In this work, we introduce a novel numerical method for solving initial value problems associated with a given differential. Our approach utilizes a spline approximation of the theoretical solution alongside the integral formulation of the analytical solution. Furthermore, we offer a rigorous proof of the method's order and provide a comprehensive stability analysis. Additionally, we showcase the effectiveness method through some examples, comparing with Taylor's methods of same order.

翻译：本文提出了一种求解给定微分方程初值问题的新型数值方法。该方法结合理论解的样条逼近与解析解的积分形式。我们给出了该方法阶数的严格证明，并进行了全面的稳定性分析。此外，通过若干算例展示了本方法的有效性，并与同阶泰勒方法进行了对比验证。