Insight to Multi-derivative Hybrid Linear Multistep Formula for directly Solving Third-order Initial Value Problems of Ordinary Differential Equations

Abstract

This article focuses on the multi-derivative hybrid linear multistep formula (MHLMF) for the numerical
solution of third-order ordinary differential equations. Power series was used as the basis function in the
derivation of the formula. An approximate solution from the basis function was interpolated at some selected off-grid points. In contrast, the third derivative of the approximate solution was located at all points in the grid and outside the grid to generate a system of linear equations to determine the unknown parameters. The derived method was examined to be consistent, convergent, and zero stable. The method was implemented to solve third order ordinary differential equations, including the Genesio equation, demonstrating that the derived methods efficiently handle nonlinear problems. Absolute errors obtained in the numerical experiments established the good performance of the proposed method when compared with other cited methods in the literature.