Special Issue "Symmetry in Ordinary and Partial Differential Equations and Applications"

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

Deadline for manuscript submissions: 30 June 2022 | Viewed by 11764

Special Issue Editor

Dr. Calogero Vetro
E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, I-90123 Palermo, Italy
Interests: difference equations; flow invariance; nonlinear regularity theory; ordinary; differential equations; partial differential equations; reduction methods; symmetry operators; weak symmetries
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

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 to 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. The methods of symmetrization are a key tool in obtaining a priori estimates of solutions to various classes of differential equations, provided that both the involved functions and the data of the problem admit some partial or fully symmetries on the framework space. In particular, comparison principles and method of moving planes up to a critical position, deserve further investigation to prove spherical or axial symmetry results for positive solutions.

This Special Issue aims to collect original and significant contributions dealing with both the theory and applications of differential equations. Also, this Special Issue may serve as a platform for the exchange of ideas between scientists of different disciplines interested in ordinary and partial differential equations and their applications.

Dr. Calogero Vetro
Guest Editor

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 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

  • difference equations
  • flow invariance
  • nonlinear regularity theory
  • ordinary differential equations
  • partial differential equations
  • reduction methods
  • symmetry operators
  • weak symmetries

Published Papers (15 papers)

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Article
Symmetries, Reductions and Exact Solutions of a Class of (2k + 2)th-Order Difference Equations with Variable Coefficients
Symmetry 2022, 14(7), 1290; https://doi.org/10.3390/sym14071290 - 21 Jun 2022
Viewed by 166
Abstract
We perform a Lie analysis of (2k+2)th-order difference equations and obtain k+1 non-trivial symmetries. We utilize these symmetries to obtain their exact solutions. Sufficient conditions for convergence of solutions are provided for some specific cases. [...] Read more.
We perform a Lie analysis of (2k+2)th-order difference equations and obtain k+1 non-trivial symmetries. We utilize these symmetries to obtain their exact solutions. Sufficient conditions for convergence of solutions are provided for some specific cases. We exemplify our theoretical analysis with some numerical examples. The results in this paper extend to some work in the recent literature. Full article
Article
On Nonlinear Biharmonic Problems on the Heisenberg Group
Symmetry 2022, 14(4), 705; https://doi.org/10.3390/sym14040705 - 31 Mar 2022
Viewed by 396
Abstract
We investigate the boundary value problem for biharmonic operators on the Heisenberg group. The inherent features of Hn make it an appropriate environment for studying symmetry rules and the interaction of analysis and geometry with manifolds. The goal of this paper is [...] Read more.
We investigate the boundary value problem for biharmonic operators on the Heisenberg group. The inherent features of Hn make it an appropriate environment for studying symmetry rules and the interaction of analysis and geometry with manifolds. The goal of this paper is to prove that a weak solution for a biharmonic operator on the Heisenberg group exists. Our key tools are a version of the Mountain Pass Theorem and the classical variational theory. This paper will be of interest to researchers who are working on biharmonic operators on Hn. Full article
Article
The Behavior and Structures of Solution of Fifth-Order Rational Recursive Sequence
Symmetry 2022, 14(4), 641; https://doi.org/10.3390/sym14040641 - 22 Mar 2022
Viewed by 374
Abstract
In this work, we aim to study some qualitative properties of higher order nonlinear difference equations. Specifically, we investigate local as well as global stability and boundedness of solutions of this equation. In addition, we will provide solutions to a number of special [...] Read more.
In this work, we aim to study some qualitative properties of higher order nonlinear difference equations. Specifically, we investigate local as well as global stability and boundedness of solutions of this equation. In addition, we will provide solutions to a number of special cases of the studied equation. Also, we present many numerical examples that support the results obtained. The importance of the results lies in completing the results in the literature, which aims to develop the theoretical side of the qualitative theory of difference equations. Full article
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Article
Positive Solutions of a Fractional Boundary Value Problem with Sequential Derivatives
Symmetry 2021, 13(8), 1489; https://doi.org/10.3390/sym13081489 - 13 Aug 2021
Cited by 2 | Viewed by 407
Abstract
We investigate the existence of positive solutions of a Riemann-Liouville fractional differential equation with sequential derivatives, a positive parameter and a nonnegative singular nonlinearity, supplemented with integral-multipoint boundary conditions which contain fractional derivatives of various orders and Riemann-Stieltjes integrals. Our general boundary conditions [...] Read more.
We investigate the existence of positive solutions of a Riemann-Liouville fractional differential equation with sequential derivatives, a positive parameter and a nonnegative singular nonlinearity, supplemented with integral-multipoint boundary conditions which contain fractional derivatives of various orders and Riemann-Stieltjes integrals. Our general boundary conditions cover some symmetry cases for the unknown function. In the proof of our main existence result, we use an application of the Krein-Rutman theorem and two theorems from the fixed point index theory. Full article
Article
Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type
Symmetry 2021, 13(5), 871; https://doi.org/10.3390/sym13050871 - 13 May 2021
Cited by 2 | Viewed by 523
Abstract
The article deals with nonlinear second-order evolutionary partial differential equations (PDEs) of the parabolic type with a reasonably general form. We consider the case of PDE degeneration when the unknown function vanishes. Similar equations in various forms arise in continuum mechanics to describe [...] Read more.
The article deals with nonlinear second-order evolutionary partial differential equations (PDEs) of the parabolic type with a reasonably general form. We consider the case of PDE degeneration when the unknown function vanishes. Similar equations in various forms arise in continuum mechanics to describe some diffusion and filtration processes as well as to model heat propagation in the case when the properties of the process depend significantly on the unknown function (concentration, temperature, etc.). One of the exciting and meaningful classes of solutions to these equations is diffusion (heat) waves, which describe the propagation of perturbations over a stationary (zero) background with a finite velocity. It is known that such effects are atypical for parabolic equations; they arise as a consequence of the degeneration mentioned above. We prove the existence theorem of piecewise analytical solutions of the considered type and construct exact solutions (ansatz). Their search reduces to the integration of Cauchy problems for second-order ODEs with a singularity in the term multiplying the highest derivative. In some special cases, the construction is brought to explicit formulas that allow us to study the properties of solutions. The case of the generalized porous medium equation turns out to be especially interesting as the constructed solution has the form of a soliton moving at a constant velocity. Full article
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Article
Analysis of an Electrical Circuit Using Two-Parameter Conformable Operator in the Caputo Sense
Symmetry 2021, 13(5), 771; https://doi.org/10.3390/sym13050771 - 29 Apr 2021
Cited by 1 | Viewed by 497
Abstract
The problem of voltage dynamics description in a circuit containing resistors, and at least two fractional order elements such as supercapacitors, supplied with constant voltage is addressed. A new operator called Conformable Derivative in the Caputo sense is used. A state solution is [...] Read more.
The problem of voltage dynamics description in a circuit containing resistors, and at least two fractional order elements such as supercapacitors, supplied with constant voltage is addressed. A new operator called Conformable Derivative in the Caputo sense is used. A state solution is proposed. The considered operator is a generalization of three derivative definitions: classical definition (integer order), Caputo fractional definition and the so-called Conformable Derivative (CFD) definition. The proposed solution based on a two-parameter Conformable Derivative in the Caputo sense is proven to be better than the classical approach or the one-parameter fractional definition. Theoretical considerations are verified experimentally. The cumulated matching error function is given and it reveals that the proposed CFD–Caputo method generates an almost two times lower error compared to the classical method. Full article
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Article
Toward a Wong–Zakai Approximation for Big Order Generators
Symmetry 2020, 12(11), 1893; https://doi.org/10.3390/sym12111893 - 18 Nov 2020
Viewed by 1191
Abstract
We give a new approximation with respect of the traditional parametrix method of the solution of a parabolic equation whose generator is of big order and under the Hoermander form. This generalizes to a higher order generator the traditional approximation of Stratonovitch diffusion [...] Read more.
We give a new approximation with respect of the traditional parametrix method of the solution of a parabolic equation whose generator is of big order and under the Hoermander form. This generalizes to a higher order generator the traditional approximation of Stratonovitch diffusion which put in relation random ordinary differential equation (the leading process is random and of finite energy. When a trajectory of it is chosen, the solution of the equation is defined) and stochastic differential equation (the leading process is random and only continuous and we cannot choose a path in the solution which is only almost surely defined). We consider simple operators where the computations can be fully performed. This approximation fits with the semi-group only and not for the full path measure in the case of a stochastic differential equation. Full article
Article
Multiquadrics without the Shape Parameter for Solving Partial Differential Equations
Symmetry 2020, 12(11), 1813; https://doi.org/10.3390/sym12111813 - 02 Nov 2020
Cited by 4 | Viewed by 790
Abstract
In this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional collocation method that assigns the RBF [...] Read more.
In this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional collocation method that assigns the RBF at each center point coinciding with an interior point, we separated the center points from the interior points, in which the center points were regarded as the fictitious sources collocated outside the domain. The interior, boundary, and source points were therefore collocated within, on, and outside the domain, respectively. Since the radial distance between the interior point and the source point was always greater than zero, the MQ and IMQ RBFs and their derivatives in the governing equation were smooth and globally infinitely differentiable. Accordingly, the shape parameter was no longer required in the MQ and IMQ RBFs. Numerical examples with the domain in symmetry and asymmetry are presented to verify the accuracy and robustness of the proposed method. The results demonstrated that the proposed method using MQ RBFs without the shape parameter acquires more accurate results than the conventional RBF collocation method with the optimum shape parameter. Additionally, it was found that the locations of the fictitious sources were not sensitive to the accuracy. Full article
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Article
Large Time Behavior for Inhomogeneous Damped Wave Equations with Nonlinear Memory
Symmetry 2020, 12(10), 1609; https://doi.org/10.3390/sym12101609 - 27 Sep 2020
Cited by 2 | Viewed by 634
Abstract
We investigate the large time behavior for the inhomogeneous damped wave equation with nonlinear memory [...] Read more.
We investigate the large time behavior for the inhomogeneous damped wave equation with nonlinear memory ϕtt(t,ω)Δϕ(t,ω)+ϕt(t,ω)=1Γ(1ρ)0t(tσ)ρ|ϕ(σ,ω)|qdσ+μ(ω),t>0, ωRN imposing the condition (ϕ(0,ω),ϕt(0,ω))=(ϕ0(ω),ϕ1(ω))inRN, where N1, q>1, 0<ρ<1, ϕiLloc1(RN), i=0,1, μLloc1(RN) and μ0. Namely, it is shown that, if ϕ0,ϕ10, μL1(RN) and RNμ(ω)dω>0, then for all q>1, the considered problem has no global weak solution. Full article
Article
On a Fractional in Time Nonlinear Schrödinger Equation with Dispersion Parameter and Absorption Coefficient
Symmetry 2020, 12(7), 1197; https://doi.org/10.3390/sym12071197 - 20 Jul 2020
Cited by 2 | Viewed by 723
Abstract
This paper is concerned with the nonexistence of global solutions to fractional in time nonlinear Schrödinger equations of the form [...] Read more.
This paper is concerned with the nonexistence of global solutions to fractional in time nonlinear Schrödinger equations of the form i α t α ω ( t , z ) + a 1 ( t ) Δ ω ( t , z ) + i α a 2 ( t ) ω ( t , z ) = ξ | ω ( t , z ) | p , ( t , z ) ( 0 , ) × R N , where N 1 , ξ C \ { 0 } and p > 1 , under suitable initial data. To establish our nonexistence theorem, we adopt the Pohozaev nonlinear capacity method, and consider the combined effects of absorption and dispersion terms. Further, we discuss in details some special cases of coefficient functions a 1 , a 2 L l o c 1 ( [ 0 , ) , R ) , and provide two illustrative examples. Full article
Article
Some Generalised Fixed Point Theorems Applied to Quantum Operations
Symmetry 2020, 12(5), 759; https://doi.org/10.3390/sym12050759 - 06 May 2020
Cited by 4 | Viewed by 1138
Abstract
In this paper, we consider an order-preserving mapping T on a complete partial b-metric space satisfying some contractive condition. We were able to show the existence and uniqueness of the fixed point of T. In the application aspect, the fidelity of [...] Read more.
In this paper, we consider an order-preserving mapping T on a complete partial b-metric space satisfying some contractive condition. We were able to show the existence and uniqueness of the fixed point of T. In the application aspect, the fidelity of quantum states was used to establish the existence of a fixed quantum state associated to an order-preserving quantum operation. The method we presented is an alternative in showing the existence of a fixed quantum state associated to quantum operations. Our method does not capitalise on the commutativity of the quantum effects with the fixed quantum state(s) (Luders’s compatibility criteria). The Luders’s compatibility criteria in higher finite dimensional spaces is rather difficult to check for any prospective fixed quantum state. Some part of our results cover the famous contractive fixed point results of Banach, Kannan and Chatterjea. Full article
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Article
Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain
Symmetry 2020, 12(3), 394; https://doi.org/10.3390/sym12030394 - 04 Mar 2020
Viewed by 843
Abstract
We study the large-time behavior of solutions to the nonlinear exterior problem [...] Read more.
We study the large-time behavior of solutions to the nonlinear exterior problem L u ( t , x ) = κ | u ( t , x ) | p , ( t , x ) ( 0 , ) × D c under the nonhomegeneous Neumann boundary condition u ν ( t , x ) = λ ( x ) , ( t , x ) ( 0 , ) × D , where L : = i t + Δ is the Schrödinger operator, D = B ( 0 , 1 ) is the open unit ball in R N , N 2 , D c = R N D , p > 1 , κ C , κ 0 , λ L 1 ( D , C ) is a nontrivial complex valued function, and ν is the outward unit normal vector on D , relative to D c . Namely, under a certain condition imposed on ( κ , λ ) , we show that if N 3 and p < p c , where p c = N N 2 , then the considered problem admits no global weak solutions. However, if N = 2 , then for all p > 1 , the problem admits no global weak solutions. The proof is based on the test function method introduced by Mitidieri and Pohozaev, and an adequate choice of the test function. Full article
Article
Lie Symmetry Analysis, Explicit Solutions and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation
Symmetry 2020, 12(1), 170; https://doi.org/10.3390/sym12010170 - 16 Jan 2020
Cited by 19 | Viewed by 1481
Abstract
In this paper, we investigate a spatially two-dimensional Burgers–Huxley equation that depicts the interaction between convection effects, diffusion transport, reaction gadget, nerve proliferation in neurophysics, as well as motion in liquid crystals. We have used the Lie symmetry method to study the vector [...] Read more.
In this paper, we investigate a spatially two-dimensional Burgers–Huxley equation that depicts the interaction between convection effects, diffusion transport, reaction gadget, nerve proliferation in neurophysics, as well as motion in liquid crystals. We have used the Lie symmetry method to study the vector fields, optimal systems of first order, symmetry reductions, and exact solutions. Furthermore, using the power series method, a set of series solutions are obtained. Finally, conservation laws are derived using optimal systems. Full article
Article
Impulsive Evolution Equations with Causal Operators
Symmetry 2020, 12(1), 48; https://doi.org/10.3390/sym12010048 - 25 Dec 2019
Viewed by 882
Abstract
In this paper, we establish sufficient conditions for the existence of mild solutions for certain impulsive evolution differential equations with causal operators in separable Banach spaces. We rely on the existence of mild solutions for the strongly continuous semigroups theory, the measure of [...] Read more.
In this paper, we establish sufficient conditions for the existence of mild solutions for certain impulsive evolution differential equations with causal operators in separable Banach spaces. We rely on the existence of mild solutions for the strongly continuous semigroups theory, the measure of noncompactness and the Schauder fixed point theorem. We consider the impulsive integro-differential evolutions equation and impulsive reaction diffusion equations (which could include symmetric kernels) as applications to illustrate our main results. Full article

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Comment
Comments on the Paper “Lie Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation”
Symmetry 2020, 12(6), 900; https://doi.org/10.3390/sym12060900 - 01 Jun 2020
Cited by 1 | Viewed by 740
Abstract
This comment is devoted to the paper “Lie Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation” (Symmery, 2020, vol.12, 170), in which several results are either incorrect, or incomplete, or misleading. Full article
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