Direct Solutions of Second-Order Ordinary Differential Equations Using Linear Multi-Step Method

Abstract

The analytical solutions to second-order ODEs often prove elusive, necessitating the development of numerical methods for their approximation. However, solving second-order ordinary differential equations (ODEs) directly with linear multi-step methods abridges the transformation process of changing the equations to a system of first order equations before solving. The aim of this study is to develop and optimize a direct solution methodology for second-order ordinary differential equations (ODEs) by comparing the step length of linear multi-step methods. Having developed a new class of continuous linear multi step methods, initial value problems of general second order ordinary differential equations was solved via collocation and interpolation technique on the Chebyshev polynomial equations. The three-step method together with the four-step method that was developed were analyzed based on the properties of linear multistep methods and were found to be zero -stable, consistent and convergent with good region of absolute stability. The new class of linear multi step methods has the advantage of evaluation of functions at different step lengths. These proposed methods were implemented on second order ordinary differential initial value problems. The performance of the new methods was tested and compared to the exact solution of two worked examples. Hence, the computed results together with its associated errors shows that the equation converges faster as the step length reduces.