Homotopy analysis method

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 158488407X 
• 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. (2004), "On the homotopy analysis method for nonlinear problems", Applied Mathematics and Computation 147 (2): 499–513, doi:10.1016/S0096-3003(02)00790-7, http://numericaltank.sjtu.edu.cn/paper/AMC-HAM-2004.pdf 
• Liao, S.J.; Tan, Y. (2007), "A general approach to obtain series solutions of nonlinear differential equations", Studies in Applied Mathematics 119 (4): 297–354, doi:10.1111/j.1467-9590.2007.00387.x, http://numericaltank.sjtu.edu.cn/paper/SAM-2007-Vol%20119%20pp%20297-355.pdf ; see also this Mathematica notebook file
• Liao, S.J. (2009), "Notes on the homotopy analysis method: some definitions and theorems", Communications in Nonlinear Science and Numerical Simulation 14 (4): 983–997, Bibcode 2009CNSNS..14..983L, doi:10.1016/j.cnsns.2008.04.013, http://numericaltank.sjtu.edu.cn/paper/LIAO-CNSNS-2009%20Notes%20on%20HAM.pdf