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Two-parameter homotopy method for nonlinear equations

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Abstract

We introduce a second auxiliary parameter into the zero-order deformation equation and propose a generalization of the homotopy analysis method. This includes the derivation of a general solution in terms of the Bell polynomials for nonlinear equations. Numerical examples show that the proposed zero-order deformation equation improves the convergence region and rate of the series solution and allows greater freedom in the selection of auxiliary operators. This facilitates the development of a homotopy iteration scheme for nonlinear equations with discontinuous or zero derivatives that are not amenable to Newton-type iteration schemes. The homotopy iteration scheme represents a generalization of conventional iteration schemes and additional examples demonstrate its applicability for a wider range of nonlinear problems.

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Authors and Affiliations

  1. Department of Ocean and Resources Engineering, University of Hawaii at Manoa, Honolulu, HI, 96822, USA

    Yongyan Wu & Kwok Fai Cheung

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  1. Yongyan Wu
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  2. Kwok Fai Cheung
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Correspondence to Kwok Fai Cheung.

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Wu, Y., Cheung, K.F. Two-parameter homotopy method for nonlinear equations. Numer Algor 53, 555–572 (2010). https://doi.org/10.1007/s11075-009-9319-7

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  • DOI: https://doi.org/10.1007/s11075-009-9319-7

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