Abstract: The homotopy analysis method (HAM) was extended based on the new definition of the Adomian Polynomials proposed by Rach in 2008. Three new kinds of Rm in HAM were defined. By analyzing we know that their expanding speeds are higher than that of the original HAM. These methods provide us more choices to deal with differential equations, especially when the original HAM does not work well. A Maple package was developed to automatically solve ordinary differential system with all of these methods conveniently.

Key words: Adomian decomposition method, Adomian polynomials, Taylor expansion series, nonlinear differential equations, homotopy analysis method, Adomian decomposition method, Adomian polynomials, Taylor expansion series, nonlinear differential equations