Symmetry

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15 pages, 343 KiB

Open AccessArticle

New Results Regarding Positive Periodic Solutions of Generalized Leslie–Gower-Type Population Models

by Axiu Shu, Xiaoliang Li and Bo Du

Symmetry 2024, 16(10), 1399; https://doi.org/10.3390/sym16101399 - 21 Oct 2024

Viewed by 855

Abstract

In this paper, we focus on the existence of positive periodic solutions of generalized Leslie–Gower-type population models. Using the topological degree, we provide sufficient conditions to demonstrate the existence of positive periodic solutions to the considered models. It is interesting that the positive [...] Read more.

In this paper, we focus on the existence of positive periodic solutions of generalized Leslie–Gower-type population models. Using the topological degree, we provide sufficient conditions to demonstrate the existence of positive periodic solutions to the considered models. It is interesting that the positive periodic solutions in this paper are general positive functions, not e exponential functions, which generalizes and improves the existing results. We note that due to the symmetrical property of periodic solutions, the results of this paper provide a deeper understanding of the periodic behavior of biological populations. Two numerical examples show the effectiveness of our main results. Full article

(This article belongs to the Special Issue The Advances of Nonlinear Equations: Mathematical Models, Symmetry and Applications)

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21 pages, 1081 KiB

Open AccessArticle

Comparative Study of Crossover Mathematical Model of Breast Cancer Based on Ψ-Caputo Derivative and Mittag-Leffler Laws: Numerical Treatments

by Nasser H. Sweilam, Seham M. Al-Mekhlafi, Waleed S. Abdel Kareem and Ghader Alqurishi

Symmetry 2024, 16(9), 1172; https://doi.org/10.3390/sym16091172 - 6 Sep 2024

Viewed by 948

Abstract

Two novel crossover models for breast cancer that incorporate

Ψ

-Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion and the crossover model for breast cancer that incorporates Atangana–Baleanu Caputo fractal variable-order fractional [...] Read more.

Two novel crossover models for breast cancer that incorporate

Ψ

-Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion and the crossover model for breast cancer that incorporates Atangana–Baleanu Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion are presented here, where we used a simple nonstandard kernel function

Ψ (t)

in the first model and a non-singular kernel in the second model. Moreover, we evaluated our models using actual statistics from Saudi Arabia. To ensure consistency with the physical model problem, the symmetry parameter

ζ

is introduced. We can obtain the fractal variable-order fractional Caputo and Caputo–Katugampola derivatives as special cases from the proposed

Ψ

-Caputo derivative. The crossover dynamics models define three alternative models: fractal variable-order fractional model, fractal fractional-order model, and variable-order fractional stochastic model over three-time intervals. The stability of the proposed model is analyzed. The

Ψ

-nonstandard finite-difference method is designed to solve fractal variable-order fractional and fractal fractional models, and the Toufik–Atangana method is used to solve the second crossover model with the non-singular kernel. Also, the nonstandard modified Euler–Maruyama method is used to study the variable-order fractional stochastic model. Numerous numerical tests and comparisons with real data were conducted to validate the methods’ efficacy and support the theoretical conclusions. Full article

(This article belongs to the Special Issue The Advances of Nonlinear Equations: Mathematical Models, Symmetry and Applications)

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16 pages, 657 KiB

Open AccessArticle

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 935

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

Open AccessArticle

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 - 8 Nov 2023

Cited by 1 | Viewed by 995

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

Open AccessArticle

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 - 4 Sep 2023

Cited by 2 | Viewed by 965

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

\pm 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

Open AccessArticle

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 - 6 Aug 2023

Cited by 10 | Viewed by 1197

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

Open AccessArticle

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 1800

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

Open AccessReview

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

Cited by 1 | Viewed by 796

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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