A novel class of Obrechkoff-type methods for solving boundary value problems

Abstract

A new family of high-order Obrechkoff-type block methods for the direct solution of second-order boundary value problems is proposed in this study. To discretize the differential equations, the methods are built using piecewise linear and constant approaches, as well as interpolation and collocation techniques based on Hermite splines of both first and higher orders. They are further developed using truncated power series expansions, achieving eighth order accuracy with favourable stability properties. The method captures key characteristics of the problem and is computationally efficient. A thorough theoretical analysis confirms that it is consistent, convergent, and zero-stable. A single illustrative example is considered and comparison with the well-known Runge–Kutta method demonstrates its superiority in accuracy and error reduction.