Two Higher-Order Single-Step Embedded Block Hybrid Integrators for First-Order Ordinary Differential Equations

In this research work, the extended Backward Differentiation Formula (EBDF) is used to develop an embedded block hybrid method (EBHM) for step numbers and for the solution of first-order initial-value problems (IVPs) in ordinary differential equations (ODEs). The methods are obtained by continuous approximation using multi-step collocation techniques. The new embedded block hybrid method for and consist of five and six discrete formulae, respectively, which are simultaneously used as integrators. Analyses of the EBHM5 and EBHM6 indicate that the methods are respectively of orders six and seven, hence consistent. In addition, both methods are zero-stable and therefore convergent. Linear stability analyses indicate that the methods are A-stable, hence suitable for stiff problems. The new methods are used to solve some numerical examples, and the absolute errors show a better rate of convergence when compared with existing methods in the literature.