Modified Homotopy Perturbation Technique for the Approximate Solution of Nonlinear Equations

Farooq Ahmed Shah

doi:10.1155/2014/787591

Abstract

We use a new modified homotopy perturbation method to suggest and analyze some new iterative methods for solving nonlinear equations. This new modification of the homotopy method is quite flexible. Various numerical examples are given to illustrate the efficiency and performance of the new methods. These new iterative methods may be viewed as an addition and generalization of the existing methods for solving nonlinear equations.

Highlights

It is well known that a wide class of problems, which arises in various branches of mathematical and engineering science, can be studied in the unified framework of nonlinear equation of the form f(x) = 0
Numerical methods for finding the approximate solutions of the nonlinear equation are being developed by using several different techniques including the Taylor series, quadrature formulas, homotopy, and decomposition techniques; see [1–14] and the references therein
One can notice that if the derivative of the function vanishes, that is, |f󸀠(x)| = 0, during the iterative process, the sequence generated by the Newton method or the methods derived in [1–9] are not defined

Summary

Introduction

We rewrite the given nonlinear equation along with the auxiliary function g(x) = 0, as an equivalent coupled system of equations using the Taylor series This approach enables us to express the given nonlinear equation as sum of linear and nonlinear equations. One can notice that if the derivative of the function vanishes, that is, |f󸀠(x)| = 0, during the iterative process, the sequence generated by the Newton method or the methods derived in [1–9] are not defined Due to such cardinal sin of division results a mathematical breakdown. This is another motivation of the paper that the derived higher order methods converge even if the derivative vanishes during the iterative process. Our results can be considered as an important improvement and refinement of the previously known results

Construction of Iterative Methods

Convergence Analysis

Numerical Results

Method

Conclusion

Conflict of Interests