Generalization of a class of uniformly optimized k -step hybrid block method for solving two-point boundary value problems

This study aims to develop, analyze and implement an efficient method for approximating two-point boundary value problems of ordinary differential equations. The method contains six and twelve implicit formulas, respectively, for the one-step and two-step schemes. The continuous approximations, using the shifted Chebyshev polynomial as the basis function, were obtained via evaluations at three different points on the selected one-step method, including two optimized hybrid points. Evaluations were carried out on six different points on the selected two-step method, including four generalized optimized hybrid points. Qualitative analysis of the method proves the proposed methods are consistent, zero-stable, convergent and have a larger region of absolute stability. The quantitative analysis shows that the methods compared favorably well and established some superiority strength with the existing methods.