Generalized Hybrid One-Step Block Method Involving Fifth Derivative for Solving Fourth-Order Ordinary Differential Equation Directly

Abstract

A general one-step three-hybrid (off-step) points block method is proposed for solving fourth-order initial value problems of ordinary differential equations directly. A power series approximate function is employed for deriving this method. The approximate function is interpolated at $\left\{x_n,x_{n+r},x_{n+s},x_{n+t}\right\}$ while its fourth and fifth derivatives are collocated at all points $\left\{x_i\right\}$, $i=\left\{\mathrm{0},r,s,t,\mathrm{1}\right\}$, in the interval of approximation. Several fourth-order initial value problems of ordinary differential equations are then solved to compare the performance of the proposed method with the derived methods. The analysis of the method reveals that the method is consistent and zero stable concluding that the method is also convergent. The numerical results demonstrate the superiority of the new method over the existing ones in terms of error.