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The essence of the homotopy analysis method. (English) Zbl 1194.34010
Summary: 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 $$h$$ cannot 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.

##### MSC:
 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. 34A45 Theoretical approximation of solutions to ordinary differential equations
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##### References:
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