HYBRID-BLOCK METHOD SOLUTION OF STIFF SYSTEMS OF INITIAL VALUE PROBLEMS USING HERMITE POLYNOMIALS AS BASIS FUNCTIONS

HYBRID-BLOCK METHOD SOLUTION OF STIFF SYSTEMS

Abstract

This study develops a hybrid-block method for solving stiff systems of initial value problems (SIVPs) using Hermite polynomials as basis functions. Stiff ODEs frequently arise in physical, biological, and engineering models and pose challenges to conventional numerical methods because of their rapidly varying dynamics. To address this, we construct a high-order hybrid block scheme that integrates Hermite interpolation, allowing simultaneous approximation of both function values and their derivatives. The resulting method forms block matrices based on Hermite polynomials, ensuring improved numerical stability and accuracy. We validate the proposed approach on several benchmark stiff problems, including nonlinear chemical kinetics and the Robertson system. Numerical experiments show that the method produces highly accurate results and maintains stability in A, outperforming existing schemes in terms of efficiency and derivative consistency. The novelty of this work lies in embedding Hermite basis functions directly into a hybrid block framework, filling a notable gap in current stiff ODE solvers. This technique provides a robust and interpretable alternative to existing symbolic or stepwise polynomial methods for the integration of rigid systems.