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 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 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.

scholarly journals Generating Optimal Eighth Order Methods for Computing Multiple Roots

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1947
Author(s):  
Deepak Kumar ◽  
Sunil Kumar ◽  
Janak Raj Sharma ◽  
Matteo d’Amore
Keyword(s):  
Real Life ◽  
Nonlinear Function ◽  
Basins Of Attraction ◽  
Computational Time ◽  
Multiple Roots ◽  
Time Stability ◽  
Academic Problems ◽  
Eighth Order ◽  
Multiple Zeros ◽  
New Family

There are a few optimal eighth order methods in literature for computing multiple zeros of a nonlinear function. Therefore, in this work our main focus is on developing a new family of optimal eighth order iterative methods for multiple zeros. The applicability of proposed methods is demonstrated on some real life and academic problems that illustrate the efficient convergence behavior. It is shown that the newly developed schemes are able to compete with other methods in terms of numerical error, convergence and computational time. Stability is also demonstrated by means of a pictorial tool, namely, basins of attraction that have the fractal-like shapes along the borders through which basins are symmetric.

scholarly journals Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 239 ◽  
Author(s):  
Ramandeep Behl ◽  
M. Salimi ◽  
M. Ferrara ◽  
S. Sharifi ◽  
Samaher Alharbi
Keyword(s):  
Iterative Methods ◽  
Chemical Engineering ◽  
Flame Temperature ◽  
Real Life ◽  
Chemical Reactor ◽  
Basins Of Attraction ◽  
Eighth Order ◽  
Present Scheme ◽  
New Family ◽  
Derivative Free

In this study, we present a new higher-order scheme without memory for simple zeros which has two major advantages. The first one is that each member of our scheme is derivative free and the second one is that the present scheme is capable of producing many new optimal family of eighth-order methods from every 4-order optimal derivative free scheme (available in the literature) whose first substep employs a Steffensen or a Steffensen-like method. In addition, the theoretical and computational properties of the present scheme are fully investigated along with the main theorem, which demonstrates the convergence order and asymptotic error constant. Moreover, the effectiveness of our scheme is tested on several real-life problems like Van der Waal’s, fractional transformation in a chemical reactor, chemical engineering, adiabatic flame temperature, etc. In comparison with the existing robust techniques, the iterative methods in the new family perform better in the considered test examples. The study of dynamics on the proposed iterative methods also confirms this fact via basins of attraction applied to a number of test functions.

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 New Ostrowski-Type Fractional Integral Inequalities via Generalized Exponential-Type Convex Functions and Applications

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1429
Author(s):  
Soubhagya Kumar Sahoo ◽  
Muhammad Tariq ◽  
Hijaz Ahmad ◽  
Jamshed Nasir ◽  
Hassen Aydi ◽  
...  
Keyword(s):  
Fractional Integral ◽  
Great Part ◽  
Real Life ◽  
Strong Relationship ◽  
Integral Inequalities ◽  
Positive Real ◽  
New Class ◽  
Life Problems ◽  
Special Cases ◽  
Mathematical Sciences

Recently, fractional calculus has been the center of attraction for researchers in mathematical sciences because of its basic definitions, properties and applications in tackling real-life problems. The main purpose of this article is to present some fractional integral inequalities of Ostrowski type for a new class of convex mapping. Specifically, n–polynomial exponentially s–convex via fractional operator are established. Additionally, we present a new Hermite–Hadamard fractional integral inequality. Some special cases of the results are discussed as well. Due to the nature of convexity theory, there exists a strong relationship between convexity and symmetry. When working on either of the concepts, it can be applied to the other one as well. Integral inequalities concerned with convexity have a lot of applications in various fields of mathematics in which symmetry has a great part to play. Finally, in applications, some new limits for special means of positive real numbers and midpoint formula are given. These new outcomes yield a few generalizations of the earlier outcomes already published in the literature.

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.

New derivative-free iterative family having optimal convergence order sixteen and its applications

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Manpreet Kaur ◽  
Sanjeev Kumar ◽  
Munish Kansal
Keyword(s):  
Eigenvalue Problems ◽  
Real Life ◽  
Nonlinear Problems ◽  
Chemical Reactor ◽  
Basins Of Attraction ◽  
Convergence Order ◽  
Content Type ◽  
New Class ◽  
Life Problems ◽  
Derivative Free

PurposeThe purpose of the article is to construct a new class of higher-order iterative techniques for solving scalar nonlinear problems.Design/methodology/approachThe scheme is generalized by using the power-mean notion. By applying Neville's interpolating technique, the methods are formulated into the derivative-free approaches. Further, to enhance the computational efficiency, the developed iterative methods have been extended to the methods with memory, with the aid of the self-accelerating parameter.FindingsIt is found that the presented family is optimal in terms of Kung and Traub conjecture as it evaluates only five functions in each iteration and attains convergence order sixteen. The proposed family is examined on some practical problems by modeling into nonlinear equations, such as chemical equilibrium problems, beam positioning problems, eigenvalue problems and fractional conversion in a chemical reactor. The obtained results confirm that the developed scheme works more adequately as compared to the existing methods from the literature. Furthermore, the basins of attraction of the different methods have been included to check the convergence in the complex plane.Originality/valueThe presented experiments show that the developed schemes are of great benefit to implement on real-life problems.

scholarly journals Some Hermite–Hadamard-Type Fractional Integral Inequalities Involving Twice-Differentiable Mappings

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2209
Author(s):  
Soubhagya Kumar Sahoo ◽  
Muhammad Tariq ◽  
Hijaz Ahmad ◽  
Ayman A. Aly ◽  
Bassem F. Felemban ◽  
...  
Keyword(s):  
Applied Sciences ◽  
Focal Point ◽  
Real Life ◽  
Integral Inequalities ◽  
Mathematical Science ◽  
Life Problems ◽  
Special Cases ◽  
Wide Range ◽  
Fractional Analysis ◽  
Almost All

The theory of fractional analysis has been a focal point of fascination for scientists in mathematical science, given its essential definitions, properties, and applications in handling real-life problems. In the last few decades, many mathematicians have shown their considerable interest in the theory of fractional calculus and convexity due to their wide range of applications in almost all branches of applied sciences, especially in numerical analysis, physics, and engineering. The objective of this article is to establish Hermite-Hadamard type integral inequalities by employing the k-Riemann-Liouville fractional operator and its refinements, whose absolute values are twice-differentiable h-convex functions. Moreover, we also present some special cases of our presented results for different types of convexities. Moreover, we also study how q-digamma functions can be applied to address the newly investigated results. Mathematical integral inequalities of this class and the arrangements associated have applications in diverse domains in which symmetry presents a salient role.

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