A CLASS OF IMPLICIT MULTI-HYBRID BLOCK METHODS FOR SOLUTION OF SPECIAL SECOND ORDER ORDINARY DELAY DIFFERENTIAL EQUATIONS

Abstract

This research considers the numerical solutions of a class of implicit Multi-Hybrid Block Methods for solution of special second order Ordinary Delay Differential Equations (ODDEs) directly without reduction to systems of first order ordinary differential equations. The methods were generated using collocation approach via a combination of power series and exponential functions. The approximate basis functions are interpolated at the first two grid points and collocated at both grid and off-grid points. The developed schemes and its derivatives were combined to form block methods to simultaneously solve second order Ordinary Delay Differential Equations (ODDEs) directly without the rigor of developing separate predictors. The developed methods were obtained for different step lengths, specifically, for k = 2, 3, 4 and 5 with generalized number of hybrid points (3k). The basic properties of the methods were examined, the methods were found to have high order of accuracy, low error constants, possesses large interval of absolute stability and hence convergent. The developed methods were applied to solve some special second order Ordinary Delay Differential Equations. The methods also solve an engineering problem namely Matheiu’s equation in order to test for the efficiency and accuracy of the methods. The results obtained were compared with some existing methods in the literature. The results obtained showed better performance in terms of accuracy when compared.