On the relationship between the homotopy analysis method and Euler transform

A new transform, namely the homotopy transform, is defined for the first time. Then, it is proved that the famous Euler transform is only a special case of the so-called homotopy transform which depends upon one non-zero auxiliary parameter h and two convergent series Sigma(+alpha)(k-1)alpha(1,K) = 1 and Sigma(+alpha)(k-1)beta(1,K) = 1. In the frame of the homotopy analysis method. a general analytic approach for highly nonlinear differential equations, the so-called homotopy transform is obtained by means of a simple example. This fact indicates that the famous Euler transform is equivalent to the homotopy analysis method in some special cases. On one side, this explains why the convergence of the series solution given by the homotopy analysis method can be guaranteed. On the other side, it also shows that the homotopy analysis method is more general and thus more powerful than the Euler transform. (C) 2009 Elsevier B.V. All rights reserved.