Some New Hybrid Block Methods for Solving Non-stiff Initial Value Problems of Ordinary Differential Equations

Abstract

In this paper, we derive some new k-step hybrid block methods (HBM) for the solution of first-order non-stiff initial value problems (IVPs) of ordinary differential equations (ODEs). A continuous hybrid multistep method (CHM) with variable coefficients is first developed using interpolation and collocation of a polynomial approximate solution. Discrete hybrid methods are then obtained by evaluating the CHM at some points of interest which give a class of discrete multistep methods that are implemented as hybrid block methods for k=3,4 yielding HBM3 and HBM4 respectively. Properties of the HBM3 and HBM4 are investigated and the results yield orders seven and nine respectively and their convergence is established. The numerical performance of the block methods on some non-stiff ODEs shows that the methods compare favourably with some existing methods.