Two-parameter homotopy method for nonlinear equations

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.

Constructing Frozen Jacobian Iterative Methods for Solving Systems of Nonlinear Equations, Associated with ODEs and PDEs Using the Homotopy Method

“…We extend the model of iterative method Equation (22), by including more free-parameters, so that we can enhance the convergence order.…”

“…Now, we define a parametric homotopy between L(y) and F(y), taking inspiration from the work in [22]. Therefore:…”

“…The auxiliary parameters can be used for different purposes. Some authors use the homotopy parameters to enhance the convergence by changing the dynamics of the iterative method and give estimates for the optimal parameters [22]. Now, we present in detail our proposals.…”

“…In 2010, Yongyan et al [22] proposed a two-parameter homotopy method for the construction of iterative methods, for solving scalar nonlinear equations. The authors in [22] use the homotopy auxiliary parameters to change the dynamics of the proposed iterative methods for solving nonlinear equations and provide the optimal values of auxiliary parameters, for enhancing the convergence speed without altering the convergence order of the iterative method.…”

“…The authors in [22] use the homotopy auxiliary parameters to change the dynamics of the proposed iterative methods for solving nonlinear equations and provide the optimal values of auxiliary parameters, for enhancing the convergence speed without altering the convergence order of the iterative method. It is worth noting that they used auxiliary parameters to change the path of converging sequences of successive approximations to obtain fast convergence.…”