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.

New technique for the approximation of the zeros of nonlinear scientific models

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Faisal Ali ◽  
Waqas Aslam ◽  
Shuliang Huang
Keyword(s):  
Mathematical Models ◽  
Graphical Analysis ◽  
Scientific Models ◽  
Complex Polynomials ◽  
Governing Equations ◽  
Science And Engineering ◽  
New Family ◽  
Engineering Sciences ◽  
Special Cases ◽  
Methods Numerical

AbstractMost of the problems in mathematical and engineering sciences can be studied in the context of nonlinear equations. In this paper, we develop a new family of iterative methods for the approximation of the zeros of mathematical models whose governing equations are nonlinear in nature. The proposed methods are based on decomposition technique due to Daftardar-Gejji and Jaffri [1]. The new family gives several iterative schemes as special cases. The convergence analysis of proposed methods is also presented. In order to determine the performance of newly developed methods, numerical as well as graphical analysis of four complex mathematical models from diverse fields of science and engineering are considered. We also consider some complex polynomials to visualize the roots through polynomiography in the context of proposed methods.

scholarly journals Regularized Mixed Variational-Like Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Saira Zainab ◽  
Eisa Al-Said
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Regularization Method ◽  
Auxiliary Principle ◽  
Iterative Regularization ◽  
Auxiliary Principle Technique ◽  
Iterative Schemes ◽  
Special Cases

We use auxiliary principle technique coupled with iterative regularization method to suggest and analyze some new iterative methods for solving mixed variational-like inequalities. The convergence analysis of these new iterative schemes is considered under some suitable conditions. Some special cases are also discussed. Our method of proofs is very simple as compared with other methods. Our results represent a significant refinement of the previously known results.

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 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 Trifunction Bihemivariational Inequalities

2021 ◽  
pp. 287-313
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Hemivariational Inequalities ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
New Class ◽  
Mild Conditions ◽  
Special Cases

In this paper, we consider a new class of hemivariational inequalities, which is called the trifunction bihemivariational inequality. We suggest and analyze some iterative methods for solving the trifunction bihemivariational inequality using the auxiliary principle technique. The convergence analysis of these iterative methods is also considered under some mild conditions. Several special cases are also considered. Results proved in this paper can be viewed as a refinement and improvement of the known results.

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 Iterative Schemes for a Class of Mixed Trifunction Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Eisa Al-Said
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Iterative Methods ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
Iterative Schemes ◽  
New Class ◽  
Special Cases

We use the auxiliary principle technique to suggest and analyze some iterative methods for solving a new class of variational inequalities, which is called the mixed trifunction variational inequality. The mixed trifunction variational inequality includes the trifunction variational inequalities and the classical variational inequalities as special cases. Convergence of these iterative methods is proved under very mild and suitable assumptions. Several special cases are also considered. Results proved in this paper continue to hold for these known and new classes of variational inequalities and its variant forms.

Some New Iterative Techniques for the Problems Involving Nonlinear Equations

2019 ◽  
Vol 17 (07) ◽  
pp. 1950037 ◽  
Author(s):  
Faisal Ali ◽  
Waqas Aslam ◽  
Arif Rafiq
Keyword(s):  
Mathematical Models ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Graphical Analysis ◽  
Decomposition Technique ◽  
Convergence Criteria ◽  
Iterative Techniques ◽  
New Methods

We develop some new iterative methods, using decomposition technique, for solving the problems which involve nonlinear equations. Importantly, these methods include the generalization of some well-known existing methods. We prove the convergence criteria of our newly proposed methods. Various test examples are considered to validate the efficiency of our new methods. We also give the numerical as well as graphical analysis for two mathematical models to endorse the performance of these methods.

scholarly journals DYNAMICAL ANALYSIS TO EXPLAIN THE NUMERICAL ANOMALIES IN THE FAMILY OF ERMAKOV-KALITLIN TYPE METHODS

2019 ◽  
Vol 24 (3) ◽  
pp. 335-350
Author(s):  
Alicia Cordero ◽  
Juan R. Torregrosa ◽  
Pura Vindel
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Dynamical Analysis ◽  
Test Functions ◽  
Iterative Schemes ◽  
New Family ◽  
The Family ◽  
The Stability ◽  
Appl Math ◽  
Parametric Class

In this paper, we study the dynamics of an iterative method based on the Ermakov-Kalitkin class of iterative schemes for solving nonlinear equations. As it was proven in ”A new family of iterative methods widening areas of convergence, Appl. Math. Comput.”, this family has the property of getting good estimations of the solution when Newton’s method fails. Moreover, the set of converging starting points for several non-polynomial test functions was plotted and they showed to be wider in the case of proposed methods than in Newton’s case, for small values of the parameter. Now, we make a complex dynamical analysis of this parametric class in order to justify the stability properties of this family.

scholarly journals Convergence analysis of the iterative methods for quasi complementarity problems

1988 ◽  
Vol 11 (2) ◽  
pp. 319-334 ◽  
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Complementarity Problems ◽  
Continuous Mapping ◽  
Point Mapping ◽  
Special Cases ◽  
Point To Point

In this paper, we consider the iterative methods for the quasi complementarity problems of the formu−m(u)≥0,   T(u)≥0,   (u−m(u),T(u))=0,wheremis a point-to-point mapping andTis a continuous mapping fromRninto itself. The algorithms considered in this paper are general and unified ones, which include many existing algorithms as special cases for solving the complementarity problems.

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