This paper proposes a new hybrid block method of order five for solving second-order ordinary differential equations directly. The method is developed using interpolation and collocation techniques. The use of the power series approximate solution as an interpolation polynomial and its second derivative as a collocation equation is considered in deriving the method. Properties of the method such as zero stability, order, consistency, convergence and region of absolute stability are investigated. The new method is then applied to solve the system of second-order ordinary differential equations and the accuracy is better when compared with the existing methods in terms of error.

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Our focus in this paper is the development, analysis and implementation of a new hybrid block method with one generalized off step point for solving second order ordinary differential equation directly. In the derivation of the method, power series is adopted as basis function to obtain the main continuous scheme through collocation and interpolations approach. Taylor method is also used together with new method to generate the non-overlapping numerical results. As required by all numerical methods, the numerical properties of the new block which include convergent and stability region are also investigated. The developed method was found to compare favourably with the existing methods in term of error.

A new one step hybrid block method with generalized three off-step points using interpolation and collocation approach for solving initial value problems of second order ordinary differential equation directly is proposed. In deriving this method, the power series used as basis function to approximate the solution is interpolated at the last two points of the three off-step points while its second derivative is collocated at all points in the selected interval. The method is proven to be zero stable, consistent, convergent and of order five. For the purpose of testing, specific points of the developed method are chosen to solve second order initial value problem directly. The numerical results obtained indicate Z. Omar and Ra'ft. Abdelrahim that the new method outperforms the previous methods in terms of accuracy.