Two Step Block Hybrid Method with Four Off-mesh Points (2BHM4) for the Direct Solutions of Second, Third, Fourth and Fifth-Order Oscillatory Differential Equations

This research addresses the computational burden and accumulated errors that can arise when higher-order ordinary differential equations are reduced to first-order systems before numerical solution. A Two-Step Block Hybrid Method with Four Off-mesh Points (2BHM4) is developed for the direct solution of second-, third-, fourth- and fifth-order initial value problems. The method is derived from a power-series basis function through interpolation and collocation, using a specific configuration of off-mesh interpolation and collocation points that yields a uniform eighth-order scheme applicable to all four orders within a single framework. The theoretical properties of the method are analysed, and the method is reported to be consistent, zero-stable and A-stable. The 2BHM4 is applied directly, without reduction to first-order systems, to a second-order cooling-of-a-body problem, a third-order oscillatory linear equation, a fourth-order oscillatory equation and a special fifth-order initial value problem. The numerical simulations show that the method generally achieves small absolute errors compared with the referenced existing methods across the four test problems, with solution curves that are nearly indistinguishable from the exact solutions. These findings indicate that 2BHM4 is an accurate direct solver for the higher-order differential equations examined, while its block structure enables simultaneous evaluation at multiple points.