Notes on the homotopy analysis method: Some definitions and theorems

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The homotopy analysis method for solving the time-fractional Fornberg-Whitham equation and comparison with Adomian's decomposition method (2013, background): “Based on homotopy of topology, the validity of the HAM is independent of whether or not there exist small parameters in the considered equation [18–22].”
A one-step optimal homotopy analysis method for nonlinear differential equations (2010, background): “(6) was published 10 years ago and even more generalized form was reported [6], most users of the HAM applied Eq.”
Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation (2009, positive): “Note that, an analytic method for strongly nonlinear problems, namely the homotopy analysis method (HAM) [9–13], was proposed by Liao in 1992, six years earlier than the homotopy perturbation method [6] and the variational iteration method [7].”
On the relationship between the homotopy analysis method and Euler transform (2010, methodology): “Thereafter, the HAM has been improved step by step [14–20] and has been widely applied in science, engineering and finance [21–29].”