scholarly journals Convergence and Stability of a Parametric Class of Iterative Schemes for Solving Nonlinear Systems

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 86
Author(s):  
Alicia Cordero ◽  
Eva G. Villalba ◽  
Juan R. Torregrosa ◽  
Paula Triguero-Navarro
Keyword(s):  
Nonlinear Systems ◽  
Dynamical Behavior ◽  
Order Convergence ◽  
Hammerstein Integral Equation ◽  
Iterative Schemes ◽  
The Third ◽  
The Family ◽  
The Stability ◽  
Theoretical Results ◽  
Parametric Class

A new parametric class of iterative schemes for solving nonlinear systems is designed. The third- or fourth-order convergence, depending on the values of the parameter being proven. The analysis of the dynamical behavior of this class in the context of scalar nonlinear equations is presented. This study gives us important information about the stability and reliability of the members of the family. The numerical results obtained by applying different elements of the family for solving the Hammerstein integral equation and the Fisher’s equation confirm the theoretical results.

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 An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior

2019 ◽  
Vol 17 (1) ◽  
pp. 1567-1598
Author(s):  
Tianbao Liu ◽  
Xiwen Qin ◽  
Qiuyue Li
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Second Derivative ◽  
Third Order ◽  
Numerical Tests ◽  
The Family ◽  
Theoretical Results ◽  
High Computational Efficiency

Abstract In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.

scholarly journals Correction: FRACTIONAL ORDER BARBALAT’S LEMMA AND ITS APPLICATIONS IN THE STABILITY OF FRACTIONAL ORDER NONLINEAR SYSTEMS

2017 ◽  
Vol 22 (4) ◽  
pp. 503-513 ◽  
Author(s):  
Fei Wang ◽  
Yongqing Yang
Keyword(s):  
Nonlinear Systems ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Caputo Fractional Derivative ◽  
Barbalat’S Lemma ◽  
Liouville Derivative ◽  
Liouville Fractional Derivative ◽  
The Stability ◽  
The Relationship ◽  
Theoretical Results

This paper investigates fractional order Barbalat’s lemma and its applications for the stability of fractional order nonlinear systems with Caputo fractional derivative at first. Then, based on the relationship between Caputo fractional derivative and Riemann-Liouville fractional derivative, fractional order Barbalat’s lemma with Riemann-Liouville derivative is derived. Furthermore, according to these results, a set of new formulations of Lyapunov-like lemmas for fractional order nonlinear systems are established. Finally, an example is presented to verify the theoretical results in this paper.

scholarly journals A New Class of Iterative Processes for Solving Nonlinear Systems by Using One Divided Differences Operator

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 776 ◽  
Author(s):  
Alicia Cordero ◽  
Cristina Jordán ◽  
Esther Sanabria ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Systems ◽  
Boundary Problem ◽  
Fourth Order ◽  
Difference Operator ◽  
Divided Difference ◽  
Order Convergence ◽  
New Class ◽  
Numerical Tests ◽  
New Family ◽  
Theoretical Results

In this manuscript, a new family of Jacobian-free iterative methods for solving nonlinear systems is presented. The fourth-order convergence for all the elements of the class is established, proving, in addition, that one element of this family has order five. The proposed methods have four steps and, in all of them, the same divided difference operator appears. Numerical problems, including systems of academic interest and the system resulting from the discretization of the boundary problem described by Fisher’s equation, are shown to compare the performance of the proposed schemes with other known ones. The numerical tests are in concordance with the theoretical results.

scholarly journals Design, Convergence and Stability of a Fourth-Order Class of Iterative Methods for Solving Nonlinear Vectorial Problems

2021 ◽  
Vol 5 (3) ◽  
pp. 125
Author(s):  
Alicia Cordero ◽  
Cristina Jordán ◽  
Esther Sanabria-Codesal ◽  
Juan R. Torregrosa
Keyword(s):  
Fourth Order ◽  
Chaotic Behavior ◽  
Basins Of Attraction ◽  
Order Convergence ◽  
Iterative Schemes ◽  
Numerical Tests ◽  
New Methods ◽  
Stable Elements ◽  
Parameter Values ◽  
Theoretical Results

A new parametric family of iterative schemes for solving nonlinear systems is presented. Fourth-order convergence is demonstrated and its stability is analyzed as a function of the parameter values. This study allows us to detect the most stable elements of the class, to find the fractals in the boundary of the basins of attraction and to reject those with chaotic behavior. Some numerical tests show the performance of the new methods, confirm the theoretical results and allow to compare the proposed schemes with other known ones.

scholarly journals Stability and Controllability of Nonlinear Systems

2020 ◽  
pp. 51-60
Author(s):  
S. T. Cotterell ◽  
I. Davies ◽  
L. C. Abraham
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Equilibrium Solution ◽  
Indirect Method ◽  
Real Life ◽  
Linearization Method ◽  
Controlled System ◽  
The Stability ◽  
Theoretical Results

This paper is aimed at establishing new stability and controllability results for nonlinear systems. The approach is to use the Lyapunov indirect method to obtain the stability of the equilibrium solution of the uncontrolled nonlinear system by applying the Jacobi’s linearization method and the controllability of the controlled system obtained by the rank criterion for properness. Example is given with a real-life application to illustrate the effectiveness of the theoretical results.

scholarly journals Semi-Local Analysis and Real Life Applications of Higher-Order Iterative Schemes for Nonlinear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 92
Author(s):  
Ramandeep Behl ◽  
Sonia Bhalla ◽  
Ioannis K. Argyros ◽  
Sanjeev Kumar
Keyword(s):  
Nonlinear Systems ◽  
Applied Mathematics ◽  
Real Life ◽  
Higher Order ◽  
Local Analysis ◽  
First Order ◽  
The Family ◽  
Lipschitz Constants ◽  
Theoretical Results ◽  
Real World Problems

Our aim is to improve the applicability of the family suggested by Bhalla et al. (Computational and Applied Mathematics, 2018) for the approximation of solutions of nonlinear systems. Semi-local convergence relies on conditions with first order derivatives and Lipschitz constants in contrast to other works requiring higher order derivatives not appearing in these schemes. Hence, the usage of these schemes is improved. Moreover, a variety of real world problems, namely, Bratu’s 1D, Bratu’s 2D and Fisher’s problems, are applied in order to inspect the utilization of the family and to test the theoretical results by adopting variable precision arithmetics in Mathematica 10. On account of these examples, it is concluded that the family is more efficient and shows better performance as compared to the existing one.

scholarly journals Generating Root-Finder Iterative Methods of Second Order: Convergence and Stability

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 55 ◽  
Author(s):  
Francisco I. Chicharro ◽  
Alicia Cordero ◽  
Neus Garrido ◽  
Juan R. Torregrosa
Keyword(s):  
Weight Function ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Weight Functions ◽  
Iterative Schemes ◽  
Numerical Tests ◽  
Low Degree ◽  
Second Order Convergence ◽  
The Stability ◽  
Low Degree Polynomials

In this paper, a simple family of one-point iterative schemes for approximating the solutions of nonlinear equations, by using the procedure of weight functions, is derived. The convergence analysis is presented, showing the sufficient conditions for the weight function. Many known schemes are members of this family for particular choices of the weight function. The dynamical behavior of one of these choices is presented, analyzing the stability of the fixed points and the critical points of the rational function obtained when the iterative expression is applied on low degree polynomials. Several numerical tests are given to compare different elements of the proposed family on non-polynomial problems.

scholarly journals Solution Bounds, Stability, and Estimation of Trapping/Stability Regions of Some Nonlinear Time-Varying Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Mark A. Pinsky ◽  
Steve Koblik
Keyword(s):  
Nonlinear Systems ◽  
Control Problems ◽  
Stability Criteria ◽  
Stability Regions ◽  
Novel Approach ◽  
Time Varying Systems ◽  
Trapping Regions ◽  
The Stability ◽  
And Control ◽  
Theoretical Results

Estimation of solution norms and stability for time-dependent nonlinear systems is ubiquitous in numerous engineering, natural science, and control problems. Yet, practically valuable results are rare in this area. This paper develops a novel approach, which bounds the solution norms, derives the corresponding stability criteria, and estimates the trapping/stability regions for some nonautonomous and nonlinear systems, which arise in various application domains. Our inferences rest on deriving a scalar differential inequality for the norms of solutions to the initial systems. Utility of the Lipschitz inequality linearizes the associated auxiliary differential equation and yields both the upper bounds for the norms of solutions and the relevant stability criteria. To refine these inferences, we introduce a nonlinear extension of the Lipschitz inequality, which improves the developed bounds and allows estimation of the stability/trapping regions for the corresponding systems. Finally, we confirm the theoretical results in representative simulations.

scholarly journals Stability Analysis and Numerical Computation of the Fractional Predator–Prey Model with the Harvesting Rate

2020 ◽  
Vol 4 (3) ◽  
pp. 35 ◽  
Author(s):  
Mehmet Yavuz ◽  
Ndolane Sene
Keyword(s):  
Arbitrary Order ◽  
Dynamical Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Uniqueness Of The Solution ◽  
Numerical Discretization ◽  
The Stability ◽  
Net Reproduction ◽  
Theoretical Results

In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral was introduced and the corresponding numerical discretization of the predator–prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense.

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