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Symmetry
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The Advances of Nonlinear Equations: Mathematical Models, Symmetry and Applications
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The Advances of Nonlinear Equations: Mathematical Models, Symmetry and Applications

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 15 October 2024 | Viewed by 4979

Special Issue Editor

Dr. Sunil Kumar

E-Mail Website
Guest Editor
Department of Mathematics, University Centre for Research and Development (UCRD), Chandigarh University, Mohali 140413, India
Interests: numerical methods; local convergence; semi-local convergence

Special Issue Information

Dear Colleagues,

Solving nonlinear equations for simple roots, multiple roots and systems is a significant task that involves many areas of science and engineering. Usually, iterative methods are used when direct methods fail to solve the problem. Iterative algorithms play a fundamental role in this regard. In this area of research, the work of many researchers has led to exponential growth in the last few years.

The main theme of this Special Issue is the development of iterative algorithms, convergence analysis, and the stability and application of new iterative schemes for solving nonlinear problems generated from real-life problems. This issue includes methods with and without memory, with derivatives or derivative-free, and an analysis of their convergence that can be local, semi-local, or global. This issue also deals with the complex dynamics of iterative methods, i.e., basin of attraction, and iterative methods to optimize nonlinear functions.

Please note that all submissions should be within the scope of the Symmetry journal.

Dr. Sunil Kumar
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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. Symmetry 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

  • optimal methods
  • iterative methods
  • order of convergence
  • nonlinear problem
  • computationally efficiency
  • derivative-free methods
  • multiple zeros
  • Frechet-derivative
  • Newton-like methods
  • local convergence
  • semi-local convergence
  • Traub–Steffensen method
  • divided differences
  • basins of attraction
  • optimimum function

Published Papers (6 papers)

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Research

Jump to: Review

16 pages, 657 KiB  
Article
A Globally Convergent Iterative Method for Matrix Sign Function and Its Application for Determining the Eigenvalues of a Matrix Pencil
by Munish Kansal, Vanita Sharma, Pallvi Sharma and Lorentz Jäntschi
Symmetry 2024, 16(4), 481; https://doi.org/10.3390/sym16040481 - 16 Apr 2024
Viewed by 387
Abstract
In this research article, we propose a new matrix iterative method with a convergence order of five for computing the sign of a complex matrix by examining the different patterns and symmetry of existing methods. Analysis of the convergence of the method was [...] Read more.
In this research article, we propose a new matrix iterative method with a convergence order of five for computing the sign of a complex matrix by examining the different patterns and symmetry of existing methods. Analysis of the convergence of the method was explored on a global scale, and attraction basins were demonstrated. In addition to this, the asymptotic stability of the scheme was explored.Then, an algorithm for determing thegeneralized eigenvalues for the case of regular matrix pencils was investigated using the matrix sign computation. We performed a series of numerical experiments using numerous matrices to confirm the usefulness and superiority of the proposed method. Full article
(This article belongs to the Special Issue The Advances of Nonlinear Equations: Mathematical Models, Symmetry and Applications)
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18 pages, 24247 KiB  
Article
Toward Enhanced Geological Analysis: A Novel Approach Based on Transmuted Semicircular Distribution
by Phani Yedlapalli, Gajula Naveen Venkata Kishore, Wadii Boulila, Anis Koubaa and Nabil Mlaiki
Symmetry 2023, 15(11), 2030; https://doi.org/10.3390/sym15112030 - 08 Nov 2023
Viewed by 653
Abstract
This paper introduces a novel semicircular distribution obtained by applying the quadratic rank transmutation map to the stereographic semicircular exponential distribution, referred to as the transmuted stereographic semicircular exponential distribution (TSSCED). This newly proposed distribution exhibits enhanced flexibility compared to the [...] Read more.
This paper introduces a novel semicircular distribution obtained by applying the quadratic rank transmutation map to the stereographic semicircular exponential distribution, referred to as the transmuted stereographic semicircular exponential distribution (TSSCED). This newly proposed distribution exhibits enhanced flexibility compared to the baseline stereographic semicircular exponential distribution (SSCEXP). We conduct a comprehensive analysis of the model’s properties and demonstrate its efficacy in data modeling through the application to a real dataset. Full article
(This article belongs to the Special Issue The Advances of Nonlinear Equations: Mathematical Models, Symmetry and Applications)
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11 pages, 1136 KiB  
Article
On a Fifth-Order Method for Multiple Roots of Nonlinear Equations
by Beny Neta
Symmetry 2023, 15(9), 1694; https://doi.org/10.3390/sym15091694 - 04 Sep 2023
Viewed by 665
Abstract
There are several measures for comparing methods for solving a single nonlinear equation. The first is the order of convergence, then the cost to achieve such rate. This cost is measured by counting the number of functions (and derivatives) evaluated at each step. [...] Read more.
There are several measures for comparing methods for solving a single nonlinear equation. The first is the order of convergence, then the cost to achieve such rate. This cost is measured by counting the number of functions (and derivatives) evaluated at each step. After that, efficiency is defined as a function of the order of convergence and cost. Lately, the idea of basin of attraction is used. This shows how far one can start and still converge to the root. It also shows the symmetry/asymmetry of the method. It was shown that even methods that show symmetry when solving polynomial equations are not so when solving nonpolynomial ones. We will see here that the Euler–Cauchy method (a member of the Laguerre family of methods for multiple roots) is best in the sense that the boundaries of the basins have no lobes. The symmetry in solving a polynomial equation having two roots at ±1 with any multiplicity is obvious. In fact, the Euler–Cauchy method converges very fast in this case. We compare one member of a family of fifth-order methods for multiple roots with several well-known lower-order and efficient methods. We will show using a basin of attraction that the fifth-order method cannot compete with most of those lower-order methods. Full article
(This article belongs to the Special Issue The Advances of Nonlinear Equations: Mathematical Models, Symmetry and Applications)
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18 pages, 848 KiB  
Article
Efficient Families of Multi-Point Iterative Methods and Their Self-Acceleration with Memory for Solving Nonlinear Equations
by G Thangkhenpau, Sunil Panday, Liviu C. Bolunduţ and Lorentz Jäntschi
Symmetry 2023, 15(8), 1546; https://doi.org/10.3390/sym15081546 - 06 Aug 2023
Cited by 7 | Viewed by 778
Abstract
In this paper, we have constructed new families of derivative-free three- and four-parametric methods with and without memory for finding the roots of nonlinear equations. Error analysis verifies that the without-memory methods are optimal as per Kung–Traub’s conjecture, with orders of convergence of [...] Read more.
In this paper, we have constructed new families of derivative-free three- and four-parametric methods with and without memory for finding the roots of nonlinear equations. Error analysis verifies that the without-memory methods are optimal as per Kung–Traub’s conjecture, with orders of convergence of 4 and 8, respectively. To further enhance their convergence capabilities, the with-memory methods incorporate accelerating parameters, elevating their convergence orders to 7.5311 and 15.5156, respectively, without introducing extra function evaluations. As such, they exhibit exceptional efficiency indices of 1.9601 and 1.9847, respectively, nearing the maximum efficiency index of 2. The convergence domains are also analysed using the basins of attraction, which exhibit symmetrical patterns and shed light on the fascinating interplay between symmetry, dynamic behaviour, the number of diverging points, and efficient root-finding methods for nonlinear equations. Numerical experiments and comparison with existing methods are carried out on some nonlinear functions, including real-world chemical engineering problems, to demonstrate the effectiveness of the new proposed methods and confirm the theoretical results. Notably, our numerical experiments reveal that the proposed methods outperform their existing counterparts, offering superior precision in computation. Full article
(This article belongs to the Special Issue The Advances of Nonlinear Equations: Mathematical Models, Symmetry and Applications)
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23 pages, 2165 KiB  
Article
Extension of King’s Iterative Scheme by Means of Memory for Nonlinear Equations
by Saima Akram, Maira Khalid, Moin-ud-Din Junjua, Shazia Altaf and Sunil Kumar
Symmetry 2023, 15(5), 1116; https://doi.org/10.3390/sym15051116 - 19 May 2023
Cited by 4 | Viewed by 1388
Abstract
We developed a new family of optimal eighth-order derivative-free iterative methods for finding simple roots of nonlinear equations based on King’s scheme and Lagrange interpolation. By incorporating four self-accelerating parameters and a weight function in a single variable, we extend the proposed family [...] Read more.
We developed a new family of optimal eighth-order derivative-free iterative methods for finding simple roots of nonlinear equations based on King’s scheme and Lagrange interpolation. By incorporating four self-accelerating parameters and a weight function in a single variable, we extend the proposed family to an efficient iterative scheme with memory. Without performing additional functional evaluations, the order of convergence is boosted from 8 to 15.51560, and the efficiency index is raised from 1.6817 to 1.9847. To compare the performance of the proposed and existing schemes, some real-world problems are selected, such as the eigenvalue problem, continuous stirred-tank reactor problem, and energy distribution for Planck’s radiation. The stability and regions of convergence of the proposed iterative schemes are investigated through graphical tools, such as 2D symmetric basins of attractions for the case of memory-based schemes and 3D stereographic projections in the case of schemes without memory. The stability analysis demonstrates that our newly developed schemes have wider symmetric regions of convergence than the existing schemes in their respective domains. Full article
(This article belongs to the Special Issue The Advances of Nonlinear Equations: Mathematical Models, Symmetry and Applications)
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Review

Jump to: Research

20 pages, 346 KiB  
Review
Ordered Patterns of (3+1)-Dimensional Hadronic Gauged Solitons in the Low-Energy Limit of Quantum Chromodynamics at a Finite Baryon Density, Their Magnetic Fields and Novel BPS Bounds
by Fabrizio Canfora, Evangelo Delgado and Luis Urrutia
Symmetry 2024, 16(5), 518; https://doi.org/10.3390/sym16050518 - 25 Apr 2024
Viewed by 199
Abstract
In this paper, we will review two analytical approaches to the construction of non-homogeneous Baryonic condensates in the low-energy limit of QCD in (3+1) dimensions. In both cases, the minimal coupling with the Maxwell U(1) gauge field can be taken [...] Read more.
In this paper, we will review two analytical approaches to the construction of non-homogeneous Baryonic condensates in the low-energy limit of QCD in (3+1) dimensions. In both cases, the minimal coupling with the Maxwell U(1) gauge field can be taken explicitly into account. The first approach (which is related to the generalization of the usual spherical hedgehog ansatz to situations without spherical symmetry at a finite Baryon density) allows for the construction of ordered arrays of Baryonic tubes and layers. When the minimal coupling of the Pions to the U(1) Maxwell gauge field is taken into account, one can show that the electromagnetic field generated by these inhomogeneous Baryonic condensates is of a force-free type (in which the electric and magnetic components have the same size). Thus, it is natural to wonder whether it is also possible to analytically describe magnetized hadronic condensates (namely, Hadronic distributions generating only a magnetic field). The idea of the second approach is to construct a novel BPS bound in the low-energy limit of QCD using the theory of the Hamilton–Jacobi equation. Such an approach allows us to derive a new topological bound which (unlike the usual one in the Skyrme model in terms of the Baryonic charge) can actually be saturated. The nicest example of this phenomenon is a BPS magnetized Baryonic layer. However, the topological charge appearing naturally in the BPS bound is a non-linear function of the Baryonic charge. Such an approach allows us to derive important physical quantities (which would be very difficult to compute with other methods), such as how much one should increase the magnetic flux in order to increase the Baryonic charge by one unit. The novel results of this work include an analysis of the extension of the Hamilton–Jacobi approach to the case in which Skyrme coupling is not negligible. We also discuss some relevant properties of the Dirac operator for quarks coupled to magnetized BPS layers. Full article
(This article belongs to the Special Issue The Advances of Nonlinear Equations: Mathematical Models, Symmetry and Applications)
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Displaying articles 1-6

Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

Title: Ordered patterns of (31)-dimensional Hadronic gauged solitons in the low-energy limit of QCD at finite Baryon density, their magnetic fields and novel BPS bounds+
Authors: Fabrizio Canfora; Evangelo Delgado; Luis Urrutia
Affiliation: Facultad de Ingeniería, Arquitectura y Diseño, Universidad San Sebastián, General Lagos 1163, Valdivia 5110693, Chile; Centro de Estudios Científicos (CECs), Avenida Arturo Prat 514, Valdivia, Chile
Abstract: In this paper we will review two analytic approaches to the construction of non-homogeneous Baryonic condensates in the low energy limit of QCD in (3+1) dimensions: in both cases, the minimal coupling with Maxwell U(1) gauge field can be taken explicitly into account. The first approach (which is related to the generalization of the usual spherical hedgehog ansatz to situations without spherical symmetry at finite Baryon density) allows the construction of ordered arrays of Baryonic tubes and layers. When the minimal coupling of the Pions to the U(1) Maxwell gauge field is taken into account, one can show that the electromagnetic field generated by these inhomogeneous Baryonic condensates is of force free type (in which the electric and magnetic components have the same size). Thus, it is natural to wonder whether it is possible to describe analytically also magnetized hadronic condensates (namely, Hadronic distributions generating only a magnetic field). The idea of the second approach is to construct a novel BPS bound in the low energy limit of QCD using the theory of Hamilton-Jacobi equation. Such an approach allows to derive a new topological bound which (unlike the usual one in the Skyrme model in terms of the Baryonic charge) can actually be saturated. The nicest example of this phenomenon is a BPS magnetized Baryonic layer. However, the topological charge appearing naturally in the BPS bound is a non-linear function of the Baryonic charge. Such an approach allows to derive important physical quantities (which would be very difficult to compute with other methods) such as how much one should increase the magnetic flux in order to increase the Baryonic charge by one unit. The novel results of this work include the analysis of the extension of the Hamilton-Jacobi approach to the case in which the Skyrme coupling is not negligible. We also discuss some relevant properties of the Dirac operator for the quarks coupled to magnetized BPS layers.

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