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Symmetry
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Symmetry and Asymmetry in Nonlinear Partial Differential Equations
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Symmetry and Asymmetry in Nonlinear Partial Differential Equations

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

Deadline for manuscript submissions: closed (31 July 2025) | Viewed by 7185

Special Issue Editors

Dr. Eugenia N. Petropoulou

E-Mail Website
Guest Editor
Department of Civil Engineering; University of Patras, 26504 Patras, Greece
Interests: differential equations; difference equations; special functions and orthogonal polynomials; functional equations and operator theory
Special Issues, Collections and Topics in MDPI journals
Dr. Michalis Xenos

E-Mail Website1 Website2
Guest Editor
Department of Mathematics, University of Ioannina, Ioannina, TK 45110 Ioannina, Greece
Interests: applied mathematics; fluid mechanics; computational fluid dynamics

Special Issue Information

Dear Colleagues,

It is well known that partial differential equations (PDEs) play a fundamental role in modelling and solving practical problems in a wide range of fields. This is accomplished either by theoretically studying a PDE and obtaining information on the qualitative characteristics of its solution or by obtaining an exact, approximate or numerical solution for it. In both cases, conservation laws and the symmetric properties of the PDE under consideration often prove to be extremely useful. Symmetry may also be taken into consideration with respect to the domain geometry or the boundary conditions that accompany a PDE. Moreover, studying the symmetry or asymmetry of a PDE and its solution may provide further insight into the application that this PDE describes.

The main aim of this Special Issue is to collect a variety of papers on nonlinear partial differential equations in which the concepts of symmetry/asymmetry are utilised or studied. All types of methods (theoretical, approximate and numerical) are welcome.

Dr. Eugenia N. Petropoulou
Dr. Michalis Xenos
Guest Editors

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

  • symmetry
  • asymmetry
  • partial differential equations
  • numerical methods
  • approximate methods
  • exact solutions

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Published Papers (7 papers)

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Research

19 pages, 1006 KB  
Article
The Swinging Sticks Pendulum: Small Perturbations Analysis
by Yundong Li, Rong Tang, Bikash Kumar Das, Marcelo F. Ciappina and Sergio Elaskar
Symmetry 2025, 17(9), 1467; https://doi.org/10.3390/sym17091467 - 5 Sep 2025
Abstract
The swinging sticks pendulum is an intriguing physical system that exemplifies the intersection of Lagrangian mechanics and chaos theory. It consists of a series of slender, interconnected metal rods, each with a counterweighted end that introduces an asymmetrical mass distribution. The rods are [...] Read more.
The swinging sticks pendulum is an intriguing physical system that exemplifies the intersection of Lagrangian mechanics and chaos theory. It consists of a series of slender, interconnected metal rods, each with a counterweighted end that introduces an asymmetrical mass distribution. The rods are arranged to pivot freely about their attachment points, enabling both rotational and translational motion. Unlike a simple pendulum, this system exhibits complex and chaotic behavior due to the interplay between its degrees of freedom. The Lagrangian formalism provides a robust framework for modeling the system’s dynamics, incorporating both rotational and translational components. The equations of motion are derived from the Euler–Lagrange equations and lack closed-form analytical solutions, necessitating the use of numerical methods. In this work, we employ the Bulirsch–Stoer method, a high-accuracy extrapolation technique based on the modified midpoint method, to solve the equations numerically. The system possesses four fixed points, each one associated with a different level of energy. The fixed point with the lowest energy level is a center, around which small perturbations are studied. The other three fixed points are unstable. The maximum energy used for the perturbations is 0.001% larger than the lowest equilibrium energy. When the system’s total energy is low, nonlinear terms in the equations can be neglected, allowing for a linearized treatment based on small-angle approximations. Under these conditions, the pendulum oscillates with small amplitudes around a stable equilibrium point. The resulting motion is analyzed using tools from nonlinear dynamics and Fourier analysis. Several trajectories are generated and examined to reveal frequency interactions and the emergence of complex dynamical behavior. When a small initial perturbation is applied to one rod, its motion is characterized by a single frequency with significantly greater amplitude and angular velocity compared to the second rod. In contrast, the second rod displayed dynamics that involved two frequencies. The present study, to the best of our knowledge, is the first attempt to describe the dynamical behavior of this pendulum. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry in Nonlinear Partial Differential Equations)
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16 pages, 1333 KB  
Article
The Role of Hidden Symmetry in Inertial Instability Dynamics
by Diana-Corina Bostan, Adrian Timofte, Florin Marian Nedeff, Valentin Nedeff, Mirela Panaite-Lehăduş and Maricel Agop
Symmetry 2025, 17(7), 994; https://doi.org/10.3390/sym17070994 - 24 Jun 2025
Viewed by 278
Abstract
Inertial instability is a key process in the dynamics of rotating and stratified fluids, which arises when the absolute vorticity of the flow becomes negative. This study explored the nonlinear behavior of inertial instability by incorporating a hidden symmetry into the equations of [...] Read more.
Inertial instability is a key process in the dynamics of rotating and stratified fluids, which arises when the absolute vorticity of the flow becomes negative. This study explored the nonlinear behavior of inertial instability by incorporating a hidden symmetry into the equations of motion governing atmospheric dynamics. The atmosphere was modeled as a complex system composed of interacting structural elements, each capable of oscillatory motion influenced by planetary rotation and geostrophic shear. By applying a symmetry-based framework rooted in projective geometry and Riccati-type transformations, we show that synchronization and structural coherence can emerge spontaneously, independent of external forcing. This hidden symmetry leads to rich dynamical behavior, including phase coupling, quasi-periodicity, and bifurcations. Our results suggest that inertial instability, beyond its classical linear interpretation, may play a significant role in organizing large-scale atmospheric patterns through internal geometric constraints. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry in Nonlinear Partial Differential Equations)
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18 pages, 283 KB  
Article
Generalized Logistic Neural Networks in Positive Linear Framework
by George A. Anastassiou
Symmetry 2025, 17(5), 746; https://doi.org/10.3390/sym17050746 - 13 May 2025
Viewed by 535
Abstract
Essential neural-network operators are interpreted as positive linear operators, and the related general theory applies to them. These operators are induced by a symmetrized density function deriving from the parametrized and deformed A-generalized logistic activation function. We are acting on the space [...] Read more.
Essential neural-network operators are interpreted as positive linear operators, and the related general theory applies to them. These operators are induced by a symmetrized density function deriving from the parametrized and deformed A-generalized logistic activation function. We are acting on the space of continuous functions on a compact interval of real line to the reals. We quantitatively study the rate of convergence of these neural -network operators to the unit operator. Our inequalities involve the modulus of continuity of the function under approximation or its derivative. We produce uniform and Lp, p≥1 approximation results via these inequalities. The convexity of functions is also used to derive more refined results. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry in Nonlinear Partial Differential Equations)
43 pages, 1877 KB  
Article
Construction of General Types of Fuzzy Implications Produced by Comparing Different t-Conorms: An Application Case Using Meteorological Data
by Athina Daniilidou, Avrilia Konguetsof and Basil Papadopoulos
Symmetry 2024, 16(12), 1633; https://doi.org/10.3390/sym16121633 - 9 Dec 2024
Viewed by 1313
Abstract
The objective of this paper is to compare a fuzzy implication produced by t-conorm probor with three other fuzzy implications constructed by t-conorms max, Einstein, and Lukasiewicz. Firstly, in methodology, six pairs of combinations of five t-conorm comparisons are performed in order to [...] Read more.
The objective of this paper is to compare a fuzzy implication produced by t-conorm probor with three other fuzzy implications constructed by t-conorms max, Einstein, and Lukasiewicz. Firstly, in methodology, six pairs of combinations of five t-conorm comparisons are performed in order to find the ranking order of five fuzzy implications. Moreover, the evaluation and calculation of the four fuzzy implications (probor, max, Einstein, and Lukasiewicz) are made using meteorological data, fuzzifying the crisp values of temperature and humidity, constructing four membership degree functions, and inserting as inputs the membership degrees of meteorological variables into the two variables of the fuzzy implications. Finally, extensive tests are made so as to find which membership degree function and which fuzzy implication receives the best and the worst results. The key findings are that the application of isosceles trapezium to the fuzzy implications of Probor and Einstein gives the best values, while fuzzy implication Lukasiewicz, although it was found to be first in the ranking order, is rejected due to unreliable results. As a result, the crucial role of these implications lies in the fact that they are non-symmetrical, i.e., there is a clear difference between the cause and the causal. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry in Nonlinear Partial Differential Equations)
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12 pages, 297 KB  
Article
Improved Fractional Differences with Kernels of Delta Mittag–Leffler and Exponential Functions
by Miguel Vivas-Cortez, Pshtiwan Othman Mohammed, Juan L. G. Guirao, Majeed A. Yousif, Ibrahim S. Ibrahim and Nejmeddine Chorfi
Symmetry 2024, 16(12), 1562; https://doi.org/10.3390/sym16121562 - 21 Nov 2024
Cited by 4 | Viewed by 1086
Abstract
Special functions have been widely used in fractional calculus, particularly for addressing the symmetric behavior of the function. This paper provides improved delta Mittag–Leffler and exponential functions to establish new types of fractional difference operators in the setting of Riemann–Liouville and Liouville–Caputo. We [...] Read more.
Special functions have been widely used in fractional calculus, particularly for addressing the symmetric behavior of the function. This paper provides improved delta Mittag–Leffler and exponential functions to establish new types of fractional difference operators in the setting of Riemann–Liouville and Liouville–Caputo. We give some properties of these discrete functions and use them as the kernel of the new fractional operators. In detail, we propose the construction of the new fractional sums and differences. We also find the Laplace transform of them. Finally, the relationship between the Riemann–Liouville and Liouville–Caputo operators are examined to verify the feasibility and effectiveness of the new fractional operators. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry in Nonlinear Partial Differential Equations)
21 pages, 4027 KB  
Article
Closed-Form Exact Solution for Free Vibration Analysis of Symmetric Functionally Graded Beams
by Lorenzo Ledda, Annalisa Greco, Ilaria Fiore and Ivo Caliò
Symmetry 2024, 16(9), 1206; https://doi.org/10.3390/sym16091206 - 13 Sep 2024
Cited by 2 | Viewed by 1747
Abstract
The dynamic stiffness method is developed to analyze the natural vibration characteristics of functionally graded beams, where material properties change continuously across the beam thickness following a symmetric law distribution. The governing equations of motion and associated natural boundary conditions for free vibration [...] Read more.
The dynamic stiffness method is developed to analyze the natural vibration characteristics of functionally graded beams, where material properties change continuously across the beam thickness following a symmetric law distribution. The governing equations of motion and associated natural boundary conditions for free vibration analysis are derived using Hamilton’s principle and closed-form exact solutions are obtained for both Euler–Bernoulli and Timoshenko beam models. The dynamic stiffness matrix, which governs the relationship between force and displacements at the beam ends, is determined. Using the Wittrick–Williams algorithm, the dynamic stiffness matrix is employed to compute natural frequencies and mode shapes. The proposed procedure is validated by comparing the obtained frequencies with those given by approximated well-known formulas. Finally, a parametric investigation is conducted by varying the geometry of the structure and the characteristic mechanical parameters of the functionally graded material. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry in Nonlinear Partial Differential Equations)
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17 pages, 281 KB  
Article
Three-Dimensional Lorentz-Invariant Velocities
by James M. Hill
Symmetry 2024, 16(9), 1133; https://doi.org/10.3390/sym16091133 - 2 Sep 2024
Cited by 2 | Viewed by 1369
Abstract
Lorentz invariance underlies special relativity, and the energy formula and relative velocity formula are well known to be invariant under a Lorentz transformation. Here, we determine the functional forms in terms of four arbitrary functions for those three dimensional velocity fields that are [...] Read more.
Lorentz invariance underlies special relativity, and the energy formula and relative velocity formula are well known to be invariant under a Lorentz transformation. Here, we determine the functional forms in terms of four arbitrary functions for those three dimensional velocity fields that are automatically invariant under the most general fully three-dimensional Lorentz transformation. For general three-dimensional motion, using rectangular Cartesian coordinates (x,y,z), we determine the first-order partial differential equations for the three velocity components u(x,y,z,t), v(x,y,z,t) and w(x,y,z,t) in the x−, y− and z−directions respectively. These partial differential equations and the associated partial differential relations connecting energy and momentum are fully compatible with the Lorentz-invariant energy–momentum relations and appear not to have been given previously in the literature. We determine the spatial and temporal dependence of the functional forms for those three-dimensional velocity fields that are automatically invariant under three-dimensional Lorentz transformations. An interesting special case gives rise to families of particle paths for which the magnitude of the velocity is the speed of light. This is indicative of the abundant possibilities existing in the “fast lane”. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry in Nonlinear Partial Differential Equations)
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