Adaptive methods for second order initial value problems

This paper deals with two-step methods of order two and order four with minimum truncation error for numerical integration of second order initial value problems. The methods depend upon a parameter p>0, and reduce to the Classical Numerov method for p=0. As p becomes very large the truncation error tends to zero. The methods are unconditionally stable when applied to the test equation. To illustrate the order, accuracy and stability of the method, the test problem and non linear undamped duffing equations are solved. The results are compared with some other well known methods and the derived methods give better results than any other existing fourth order methods.