scholarly journals Efficient Three-Step Class of Eighth-Order Multiple Root Solvers and Their Dynamics

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 837
Author(s):  
R. A. Alharbey ◽  
Munish Kansal ◽  
Ramandeep Behl ◽  
J. A. Tenreiro Machado
Keyword(s):  
General Class ◽  
Real Life ◽  
Multiple Root ◽  
Order Convergence ◽  
Eighth Order ◽  
Dynamical Study ◽  
New Methods ◽  
Life Problems ◽  
Special Cases ◽  
New Strategy

This article proposes a wide general class of optimal eighth-order techniques for approximating multiple zeros of scalar nonlinear equations. The new strategy adopts a weight function with an approach involving the function-to-function ratio. An extensive convergence analysis is performed for the eighth-order convergence of the algorithm. It is verified that some of the existing techniques are special cases of the new scheme. The algorithms are tested in several real-life problems to check their accuracy and applicability. The results of the dynamical study confirm that the new methods are more stable and accurate than the existing schemes.

An Optimal Reconstruction of Chebyshev–Halley-Type Methods with Local Convergence Analysis

2019 ◽  
Vol 17 (05) ◽  
pp. 1940017
Author(s):  
Ali Saleh Alshomrani ◽  
Ioannis K. Argyros ◽  
Ramandeep Behl
Keyword(s):  
Local Convergence ◽  
Real Life ◽  
Eighth Order ◽  
Dynamical Study ◽  
First Order ◽  
Life Problems ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions ◽  
Scalar Equations ◽  
Basins Of Attractions

Our principle aim in this paper is to present a new reconstruction of classical Chebyshev–Halley schemes having optimal fourth and eighth-order of convergence for all parameters [Formula: see text] unlike in the earlier studies. In addition, we analyze the local convergence of them by using hypotheses requiring the first-order derivative of the involved function [Formula: see text] and the Lipschitz conditions. In addition, we also formulate their theoretical radius of convergence. Several numerical examples originated from real life problems demonstrate that they are applicable to a broad range of scalar equations, where previous studies cannot be used. Finally, a dynamical study of them also demonstrates that bigger and more promising basins of attractions are obtained.

scholarly journals An Iteration Function Having Optimal Eighth-Order of Convergence for Multiple Roots and Local Convergence

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1419
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros ◽  
Michael Argyros ◽  
Mehdi Salimi ◽  
Arwa Jeza Alsolami
Keyword(s):  
Quantum Physics ◽  
Real Life ◽  
Chemical Reactor ◽  
Theoretical Models ◽  
Eighth Order ◽  
Multiple Zeros ◽  
Stirred Reactor ◽  
Life Problems ◽  
Special Cases ◽  
Iteration Functions

In the study of dynamics of physical systems an important role is played by symmetry principles. As an example in classical physics, symmetry plays a role in quantum physics, turbulence and similar theoretical models. We end up having to deal with an equation whose solution we desire to be in a closed form. But obtaining a solution in such form is achieved only in special cases. Hence, we resort to iterative schemes. There is where the novelty of our study lies, as well as our motivation for writing it. We have a very limited literature with eighth-order convergent iteration functions that can handle multiple zeros m≥1. Therefore, we suggest an eighth-order scheme for multiple zeros having optimal convergence along with fast convergence and uncomplicated structure. We develop an extensive convergence study in the main theorem that illustrates eighth-order convergence of our scheme. Finally, the applicability and comparison was illustrated on real life problems, e.g., Van der Waal’s equation of state, Chemical reactor with fractional conversion, continuous stirred reactor and multi-factor problems, etc., with existing schemes. These examples further show the superiority of our schemes over the earlier ones.

scholarly journals Local Convergence Balls for Nonlinear Problems with Multiplicity and Their Extension to Eighth-Order Convergence

2019 ◽  
Vol 2019 ◽  
pp. 1-17
Author(s):  
Ramandeep Behl ◽  
Eulalia Martínez ◽  
Fabricio Cevallos ◽  
Ali S. Alshomrani
Keyword(s):  
Local Convergence ◽  
Chemical Engineering ◽  
Real Life ◽  
Nonlinear Problems ◽  
Engineering Problem ◽  
Test Problems ◽  
Order Convergence ◽  
Eighth Order ◽  
Life Problems ◽  
Fourth Order Method

The main contribution of this study is to present a new optimal eighth-order scheme for locating zeros with multiplicity m≥1. An extensive convergence analysis is presented with the main theorem in order to demonstrate the optimal eighth-order convergence of the proposed scheme. Moreover, a local convergence study for the optimal fourth-order method defined by the first two steps of the new method is presented, allowing us to obtain the radius of the local convergence ball. Finally, numerical tests on some real-life problems, such as a Van der Waals equation of state, a conversion chemical engineering problem, and two standard academic test problems, are presented, which confirm the theoretical results established in this paper and the efficiency of this proposed iterative method. We observed from the numerical experiments that our proposed iterative methods have good values for convergence radii. Further, they not only have faster convergence towards the desired zero of the involved function but also have both smaller residual error and a smaller difference between two consecutive iterations than current existing techniques.

scholarly journals Explicit Integrator of Runge-Kutta Type for Direct Solution of u(4)=f(x,u,u′,u′′)

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 246 ◽  
Author(s):  
Nizam Ghawadri ◽  
Norazak Senu ◽  
Firas Fawzi ◽  
Fudziah Ismail ◽  
Zarina Ibrahim
Keyword(s):  
Real Life ◽  
Order Ordinary Differential Equation ◽  
New Techniques ◽  
First Order ◽  
New Methods ◽  
Life Problems ◽  
Algebraic Order ◽  
Runge Kutta ◽  
Order Four ◽  
Primary Contribution

The primary contribution of this work is to develop direct processes of explicit Runge-Kutta type (RKT) as solutions for any fourth-order ordinary differential equation (ODEs) of the structure u ( 4 ) = f ( x , u , u ′ , u ′ ′ ) and denoted as RKTF method. We presented the associated B-series and quad-colored tree theory with the aim of deriving the prerequisites of the said order. Depending on the order conditions, the method with algebraic order four with a three-stage and order five with a four-stage denoted as RKTF4 and RKTF5 are discussed, respectively. Numerical outcomes are offered to interpret the accuracy and efficacy of the new techniques via comparisons with various currently available RK techniques after converting the problems into a system of first-order ODE systems. Application of the new methods in real-life problems in ship dynamics is discussed.

scholarly journals An Efficient Family of Optimal Eighth-Order Multiple Root Finders

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 310 ◽  
Author(s):  
Fiza Zafar ◽  
Alicia Cordero ◽  
Juan Torregrosa
Keyword(s):  
Life Science ◽  
Real Life ◽  
Multiple Root ◽  
Dynamical Analysis ◽  
Multiple Roots ◽  
Eighth Order ◽  
Science And Engineering ◽  
New Family ◽  
Modified Newton’S Method ◽  
Numerical Performance

Finding a repeated zero for a nonlinear equation f ( x ) = 0 , f : I ⊆ R → R has always been of much interest and attention due to its wide applications in many fields of science and engineering. Modified Newton’s method is usually applied to solve this kind of problems. Keeping in view that very few optimal higher-order convergent methods exist for multiple roots, we present a new family of optimal eighth-order convergent iterative methods for multiple roots with known multiplicity involving a multivariate weight function. The numerical performance of the proposed methods is analyzed extensively along with the basins of attractions. Real life models from life science, engineering, and physics are considered for the sake of comparison. The numerical experiments and dynamical analysis show that our proposed methods are efficient for determining multiple roots of nonlinear equations.

scholarly journals An Optimal Eighth-Order Family of Iterative Methods for Multiple Roots

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 672 ◽  
Author(s):  
Saima Akram ◽  
Fiza Zafar ◽  
Nusrat Yasmin
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Multiple Root ◽  
Multiple Roots ◽  
Order Convergence ◽  
Eighth Order ◽  
Root Finding ◽  
New Family ◽  
Two Parameters ◽  
Theoretical Results

In this paper, we introduce a new family of efficient and optimal iterative methods for finding multiple roots of nonlinear equations with known multiplicity ( m ≥ 1 ) . We use the weight function approach involving one and two parameters to develop the new family. A comprehensive convergence analysis is studied to demonstrate the optimal eighth-order convergence of the suggested scheme. Finally, numerical and dynamical tests are presented, which validates the theoretical results formulated in this paper and illustrates that the suggested family is efficient among the domain of multiple root finding methods.

scholarly journals A stable class of modified Newton-like methods for multiple roots and their dynamics

2020 ◽  
Vol 21 (6) ◽  
pp. 603-621
Author(s):  
Munish Kansal ◽  
Alicia Cordero ◽  
Juan R. Torregrosa ◽  
Sonia Bhalla
Keyword(s):  
Real Life ◽  
Chemical Reactor ◽  
Function Technique ◽  
Continuous Stirred Tank Reactor ◽  
Parallel Plates ◽  
Nonlinear Functions ◽  
Eighth Order ◽  
Multiple Zeros ◽  
Life Problems ◽  
Global Carbon

AbstractThere have appeared in the literature a lot of optimal eighth-order iterative methods for approximating simple zeros of nonlinear functions. Although, the similar ideas can be extended for the case of multiple zeros but the main drawback is that the order of convergence and computational efficiency reduce dramatically. Therefore, in order to retain the accuracy and convergence order, several optimal and non-optimal modifications have been proposed in the literature. But, as far as we know, there are limited number of optimal eighth-order methods that can handle the case of multiple zeros. With this aim, a wide general class of optimal eighth-order methods for multiple zeros with known multiplicity is brought forward, which is based on weight function technique involving function-to-function ratio. An extensive convergence analysis is demonstrated to establish the eighth-order of the developed methods. The numerical experiments considered the superiority of the new methods for solving concrete variety of real life problems coming from different disciplines such as trajectory of an electron in the air gap between two parallel plates, the fractional conversion in a chemical reactor, continuous stirred tank reactor problem, Planck’s radiation law problem, which calculates the energy density within an isothermal blackbody and the problem arising from global carbon dioxide model in ocean chemistry, in comparison with methods of similar characteristics appeared in the literature.

An Optimal Eighth-Order Scheme for Multiple Zeros of Univariate Functions

2019 ◽  
Vol 16 (04) ◽  
pp. 1843002 ◽  
Author(s):  
Ramandeep Behl ◽  
Fiza Zafar ◽  
Ali Saleh Alshormani ◽  
Moin-Ud-Din Junjua ◽  
Nusrat Yasmin
Keyword(s):  
Order Of Convergence ◽  
Real Life ◽  
Eighth Order ◽  
Multiple Zeros ◽  
Iterative Schemes ◽  
Life Problems ◽  
Order Scheme ◽  
Multiplicity Formula ◽  
First Time ◽  
Better Than

We construct an optimal eighth-order scheme which will work for multiple zeros with multiplicity [Formula: see text], for the first time. Earlier, the maximum convergence order of multi-point iterative schemes was six for multiple zeros in the available literature. So, the main contribution of this study is to present a new higher-order and as well as optimal scheme for multiple zeros for the first time. In addition, we present an extensive convergence analysis with the main theorem which confirms theoretically eighth-order convergence of the proposed scheme. Moreover, we consider several real life problems which contain simple as well as multiple zeros in order to compare with the existing robust iterative schemes. Finally, we conclude on the basis of obtained numerical results that our iterative methods perform far better than the existing methods in terms of residual error, computational order of convergence and difference between the two consecutive iterations.

scholarly journals A New Class of Halley’s Method with Third-Order Convergence for Solving Nonlinear Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Mohammed Barrada ◽  
Mariya Ouaissa ◽  
Yassine Rhazali ◽  
Mariyam Ouaissa
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Order Convergence ◽  
Eighth Order ◽  
Third Order ◽  
Numerical Examples ◽  
New Class ◽  
New Methods ◽  
New Family ◽  
Number Of Iterations

In this paper, we present a new family of methods for finding simple roots of nonlinear equations. The convergence analysis shows that the order of convergence of all these methods is three. The originality of this family lies in the fact that these sequences are defined by an explicit expression which depends on a parameter p where p is a nonnegative integer. A first study on the global convergence of these methods is performed. The power of this family is illustrated analytically by justifying that, under certain conditions, the method convergence’s speed increases with the parameter p. This family’s efficiency is tested on a number of numerical examples. It is observed that our new methods take less number of iterations than many other third-order methods. In comparison with the methods of the sixth and eighth order, the new ones behave similarly in the examples considered.

scholarly journals New Highly Efficient Families of Higher-Order Methods for Simple Roots, Permittingf′(xn)=0

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Ramandeep Behl ◽  
V. Kanwar
Keyword(s):  
Higher Order ◽  
The Other ◽  
Robust Methods ◽  
Challenging Problem ◽  
Order Convergence ◽  
Dynamical Study ◽  
Globally Convergent ◽  
Special Cases ◽  
Globally Convergent Methods ◽  
Ostrowski’S Method

Construction of higher-order optimal and globally convergent methods for computing simple roots of nonlinear equations is an earliest and challenging problem in numerical analysis. Therefore, the aim of this paper is to present optimal and globally convergent families of King's method and Ostrowski's method having biquadratic and eight-order convergence, respectively, permittingf′(x)=0in the vicinity of the required root. Fourth-order King's family and Ostrowski's method can be seen as special cases of our proposed scheme. All the methods considered here are found to be more effective to the similar robust methods available in the literature. In their dynamical study, it has been observed that the proposed methods have equal or better stability and robustness as compared to the other methods.

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