Revolutionizing Numerical Approximations: A Novel Higher-Order Implicit Method vs. Runge-Kutta for Initial Value Problems

This work is dedicated to advancing the approximation of initial value problems through the introduction of an innovative and superior method inspired by Taylor’s approach. Specifically, we present an enhanced variant achieved by accelerating the expansion of the Obreschkoff formula. This results in a higher-order implicit corrected method that outperforms Rung–Kutta’s (RK) method in terms of accuracy. We derive an error bound for the Obreschkoff higher-order method, showcasing its stability, convergence, and greater efficiency than the conventional RK method. To substantiate our claims, numerical experiments are provided, highlighting the exceptional efficacy of our proposed method over the traditional RK method.