Font Size: 

In this paper, we pay more attention to the embedding parameter h, which has an influence on the convergence region of solution series in Homotopy analysis method (HAM). We use some theorems to give the concrete influence and proof. Then we introduce a new modified method of HAM called the Piecewise homotopy analysis method (PHAM). Futhermore, examples such as NLS equation, Ricatti equation and Duffing equation are presented to illustrate the main results.

Homotopy analysis method; the convergence region; Cauchy-Kowalevskaya theorem; Piecewise homotopy analysis method