model and a comparison has been drawn between their solutions. Boundary layers equations in non dimensional form are solved analytically by the Laplace transform technique. Exact solutions for velocity and temperature are obtained in terms of Wrights function. The expressions for rate of heat transfer in both cases are also determined. Solutions for integer order derivatives are obtained as limiting case. Numerical computations were made through software Mathcad and observed some physical aspects of fractional and material parameters are presented. It is found that the rate of heat transfer of Caputo Fabrizio model have higher values than Caputo one as we increased the value of fractional parameter and fractional fluids tend to superpose to that of ordinary fluid.

Abstract: In this paper, we suggest a new ninth order predictor-corrector iterative method to solve nonlinear equations. It is also shown that this new iterative method has convergence of order nine and has efficiency index 1.7321. Moreover, some examples are given to check its validity and efficiency. Finally, we present polynomiographs for some complex polynomials via our new method.