On Some Comparison of the Numerical Methods Applied to Solve ODEs, Volterra Integral and Integro-Differential Equations

Abstract

The many problems of the different fields of nature are reduce to solve initialvalue problem for the both Ordinary Differential Equation and Volterra integro-differential equation and also to solve Volterra integral equation. And for solving theser problems in usually are used the numerical methods, which are related with the development of computer sciences. In among of them, the multistep multiderivative methods are very developing. Therefor, any result in this area is of interest in always. The ordinary multistep method and multistep second derivative
methods fundamentally investigated by Dahlquist. The Dahlquist theory development by Ibrahimov, who have receive similarly results for the advanced multistep methods and advanced multistep second derivative methods. Here by development of these results, have considered to
investigation of the multistep third derivative methods with constant coefficients. Somebody can be take the results receiving for the estimation of the accuracy for stable and unstable multistep multiderivative methods with third derivative as the simple generalization of the known results.
Here have shown that this is not correct. Noted that multistep multiderivative Methods with third derivative here investigated in fool form, found some connection between degree and order for these methods in the cases: stable and in stable. Constructed some concrete methods of multistep third derivative types and recommended some way for the application of these methods to solve some problems.