Journal of Prime Research in Mathematics

Abstract: A recently developed direct method known as the new auxiliary method shown significant improvement for solving non-leaner partial differential equations (PDEs) and gives more exact solutions compared to the traditional direct method. In this paper, we used the new auxiliary method for solitary wave solutions of the generalized Burgers Huxley equation (B-HE). The new auxiliary method is a very powerful, felicitous, effective method to get solitary wave solutions of PDEs.

Abstract: In this paper, we compute the general results in the form of the upper bounds of the Zagreb indices based on connection number called as first Zagreb connection index, second Zagreb connection index and modified first Zagreb connection index for the resultant graphs which are obtained by applying the product-related operations such as disjunction (co-normal product) and symmetric difference. At the end, some applications of the obtained results with a comparison between exact and computed values of the aforesaid Zagreb connection indices for the particular classes of alkanes are also included.

Abstract: In this manuscript, we construct a general family of optimal derivative free iterative methods by using rational interpolation. This family is further extended to a family of with-memory methods with increased order of convergence by employing two free parameters. At each iterative step, we use a suitable variation of the free parameters. These parameters are computed by using the information from current and previous iterations so that the convergence order of the existing family is increased from \(2^{n}\) to \(2^{n}+2^{n-1}+2^{n-2}\) without using any additional function evaluations. To check the performance of newly developed iterative schemes with and without memory, an extensive comparison with the existing with- and without memory methods is done by taking some real world problems and standard nonlinear functions. Numerical experiments illustrate that the proposed family of methods with-memory retain better computational efficiency and fast convergence speed as compared to existing with- and without memory methods. The performance of the methods is also analyzed visually by using complex plane. Numerical and dynamical comparisons confirm that the proposed families of with and without memory methods have better efficiency, convergence regions and speed in contrast with the existing methods of the same kind.

Abstract: This work considers a generalized class of biased estimators for the estimation of the unknown population mean of the variable of interest accompanying the issue of non-response in the study and in the auxiliary variables. The asymptotic bias and the asymptotic variance of the suggested class are acquired, up to the first degree of approximation and, compared with the linear regression estimator. The efficiency of the suggested estimators while comparing with the linear regression estimator and some other existing estimators are studied regarding percent relative efficiency (PRE). Furthermore, a simulation study also affirms the excellence of the considered class of estimators.

Abstract: In this paper, we prove \(\psi (A_{1}(4))< \psi(G)\), \( \forall \) groups which are not simple with order sixty, \( A_{1}(4) \) is Chevalley group (Linear group) of order 60. Also we prove that \( \psi (A_{2}(2))< \psi(G)\) using higher order non-simple groups of order 168.

Abstract: In this paper, we present a general family of optimal derivative free iterative methods of arbitrary high order for solving nonlinear equations by using Lagrange interpolation. The special cases of this family with optimal order of convergence two, four, eight and sixteen are obtained. These methods do not need the Newton’s or Steffensen’s iterations in the first step of their iterative schemes. The advantage of the new schemes is that they are also extendable to the iterative methods with-memory. Numerical experiments and polynomiographs are presented to confirm the theoretical results and to compare the new iterative methods with other well known methods of similar kind.