Implicit One-Step Block Hybrid Third-Derivative Method for the Direct Solution of Initial Value Problems of Second-Order Ordinary Differential Equations

Abstract

A new one-step block method with generalized three hybrid points for solving initial value problems of second-order ordinary differential equations directly is proposed. In deriving this method, a power series approximate function is interpolated at $\{x_n,x_{n+r}\}$ while its second and third derivatives are collocated at all points $\{x_n,x_{n+r},x_{n+s},x_{n+t},x_{n+\mathrm{1}}\}$ in the given interval. The proposed method is then tested on initial value problems of second-order ordinary differential equations solved by other methods previously. The numerical results confirm the superiority of the new method to the existing methods in terms of accuracy.