The automated derivation of analytic approximating solutions for periodic nonlinear oscillation

The homotopy analysis method(HAM) is a new technique, which is independent of any small or large parameters. More importantly, unlike traditional techniques, the HAM provide an easy way to control and adjust the convergence region and rate of obtained series solutions. In this paper, based on the HAM together with the homotopy-Padé technique, we developed a Maple package ADHO, which can entirely automatically output accurate analytic approximating solutions of nonlinear oscillatory systems. Meanwhile, ADHO can also give out approximations of several quantities with great physical significance in input system, such as frequency and mean of motion, etc. With a view to meeting different needs for different readers, the output of the package is in various ways. The effectiveness of ADHO is demonstrated by applications to a variety of nonlinear oscillation equations. For the most majority of examples considered the running time is no more than 500 seconds.