New Fourth and Fifth-Order Iterative Methods for Solving Nonlinear Equations

In this paper, we establish two new iterative methods of order four and five by using modified homotopy perturbation technique. We also present the convergence analysis of these iterative methods. To assess the validity and performance of these iterative methods, we have applied to solve some nonlinear problems.


Introduction
Consider the single variable nonlinear equation Finding the zeros (1) is an interesting and very ancient problem in numerical analysis.Newton and fixed point iterative methods are very old methods for solving nonlinear equations.Newton method is quadratically convergent where as fixed point method is linear convergent.Many modifications have been made in Newton's method to get cubically convergent iterative methods.Many higher order iterative methods have been established to approximate the solution of (1) by using different techniques including Taylor's series, quadrature rules, Adomain decomposition, homotopy perturbation, Gejji and Jafari decomposition, Noor decomposition, see the refrences [1]- [8].Initialty, we do not put any restrictions on the original function f.In fixed point method, we rewrite ( ) 0 f x = as ( ) ii there exist , such that 1 for all , .a b g x x a b λ

≤ < ∈
We shall establish fourth and fifth order iterative methods using modified homotopy perturbation technique.The order of convergence of a sequence of approximation is defined as; Definition 1 [9] Let the sequence { } n x converges to α .If there is a positive integer p and real number C such that ( ) Then p is order of convergence.Theorem 1 (see [6]).Suppose that is of order m.

Development of New Methods
Consider the nonlinear equation We can rewrite the above equation as ( ).
x g x = We suppose that α is a root of (2) and γ is initial guess close to α .We can rewrite (3) by using Taylor's expansion as: where We can rewrite (4) as It can be written in the form ( ) where and From (5), we see that ( ) ( ) We shall decompose the nonlinear operator ( ) N x by using modified homotopy perturbation technique.For this, we construct a homotopy where p is embedding parameter and m is unknown real number.The embedding parameter p is monotonically increases from zero to unity as the trivial problem ( ) is continuously deformed the original problem The basic assumption of modified HPM is that the solution x of (10) can be expressed as a power series in p in the following form The approximate solution of ( 2) can be obtained as Lim .
The convergence of the infinite series (13) has been proved by He [10].For the application of modified HPM to (2), we can rewrite (10) by expanding ( ) By substituting (13) in (15), we have By equating the coefficients of like powers of p, we have ( ) ( ) We find the value of unknown parameter m such that 2 0. x = From (17), we have ( ) By putting value of 1 x and 2 x in (18) yields ( ) ( ) ( ) Substitution of (20) in (17) yields ( ) ( ) This formulation allows us to form the following iterative method.Algorithm 2 For any initial value 0 x , we compute the approximation solution x + , by the iterative method.
( ) ( ) ( ) which is mainly due to Shin et al. [9] and has quadratic convergence. When From this formulation, we suggest the following iterative method.Algorithm 3 For any initial value 0 x , we compute the approximation solution x + , by the iterative method.Predictor step: From this formulation, we suggest the iteration scheme as follows.Algorithm 4 For any initial value 0 x , we compute the approximation solution x + , by the iterative method.Predictor step: y g y g y g y y g y x g y g x g y

Convergence Analysis
In this section, we present the convergence analysis of algorithm 3 and algorithm 4 established in this paper.Theorem 5 Let : f I ⊂ →   for an open interval I and consider that the nonlinear equation ( ) Since α is the root of ( ) 0 f x = and ( ) x g x = is the functional equation of ( ) . From (20), using Maple software, we have ( )

Conclusion
In this paper, we have developed two new iterative methods of order four and five for the solution of nonlinear equations based on homotopy perturbation method.To derive these iteration schemes, we have used a very simple technique.Convergence analysis is also discussed.To check convergence, performance and validity, we have applied these iterative methods to solve some nonlinear equations.From Table 1, we see the validity and efficiency of these iterative methods as compared with other methods.Thus our newly established iterative methods are interesting and reliable alternative methods of existing methods in literature of order four and order five for solving nonlinear equations under consideration.Also our methods converge faster than existing methods of order four and five such as Noor [4] and Javidi [8].