Hybrid Block Numerical Algorithm for Direct Solutions of Ordinary Differential Equations of the Third and Fourth Orders

This article examines a single, effective hybrid block numerical technique for solving initial value problems of orders three and four. The novel technique was derived by using an interpolating polynomial that was the power series approximation solution, together with an interpolation and collocation approach. The block's numerical integrators were obtained by assessing the non-interpolating points in the chosen integration interval. The theoretical analysis of the techniques was examined in line with the already established rules, and the outcome shows that it is zero-stable and convergence. When the block's efficiency was put to the test on a few third- and fourth-order ordinary differential equations, the precision of the findings outperformed those of previous approaches.