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.
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