An Extension to Implicit Second Derivative Method with Variable Step Number for Direct Solution of Second Order Ordinary Differential Equations
Olumurewa O.K. *
Department of Mathematical Sciences, Federal University of Technology, Akure, Ondo State, Nigeria.
Orunko O.E.
Department of Mathematical Sciences, Federal University of Technology, Akure, Ondo State, Nigeria.
Olaleye J.T.
Department of Mathematical Sciences, Federal University of Technology, Akure, Ondo State, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
This study developed a high – order numerical method, specifically, an implicit second derivative multistep method for direct solution of Initial Value Problems (IVPs) of second order Ordinary Differential Equations (ODEs).
Development of variants of the second derivative multistep method required order of derivative (l) to be made constant at 2 while the step – number (k) was varied from 1 – 3. Taylors series was adopted as the basis function and the derived methods were verified for accuracy, consistency, stability and convergence. Computational efficiency of the derived methods was verified using standard benchmark IVPs of second order ODEs.
Verification of the basic properties showed that the derived methods are accurate, zero - stable, consistent and convergent. Error analysis between numerical and exact solutions revealed the level of accuracy of the derived methods, which showed that three – step second derivative (k=3, l=2) method had better accuracy than one - step second derivative method (k=1, l=2) and two – step second derivative method (k=2, l=2) while the two – step second derivative method (k=2, l=2) had better accuracy than one - step second derivative method (k=1, l=2) and also competes favorably with three – step second derivative method (k=3, l=2). The results showed that accuracy of the methods increased as the step - number (k) increased, hence three – step second derivative (k=3, l=2) method was recommended for direct solution of second order IVPs.
Keywords: Implicit, derivative, step – number, order, stability, direct solution