The Modified Taylor Method for Approximating Solutions of Initial Value Problems

This study presents an enhanced variant of Taylor's method for approximating initial value problems, introducing a higher-order implicit corrected method achieved by optimizing the Taylor series expansion. The proposed method improves precision and efficiency compared to traditional Taylor and Runge-Kutta methods. The originality of this approach lies in its novel acceleration of the Taylor series, which provides higher accuracy and better performance. The methodologies employed include deriving rigorous stability and convergence proofs, as well as conducting extensive numerical experiments. These experiments compare the proposed method to conventional techniques, highlighting its superior accuracy despite a higher computational cost. An error bound is derived to quantify the method's precision, and detailed comparisons with traditional methods underscore the enhanced performance of the proposed approach. The results demonstrate that, although the method requires more computational resources, it achieves significant improvements in approximation accuracy. This work contributes a refined numerical technique with proven advantages, offering valuable insights for researchers dealing with initial value problems in numerical analysis.