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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.

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 con- vergent 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 estab- lished 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 [

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 [

Then p is order of convergence.

Theorem 1 (see [

Consider the nonlinear equation

We can rewrite the above equation as

We suppose that

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

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

The convergence of the infinite series (13) has been proved by He [

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

From (17), we have

By putting value of

Substitution of (20) in (17) yields

From (19), we get

From (16), we have

when

This formulation allows us to form the following iterative method.

Algorithm 2 For any initial value

which is mainly due to Shin et al. [

When

From this formulation, we suggest the following iterative method.

Algorithm 3 For any initial value

Predictor step:

Corrector step:

When

From this formulation, we suggest the iteration scheme as follows.

Algorithm 4 For any initial value

Predictor step:

Corrector step:

In this section, we present the convergence analysis of algorithm 3 and algorithm 4 established in this paper.

Theorem 5 Let

Proof. Let

Since α is the root of

Hence, by theorem 1, the algorithm 3 has fourth order convergence.

Theorem 6 Let

Proof. Let

Since α is the root of

Examples | Functional eq. | IT | ||
---|---|---|---|---|

CM | 5 | −1.404491648315341226350868178 | ||

NR | 5 | −1.404491648315341226350868176 | ||

J_{1} | 3 | −1.404491648215341226035086891 | ||

J_{2} | 3 | −1.404491648215341242094290841 | ||

S_{1} | 3 | −1.404491648215341226035086818 | ||

S_{2} | 3 | −1.404491648215341226035086818 | ||

CM | 4 | 0.257530285439860760455367304 | ||

NR | 5 | 0.257530285439860760455367306 | ||

J_{1} | 3 | 0.257530285439860760455367304 | ||

J_{2} | 3 | 0.257530285439860760455367303 | ||

S_{1} | 4 | 0.257530285439860760455367305 | ||

S_{2} | 4 | 0.257530285439860760455367305 | ||

CM | 4 | 0.739085133215160641655372087 | 0 | |

NR | 4 | 0.739085133215160641655372089 | ||

J_{1} | 3 | 0.739085133215160641655312087 | ||

J_{2} | 3 | 0.739085133215160641655312087 | ||

S_{1} | 3 | 0.739085133215160641655312087 | ||

S_{2} | 3 | 0.739085133215160641655312087 | ||

CM | 5 | 2 | 0 | |

NR | 5 | 2.000000000000000000000000008 | ||

J_{1} | 4 | 2.000000000000000000000000000 | ||

J_{2} | 4 | 2.000000000000000000000000000 | ||

S_{1} | 3 | 2.000000000000000000000000000 | ||

S_{2} | 3 | 2.000000000000000000000000000 | ||

CM | 5 | 2.154434690031883721759235667 | ||

NR | 5 | 2.154434690031883721759235663 | ||

J_{1} | 3 | 2.154434690031883721759293567 | ||

J_{2} | 3 | 2.154434690031883721759293566 | ||

S_{1} | 3 | 2.1544346900318837217592935665 | ||

S_{2} | 3 | 2.1544346900318837217592935665 | ||

CM | 8 | 3.0000000000000000000000000003 | ||

NR | 7 | 3.0000000000000000000000000006 | ||

J_{1} | 7 | 3.0000000000000000000000000000 | ||

J_{2} | 6 | 3.0000000000000000000000000377 | ||

S_{1} | 3 | 3.0000000000000000000000000000 | ||

S_{2} | 3 | 3.0000000000000000000001629758 |

Hence, by theorem 1, the algorithm 4 has fifth order convergence.

In this section, we shall solve some nonlinear equations to illustrate the efficiency of the newly developed fourth and fifth order iterative methods by using algorithm 3 (S_{1}) and algorithm 4 (S_{2}) in this paper. We shall make comparison with four and fifth order iterative methods established earlier such as the method of Chun (CM) [_{1}) and algorithm 2.2 (J_{2}) of Javidi [

The examples are same as in Chun [

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

MuhammadSaqib,MuhammadIqbal,ShahidAli,TariqIsmaeel,11, (2015) New Fourth and Fifth-Order Iterative Methods for Solving Nonlinear Equations. Applied Mathematics,06,1220-1227. doi: 10.4236/am.2015.68114