Homotopy analysis method: Difference between revisions

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The '''homotopy analysis method''' (HAM) isaims proposedto by Shi-Jun Liaosolve [http://numericaltank.sjtu.edu.cn/[nonlinear]](1992) who[[ordinary isdifferential currentlyequation]]s aand professor[[partial indifferential Shanghai Jiao Tong University,equation]]s Shanghai, Chinaanalytically. ThisThe method aims to solve the nonlinear ODE/PDE analytically. It distinguishes itself from other [[Mathematical analysis|analytical methods]] in the following four aspects. First, it is a [[series (mathematics)|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 [[Aleksandr Lyapunov|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 [[basis function|base function]] of the desired solution. Fourthly, the HAM can be combined with many other [[mathematics|mathematical]] methods, suchmethods—such as the [[numerical methodsmethod]]s, the series expansion methodmethods, the [[integral transform]] methods and so forth.<br /><br />

The method was devised by Shi-Jun Liao in 1992.[http://numericaltank.sjtu.edu.cn/]

== Reference ==

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

== References ==

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

[1]*{{citation | last=Liao, | first=S.J. | title=The proposed homotopy analysis technique for the solution of nonlinear problems. | publisher=PhD thesis, Shanghai Jiao Tong University, | year=1992<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 />

[2]*{{citation | last=Liao, | first=S.J. | title=Beyond Perturbation: Introduction to the Homotopy Analysis Method. | publisher=Chapman & Hall/ CRC Press, | ___location=Boca Raton, | year=2003<br />}}

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

[3]*{{citation | last=Liao, | first=S.J. | title=An explicit, totally analytic approximation of Blasius’ viscous flow problems. | journal=International Journal of Non-Linear Mechanics, | volume=34: | issue=4, 759-778,| pages=759–778 | year=1999<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]*{{citation | last=Liao, | first=S.J. Notes| ontitle=On the homotopy analysis method: somefor definitionsnonlinear andproblems theorems,| Communicationsjournal=Applied inMathematics Nonlinearand ScienceComputation and| volume=147 | issue=2 Numerical| Simulation,pages=499–513 14,| 983-997,year=2004 2009}}

[5]*{{citation | last1=Liao, | first1=S.J. and| last2=Tan, | first2=Y. | title=A general approach to obtain series solutions of nonlinear differential equations, | journal=Studies in Applied Mathematics, | volume=119, 297-254,| pages=297–254 | year=2007<br />}}

*{{citation | last=Liao | first=S.J. | title=Notes on the homotopy analysis method: some definitions and theorems | journal=Communications in Nonlinear Science and Numerical Simulation | volume=14 | pages=983–997 | year=2009 }}

[[Category:Asymptotic analysis]]