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<p>In this work, we present a method aimed at increasing the order of Taylor methods for the numerical solution of initial value problems (IVPs). The developed methodology is based on the application of splines approximation of the theoretical solution in conjunction with the integral formulation of an IVP, resulting in an increased order of approximation for Taylor methods. Furthermore, we offer a rigorous proof of the method’s order and provide a comprehensive stability analysis. Additionally, we showcase the effectiveness method through several examples, comparing with Taylor and Multistep Adams methods of same order.</p>

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Enhancing Taylor methods for solving IVPs through the spline-integral operator

  • 期刊论文

  • 发表日期：

    2025年4月24日

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  • Gustavo H. O. Salgado,
  • João P. R. Romanelli

摘要

In this work, we present a method aimed at increasing the order of Taylor methods for the numerical solution of initial value problems (IVPs). The developed methodology is based on the application of splines approximation of the theoretical solution in conjunction with the integral formulation of an IVP, resulting in an increased order of approximation for Taylor methods. Furthermore, we offer a rigorous proof of the method’s order and provide a comprehensive stability analysis. Additionally, we showcase the effectiveness method through several examples, comparing with Taylor and Multistep Adams methods of same order.