Asian Research Journal of Mathematics

This study develops a fifth-order fully hybrid block method for the direct numerical solution of third-order initial value problems. A continuous approximation is constructed using interpolation and collocation, and three intra-step hybrid points are incorporated into the block formulation. The resulting discrete schemes approximate the solution together with its first and second derivatives without reducing the governing third-order equation to an equivalent system of first-order equations. The principal numerical properties of the method are examined through local truncation-error analysis, consistency, zerostability, convergence, and absolute-stability considerations. The local truncation-error analysis establishes a uniform order of five for the developed block method. The corresponding difference formulation satisfies the root condition required for zero-stability; together with consistency, this establishes convergence. The stability behaviour is further investigated through a standard third-order test equation. Four numerical initial value problems are then used to assess computational performance at a fixed step size. Across the reported test problems, the proposed method produces absolute errors smaller than those of the comparator methods presented in the manuscript, including comparisons with methods of equal or higher formal order. These results indicate that the proposed fully hybrid formulation provides accurate numerical approximations for the class of third-order initial value problems considered in this study.