scholarly journals A Variant of Chebyshev’s Method with 3αth-Order of Convergence by Using Fractional Derivatives

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1017 ◽  
Author(s):  
Alicia Cordero ◽  
Ivan Girona ◽  
Juan R. Torregrosa
Keyword(s):  
Iterative Methods ◽  
Fractional Derivatives ◽  
Order Of Convergence ◽  
Iterative Scheme ◽  
Test Problems ◽  
Robust Performance ◽  
Symmetric Duality ◽  
Numerical Examples ◽  
Chebyshev’S Method ◽  
Left And Right

In this manuscript, we propose several iterative methods for solving nonlinear equations whose common origin is the classical Chebyshev’s method, using fractional derivatives in their iterative expressions. Due to the symmetric duality of left and right derivatives, we work with right-hand side Caputo and Riemann–Liouville fractional derivatives. To increase as much as possible the order of convergence of the iterative scheme, some improvements are made, resulting in one of them being of 3 α -th order. Some numerical examples are provided, along with an study of the dependence on initial estimations on several test problems. This results in a robust performance for values of α close to one and almost any initial estimation.

scholarly journals A Family of Fourteenth-Order Convergent Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fiza Zafar ◽  
Gulshan Bibi
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Numerical Examples ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
Specific Step

We present a family of fourteenth-order convergent iterative methods for solving nonlinear equations involving a specific step which when combined with any two-step iterative method raises the convergence order by n+10, if n is the order of convergence of the two-step iterative method. This new class include four evaluations of function and one evaluation of the first derivative per iteration. Therefore, the efficiency index of this family is 141/5 =1.695218203. Several numerical examples are given to show that the new methods of this family are comparable with the existing methods.

scholarly journals Two Higher Order Iterative Methods for Solving Nonlinear Equations

2018 ◽  
Vol 14 (1) ◽  
pp. 179-187
Author(s):  
Jivandhar Jnawali ◽  
Chet Raj Bhatta
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Higher Order ◽  
Secant Method ◽  
Subject Classification ◽  
Numerical Examples ◽  
Ams Subject Classification

 The main purpose of this paper is to derive two higher order iterative methods for solving nonlinear equations as variants of Mir, Ayub and Rafiq method. These methods are free from higher order derivatives. We obtain these methods by amalgamating Mir, Ayub and Rafiq method with standard secant method and modified secant method given by Amat and Busquier. The order of convergence of new variants are four and six. Also, numerical examples are given to compare the performance of newly introduced methods with the similar existing methods. 2010 AMS Subject Classification: 65H05 Journal of the Institute of Engineering, 2018, 14(1): 179-187

scholarly journals New Modified Newton Type Iterative Methods

2021 ◽  
Vol 2 (1) ◽  
pp. 17-24
Author(s):  
Jivandhar Jnawali
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Single Variable ◽  
Numerical Examples

In this work, we present two Newton type iterative methods for finding the solution of nonlinear equations of single variable. One is obtained as variant of McDougall and Wotherspoon method, and another is obtained by amalgamation of Potra and Pta’k method and our newly introduced method. The order of convergence of these methods are 1 + √2 and 3.5615. Some numerical examples are given to compare the performance of these methods with some similar existing methods.

scholarly journals Novel Computational Iterative Methods with Optimal Order for Nonlinear Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
F. Soleymani
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
General Class ◽  
Analytical Study ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Numerical Examples ◽  
First Order

This paper contributes a very general class of two-point iterative methods without memory for solving nonlinear equations. The class of methods is developed using weight function approach. Per iteration, each method of the class includes two evaluations of the function and one of its first-order derivative. The analytical study of the main theorem is presented in detail to show the fourth order of convergence. Furthermore, it is discussed that many of the existing fourth-order methods without memory are members from this developed class. Finally, numerical examples are taken into account to manifest the accuracy of the derived methods.

scholarly journals Multipoint Fractional Iterative Methods with (2α + 1)th-Order of Convergence for Solving Nonlinear Problems

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 452
Author(s):  
Giro Candelario ◽  
Alicia Cordero ◽  
Juan R. Torregrosa
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Newton Method ◽  
Fractional Derivatives ◽  
Order Of Convergence ◽  
Recent Literature ◽  
Nonlinear Problems ◽  
Type Method ◽  
Numerical Tests ◽  
Multipoint Method

In the recent literature, some fractional one-point Newton-type methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper, we introduce a new fractional Newton-type method with order of convergence α + 1 and compare it with the existing fractional Newton method with order 2 α . Moreover, we also introduce a multipoint fractional Traub-type method with order 2 α + 1 and compare its performance with that of its first step. Some numerical tests and analysis of the dependence on the initial estimations are made for each case, including a comparison with classical Newton ( α = 1 of the first step of the class) and classical Traub’s scheme ( α = 1 of fractional proposed multipoint method). In this comparison, some cases are found where classical Newton and Traub’s methods do not converge and the proposed methods do, among other advantages.

scholarly journals New Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction

2022 ◽  
Vol 21 ◽  
pp. 9-16
Author(s):  
O. Ababneh
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Newton's Method ◽  
Order Of Convergence ◽  
Iterative Algorithms ◽  
Basins Of Attraction ◽  
Second Derivative ◽  
Numerical Examples ◽  
Convergence Results ◽  
Modified Newton’S Method

The purpose of this paper is to propose new modified Newton’s method for solving nonlinear equations and free from second derivative. Convergence results show that the order of convergence is four. Several numerical examples are given to illustrate that the new iterative algorithms are effective.In the end, we present the basins of attraction to observe the fractal behavior and dynamical aspects of the proposed algorithms.

scholarly journals Extended local convergence for Newton-type solver under weak conditions

2021 ◽  
Vol 66 (4) ◽  
pp. 757-768
Author(s):  
Ioannis K. Argyros ◽  
◽  
Santhosh George ◽  
Kedarnath Senapati ◽  
◽  
...  
Keyword(s):  
Local Convergence ◽  
Order Of Convergence ◽  
Basins Of Attraction ◽  
Dimensional Euclidean Space ◽  
Test Problems ◽  
Eighth Order ◽  
Convergence Conditions ◽  
Numerical Examples ◽  
Special Case ◽  
Methods Numerical

We present the local convergence of a Newton-type solver for equations involving Banach space valued operators. The eighth order of convergence was shown earlier in the special case of the k-dimensional Euclidean space, using hypotheses up to the eighth derivative although these derivatives do not appear in the method. We show convergence using only the rst derivative. This way we extend the applicability of the methods. Numerical examples are used to show the convergence conditions. Finally, the basins of attraction of the method, on some test problems are presented.

scholarly journals Numerical Algorithms for Finding Zeros of Nonlinear Equations and Their Dynamical Aspects

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Amir Naseem ◽  
M. A. Rehman ◽  
Thabet Abdeljawad ◽  
Francisco Balibrea
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Numerical Algorithms ◽  
One Dimension ◽  
Second Derivative ◽  
Numerical Examples ◽  
Derivative Free ◽  
Interpolation Technique ◽  
Taylor’S Series

In this paper, we developed two new numerical algorithms for finding zeros of nonlinear equations in one dimension and one of them is second derivative free which has been removed using the interpolation technique. We derive these algorithms with the help of Taylor’s series expansion and Golbabai and Javidi’s method. The convergence analysis of these algorithms is discussed. It is established that the newly developed algorithms have sixth order of convergence. Several numerical examples have been solved which prove the better efficiency of these algorithms as compared to other well-known iterative methods of the same kind. Finally, the comparison of polynomiographs generated by other well-known iterative methods with our developed algorithms has been made which reflects their dynamical aspects.

scholarly journals A New Three-Step Class of Iterative Methods for Solving Nonlinear Systems

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1221 ◽  
Author(s):  
Raudys R. Capdevila ◽  
Alicia Cordero ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Systems ◽  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Numerical Examples ◽  
New Class ◽  
And Performance

In this work, a new class of iterative methods for solving nonlinear equations is presented and also its extension for nonlinear systems of equations. This family is developed by using a scalar and matrix weight function procedure, respectively, getting sixth-order of convergence in both cases. Several numerical examples are given to illustrate the efficiency and performance of the proposed methods.

scholarly journals A study on multiterm hybrid multi-order fractional boundary value problem coupled with its stability analysis of Ulam–Hyers type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ahmed Nouara ◽  
Abdelkader Amara ◽  
Eva Kaslik ◽  
Sina Etemad ◽  
Shahram Rezapour ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Boundary Value Problem ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Research Work ◽  
Existence Results ◽  
Fractional Boundary Value Problem ◽  
Numerical Examples ◽  
Fractional Derivatives And Integrals ◽  
Fractional Differential

AbstractIn this research work, a newly-proposed multiterm hybrid multi-order fractional boundary value problem is studied. The existence results for the supposed hybrid fractional differential equation that involves Riemann–Liouville fractional derivatives and integrals of multi-orders type are derived using Dhage’s technique, which deals with a composition of three operators. After that, its stability analysis of Ulam–Hyers type and the relevant generalizations are checked. Some illustrative numerical examples are provided at the end to illustrate and validate our obtained results.

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