Development of an Efficient Optimized Hybrid One Step Methods for Solving Fourth-Order Initial Value Problems (IVPs)

This paper develops an efficient one-step optimized block hybrid method for solving fourth-order initial value problems (IVPs) of ordinary differential equations. The approach is based on an approximate power series as the basis function, using its fourth derivative as the collocating equation, together with interpolation of the lower-order derivatives. Two hybrid points within the integration interval are introduced and selected optimally by minimizing the local truncation error, leading to an optimized block one-step scheme (OBOS). The basic properties of the method are analyzed, and it is shown to be of order at least four, zero-stable, consistent, and therefore convergent. Numerical experiments on several standard fourth-order IVPs demonstrate that the proposed method attains higher accuracy and effectiveness than some existing direct and reduction-based schemes, confirming its superiority in terms of error reduction.