A CLASS OF CONTINUOUS y-function HYBRID METHODS FOR DIRECT SOLUTION OF DIFFERENTIAL EQUATIONS

Abstract

The aim of this study is to develop a class of continuous hybrid numerical methods for direct solution of general second-order initial value problem of ordinary differential equations. The approach adopted in this work is by interpolation and collocation of a basis function and its corresponding differential system respectively. Interpolation of the basis function was done at both grid and off-grid points while the differential systems are collocated at all grid points for different step-number k. This is to ensure that the hybrid points are at the y  function . Substitution of the unknown parameters into the basis function and simplification of the resulting equation produced the required continuous, consistent and symmetric hybrid methods. Attempts were made to derive predictors of the same order with the methods to circumvent the inherent disadvantage of predictor methods of lower order. The methods were applied to solve linear and nonlinear second-order initial value problems directly. Errors in the results obtained were compared with those of the existing methods of the same and even of higher order. The comparison shows that the accuracy of the new methods is better than the existing methods.