The essence of the homotopy analysis method

with the convergence region |t -t 0 | < ρ 0 . Through introducing an approaching function, Liao gives the following so-called generalized Taylor series solution to the nonlinear equation considered,

where the approaching function µ m,n (h) satisfies lim m→∞ µ m,n (h) = 1 when n ≥ 1. Liao points out that the generalized Taylor series provides a way to control and adjust the convergence region through an auxiliary parameter h such that the homotopy analysis method is particularly suitable for problems with strong nonlinearity [3]. However, the mathematical meaning of the parameter h is still unknown. This is a serious shortcoming of the homotopy analysis method.

In the present paper, we point out that the so-called generalized Taylor series at the initial point t 0 is just the usual Taylor expansion of f (t) at another point. Our results give the meaning of h in the generalized Taylor expansion, and also uncover the essence of the homotopy analysis method. We study two examples in detail. Especially, using the series expansion method to solve the Blasius equation, we give the same result with the Liao’s generalized Taylor series solution by the homotopy analysis method.

In the book [13], Liao had studied a simple example f (t) = 1 1+t to illustrate the generalized Taylor series. He gave the following generalized Taylor expansion,

with the convergence region

where

Liao concludes that the series (3) has enlarged the convergence region of the Taylor series of f at the point t = 0 and calls it the generalized Taylor series.

We next analyze the essence of the generalized Taylor expansion in detail. We prove that the above generalized Taylor series (3) is just the usual Taylor expansion at the point t 0 = -1 -1 h . Indeed, the Taylor series of f at the point

with the convergence region |t| < 1. The Taylor series at the point t 0 is

with the convergence region |t -t 0 | < |1 + t 0 |. If we take t 0 = -1 -1 h , the above expression (7) becomes

which is just the generalized Taylor series (3).

When

Correspondingly, the convergence region |t -

Therefore, when t 0 > -1 2 and -1 < t < 1 + 2t 0 , that is, -2 < h < 0 and -1 < t < -1 -2 h = 2 |h| -1, the right side of expression (3) is just the Taylor series of f at the point t 0 = -1 -1 h . From our discussion and computation, it is easy to see that as the Taylor series of f at the point t 0 the convergence region of the series (3) depends on the point t 0 . When t 0 → +∞, that is, h → 0, the convergence region (-1, 2t 0 + 1) of the series (3) becomes (-1, +∞) naturally. In other words, t 0 can be used to control and adjust the convergence region.

Our result gives the meaning of the auxiliary parameter h, and hence uncovers the essence of the generalized Taylor expansion as the kernel of the homotopy analysis method. In general, we have the following conclusion: the generalized Taylor series at the initial point t 0 is only the usual Taylor expansion of f (t) at another point t 1 = (η 0 -t 0 )(1 + 1 h ) + t 0 , where the real number η 0 satisfies |η 0 -t 1 | = |ξ 0 -t 1 |, and ξ 0 is the nearest singular point of f (t) from t 1 . Indeed, let ξ k ( k = 1, 2, • • • ) be all singular points of a function f which is an analytic function at the point t 1 and the convergence radius be ρ

When t 0 belongs to the convergence region of the Taylor expansion of f at the point t 1 , we can represent the Taylor series at the point t 1 by the Taylor series at the point t 0 as follows,

where the approaching function

satisfies lim m→∞ µ m,n (f ; t 0 , t 1 ) = 1. Expression (11) means that the usual Taylor expansion at the point t 1 can be represented by the so-called generalized Taylor expansion at the point t 0 . Let ξ 0 be the nearest singular point from t 1 and h = ξ0-t0 t1-ξ0 , respectively, we have

It is just the convergence region of another generalized Taylor series given by Liao in Ref. [14] where the corresponding approaching function is

(1 + h) k which doesn’t depends on f . This means that these two series have the same convergence region, and hence our Taylor expansion is equivalent to the Liao’s generalized Taylor expansion. If ξ 0 is not a real number, let h = η0-t0 t1-η0 respectively we have t 1 = (η 0 -t 0 )(1 + 1 h ) + t 0 where the real number η 0 satisfies |t 1 -

For example, we take f (t) = 1 1+t 2 and t 0 = 0. Therefore, we have ξ 0 = ±i and η 0 = t 1 ± t 2 1 + 1. In order

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