an:05949937
Zbl 1221.65281
Liang, Songxin; Jeffrey, David J.
Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation
EN
Commun. Nonlinear Sci. Numer. Simul. 14, No. 12, 4057-4064 (2009).
00286509
2009
j
65M99
homotopy analysis method (HAM); analytical solution; convergence; symbolic computation
Summary: In this paper, the homotopy analysis method (HAM) proposed by Liao in 1992 and the homotopy perturbation method (HPM) proposed by He in 1998 are compared through an evolution equation used as the second example in a recent paper by \textit{D. D. Ganji, H. Tari} and \textit{M. B. Jooybari} [Comput. Math. Appl. 54, No. 7--8, 1018--1027 (2007; Zbl 1141.65384)]. It is found that the HPM is a special case of the HAM when \(\hbar=-1\). However, the HPM solution is divergent for all \(x\) and \(t\) except \(t=0\). It is also found that the solution given by the variational iteration method (VIM) is divergent too. On the other hand, using the HAM, one obtains convergent series solutions which agree well with the exact solution. This example illustrates that it is very important to investigate the convergence of approximation series. Otherwise, one might get useless results.
Zbl 1141.65384