On Some Comparisons of Multistep Methods and Their Applications to Solve Ordinary Differential Equations

The significance of Multistep Methods with constant coefficients and their application in addressing various Natural Science issues is universally acknowledged. Dahlquist conducted foundational research on these methods. Building on this, this text outlines certain developments in these theories and their use in solving Ordinary Differential, Volterra Integral, and Volterra Integro-Differential Equations. Advanced (forward-jumping) methods are examined, with a comparison made between the outcomes of these methods and those established by Dahlquist. Additionally, the study focuses on advanced second derivative multistep methods, demonstrating that the stable variants of these advanced methods yield greater accuracy. Furthermore, the research identifies the maximum achievable degree for the advanced methods. The constructed methods have been utilized to tackle model problems, and the resulting findings are presented here for illustration.