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Notes on the homotopy analysis method: some definitions and theorems. (English) Zbl 1221.65126
Summary: We describe, very briefly, the basic ideas and current developments of the homotopy analysis method, an analytic approach to get convergent series solutions of strongly nonlinear problems, which recently attracts interests of more and more researchers. Definitions of some new concepts such as the homotopy-derivative, the convergence-control parameter and so on, are given to redescribe the method more rigorously. Some lemmas and theorems about the homotopy-derivative and the deformation equation are proved. Besides, a few open questions are discussed, and a hypothesis is put forward for future studies.

MSC:
65H99Nonlinear algebraic or transcendental equations
35A25Other special methods (PDE)
35C10Series solutions of PDE