Linear Hybrid Multistep Block Method for Direct Solution of Initial Value Problems of Third Order Ordinary Differential Equations

Abstract

In this article, we focus on linear hybrid multistep method for direct solution of initial value problems of third order ordinary differential equations without reduction to system of first-order ordinary differential equations. The derivation of the method involved using collocation and interpolation techniques with power series as basis function to produce a system of linear equations. The unknown parameters in the system of equations were obtained through the Gaussian elimination technique. The values of the determined parameters were then substituted and evaluated at different grid and off-grid points to produce the required continuous block method. The discrete scheme obtained from the method is self-starting with improved accuracy and a larger interval of absolute stability. Basic properties of the method were investigated. The results showed that the method is zero stable, consistent , convergent and of order seven. The performance of the method was tested by solving linear and nonlinear problems of general third order ordinary differential equations. The result were found to compare favourably with some existing methods in literature.