The New Way to Solve Physical Problems Described by ODE of the Second Order with the Special Structure

Abstract

In the last decade, many researchers have studied extensively theoretical and practical problems of natural sciences using ODEs as a means to analyze and understand them. Specifically, second-order ODEs with special complex structures provide the necessary tools to construct mathematical models for several physical - and other- processes such as the Schturm-Liouville, Schrölinger, Population, etc. As a result, it is of great importance to construct special stable methods of a higher order as a means to solve differential equations. One of the most important efficiency methods for solving these problems is the Stёrmer-Verlet method which consists of hybrid methods with constant coefficients. In this paper, we expand on recent studies that prove that the hybrid methods are more precise than the Stёrmer-Verlet method while investigating the convergence variable. This paper aims to prove the existence of a new, stable hybrid method using a special structure of degree(p)=3k+2, where k is the order of the multistep methods. Lastly, we also provide a detailed mathematical explanation of how to construct stable methods on the intersection of multistep and hybrid methods having a degree(p)≤3k+3.