scholarly journals Fixed Point Root-Finding Methods of Fourth-Order of Convergence

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 769 ◽  
Author(s):  
Alicia Cordero ◽  
Lucía Guasp ◽  
Juan R. Torregrosa
Keyword(s):  
Weight Function ◽  
Dynamical Behavior ◽  
Nonlinear Problems ◽  
Dynamical Analysis ◽  
Function Technique ◽  
Weight Functions ◽  
Good Tool ◽  
Quadratic Polynomials ◽  
Root Finding ◽  
The Stability

In this manuscript, by using the weight-function technique, a new class of iterative methods for solving nonlinear problems is constructed, which includes many known schemes that can be obtained by choosing different weight functions. This weight function, depending on two different evaluations of the derivative, is the unique difference between the two steps of each method, which is unusual. As it is proven that all the members of the class are optimal methods in the sense of Kung-Traub’s conjecture, the dynamical analysis is a good tool to determine the best elements of the family in terms of stability. Therefore, the dynamical behavior of this class on quadratic polynomials is studied in this work. We analyze the stability of the presented family from the multipliers of the fixed points and critical points, along with their associated parameter planes. In addition, this study enables us to select the members of the class with good stability properties.

scholarly journals A Family of Multiple-Root Finding Iterative Methods Based on Weight Functions

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2194
Author(s):  
Francisco I. Chicharro ◽  
Rafael A. Contreras ◽  
Neus Garrido
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Numerical Test ◽  
Multiple Root ◽  
Initial Guess ◽  
Dynamical Analysis ◽  
Weight Functions ◽  
Root Finding ◽  
Wide Range ◽  
The Family

A straightforward family of one-point multiple-root iterative methods is introduced. The family is generated using the technique of weight functions. The order of convergence of the family is determined in its convergence analysis, which shows the constraints that the weight function must satisfy to achieve order three. In this sense, a family of iterative methods can be obtained with a suitable design of the weight function. That is, an iterative algorithm that depends on one or more parameters is designed. This family of iterative methods, starting with proper initial estimations, generates a sequence of approximations to the solution of a problem. A dynamical analysis is also included in the manuscript to study the long-term behavior of the family depending on the parameter value and the initial guess considered. This analysis reveals the good properties of the family for a wide range of values of the parameter. In addition, a numerical test on academic and engineering multiple-root functions is performed.

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.

Dynamical analysis for cylindrical geometry in non-minimally coupled f(R,T) gravity

2021 ◽  
Author(s):  
M. Z. Bhatti ◽  
Z. Yousaf ◽  
M. Yousaf
Keyword(s):  
Perturbation Technique ◽  
Dynamical Behavior ◽  
Momentum Tensor ◽  
Dynamical Analysis ◽  
Dynamical Equations ◽  
Physical Constraints ◽  
Cylindrical System ◽  
Modified Gravity Theory ◽  
The Stability ◽  
Tensor Formula

This paper aims to investigate the stability constraints under the influence of particular modified gravity theory [Formula: see text], i.e. [Formula: see text] gravity in which the Lagrangian is a varying function of [Formula: see text] and trace of energy momentum tensor ([Formula: see text]). We examine stable behavior for compact cylindrical star having anisotropic symmetric configuration. We establish dynamical equations as well as equations of continuity in the background of this particular non-minimal coupled [Formula: see text]. We utilize perturbation technique which will be applied on geometrical as well as material physical quantities to constitute collapse equation. We continue this significant investigation to understand the dynamical behavior of considered cylindrical system under non-minimal coupled [Formula: see text] functional, i.e. [Formula: see text]. This gravitational function gives compatible findings only for [Formula: see text], also [Formula: see text] and [Formula: see text] considered in this astrophysical model as coupling entity. This model contains [Formula: see text] which is constant entity, having the values of order of the effective Ricci scalar [Formula: see text]. Furthermore, we impose some physical constraints to determine and maintain the stability criteria by establishing the expression of adiabatic index, i.e. [Formula: see text] for cylindrical anisotropic configuration, in Newtonian [Formula: see text] and post-Newtonian ([Formula: see text]) eras.

COMPLEX DYNAMICAL ANALYSIS OF A COUPLED NETWORK FROM INNATE IMMUNE RESPONSES

2013 ◽  
Vol 23 (11) ◽  
pp. 1350180 ◽  
Author(s):  
JINYING TAN ◽  
XIUFEN ZOU
Keyword(s):  
Immune Responses ◽  
Negative Feedback ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Innate Immune ◽  
Dynamical Analysis ◽  
Innate Immune Responses ◽  
Dynamical Behaviors ◽  
Positive And Negative Feedback ◽  
The Stability

In this paper, we investigate the complex dynamical behaviors of a biological network that is derived from innate immune responses and that couples positive and negative feedback loops. The stability conditions of the non-negative equilibrium points (EPs) of the system are obtained, using the theory of dynamical systems, and we deduce that no more than three stable EPs exist in this system. Through bifurcation analysis and numerical simulations, we find that the system presents rich dynamical behaviors, such as monostability, bistability and oscillations. These results reveal how positive and negative feedback cooperatively regulate the dynamical behavior of the system.

scholarly journals Study of Dynamical Behavior and Stability of Iterative Methods for Nonlinear Equation with Applications in Engineering

2020 ◽  
Vol 2020 ◽  
pp. 1-20
Author(s):  
Naila Rafiq ◽  
Saima Akram ◽  
Nazir Ahmad Mir ◽  
Mudassir Shams
Keyword(s):  
Nonlinear Equation ◽  
Iterative Methods ◽  
Dynamical Behavior ◽  
Numerical Test ◽  
Computational Cost ◽  
Basins Of Attraction ◽  
Single Root ◽  
Root Finding ◽  
The Stability

In this article, we first construct a family of optimal 2-step iterative methods for finding a single root of the nonlinear equation using the procedure of weight function. We then extend these methods for determining all roots simultaneously. Convergence analysis is presented for both cases to show that the order of convergence is 4 in case of the single-root finding method and is 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The dynamical behavior is presented to analyze the stability of fixed and critical points of the rational operator of one-point iterative methods. The computational cost, basins of attraction, efficiency, log of the residual, and numerical test examples show that the newly constructed methods are more efficient as compared with the existing methods in the literature.

scholarly journals Dynamical Analysis of an Interleaved Single Inductor TITO Switching Regulator

2009 ◽  
Vol 2009 ◽  
pp. 1-19 ◽  
Author(s):  
Abdelali El Aroudi ◽  
Vanessa Moreno-Font ◽  
Luis Benadero
Keyword(s):  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Dynamical Behavior ◽  
Current Mode ◽  
Operating Conditions ◽  
Dynamical Analysis ◽  
Time Intervals ◽  
Switching Regulator ◽  
Duty Cycles ◽  
The Stability

We study the dynamical behavior of a single inductor two inputs two outputs (SITITO) power electronics DC-DC converter under a current mode control in a PWM interleaved scheme. This system is able to regulate two, generally one positive and one negative, voltages (outputs). The regulation of the outputs is carried out by the modulation of two time intervals within a switching cycle. The value of the regulated voltages is related to both duty cycles (inputs). The stability of the whole nonlinear system is therefore studied without any decoupling. Under certain operating conditions, the dynamical behavior of the system can be modeled by a piecewise linear (PWL) map, which is used to investigate the stability in the parameter space and to detect possible subharmonic oscillations and chaotic behavior. These results are confirmed by numerical one dimensional and two-dimensional bifurcation diagrams and some experimental measurements from a laboratory prototype.

scholarly journals The stability of pure weights under conditioning

1974 ◽  
Vol 15 (1) ◽  
pp. 5-12 ◽  
Author(s):  
D. J. Foulis ◽  
C. H. Randall
Keyword(s):  
Weight Function ◽  
Stochastic Models ◽  
Convex Set ◽  
Extreme Points ◽  
Weight Functions ◽  
The Stability ◽  
Universe Of Discourse

In [1], we showed how a collection of physical operations or experiments could be represented by a nonempty set of nonempty sets satisfying certain conditions (irredundancy and coherence) and we called such sets . We also introduced “complete stochastic models” for the empirical universe of discourse represented by such a manual , namely, the so-called weight functions for . These weight functions form a convex set the extreme points of which are called pure weights. We also showed that there is a so-called logic ∏() affiliated with a manual and that each weight function for induces a state on this logic.

scholarly journals Stability of combustion waves in a simplified gas–solid combustion model in porous media

2018 ◽  
Vol 376 (2117) ◽  
pp. 20170185 ◽  
Author(s):  
Fatih Ozbag ◽  
Stephen Schecter
Keyword(s):  
Weight Function ◽  
Essential Spectrum ◽  
Parabolic Systems ◽  
Travelling Wave ◽  
Weight Functions ◽  
Combustion Model ◽  
Combustion Waves ◽  
Linearized System ◽  
The Stability ◽  
Linearized Operator

We study the stability of the combustion waves that occur in a simplified model for injection of air into a porous medium that initially contains some solid fuel. We determine the essential spectrum of the linearized system at a travelling wave. For certain waves, we are able to use a weight function to stabilize the essential spectrum. We perform a numerical computation of the Evans function to show that some of these waves have no unstable discrete spectrum. The system is partly parabolic, so the linearized operator is not sectorial, and the weight function decays at one end. We use an extension of a recent result about partly parabolic systems that are stabilized by such weight functions to show nonlinear stability. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

scholarly journals Dynamical analysis of a giving up smoking model with time delay

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zizhen Zhang ◽  
Ruibin Wei ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

scholarly journals Nonlinear Dynamics and Stability Analysis of a Three-Cell Flying Capacitor DC-DC Converter

2021 ◽  
Vol 11 (4) ◽  
pp. 1395
Author(s):  
Abdelali El Aroudi ◽  
Natalia Cañas-Estrada ◽  
Mohamed Debbat ◽  
Mohamed Al-Numay
Keyword(s):  
Steady State ◽  
Stability Analysis ◽  
Monodromy Matrix ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Simple Expression ◽  
Buck Converter ◽  
State Variables ◽  
Bifurcation Phenomena ◽  
The Stability

This paper presents a study of the nonlinear dynamic behavior a flying capacitor four-level three-cell DC-DC buck converter. Its stability analysis is performed and its stability boundaries is determined in the multi-dimensional paramertic space. First, the switched model of the converter is presented. Then, a discrete-time controller for the converter is proposed. The controller is is responsible for both balancing the flying capacitor voltages from one hand and for output current regulation. Simulation results from the switched model of the converter under the proposed controller are presented. The results show that the system may undergo bifurcation phenomena and period doubling route to chaos when some system parameters are varied. One-dimensional bifurcation diagrams are computed and used to explore the possible dynamical behavior of the system. By using Floquet theory and Filippov method to derive the monodromy matrix, the bifurcation behavior observed in the converter is accurately predicted. Based on justified and realistic approximations of the system state variables waveforms, simple and accurate expressions for these steady-state values and the monodromy matrix are derived and validated. The simple expression of the steady-state operation and the monodromy matrix allow to analytically predict the onset of instability in the system and the stability region in the parametric space is determined. Numerical simulations from the exact switched model validate the theoretical predictions.

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