scholarly journals Iteration Methods with an Auxiliary Function for Nonlinear Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Faisal Ali ◽  
Waqas Aslam ◽  
Imran Khalid ◽  
Akbar Nadeem
Keyword(s):  
Nonlinear Equation ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Auxiliary Function ◽  
New Family ◽  
Special Cases ◽  
Engineering Models ◽  
Iteration Methods ◽  
Scientific Engineering ◽  
Taylor’S Series

Various iterative methods have been introduced by involving Taylor’s series on the auxiliary function g x to solve the nonlinear equation f x = 0 . In this paper, we introduce the expansion of g x with the inclusion of weights w i such that ∑ i = 1 p w i = 1 and knots τ i ∈ 0,1 in order to develop a new family of iterative methods. The methods proposed in the present paper are applicable for different choices of auxiliary function g x , and some already known methods can be viewed as the special cases of these methods. We consider the diverse scientific/engineering models to demonstrate the efficiency of the proposed methods.

scholarly journals Novel higher order iterative schemes based on the $ q- $Calculus for solving nonlinear equations

2021 ◽  
Vol 7 (3) ◽  
pp. 3524-3553
Author(s):  
Gul Sana ◽  
◽  
Muhmmad Aslam Noor ◽  
Dumitru Baleanu ◽  
◽  
...  
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Approximate Solutions ◽  
Auxiliary Function ◽  
Quantum Calculus ◽  
New Methods ◽  
Classical Calculus ◽  
Mathematical Sciences ◽  
Modern Era ◽  
Taylor’S Series

<abstract><p>The conventional infinitesimal calculus that concentrates on the idea of navigating the $ q- $symmetrical outcomes free from the limits is known as Quantum calculus (or $ q- $calculus). It focuses on the logical rationalization of differentiation and integration operations. Quantum calculus arouses interest in the modern era due to its broad range of applications in diversified disciplines of the mathematical sciences. In this paper, we instigate the analysis of Quantum calculus on the iterative methods for solving one-variable nonlinear equations. We introduce the new iterative methods called $ q- $iterative methods by employing the $ q- $analogue of Taylor's series together with the inclusion of an auxiliary function. We also investigate the convergence order of our newly suggested methods. Multiple numerical examples are utilized to demonstrate the performance of new methods with an acceptable accuracy. In addition, approximate solutions obtained are comparable to the analogous solutions in the classical calculus when the quantum parameter $ q $ tends to one. Furthermore, a potential correlation is established by uniting the $ q- $iterative methods and traditional iterative methods.</p></abstract>

scholarly journals A New Family of High-Order Ehrlich-Type Iterative Methods

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1855 ◽  
Author(s):  
Petko D. Proinov ◽  
Maria T. Vasileva
Keyword(s):  
Iterative Methods ◽  
Local Convergence ◽  
Semilocal Convergence ◽  
High Order ◽  
Convergence Theorems ◽  
New Family ◽  
Special Cases ◽  
Iteration Functions ◽  
Iteration Function ◽  
Local Convergence Analysis

One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.

scholarly journals New Family of Iterative Methods for Solving Nonlinear Models

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Faisal Ali ◽  
Waqas Aslam ◽  
Kashif Ali ◽  
Muhammad Adnan Anwar ◽  
Akbar Nadeem
Keyword(s):  
Mathematical Models ◽  
Convergence Analysis ◽  
Iterative Methods ◽  
Nonlinear Models ◽  
Graphical Analysis ◽  
Governing Equations ◽  
Iterative Schemes ◽  
New Family ◽  
Special Cases ◽  
Improved Performance

We introduce a new family of iterative methods for solving mathematical models whose governing equations are nonlinear in nature. The new family gives several iterative schemes as special cases. We also give the convergence analysis of our proposed methods. In order to demonstrate the improved performance of newly developed methods, we consider some nonlinear equations along with two complex mathematical models. The graphical analysis for these models is also presented.

scholarly journals New Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Rajinder Thukral
Keyword(s):  
Nonlinear Equations ◽  
Convergence Order ◽  
Order Derivative ◽  
Optimal Order ◽  
Eighth Order ◽  
Optimal Order Of Convergence ◽  
New Family ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Iteration Methods

A new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based onnevaluations could achieve optimal convergence order of . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for . Numerical comparisons are made to demonstrate the performance of the methods presented.

scholarly journals Generating Function Approach to the Derivation of Higher-Order Iterative Methods for Solving Nonlinear Equations

2018 ◽  
Vol 173 ◽  
pp. 03024
Author(s):  
Tugal Zhanlav ◽  
Ochbadrakh Chuluunbaatar ◽  
Vandandoo Ulziibayar
Keyword(s):  
Iterative Methods ◽  
Generating Function ◽  
Generating Functions ◽  
Order Of Convergence ◽  
Sufficient Conditions ◽  
Optimal Order ◽  
Eighth Order ◽  
New Family ◽  
Special Cases ◽  
Necessary And Sufficient

In this paper we propose a generating function method for constructing new two and three-point iterations withp(p= 4, 8) order of convergence. This approach allows us to derive a new family of optimal order iterative methods that include well known methods as special cases. Necessary and sufficient conditions forp-th (p= 4, 8) order convergence of the proposed iterations are given in terms of parameters τnand αn. We also propose some generating functions for τnand αn. We develop a unified representation of all optimal eighth-order methods. The order of convergence of the proposed methods is confirmed by numerical experiments.

scholarly journals On Iterative Methods for Solving Nonlinear Equations in Quantum Calculus

2021 ◽  
Vol 5 (3) ◽  
pp. 60
Author(s):  
Gul Sana ◽  
Pshtiwan Othman Mohammed ◽  
Dong Yun Shin ◽  
Muhmmad Aslam Noor ◽  
Mohammad Salem Oudat
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Coupled System ◽  
Numerical Examples ◽  
Quantum Calculus ◽  
Acceptable Accuracy ◽  
System Technique ◽  
Taylor’S Series

Quantum calculus (also known as the q-calculus) is a technique that is similar to traditional calculus, but focuses on the concept of deriving q-analogous results without the use of the limits. In this paper, we suggest and analyze some new q-iterative methods by using the q-analogue of the Taylor’s series and the coupled system technique. In the domain of q-calculus, we determine the convergence of our proposed q-algorithms. Numerical examples demonstrate that the new q-iterative methods can generate solutions to the nonlinear equations with acceptable accuracy. These newly established methods also exhibit predictability. Furthermore, an analogy is settled between the well known classical methods and our proposed q-Iterative methods.

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