New efficient numerical model for solving second, third and fourth order ordinary differential equations directly

Abstract

This article presents a two-step hybrid linear multistep block method for solving second, third and fourth order initial value problems of ordinary differential equations directly. The derivation of the method was done using collocation and interpolation techniques, while approximated power series was used as an interpolating 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 basic properties of the developed method, that is, order, error constant, zero stability, region of absolute stability, convergence and consistency of the method were properly investigated. The numerical results demonstrated that the scheme developed handles: second, third and fourth order ordinary differential equations efficiently and accurately when compared with existing methods. The proposed method takes away the burden of developing a separate method for the solution of second, third and fourth order initial value problem of ordinary differential equations.