Direct Solution of 𝒚′′(𝒙) = 𝒇(𝒙, 𝒚, 𝒚′) Using Four Points Block-Hybrid Linear Multistep Method of Order Seven with Applications

Abstract

The study aims to construct an implicit block hybrid method with four points to tackle general second order initial value problems of ordinary differential equations (ODEs) directly. Power series is used as the basis function to obtain the proposed method which involved the first and second derivatives of 𝑓(𝑥, 𝑦, 𝑦′). From the investigation done, it was found that the proposed method is consistent and zerostable, hence it is convergent. The proposed method’s efficiency was obtained and a comparison was made in terms of accuracy to some existing methods with similar order and the ones higher than it. The new proposed method is able to solve linear, nonlinear and systems of equations of general second order Initial Value Problems and outperformed existing methods with impressive results. Applications of the proposed method to a real-life problem is discussed.