A class of continuous methods for general second order initial value problems in ordinary differential equations

In this paper we propose a class of continuous methods for the solution of initial value problems of general second order ordinary differential equations . The procedure which yields a solution matrix equation for different stepnumber k≥2 is based on collocation of the differential systems at the grid points. For n=2, three discrete schemes of order four for k=2 and k=3, and of order five for k=4 are recovered from the methods. Numerical examples are given to demonstrate and compare the efficiency of the methods for the stepnumbers k=2 and k=3 respectively.