Special Issue "Symmetry in Nonlinear Equations: Mathematical Models, Methods and Applications"

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

Deadline for manuscript submissions: closed (31 October 2021) | Viewed by 21256

Special Issue Editors

1. Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy
2. Department of Mathematics, Faculty of Science, Hadhramout University, Hadhramout 50512, Yemen
Interests: qualitative theory; ordinary differential equations; functional differential equations; dynamical systems; mathematical modeling
Special Issues, Collections and Topics in MDPI journals
Department of Mathematics and Computer Science, Transilvania University of Brasov, 500093 Brasov, Romania
Interests: differential equations; partial differential equations; equations of evolution; integral equations; mixed initial-boundary value problems for PDE; termoelasticity; media with microstretch; environments goals; nonlinear problems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Neutral differential equations (NDEs) are differential equations with delays, where the delays can appear in both the state variables and their time derivatives.

Ordinary and partial differential equations are universally recognized as powerful tools to model and solve practical problems involving nonlinear phenomena. In particular, we mention physical processes as problems in elasticity theory, where we deal with composites made of two different materials with different hardening exponents.

Therefore, the theory of differential equations has been successfully applied to establish the existence and multiplicity of solutions of boundary value problems via direct methods, minimax theorems, variational methods, and topological methods. If possible, one looks towards solutions in special forms by using the symmetries of the driving equation. This also leads to the study of the difference counterparts of such equations to provide exact or approximate solutions. We mention the reduction methods for establishing exact solutions as solutions of lower-dimensional equations.

This Special Issue will accept high-quality papers with original research results in theoretical research, and recent progress in the study of applied problems in science and technology.

Dr. Omar Bazighifan
Prof. Dr. Marin Marin
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. 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

  • Symmetry operators
  • Mathematical modeling techniques
  • Qualitative properties of solutions
  • Oscillation theory
  • Delay differential equations
  • Ordinary differential/ difference equations
  • Partial/Functional equations
  • Fractional differential and integral calculus
  • Applications

Published Papers (14 papers)

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Research

Article
Stability Analysis of a Diffusive Three-Species Ecological System with Time Delays
Symmetry 2021, 13(11), 2217; https://doi.org/10.3390/sym13112217 - 19 Nov 2021
Cited by 5 | Viewed by 1227
Abstract
In this study, the dynamics of a diffusive Lotka–Volterra three-species system with delays were explored. By employing the Galerkin Method, which generates semi-analytical solutions, a partial differential equation system was approximated through mathematical modeling with delay differential equations. Steady-state curves and Hopf bifurcation [...] Read more.
In this study, the dynamics of a diffusive Lotka–Volterra three-species system with delays were explored. By employing the Galerkin Method, which generates semi-analytical solutions, a partial differential equation system was approximated through mathematical modeling with delay differential equations. Steady-state curves and Hopf bifurcation maps were created and discussed in detail. The effects of the growth rate of prey and the mortality rate of the predator and top predator on the system’s stability were demonstrated. Increase in the growth rate of prey destabilised the system, whilst increase in the mortality rate of predator and top predator stabilised it. The increase in the growth rate of prey likely allowed the occurrence of chaotic solutions in the system. Additionally, the effects of hunting and maturation delays of the species were examined. Small delay responses stabilised the system, whilst great delays destabilised it. Moreover, the effects of the diffusion coefficients of the species were investigated. Alteration of the diffusion coefficients rendered the system permanent or extinct. Full article
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Article
A Variety of Dynamic Steffensen-Type Inequalities on a General Time Scale
Symmetry 2021, 13(9), 1738; https://doi.org/10.3390/sym13091738 - 18 Sep 2021
Cited by 8 | Viewed by 931
Abstract
This work is motivated by the work of Josip Pečarić in 2013 and 1982 and the work of Srivastava in 2017. By the utilization of the diamond-α dynamic inequalities, which are defined as a linear mixture of the delta and nabla integrals, [...] Read more.
This work is motivated by the work of Josip Pečarić in 2013 and 1982 and the work of Srivastava in 2017. By the utilization of the diamond-α dynamic inequalities, which are defined as a linear mixture of the delta and nabla integrals, we present and prove very important generalized results of diamond-α Steffensen-type inequalities on a general time scale. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. Full article
Article
Hermite–Hadamard Type Inequalities Involving k-Fractional Operator for (h¯,m)-Convex Functions
Symmetry 2021, 13(9), 1686; https://doi.org/10.3390/sym13091686 - 13 Sep 2021
Cited by 32 | Viewed by 1686
Abstract
The principal motivation of this paper is to establish a new integral equality related to k-Riemann Liouville fractional operator. Employing this equality, we present several new inequalities for twice differentiable convex functions that are associated with Hermite–Hadamard integral inequality. Additionally, some novel [...] Read more.
The principal motivation of this paper is to establish a new integral equality related to k-Riemann Liouville fractional operator. Employing this equality, we present several new inequalities for twice differentiable convex functions that are associated with Hermite–Hadamard integral inequality. Additionally, some novel cases of the established results for different kinds of convex functions are derived. This fractional integral sums up Riemann–Liouville and Hermite–Hadamard’s inequality, which have a symmetric property. Scientific inequalities of this nature and, particularly, the methods included have applications in different fields in which symmetry plays a notable role. Finally, applications of q-digamma and q-polygamma special functions are presented. Full article
Article
On the Qualitative Behavior of Third-Order Differential Equations with a Neutral Term
Symmetry 2021, 13(7), 1287; https://doi.org/10.3390/sym13071287 - 17 Jul 2021
Cited by 4 | Viewed by 1049
Abstract
In this paper, we analyze the asymptotic behavior of solutions to a class of third-order neutral differential equations. Using different methods, we obtain some new results concerning the oscillation of this type of equation. Our new results complement related contributions to the subject. [...] Read more.
In this paper, we analyze the asymptotic behavior of solutions to a class of third-order neutral differential equations. Using different methods, we obtain some new results concerning the oscillation of this type of equation. Our new results complement related contributions to the subject. The symmetry plays a important and fundamental role in the study of oscillation of solutions to these equations. An example is presented in order to clarify the main results. Full article
Article
New Oscillation Criteria for Neutral Delay Differential Equations of Fourth-Order
Symmetry 2021, 13(7), 1277; https://doi.org/10.3390/sym13071277 - 16 Jul 2021
Cited by 2 | Viewed by 1120
Abstract
New oscillatory properties for the oscillation of solutions to a class of fourth-order delay differential equations with several deviating arguments are established, which extend and generalize related results in previous studies. Some oscillation results are established by using the Riccati technique under the [...] Read more.
New oscillatory properties for the oscillation of solutions to a class of fourth-order delay differential equations with several deviating arguments are established, which extend and generalize related results in previous studies. Some oscillation results are established by using the Riccati technique under the case of canonical coefficients. The symmetry plays an important and fundamental role in the study of the oscillation of solutions of the equations. Examples are given to prove the significance of the new theorems. Full article
Article
Abundant Traveling Wave and Numerical Solutions of Weakly Dispersive Long Waves Model
Symmetry 2021, 13(6), 1085; https://doi.org/10.3390/sym13061085 - 17 Jun 2021
Cited by 23 | Viewed by 1718
Abstract
In this article, plenty of wave solutions of the (2 + 1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony ((2 + 1)-D KP-BBM) model are constructed by employing two recent analytical schemes (a modified direct algebraic (MDA) method and modified Kudryashov (MK) method). From the point of view of [...] Read more.
In this article, plenty of wave solutions of the (2 + 1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony ((2 + 1)-D KP-BBM) model are constructed by employing two recent analytical schemes (a modified direct algebraic (MDA) method and modified Kudryashov (MK) method). From the point of view of group theory, the proposed analytical methods in our article are based on symmetry, and effectively solve those problems which actually possess explicit or implicit symmetry. This model is a vital model in shallow water phenomena where it demonstrates the wave surface propagating in both directions. The obtained analytical solutions are explained by plotting them through 3D, 2D, and contour sketches. These solutions’ accuracy is also tested by calculating the absolute error between them and evaluated numerical results by the Adomian decomposition (AD) method and variational iteration (VI) method. The considered numerical schemes were applied based on constructed initial and boundary conditions through the obtained analytical solutions via the MDA, and MK methods which show the synchronization between computational and numerical obtained solutions. This coincidence between the obtained solutions is explained through two-dimensional and distribution plots. The applied methods’ symmetry is shown through comparing their obtained results and showing the matching between both obtained solutions (analytical and numerical). Full article
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Article
Bright–Dark Soliton Waves’ Dynamics in Pseudo Spherical Surfaces through the Nonlinear Kaup–Kupershmidt Equation
Symmetry 2021, 13(6), 963; https://doi.org/10.3390/sym13060963 - 28 May 2021
Cited by 38 | Viewed by 2578
Abstract
The soliton waves’ physical behavior on the pseudo spherical surfaces is studied through the analytical solutions of the nonlinear (1+1)–dimensional Kaup–Kupershmidt (KK) equation. This model is named after Boris Abram Kupershmidt and David J. Kaup. This model has been used in [...] Read more.
The soliton waves’ physical behavior on the pseudo spherical surfaces is studied through the analytical solutions of the nonlinear (1+1)–dimensional Kaup–Kupershmidt (KK) equation. This model is named after Boris Abram Kupershmidt and David J. Kaup. This model has been used in various branches such as fluid dynamics, nonlinear optics, and plasma physics. The model’s computational solutions are obtained by employing two recent analytical methods. Additionally, the solutions’ accuracy is checked by comparing the analytical and approximate solutions. The soliton waves’ characterizations are illustrated by some sketches such as polar, spherical, contour, two, and three-dimensional plots. The paper’s novelty is shown by comparing our obtained solutions with those previously published of the considered model. Full article
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Article
An Oscillation Criterion of Nonlinear Differential Equations with Advanced Term
Symmetry 2021, 13(5), 843; https://doi.org/10.3390/sym13050843 - 10 May 2021
Cited by 8 | Viewed by 1177
Abstract
The aim of the present paper is to provide oscillation conditions for fourth-order damped differential equations with advanced term. By using the Riccati technique, some new oscillation criteria, which ensure that every solution oscillates, are established. In fact, the obtained results extend, unify [...] Read more.
The aim of the present paper is to provide oscillation conditions for fourth-order damped differential equations with advanced term. By using the Riccati technique, some new oscillation criteria, which ensure that every solution oscillates, are established. In fact, the obtained results extend, unify and correlate many of the existing results in the literature. Furthermore, two examples with specific parameter values are provided to confirm our results. Full article
Article
The Properties of Eigenvalues and Eigenfunctions for Nonlocal Sturm–Liouville Problems
Symmetry 2021, 13(5), 820; https://doi.org/10.3390/sym13050820 - 07 May 2021
Cited by 1 | Viewed by 1141
Abstract
The present paper is concerned with the spectral theory of nonlocal Sturm–Liouville eigenvalue problems on a finite interval. The continuity, differentiability and comparison results of eigenvalues with respect to the nonlocal potentials are studied, and the oscillation properties of eigenfunctions are investigated. The [...] Read more.
The present paper is concerned with the spectral theory of nonlocal Sturm–Liouville eigenvalue problems on a finite interval. The continuity, differentiability and comparison results of eigenvalues with respect to the nonlocal potentials are studied, and the oscillation properties of eigenfunctions are investigated. The comparison result of eigenvalues and the oscillation properties of eigenfunctions indicate that the spectral properties of nonlocal problems are very different from those of classical Sturm–Liouville problems. Some examples are given to explain this essential difference. Full article
Article
Feedback Control for a Diffusive and Delayed Brusselator Model: Semi-Analytical Solutions
Symmetry 2021, 13(4), 725; https://doi.org/10.3390/sym13040725 - 20 Apr 2021
Cited by 9 | Viewed by 1573
Abstract
This paper describes the stability and Hopf bifurcation analysis of the Brusselator system with delayed feedback control in the single domain of a reaction–diffusion cell. The Galerkin analytical technique is used to present a system equation composed of ordinary differential equations. The condition [...] Read more.
This paper describes the stability and Hopf bifurcation analysis of the Brusselator system with delayed feedback control in the single domain of a reaction–diffusion cell. The Galerkin analytical technique is used to present a system equation composed of ordinary differential equations. The condition able to determine the Hopf bifurcation point is found. Full maps of the Hopf bifurcation regions for the interacting chemical species are shown and discussed, indicating that the time delay, feedback control, and diffusion parameters can play a significant and important role in the stability dynamics of the two concentration reactants in the system. As a result, these parameters can be changed to destabilize the model. The results show that the Hopf bifurcation points for chemical control increase as the feedback parameters increase, whereas the Hopf bifurcation points decrease when the diffusion parameters increase. Bifurcation diagrams with examples of periodic oscillation and phase-plane maps are provided to confirm all the outcomes calculated in the model. The benefits and accuracy of this work show that there is excellent agreement between the analytical results and numerical simulation scheme for all the figures and examples that are illustrated. Full article
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Article
Second-Order Impulsive Delay Differential Systems: Necessary and Sufficient Conditions for Oscillatory or Asymptotic Behavior
Symmetry 2021, 13(4), 722; https://doi.org/10.3390/sym13040722 - 19 Apr 2021
Cited by 10 | Viewed by 1634
Abstract
In this work, we aimed to obtain sufficient and necessary conditions for the oscillatory or asymptotic behavior of an impulsive differential system. It is easy to notice that most works that study the oscillation are concerned only with sufficient conditions and without impulses, [...] Read more.
In this work, we aimed to obtain sufficient and necessary conditions for the oscillatory or asymptotic behavior of an impulsive differential system. It is easy to notice that most works that study the oscillation are concerned only with sufficient conditions and without impulses, so our results extend and complement previous results in the literature. Further, we provide two examples to illustrate the main results. Full article
Article
Oscillation Results of Emden–Fowler-Type Differential Equations
Symmetry 2021, 13(3), 410; https://doi.org/10.3390/sym13030410 - 03 Mar 2021
Viewed by 999
Abstract
In this article, we obtain oscillation conditions for second-order differential equation with neutral term. Our results extend, improve, and simplify some known results for neutral delay differential equations. Several effective and illustrative implementations are provided. Full article
Article
New Theorems for Oscillations to Differential Equations with Mixed Delays
Symmetry 2021, 13(3), 367; https://doi.org/10.3390/sym13030367 - 25 Feb 2021
Cited by 10 | Viewed by 1331
Abstract
The oscillation of differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of this equations. The purpose of this article is to establish new [...] Read more.
The oscillation of differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of this equations. The purpose of this article is to establish new oscillatory properties which describe both the necessary and sufficient conditions for a class of nonlinear second-order differential equations with neutral term and mixed delays of the form p(ι)w(ι)α+r(ι)uβ(ν(ι))=0,ιι0 where w(ι)=u(ι)+q(ι)u(ζ(ι)). Furthermore, examining the validity of the proposed criteria has been demonstrated via particular examples. Full article
Article
More Effective Conditions for Oscillatory Properties of Differential Equations
Symmetry 2021, 13(2), 278; https://doi.org/10.3390/sym13020278 - 06 Feb 2021
Cited by 6 | Viewed by 1260
Abstract
In this work, we present several oscillation criteria for higher-order nonlinear delay differential equation with middle term. Our approach is based on the use of Riccati substitution, the integral averaging technique and the comparison technique. The symmetry contributes to deciding the right way [...] Read more.
In this work, we present several oscillation criteria for higher-order nonlinear delay differential equation with middle term. Our approach is based on the use of Riccati substitution, the integral averaging technique and the comparison technique. The symmetry contributes to deciding the right way to study oscillation of solutions of this equations. Our results unify and improve some known results for differential equations with middle term. Some illustrative examples are provided. Full article
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