Surpassing taylor method: the generalized taylor method for solving initial value problems

Abstract

This study presents an advancement in the Taylor method through the development of an enhanced, higher-order variant that accelerates series expansion, yielding a refined, implicit
formulation with improved accuracy. Both stability and convergence properties are thoroughly analyzed. While this method demonstrates enhanced efficiency in comparison to conventional
Taylor and fourth-order Runge-Kutta (RK4) methods, the improvements are presented as relative advancements rather than definitive superiority. This refined approach provides a valuable addition to numerical methods for solving initial value problems with greater precision and reliability.