Homotopy analysis method

The two dashed paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy.

The homotopy analysis method (HAM) is a semi-analytical technique to solve nonlinear ordinary/partial differential equations. The homotopy analysis method employs the concept of the homotopy from topology to generate a convergent series solution for nonlinear systems. This is enabled by utilizing a homotopy-Maclaurin series to deal with the nonlinearities in the system.

The HAM was first devised in 1992 by Liao Shijun of Shanghai Jiaotong University in his PhD dissertation[1] and further modified[2] in 1997 to introduce a non-zero auxiliary parameter, referred to as the convergence-control parameter, c0, to construct a homotopy on a differential system in general form.[3] The convergence-control parameter is a non-physical variable that provides a simple way to verify and enforce convergence of a solution series. The capability of the HAM to naturally show convergence of the series solution is unusual in analytical and semi-analytic approaches to nonlinear partial differential equations.

  1. ^ Liao, S.J. (1992), The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University
  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
  3. ^ Liao, S.J. (2003), Beyond Perturbation: Introduction to the Homotopy Analysis Method, Boca Raton: Chapman & Hall/ CRC Press, ISBN 978-1-58488-407-1[1]