There are dozens of block methods in literature intended for solving second order initial-value problems. This article aimed at the analysis of the efficiency of k-step block methods for directly solving general second-order initial-value problems. Each of these methods consists of a set of $2k$ multi-step formulas (although we will see that this number can be reduced to $k+1$ in case of a special equation) that provides approximate solutions at k grid points at once. The usual way to obtain these formulas is by using collocation and interpolation at different points, which are not all necessarily in the mesh (it may also be considered intra-step or off-step points). An important issue is that for each k, all of them are essentially the same method, although they can adopt different formulations. Nevertheless, the performance of those formulations is not the same. The analysis of the methods presented give some clues as how to select the most appropriate ones in terms of computational efficiency. The numerical experiments show that using the proposed formulations, the computing time can be reduced to less than half.

A Block of hybrid method with three off-step points based is presented in this work for direct approximation of solution of third-order Initial and Boundary Value Problems (IVPs and BVPs). This off-step points are formulated such that they exist only on a single step at a time. Hence, these points are shifted to three positions respectively in order to obtain three different integrators for computational analysis. These analysis includes; order of the methods, consistency, stability and convergence, global error, number of functions evaluation and CPU time. The superiority of these methods over existing methods is established numerically on different test problems in literature

Fifth-order Korteweg-de Vries (KdV) equations, arise in modeling waves phenomena such as the propagation of shallow water waves over a flat surface, gravity-capillary waves and sound waves in plasmas. In this work, a one-step block hybrid linear multistep method was derived using the collocation technique, to solve fifth-order KdV models via the Method of Line (MoL). The consistency, stability and convergence of the method were established. The efficiency of the method can be seen from comparison of the exact solutions of problems and other methods cited from literature.

This study employs Shifted Vieta-Lucas Polynomials using the variational iteration approach to numerically resolve sixth and seventh order Boundary Value Problems (BVPs), The proposed method in the study is used, with the trial functions for the approximation being the shifted Vieta Lucas polynomials generated for the given boundary value problem. The study provides an efficient and accurate solutions to higher-order BVPs using a combination of Shifted Vieta-Lucas Polynomials and the variational iteration approach and finds applications in mathematics and other fields of sciences and engineering.

The search for efficient higher order methods is a constant goal in numerical analysis. In this paper, a higher order two‐step hybrid block method is presented to directly solve second‐order initial value problems in ordinary differential equations. In addition to the higher order, the proposed method has been formulated in variable step‐size mode to extract its best performance. Comparisons with other methods in the literature show the good accuracy it can provide. Theoretical aspects such as linear stability and convergence analysis are also discussed.