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".

Deadline for manuscript submissions: 29 March 2024 | Viewed by 37218

Special Issue Editor

Associate Professor, Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 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

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

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

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Published Papers (20 papers)

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Editorial

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3 pages, 161 KiB  
Editorial
Symmetry in Ordinary and Partial Differential Equations and Applications
by Calogero Vetro
Symmetry 2023, 15(7), 1425; https://doi.org/10.3390/sym15071425 - 15 Jul 2023
Viewed by 665
Abstract
This Special Issue of the journal Symmetry is dedicated to recent progress in the field of nonlinear differential problems [...] Full article

Research

Jump to: Editorial, Other

20 pages, 339 KiB  
Article
How to Pose Problems on Periodicity and Teaching with Problem Posing
by Guoqiang Dang, Yufeng Guo and Kai Li
Symmetry 2023, 15(9), 1716; https://doi.org/10.3390/sym15091716 - 07 Sep 2023
Viewed by 747
Abstract
Research on how to pose good problems in mathematical science is rarely touched. Inspired by Kilpatrick’s “Where do good problems come from?”, the current research investigates the problem of the specific problem posed by mathematicians in mathematical sciences. We select a recent mathematical [...] Read more.
Research on how to pose good problems in mathematical science is rarely touched. Inspired by Kilpatrick’s “Where do good problems come from?”, the current research investigates the problem of the specific problem posed by mathematicians in mathematical sciences. We select a recent mathematical conjecture of Yang related to periodic functions in the field of functions of one complex variable. These problems are extended to complex differential equations, difference equations, differential-difference equations, etc. Through mathematical analysis, we try to reproduce the effective strategies or techniques used by mathematicians in posing these new problems. The results show that mathematicians often use generalization, constraint manipulation, and specialization when they pose new mathematical problems. Conversely, goal manipulation and targeting a particular solution are rarely used. The results of the study may have a potential impact and promotion on implementing problem-posing teaching in primary and secondary schools. Accordingly, teachers and students can be encouraged to think like mathematicians, posing better problems and learning mathematics better. Then, we give some examples of mathematical teaching at the high school level using problem-posing strategies, which are frequently employed by mathematicians or mathematical researchers, and demonstrate how these strategies work. Therefore, this is a pioneering research that integrates the mathematical problem posing by mathematicians and the mathematical problem posing by elementary and secondary school math teachers and students. In addition, applying constraint manipulation and analogical reasoning, we present four unsolved mathematical problems, including three problems of complex difference-related periodic functions and one problem with complex difference equations. Full article
12 pages, 298 KiB  
Article
Positive Solutions for a Fractional Differential Equation with Sequential Derivatives and Nonlocal Boundary Conditions
by Alexandru Tudorache and Rodica Luca
Symmetry 2022, 14(9), 1779; https://doi.org/10.3390/sym14091779 - 26 Aug 2022
Cited by 5 | Viewed by 1081
Abstract
We study the existence of positive solutions for a Riemann–Liouville fractional differential equation with sequential derivatives, a positive parameter and a sign-changing singular nonlinearity, subject to nonlocal boundary conditions containing varied fractional derivatives and general Riemann–Stieltjes integrals. We also present the associated Green [...] Read more.
We study the existence of positive solutions for a Riemann–Liouville fractional differential equation with sequential derivatives, a positive parameter and a sign-changing singular nonlinearity, subject to nonlocal boundary conditions containing varied fractional derivatives and general Riemann–Stieltjes integrals. We also present the associated Green functions and some of their properties. In the proof of the main results, we apply the Guo–Krasnosel’skii fixed point theorem. Two examples are finally given that illustrate our results. Full article
19 pages, 346 KiB  
Article
Conjugation Conditions for Systems of Differential Equations of Different Orders on a Star Graph
by Baltabek Kanguzhin and Gauhar Auzerkhan
Symmetry 2022, 14(9), 1761; https://doi.org/10.3390/sym14091761 - 23 Aug 2022
Cited by 2 | Viewed by 1180
Abstract
In this paper, a one-dimensional mathematical model for investigating the vibrations of structures consisting of elastic and weakly curved rods is proposed. The three-dimensional structure is replaced by a limit graph, on each arc of which a system of three differential equations is [...] Read more.
In this paper, a one-dimensional mathematical model for investigating the vibrations of structures consisting of elastic and weakly curved rods is proposed. The three-dimensional structure is replaced by a limit graph, on each arc of which a system of three differential equations is written out. The differential equations describe the longitudinal and transverse vibrations of an elastic rod, taking into account the influence of longitudinal and transverse vibrations on each other. Describing conjugation conditions at joints of four or more rods is an important problem. This article assumes new conjugation conditions that guarantee the all-around decidability and symmetry of the resulting boundary value problems for systems of differential equations on a star graph. In addition, the paper proposes a physical interpretation of the conjugation conditions found. Thus, the work presents one more area of knowledge where symmetry phenomena occur. The symmetry here is manifested in the preservation of conjugation conditions when passing to the conjugate operator. Full article
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17 pages, 1717 KiB  
Article
Numerical Integration of Stiff Differential Systems Using Non-Fixed Step-Size Strategy
by Joshua Sunday, Ali Shokri, Joshua Amawa Kwanamu and Kamsing Nonlaopon
Symmetry 2022, 14(8), 1575; https://doi.org/10.3390/sym14081575 - 30 Jul 2022
Cited by 10 | Viewed by 1462
Abstract
Over the years, researches have shown that fixed (constant) step-size methods have been efficient in integrating a stiff differential system. It has however been observed that for some stiff differential systems, non-fixed (variable) step-size methods are required for efficiency and for accuracy to [...] Read more.
Over the years, researches have shown that fixed (constant) step-size methods have been efficient in integrating a stiff differential system. It has however been observed that for some stiff differential systems, non-fixed (variable) step-size methods are required for efficiency and for accuracy to be attained. This is because such systems have solution components that decay rapidly and/or slowly than others over a given integration interval. In order to curb this challenge, there is a need to propose a method that can vary the step size within a defined integration interval. This challenge motivated the development of Non-Fixed Step-Size Algorithm (NFSSA) using the Lagrange interpolation polynomial as a basis function via integration at selected limits. The NFSSA is capable of integrating highly stiff differential systems in both small and large intervals and is also efficient in terms of economy of computer time. The validation of properties of the proposed algorithm which include order, consistence, zero-stability, convergence, and region of absolute stability were further carried out. The algorithm was then applied to solve some samples mildly and highly stiff differential systems and the results generated were compared with those of some existing methods in terms of the total number of steps taken, number of function evaluation, number of failure/rejected steps, maximum errors, absolute errors, approximate solutions and execution time. The results obtained clearly showed that the NFSSA performed better than the existing ones with which we compared our results including the inbuilt MATLAB stiff solver, ode 15s. The results were also computationally reliable over long intervals and accurate on the abscissae points which they step on. Full article
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14 pages, 449 KiB  
Article
Symmetries, Reductions and Exact Solutions of a Class of (2k + 2)th-Order Difference Equations with Variable Coefficients
by Mensah Folly-Gbetoula
Symmetry 2022, 14(7), 1290; https://doi.org/10.3390/sym14071290 - 21 Jun 2022
Cited by 1 | Viewed by 1134
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
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10 pages, 274 KiB  
Article
On Nonlinear Biharmonic Problems on the Heisenberg Group
by Jiabin Zuo, Said Taarabti, Tianqing An and Dušan D. Repovš
Symmetry 2022, 14(4), 705; https://doi.org/10.3390/sym14040705 - 31 Mar 2022
Cited by 2 | Viewed by 1515
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
18 pages, 447 KiB  
Article
The Behavior and Structures of Solution of Fifth-Order Rational Recursive Sequence
by Elsayed M. Elsayed, Badriah S. Aloufi and Osama Moaaz
Symmetry 2022, 14(4), 641; https://doi.org/10.3390/sym14040641 - 22 Mar 2022
Cited by 7 | Viewed by 1495
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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10 pages, 287 KiB  
Article
Positive Solutions of a Fractional Boundary Value Problem with Sequential Derivatives
by Alexandru Tudorache and Rodica Luca
Symmetry 2021, 13(8), 1489; https://doi.org/10.3390/sym13081489 - 13 Aug 2021
Cited by 5 | Viewed by 1269
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
16 pages, 366 KiB  
Article
Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type
by Alexander Kazakov
Symmetry 2021, 13(5), 871; https://doi.org/10.3390/sym13050871 - 13 May 2021
Cited by 7 | Viewed by 1923
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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11 pages, 1755 KiB  
Article
Analysis of an Electrical Circuit Using Two-Parameter Conformable Operator in the Caputo Sense
by Ewa Piotrowska and Łukasz Sajewski
Symmetry 2021, 13(5), 771; https://doi.org/10.3390/sym13050771 - 29 Apr 2021
Cited by 7 | Viewed by 1787
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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10 pages, 230 KiB  
Article
Toward a Wong–Zakai Approximation for Big Order Generators
by Rémi Léandre
Symmetry 2020, 12(11), 1893; https://doi.org/10.3390/sym12111893 - 18 Nov 2020
Cited by 2 | Viewed by 2218
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
20 pages, 10063 KiB  
Article
Multiquadrics without the Shape Parameter for Solving Partial Differential Equations
by Cheng-Yu Ku, Chih-Yu Liu, Jing-En Xiao and Shih-Meng Hsu
Symmetry 2020, 12(11), 1813; https://doi.org/10.3390/sym12111813 - 02 Nov 2020
Cited by 7 | Viewed by 2244
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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12 pages, 288 KiB  
Article
Large Time Behavior for Inhomogeneous Damped Wave Equations with Nonlinear Memory
by Mohamed Jleli, Bessem Samet and Calogero Vetro
Symmetry 2020, 12(10), 1609; https://doi.org/10.3390/sym12101609 - 27 Sep 2020
Cited by 4 | Viewed by 1595
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
12 pages, 287 KiB  
Article
On a Fractional in Time Nonlinear Schrödinger Equation with Dispersion Parameter and Absorption Coefficient
by Mohamed Jleli, Bessem Samet and Calogero Vetro
Symmetry 2020, 12(7), 1197; https://doi.org/10.3390/sym12071197 - 20 Jul 2020
Cited by 4 | Viewed by 1765
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
16 pages, 323 KiB  
Article
Some Generalised Fixed Point Theorems Applied to Quantum Operations
by Umar Batsari Yusuf, Poom Kumam and Sikarin Yoo-Kong
Symmetry 2020, 12(5), 759; https://doi.org/10.3390/sym12050759 - 06 May 2020
Cited by 6 | Viewed by 2850
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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9 pages, 246 KiB  
Article
Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain
by Awatif Alqahtani, Mohamed Jleli, Bessem Samet and Calogero Vetro
Symmetry 2020, 12(3), 394; https://doi.org/10.3390/sym12030394 - 04 Mar 2020
Cited by 1 | Viewed by 1924
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
13 pages, 284 KiB  
Article
Lie Symmetry Analysis, Explicit Solutions and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation
by Amjad Hussain, Shahida Bano, Ilyas Khan, Dumitru Baleanu and Kottakkaran Sooppy Nisar
Symmetry 2020, 12(1), 170; https://doi.org/10.3390/sym12010170 - 16 Jan 2020
Cited by 30 | Viewed by 3717
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
19 pages, 345 KiB  
Article
Impulsive Evolution Equations with Causal Operators
by Tahira Jabeen, Ravi P. Agarwal, Vasile Lupulescu and Donal O’Regan
Symmetry 2020, 12(1), 48; https://doi.org/10.3390/sym12010048 - 25 Dec 2019
Cited by 3 | Viewed by 2022
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

Other

Jump to: Editorial, Research

7 pages, 232 KiB  
Comment
Comments on the Paper “Lie Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation”
by Roman Cherniha
Symmetry 2020, 12(6), 900; https://doi.org/10.3390/sym12060900 - 01 Jun 2020
Cited by 2 | Viewed by 2034
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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