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We introduce here a new two-step derivate-free inverse simultaneous iterative method for estimating all roots of nonlinear equation. It is proved that convergence order of the newly constructed method is four. Lower bound of the convergence order is determined using Mathematica and verified with theoretical local convergence order of the method introduced. Some nonlinear models which are taken from physical and engineering sciences as numerical test examples to demonstrate the performance and efficiency of the newly constructed modified inverse simultaneous methods as compared to classical methods existing in literature are presented. Dynamical planes and residual graphs are drawn using MATLAB to elaborate efficiency, robustness, and authentication in its domain.

A wide range of problems in physical and engineering sciences can be formulated as a nonlinear equation:

The most ancient and popular iterative technique for approximating single roots of (

Nedzibove et al., in [

In the last few years, lot of work has been carried out on numerical iterative methods which approximate single root at a time of (

Among derivative-free simultaneous methods, Weierstrass–Dochive [

Nedzibove [

The main aim of this paper is to construct a two-step inverse method of convergence order four.

We modify the Weierstrass method (

Thus, method (

We prove here that convergence order of the IWM2 method is four.

Let

Let

Using Theorem

Let

Then, there exists

Let

Thus,

Using (

Thus, from inequality (

For a fixed point

Let

Consider

Thus, we obtain

Using the expression

If we assume all errors are of the same order, i.e.,

From second-step of IWM2, we have

Thus, we obtain

As from the above argument

If we assume all errors are of the same order, i.e.,

Hence, the theorem is proved.

Consider

The lower bound of the convergence obtained until the first nonzero element of the row is found. The Mathematica code is given for each of the consider methods as follows.

Weierstrass–Dochive Method (WDK):

Modified Inverse Weierstrass Method:

WDK2 Method:

IWM2 Method:

The elapsed time from Table

Elapsed time in seconds.

Method | WDK | IWDK | WDK2 | IWM2 |
---|---|---|---|---|

0.12937 | 0.142207 | 0.323190 | 0.107267 | |

0.160921 | 0.23889 | 0.431936 | 0.153851 |

(a), (b), (c), and (d) show basins of attraction for nonlinear function

The elapsed time from Table

(a), (b), (c), and (d) show basins of attraction for nonlinear function

Computational time in seconds of WDK2 and IWM2 for nonlinear function

Some nonlinear models from engineering and physical sciences are considered to illustrate the performance and efficiency of WDK2 and IWM2 using CAS Maple 18 with 64 digits floating point arithmetic for all computer calculations. We approximate the roots of (

Simultaneous finding of all roots.

Method | ||||
---|---|---|---|---|

WDK2 | 0.0 | 0.0 | 6.8 | 6.8 |

IWM2 | 0.0 | 0.0 | 1.2 | 2.4 |

Simultaneous finding of all roots.

Method | |||
---|---|---|---|

WDK2 | 8036.0 | 8036.0 | 20.2 |

IWM2 | 4.9 | 4.9 | 1.7 |

Simultaneous finding of all roots.

Method | ||||
---|---|---|---|---|

WDK2 | 0.2 | 0.4 | 0.5 | 0.7 |

IWM2 | 4.8 | 9.4 | 0.001 | 0.004 |

Simultaneous finding of all roots.

Methods | |||
---|---|---|---|

WDK2 | 9.3 | 9.3 | 7.5 |

IWM2 | 3.9 | 7.0 | 1.3 |

In this section, we discuss some applications in engineering.

Fractional Conversion.

As expression described in [

The exact roots of (

The initial calculated values of (

Table

Van der Waal’s Fluid Model.

A Van der Waals fluid is the one which satisfies the equation of state:

The exact roots of (

The initial calculated values of (

Table

Continuous Stirred Tank Reactor (CSTR).

An isothermal stirred tank reactor (CSTR) is considered here. Items A and

For a simple feedback control system, this problem was first tested by Douglas (see [

The transfer function has the four negative real roots, i.e.,

The initial calculated values of (

Table

(see [

Consider the Predator-Prey model in which the predation rate is denoted by

Taking

The exact roots of (

The initial estimates for

Table

In this work, new two-step derivative-free inverse iterative methods of convergence order 4 for the simultaneous approximations of all roots of a nonlinear equation (

Error graph of WDK2 and IWM2 for

Error graph of WDK2 and IWM2 for

Error graph of WDK2 and IWM2 for

Error graph of WDK2 and IWM2 for

No data were used to support this study.

The authors declare that they have no conflicts of interest regarding the publication of this article.

All authors’ contributed equally in the preparation of this manuscript.