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Axioms
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Numerical Methods for Solving Nonlinear Equations and Systems: Convergence and...
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Numerical Methods for Solving Nonlinear Equations and Systems: Convergence and Stability

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (30 October 2019) | Viewed by 28568

Special Issue Editors

Prof. Dr. Juan Ramón Torregrosa Sánchez

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Guest Editor
Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, 46022 València, Spain
Interests: iterative processes; matrix analysis; numerical analysis
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Alicia Cordero

E-Mail Website
Guest Editor
School of Telecommunications Engineering, Universitat Politècnica de València, 46022 Valencia, Spain
Interests: numerical analysis; iterative methods; nonlinear problems; discrete dynamics, real and complex
Special Issues, Collections and Topics in MDPI journals
Dr. Ramandeep Behl

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Guest Editor
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Dr. Fiza Zafar

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Guest Editor
Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, 60800 Multan, Pakistan

Special Issue Information

Dear Colleagues,

Solving nonlinear equations and systems is a non-trivial task that involves many areas of science and technology. Usually it is not affordable in a direct way and iterative algorithms play a fundamental role in solving nonlinear equations and systems. This is an area of research that has experienced exponential growth in the last years.

The main theme of this Special Issue is the design, analysis of convergence and stability and application to practical problems of new iterative schemes for solving nonlinear problems. This includes methods with and without memory, with derivatives or derivative-free, and the real or complex dynamics associated with them and an analysis of their convergence that can be local, semilocal or global. High quality submissions on related topics are also welcome.

Prof. Dr. Juan R. Torregrosa
Prof. Dr. Alicia Cordero
Dr. Ramandeep Behl
Dr. Fiza Zafar
Guest Editors

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Nonlinear problems
  • Iterative methods
  • Convergence
  • Efficiency
  • Chaotic behavior
  • Complex or real dynamics

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Related Special Issue

Published Papers (7 papers)

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Research

16 pages, 2626 KiB  
Article
Numerical Solutions of Coupled Burgers’ Equations
by Hijaz Ahmad, Tufail A. Khan and Clemente Cesarano
Axioms 2019, 8(4), 119; https://doi.org/10.3390/axioms8040119 - 23 Oct 2019
Cited by 64 | Viewed by 6206
Abstract
In this article, two new modified variational iteration algorithms are investigated for the numerical solution of coupled Burgers’ equations. These modifications are made with the help of auxiliary parameters to speed up the convergence rate of the series solutions. Three numerical test problems [...] Read more.
In this article, two new modified variational iteration algorithms are investigated for the numerical solution of coupled Burgers’ equations. These modifications are made with the help of auxiliary parameters to speed up the convergence rate of the series solutions. Three numerical test problems are given to judge the behavior of the modified algorithms, and error norms are used to evaluate the accuracy of the method. Numerical simulations are carried out for different values of parameters. The results are also compared with the existing methods in the literature. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Nonlinear Equations and Systems: Convergence and Stability)
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16 pages, 1956 KiB  
Article
Mathematical and Numerical Modeling of On-Threshold Modes of 2-D Microcavity Lasers with Piercing Holes
by Alexander O. Spiridonov, Evgenii M. Karchevskii and Alexander I. Nosich
Axioms 2019, 8(3), 101; https://doi.org/10.3390/axioms8030101 - 1 Sep 2019
Cited by 16 | Viewed by 3268
Abstract
This study considers the mathematical analysis framework aimed at the adequate description of the modes of lasers on the threshold of non-attenuated in time light emission. The lasers are viewed as open dielectric resonators equipped with active regions, filled in with gain material. [...] Read more.
This study considers the mathematical analysis framework aimed at the adequate description of the modes of lasers on the threshold of non-attenuated in time light emission. The lasers are viewed as open dielectric resonators equipped with active regions, filled in with gain material. We introduce a generalized complex-frequency eigenvalue problem for such cavities and prove important properties of the spectrum of its eigensolutions. This involves reduction of the problem to the set of the Muller boundary integral equations and their discretization with the Nystrom technique. Embedded into this general framework is the application-oriented lasing eigenvalue problem, where the real emission frequencies and the threshold gain values together form two-component eigenvalues. As an example of on-threshold mode study, we present numerical results related to the two-dimensional laser shaped as an active equilateral triangle with a round piercing hole. It is demonstrated that the threshold of lasing and the directivity of light emission, for each mode, can be efficiently manipulated with the aid of the size and, especially, the placement of the piercing hole, while the frequency of emission remains largely intact. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Nonlinear Equations and Systems: Convergence and Stability)
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14 pages, 1334 KiB  
Article
Generating Root-Finder Iterative Methods of Second Order: Convergence and Stability
by Francisco I. Chicharro, Alicia Cordero, Neus Garrido and Juan R. Torregrosa
Axioms 2019, 8(2), 55; https://doi.org/10.3390/axioms8020055 - 6 May 2019
Cited by 11 | Viewed by 3631
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Nonlinear Equations and Systems: Convergence and Stability)
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15 pages, 1336 KiB  
Article
Stability Anomalies of Some Jacobian-Free Iterative Methods of High Order of Convergence
by Alicia Cordero, Javier G. Maimó, Juan R. Torregrosa and María P. Vassileva
Axioms 2019, 8(2), 51; https://doi.org/10.3390/axioms8020051 - 25 Apr 2019
Viewed by 3007
Abstract
In this manuscript, we design two classes of parametric iterative schemes to solve nonlinear problems that do not need to evaluate Jacobian matrices and need to solve three linear systems per iteration with the same divided difference operator as the coefficient matrix. The [...] Read more.
In this manuscript, we design two classes of parametric iterative schemes to solve nonlinear problems that do not need to evaluate Jacobian matrices and need to solve three linear systems per iteration with the same divided difference operator as the coefficient matrix. The stability performance of the classes is analyzed on a quadratic polynomial system, and it is shown that for many values of the parameter, only convergence to the roots of the problem exists. Finally, we check the performance of these methods on some test problems to confirm the theoretical results. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Nonlinear Equations and Systems: Convergence and Stability)
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17 pages, 365 KiB  
Article
Efficient Two-Step Fifth-Order and Its Higher-Order Algorithms for Solving Nonlinear Systems with Applications
by Parimala Sivakumar and Jayakumar Jayaraman
Axioms 2019, 8(2), 37; https://doi.org/10.3390/axioms8020037 - 1 Apr 2019
Cited by 5 | Viewed by 2894
Abstract
This manuscript presents a new two-step weighted Newton’s algorithm with convergence order five for approximating solutions of system of nonlinear equations. This algorithm needs evaluation of two vector functions and two Frechet derivatives per iteration. Furthermore, it is improved into a general multi-step [...] Read more.
This manuscript presents a new two-step weighted Newton’s algorithm with convergence order five for approximating solutions of system of nonlinear equations. This algorithm needs evaluation of two vector functions and two Frechet derivatives per iteration. Furthermore, it is improved into a general multi-step algorithm with one more vector function evaluation per step, with convergence order 3 k + 5 , k ≥ 1 . Error analysis providing order of convergence of the algorithms and their computational efficiency are discussed based on the computational cost. Numerical implementation through some test problems are included, and comparison with well-known equivalent algorithms are presented. To verify the applicability of the proposed algorithms, we have implemented them on 1-D and 2-D Bratu problems. The presented algorithms perform better than many existing algorithms and are equivalent to a few available algorithms. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Nonlinear Equations and Systems: Convergence and Stability)
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14 pages, 40736 KiB  
Article
Complex Soliton Solutions to the Gilson–Pickering Model
by Haci Mehmet Baskonus
Axioms 2019, 8(1), 18; https://doi.org/10.3390/axioms8010018 - 1 Feb 2019
Cited by 60 | Viewed by 5102
Abstract
In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson–Pickering model is applied. A set of new complex soliton solutions to the Gilson–Pickering model are successfully constructed. In addition, 2D and 3D [...] Read more.
In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson–Pickering model is applied. A set of new complex soliton solutions to the Gilson–Pickering model are successfully constructed. In addition, 2D and 3D graphs and contour simulations to the complex soliton solutions are plotted with the help of computational programs. Finally, at the end of the manuscript a conclusion about new complex soliton solutions is given. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Nonlinear Equations and Systems: Convergence and Stability)
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11 pages, 239 KiB  
Article
Interval Methods with Fifth Order of Convergence for Solving Nonlinear Scalar Equations
by Ivan G. Ivanov and Tonya Mateva
Axioms 2019, 8(1), 15; https://doi.org/10.3390/axioms8010015 - 31 Jan 2019
Cited by 3 | Viewed by 3144
Abstract
In this paper, based on Kou’s classical iterative methods with fifth-order of convergence, we propose new interval iterative methods for computing a real root of the nonlinear scalar equations. Some numerical experiments have executed with the program INTLAB in order to confirm the [...] Read more.
In this paper, based on Kou’s classical iterative methods with fifth-order of convergence, we propose new interval iterative methods for computing a real root of the nonlinear scalar equations. Some numerical experiments have executed with the program INTLAB in order to confirm the theoretical results. The computational results have described and compared with Newton’s interval method, Ostrowski’s interval method and Ostrowski’s modified interval method. We conclude that the proposed interval schemes are effective and they are comparable to the classical interval methods. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Nonlinear Equations and Systems: Convergence and Stability)
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