Hermite Polynomial-based Methods for Optimal Order Approximation of First-order Ordinary Differential Equations

This study investigates the continuous linear multistep techniques utilized for solving first-order initial value problems in ordinary differential equations. Specifically, the study focuses on step k = 9, utilizing Hermite polynomials as basis functions. This study effectively constructs the Adams-Bashforth, Adams-Moulton, and optimal order methods by applying collocation and interpolation methodologies. The methods are thoroughly examined using various numerical instances to demonstrate their efficacy and validity. Notably, the optimal order method exhibits superior accuracy and efficiency compared to the traditional Adams-Bashforth and Adams-Moulton methods. The research results contribute novel and improved methodologies for solving initial value problems in differential equations, which have extensive applications across diverse mathematical and scientific domains.