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On the relationship between the homotopy analysis method and Euler transform. (English) Zbl 1221.65206
Summary: 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 $$\hbar$$ and two convergent series. 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 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.

##### MSC:
 65L99 Numerical methods for ordinary differential equations 34E10 Perturbations, asymptotics of solutions to ordinary differential equations 34A45 Theoretical approximation of solutions to ordinary differential equations 41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) 40G05 Cesàro, Euler, Nörlund and Hausdorff methods
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