Backward One-Step Block Hybrid Numerical Method for Solving Second-Order Initial Value Problems

This study develops a backward hybrid block method for solving oscillatory second-order initial value problems (IVPs), which arise frequently in engineering and the physical sciences. Existing hybrid methods often struggle with stiff-oscillatory systems, lacking the robustness needed for accurate and stable solutions. To overcome this limitation, the proposed method incorporates backward-step information to significantly improve both accuracy and stability. The derivation employs multistep collocation and interpolation techniques, with the Chebyshev polynomial of the first kind serving as the basis function. This choice of basis function is particularly effective for approximating oscillatory behavior. Unknown parameters are efficiently determined using Gaussian elimination. The resulting continuous scheme is then evaluated at selected points to obtain the discrete block method. A detailed theoretical analysis establishes the method's order, error constant, consistency, and zero-stability, collectively confirming its convergence. To validate its performance, the method is applied to standard oscillatory test problems. Numerical results demonstrate that the proposed method achieves superior accuracy compared to existing methods in the literature. The findings confirm that this backward-step hybrid block method offers a robust, reliable, and computationally efficient solution for oscillatory IVPs. Its accuracy and stability make it a valuable tool for researchers and practitioners dealing with oscillatory problems in engineering and the physical sciences.