DOI: https://doi.org/10.36347/sjpms.2026.v13i04.006

Pages: 203-217

This study presents series-based and exponentially-fitted three-point optimized hybrid numerical methods for the direct resolution of general third-order ordinary differential equations. The approaches are formulated by transforming the governing problems into second-kind Volterra integral equations, thus obviating the necessity for reduction to analogous first-order systems. Power series and exponential fitting techniques are incorporated as basis functions to improve accuracy and stability, particularly for stiff problems. Theoretical analysis establishes the order of accuracy, local truncation error, consistency, and zero-stability of the schemes, while the regions of absolute stability are also investigated. Numerical studies on specific benchmark issues indicate that the proposed methods attain superior accuracy and enhanced computational efficiency relative to traditional approaches. The results validate the efficacy and robustness of the optimized hybrid Volterra integral framework for addressing general third-order ordinary differential equations.