We propose a new self-starting sixth-order hybrid block linear multistep method using backward differentiation formula for direct solution of third-order differential equations with either initial conditions or boundary conditions. The method used collocation and interpolation techniques with three off-step points and five-step points, choosing power series as the basis function. The convergence of the method is established, and three numerical experiments of initial and boundary value problems are used to demonstrate the efficiency of the proposed method. The numerical results in Tables and Figures show the efficiency of the method. Furthermore, the numerical method outperformed the results from existing literature in terms of accuracy as evident in the results of absolute errors produced. تفاصيل المقالة

In this paper, we propose some necessary conditions for convergence of Triple Accelerated Over-Relaxation (TAOR) method with respect to $M-$ coefficient matrices. The theoretical approach for the proofs is analyzed through some standard procedures in the literature. Some numerical experiments are performed to show the efficiency of our approach, and the results obtained compared favourably with those obtained through the existing methods in terms of spectral radii of their iteration matrices. تفاصيل المقالة