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 entirely independent of small physical parameters. Thus it is applicable for not only weakly but also strongly nonlinear problems. Secondly, the HAM is an 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 Shijun Liao in 1992.

More details, please login LIAO's homepage:http://numericaltank.sjtu.edu.cn/[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[2]
• 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) [3]
• Liao, S.J. (1999), "A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate", J. Fluid Mechanics, 385: 101–128[4]
• Liao, S.J.; Campo, A. (2002), "Analytic solutions of the temperature distribution in Blasius viscous flow problems", J. Fluid Mechanics, 453: 411–425[5]
• Liao, S.J. (2003), "On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet", J. Fluid Mechanics, 488: 189–212[6]
• 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: 2003–2016[7]
• 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
• 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;
• 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
• 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", Physics of Fluids, 22 (5)[8]
• Li, Y.J.; Nohara, B.T.; Liao, S.J. (2010), "Series solutions of coupled Van der Pol equation by means of homotopy analysis method", J. Mathematical Physics, 51 (063517)[9]