Numerical Method for Direct Solution of Bettis and Stiefel Second-Order Oscillatory Differential Equations

This paper introduces a new numerical method for directly solving Bettis and Stiefel second-order oscillatory differential equations, which find extensive use in modeling slowly varying amplitude and phase systems, particularly in physics, engineering, and space orbit mechanics.The majority of conventional numerical methods are found lacking in efficiently solving the above category of equations due to instability and inefficiency in the process of reducing second-order systems to first-order systems.To overcome these restrictions, a recurring hybrid block strategy was constructed with the power series as a base function so that multiple points could be computed at once without system reduction.The technique was tested for its chief numerical properties, including order, consistency, and zero-stability, and it was proved that convergence had a clearly defined region of absolute stability.Numerical examples of Bettis and Stiefel equations demonstrated perfect agreement between the calculated and exact solutions with greater accuracy than current methods.The findings confirm the effectiveness, reliability, and high accuracy of the proposed method for the direct solution of second-order oscillatory systems.