Abstract
We developed a one step-three hybrid point constant order predictor corrector method for the solution of general third order initial value problems. The method was developed using method of interpolation and collocation of power series approximate solution to generate a continuous linear multistep method which was evaluated at some selected grid point to give the discrete linear multistep method. The predictors are implemented in block method while the corrector gave the solution at an overlapping interval. The basic properties of both the corrector and the predictors were investigated and found to be zero stable, consistent and convergent. The region of absolute stability was also investigated. The efficiency of the derived method was tested on some numerical examples and found to compete favourably with the existing methods.
Highlights
This paper considers the numerical solution to the general third order initial value problems of the form
Problem III is a stiff problem solved by Awoyemi and Idowu [13], where a hybrid method of order seven implemented in predictor corrector method was proposed
We have developed a one step, three hybrid points method implemented in constant order block predictor corrector method in this paper
Summary
F is continuous and satisfies the uniqueness theorem given by Awoyemi et al [1]. Direct method for the solution of higher order ordinary differential equations has been established in literature to be better than the method of reduction to system of first order ordinary differential equations [15]. It should be reminded that Milne in 1953 first developed block method to serve as a predictor to a predictor-corrector algorithm before it was later adopted as a full method [5,7,8,9,10,11] revisited the Milne approach and they concluded that though the method is expensive than the block method but gives better result than the block method They tagged Milne’s approach as constant order predictor-corrector method. We combine the unique properties of hybrid method and the constant order predictor-corrector method to develop a new numerical scheme for the solution of third order initial value problems of ordinary differential equations.
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