New class of rational multistep methods for solving initial value problem

There exists initial value problem whose solution possesses singularity. studies show that conventional numerical method such as multistep method fail woefully near the singular point when solving problem whose solution possesses singularity. this is because a multistep method is based on the local representation of polynomial of the theoretical solution of an initial value problem. therefore,a natural step would appear to be the replacement of the polynomial function for a multistep method,by a rational function due to its smooth behaviour in the neighbourhood of singularity. in this paper,we have developed a new class of twostep numerical methods that are based on rational functions in solving general initial value problem and problem whose solution possesses singularity. these new methods are called rational multistep methods. the developments of these rational multistep methods,as well as the local truncation error and stability analysis for each rational multistep method are presented. we have found out that only the second order,third order and fourth order rational multistep methods are a-stable. numerical experiments have showed that all newly developed rational multistep methods presented in this paper are suitable to solve general initial value problem,stiff problem and problem whose solution possesses singularity.