×

Homotopy analysis method: A new analytical technique for nonlinear problems. (English) Zbl 0927.65069

Summary: The basic ideas of a new kind of analytical technique, namely the homotopy analysis method (HAM), are simply described. Different from perturbation techniques, the HAM does not depend on whether or not there exist small parameters in the nonlinear equations under consideration. Therefore, it provides us with a powerful tool to analyse strongly nonlinear problems. A simple but typical example is used to illustrate the validity and the great potential of the HAM. Moreover, a pure mathematical theorem, namely the general Taylor theorem, is given in appendix, which provides us with some rational knowledge for the validity of this new analytical technique.

MSC:

65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Liao, S. J., The Homotopy Analysis Method and its applications in mechanics, (Ph. D. Dissertation (in English) (1992), Shanghai Jiao Tong Univ)
[2] Liao, S. J., A kind of linear invariance under homotopy and some simple applications of it in mechanics, (Bericht Nr. 520 (1992), Institut fuer Schiffbau der Universitaet Hamburg)
[3] Liao, S. J., J. Appl. Mech., 14, 1173-1191 (1992) · Zbl 0754.76065
[4] Liao, S. J., J. Ship Research, 36, 30-37 (1992)
[5] Liao, S. J., A kind of approximate solution technique which does not depend upon small parameters: a special example, Int. J. Non-linear Mech., 30, 371-380 (1995) · Zbl 0837.76073
[7] Liao, S. J., Boundary Elements XVII, ((1995), Computational Mechanics Publications: Computational Mechanics Publications Southampton), 67-74 · Zbl 0839.65132
[8] Liao, S. J., Int. J. Numer. Methods Fluids, 23, 739-751 (1996) · Zbl 0882.65108
[9] Liao, S. J.; Chwang, A. T., Int. J. Numer. Methods Fluids, 23, 467-483 (1996) · Zbl 0863.76037
[10] Liao, S. J., Homotopy Analysis Method and its applications in mathematics, J. Basic Sci. Eng., 5, 2, 111-125 (1997)
[11] Liao, S. J., The common ground of all numerical and analytical technique for solving nonlinear problems, Commun. Nonlinear Sci. Nonlinear Simul., 1, 4, 26-30 (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.