Titlebook: Homotopy Analysis Method in Nonlinear Differential Equations; Shijun Liao Book 2012 Higher Education Press,Beijng and Springer-Verlag GmbH

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Basic Ideas of the Homotopy Analysis Method the concept of the homotopy, the flexibility of constructing equations for continuous variations, the way to guarantee convergence of solution series, the essence of the convergence-control parameter .., the methods to accelerate convergence, and so on. The corresponding Mathematica codes are given

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Optimal Homotopy Analysis Method, which logically contains the basic optimal HAM with only one convergence-control parameter and also the optimal HAM with an infinite number of parameters. It is found that approximations given by the optimal HAMs converge fast in general. Especially, the basic optimal HAM mostly gives good enough

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Systematic Descriptions and Related Theoremshomotopy-derivative operator and deformation equations are proved, which are helpful to gain high-order approximations. Some theorems of convergence are proved, and the methods to control and accelerate convergence are generally described. A few of open questions are discussed.

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Nonlinear Boundary-value Problems with Multiple Solutionsthematica package BVPh (version 1.0) for .th-order nonlinear boundary-value equations . in a finite interval 0≤.≤., subject to the . linear boundary conditions ., (1≤.≤.), where . is a .th-order nonlinear differential operator, . is a linear operator, γ. is a constant, respectively. Especially, the