Asian Research Journal of Mathematics

This study presents a second-derivative hybrid block method (SDHBM) with variational stability parameters for the direct numerical solution of second-order initial value problems in ordinary differential equations. The method is derived through interpolation and collocation using a power-series approximation, which is evaluated at selected grid and off-grid points to obtain the continuous and discrete schemes. Variational stability parameters are incorporated into the second characteristic component of the block method to provide flexibility in the method coefficients and to permit different stability regions through suitable parameter choices. The analytical properties of the method are examined in terms of order, error constants, consistency, zero-stability and absolute stability. The resulting scheme has orders seven and six across its components, is consistent and satisfies the root condition required for zero-stability. Stability behaviour is investigated using the boundary locus technique, and the resulting stability plots indicate A(α)-stability, A-stability and stability within a defined region for selected parameters. The method is tested on five second-order initial value problems, including four single equations and one system of equations. Numerical results are compared with results from existing methods reported in the literature. The comparisons show that the proposed SDHBM produces small absolute errors and is either more accurate than or competitive with the referenced methods for the selected examples.