同伦分析方法的本质
The essence of the homotopy analysis method
AI总结：
本文通过揭示辅助参数h与泰勒展开点t1的关系，阐明了同伦分析方法的本质，并利用不同点的级数展开方法得到了与Liao方法相同的Blasius方程级数解。
摘要：
包含一个秘密辅助参数$h$的广义泰勒展开可以控制和调整级数的收敛区域，这是Liao提出的同伦分析方法的基础。$h$的秘密无法在同伦分析方法的框架内理解，这是Liao方法的一个严重缺陷。我们解决了这个问题。通过对一个简单例子的详细研究，我们表明广义泰勒展开只是在不同点$t_1$处的通常泰勒展开。我们证明了$h$与$t_1$之间存在关系，这揭示了$h$的意义以及同伦分析方法的本质。作为一个重要例子，我们研究了Blasius方程的级数解。利用不同点的级数展开方法，我们得到了与Liao的同伦分析方法给出的相同结果。
英文摘要：
The generalized Taylor expansion including a secret auxiliary parameter $h$ which can control and adjust the convergence region of the series is the foundation of the homotopy analysis method proposed by Liao. The secret of $h$ can't be understood in the frame of the homotopy analysis method. This is a serious shortcoming of Liao's method. We solve the problem. Through a detailed study of a simple example, we show that the generalized Taylor expansion is just the usual Taylor's expansion at different point $t_1$. We prove that there is a relationship between $h$ and $t_1$, which reveals the meaning of $h$ and the essence of the homotopy analysis method. As an important example, we study the series solution of the Blasius equation. Using the series expansion method at different points, we obtain the same result with liao's solution given by the homotopy analysis method.
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