Homotopy analysis method: Difference between revisions

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The ''homotopy analysis method'' (HAM) is proposed by Shi-Jun Liao [http://numericaltank.sjtu.edu.cn/](1992) who is currently a professor in Shanghai Jiao Tong University, Shanghai, China. TheThis method aims to solve the nonlinear ODE/PDE analytically. It 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 us with a simple way to ensure the convergence of the solution; also it provides us with great freedom to choose the base function of the desired solution. Fourthly, the HAM can be combined with many other mathematical methods, such as the numerical methods, the series expansion method, the integral transform methods and so forth.<br /><br />

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The homotopy analysis method (HAM) is proposed by Shi-Jun Liao (1992) who is currently a professor in Shanghai Jiao Tong University. The method aims to solve the nonlinear ODE/PDE analytically. It 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 us with a simple way to ensure the convergence of the solution; also it provides us with great freedom to choose the base function of the desired solution. Fourthly, the HAM can be combined with many other mathematical methods, such as the numerical methods, the series expansion method, the integral transform methods and so forth.

Reference

[1] Liao, S.J. The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University, 1992

[2] Liao, S.J. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/ CRC Press, Boca Raton, 2003

[3] Liao, S.J. An explicit, totally analytic approximation of Blasius’ viscous flow problems. International Journal of Non-Linear Mechanics, 34:4, 759-778, 1999

[4] Liao, S.J. On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation, 147:2, 499-513, 2004

[5] Liao, S.J. and Tan, Y. A general approach to obtain series solutions of nonlinear differential equations, Studies in Applied Mathematics, 119, 297-254, 2007

== Reference ==

[1] Liao, S.J. The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University, 1992<br />

[2] Liao, S.J. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/ CRC Press, Boca Raton, 2003<br />

[3] Liao, S.J. An explicit, totally analytic approximation of Blasius’ viscous flow problems. International Journal of Non-Linear Mechanics, 34:4, 759-778, 1999<br />

[4] Liao, S.J. On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation, 147:2, 499-513, 2004<br />

[5] Liao, S.J. and Tan, Y. A general approach to obtain series solutions of nonlinear differential equations, Studies in Applied Mathematics, 119, 297-254, 2007<br />

[6] Liao, S.J. Notes on the homotopy analysis method: some definitions and theorems, Communications in Nonlinear Science and Numerical Simulation, 14, 983-997, 2009