The homotopy analysis method (HAM) aims to solve nonlinear ordinary differential equations and partial differential equations analytically. The method distinguishes itself from other analytical methods in the following four aspects. First, it is a series expansion method but it is independent of small physical parameters at all. Thus it is applicable for not only weakly but also strongly nonlinear problems. Secondly, the HAM is a unified method for the Lyapunov artificial small parameter method, the delta expansion method and the Adomian decomposition method. Thirdly, the HAM provides a simple way to ensure the convergence of the solution; also it provides freedom to choose the base function of the desired solution. Fourthly, the HAM can be combined with many other mathematical methods—such as numerical methods, series expansion methods, integral transform methods and so forth.
The method was devised by Shi-Jun Liao in 1992.[1]
References
- Liao, S.J. (1992), The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University
- Liao, S.J. (2003), Beyond Perturbation: Introduction to the Homotopy Analysis Method, Boca Raton: Chapman & Hall/ CRC Press, ISBN 1-58488-407-X
- Liao, S.J. (2012), Homotopy Analysis Method in Nonlinear Differential Equation, Berlin & Beijing: Springer & Higher Education Press, ISBN 978-3642251313(3642251315)
{{citation}}: Check|isbn=value: invalid character (help)For Details see [2] - Liao, S.J. (1999), "An explicit, totally analytic approximation of Blasius' viscous flow problems", International Journal of Non-Linear Mechanics, 34 (4): 759–778, Bibcode:1999IJNLM..34..759L, doi:10.1016/S0020-7462(98)00056-0
- Liao, S.J. (2010), "An optimal homotopy-analysis approach for strongly nonlinear differential equations", Communications in Nonlinear Science and Numerical Simulation, 15: 2003–2016Download from site [3]
- Liao, S.J. (2004), "On the homotopy analysis method for nonlinear problems" (PDF), Applied Mathematics and Computation, 147 (2): 499–513, doi:10.1016/S0096-3003(02)00790-7
- Liao, S.J.; Tan, Y. (2007), "A general approach to obtain series solutions of nonlinear differential equations" (PDF), Studies in Applied Mathematics, 119 (4): 297–354, doi:10.1111/j.1467-9590.2007.00387.x; see also this Mathematica notebook file
- Liao, S.J. (2009), "Notes on the homotopy analysis method: some definitions and theorems" (PDF), Communications in Nonlinear Science and Numerical Simulation, 14 (4): 983–997, Bibcode:2009CNSNS..14..983L, doi:10.1016/j.cnsns.2008.04.013
- Xu, H.; Lin, Z.L.; Liao, S.J.; Wu, J.Z.; Majdalani, J. (2010), "Homotopy-based solutions of the Navier–Stokes equations for a porous channel with orthogonally moving walls" (PDF), Physics of Fluids, 22 (5); see also this [4]
- 廖, 世俊 (2009), "同伦分析方法：求解强非线性问题的一个新途径", 科学观察, 4 (5): 48–49