A new numerical model for the direct solution of higher-order ordinary differential equations

A unique continuous three-step hybrid block method for the solution of second, third, and fourth-order initial value problems with constant step size was proposed in this research. A linear multi-step collocation approach was applied in the derivation of the new method with the use of a power series approximate solution as an interpolation polynomial. The fourth derivative of the power series was collocated at the entire grid and off-grid points, while the fifth and sixth derivatives of the polynomial were collocated at the endpoint only. The numerical integrators that formed the block were also derived by evaluating the continuous scheme along with its derivatives at the non-interpolating points within the selected interval of the integration. The basic properties of the developed method were properly investigated. The comparison of the results with the existing methods showed that the new block method was better in accuracy than the existing methods.