A Pair of Three-Step Hybrid Block Methods for the Solutions of Linear and Nonlinear First-Order Systems

Abstract

In this research paper, a pair of three-step hybrid block methods is derived for the solutions of linear and nonlinear first-order systems. The derivation is carried out with the aid of collocation and interpolation technique and the adoption of power series as basis function. The first and second three-step hybrid block methods are derived by incorporating a single and double off-grid point(s) respectively within the three-step integration interval. The methods derived were then applied on some linear and nonlinear first-order systems to test their accuracy and efficiency. The results obtained show that the three-step hybrid block method with two off-grid points performed better than the three-step hybrid block method with one off-grid point. It was also clear from the results obtained that the two methods derived performed better than the existing methods with which we compared our results. We further analyzed the basic properties of the methods derived. These properties include zero-stability, consistence, convergence and region of absolute stability.