Formation of Hybrid Block Method of Higher Step-Sizes, Through the Continuous Multi-Step Collocation

Abstract

We construct a self-starting Simpson’s type block hybrid method (BHM) consisting of very closely accurate members each of order p=q+2 as a block. The higher order members of each were obtained by increasing the number k in the multi-step collocation (MC) used to derive the k-step continuous formula (k≥2) through the aid of MAPLE soft wire program. The stability analysis is poorer as the step size k increases, as expected with the linear multi-step methods (lmms), discrete or continuous. In this paper we identify a continuous hybrid block schemes (CHBS) through the addition of one off-mesh collocation points in the MC. The (CHBS) is evaluated along with it’s first derivative where necessary to give continuous hybrid block schemes for a simultaneous application to the stiff ordinary differential equations (SODEs} with initial, boundary or mixed conditions.