On the Modification of Simpson Method and their Application to Solve Some Problems from the Enviromental

Abstract

Recently, many scientists have been engaged in research differential equations, which have used in solving some problems from the environment.  The increase in pollution volumes has recently ensures the relevance of research in this area. Therefore here consider investigation of numerical solution to solve initial-value problems for Ordinary Differential Equations by the well-known Simpson method. This method is stable and has the order of accuracy   and the number of mesh points used in its construction equal to 3(three). However, the stability Region of this method consists of one point, which is called the origin of the coordinate system. Therefore, the expansion of the stability region for this method is relevant. Her have constructed Simpson-type methods, which have an extended region of stability. For the construct, more exact methods are suggested to use advanced or forward-jumping methods. And also have shown that by using an advanced method, one can been construct A-stable methods with the order of accuracy , in the case when the number of mesh points is equal to three．