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".

Deadline for manuscript submissions: 30 June 2021.

Special Issue Editor

Dr. Calogero Vetro
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 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.

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

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 papers will be 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 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 (9 papers)

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Open AccessFeature PaperArticle
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 521
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
Open AccessArticle
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 1 | Viewed by 389
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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Open AccessArticle
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
Viewed by 414
Abstract
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( [...] 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
Open AccessArticle
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 1 | Viewed by 446
Abstract
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 [...] 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
Open AccessArticle
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 1 | Viewed by 781
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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Open AccessArticle
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 600
Abstract
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 , ) × [...] 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
Open AccessArticle
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 8 | Viewed by 837
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
Open AccessArticle
Impulsive Evolution Equations with Causal Operators
Symmetry 2020, 12(1), 48; https://doi.org/10.3390/sym12010048 - 25 Dec 2019
Viewed by 601
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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Open AccessComment
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
Viewed by 435
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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