On some comparison of the Adams-Moulton, Adams-Bashforth methods with the methods of fractional step type

The numerical solution of the Ordinary Differential Equations is investigated starting from the Euler method. This method has been developed by Adams-Moulton, Adams-Bashforth, and Runge-Kutta. Among numerical methods, Euler’s method, Trapezoidal method, Midpoint, and Simpson's method are very popular. But the structure of the Midpoint method differs from the others in that it uses a fractional step-size. Typically methods with a fractional step-size are more accurate. Here by using methods are constructed some methods, which are more accurate than the Adam’s Methods. Methods with the fractional step-size have been developed in two directions. One of them is using fractional step-size as a rational number, and the other uses fractional step-size as an irrational and rational number. In usual methods with the irrational step-size are called the hybrid methods, which have been investigated starting from the work of J.Butcher (1965) and Gear (1965). But methods with the fractional steps have been investigated by the Academician Yanenko. Recently there are many works of the different authors dedicated to investigation of the hybrid methods. Here investigated the intersection of these directions by using Adams Methods with the fractional step. Constructed methods in the intersection of these methods and have shown the advantages of these methods.