Note on Studying Multistep Multiderivative Methods and it’s Application to Solve Ordinary Differential Equations

Abstract

As is known Multistep Multiderivative Methods are investigated for a very long time. Recently some modifications of these methods have been constructed. These methods are successfully applied to solve initial-value problems for the Ordinary Differential and Volterra integro-differential Equations of any order and also to solving Volterra integral equations. Here,  have investigated application of the Multistep Third Derivative Method to solve initial value problems for the ODEs of first, second and third orders. Shown that the Multistep  Methods with the constant coefficients can be presented in different forms, depending on the object of research. By study these problems have found some necessary conditions for  the stability of the above noted Multistep Multiderivative Methods. It is known that one and the same numerical methods can be applied to solve initial-value problems for ODEs of the different orders. Here, it has been proven that in the application of one and the same Multistep Multi Derivative Method to solving initial-value problems for the ODEs of the same orders, the maximal value for its accuracy depends from the values of some coefficients can be receive different values. This idea is illustrated by using the initial-value problem for Ordinary Differential Equations with special structure. A classic representative of such problems is the initial value problem for the ODEs of the second order. Here, fully studied this problem, and have investigated the extended version of the named problem. And also have considered investigating some relation between different types of  Multistep Multiderivative Methods with constant coefficients. By using these relations, have defined some connection for the degree and order of stable methods.