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Liao, Shijun

Notes on the homotopy analysis method: some definitions and theorems. (English) Zbl 1221.65126

Commun. Nonlinear Sci. Numer. Simul. 14, No. 4, 983-997 (2009).
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.

Cited in 4 Reviews
Cited in 243 Documents

MSC:

65H99 Nonlinear algebraic or transcendental equations
35A25 Other special methods applied to PDEs
35C10 Series solutions to PDEs

Keywords:

nonlinear equations; series solution; homotopy analysis method
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\textit{S. Liao}, Commun. Nonlinear Sci. Numer. Simul. 14, No. 4, 983--997 (2009; Zbl 1221.65126)
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References:

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