Symmetry

Editorial

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3 pages, 161 KiB

Open AccessEditorial

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 975

Abstract

This Special Issue of the journal Symmetry is dedicated to recent progress in the field of nonlinear differential problems [...] Full article

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

Research

Jump to: Editorial, Other

20 pages, 339 KiB

Open AccessArticle

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 - 7 Sep 2023

Cited by 1 | Viewed by 1300

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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

12 pages, 298 KiB

Open AccessArticle

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 1604

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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

19 pages, 346 KiB

Open AccessArticle

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 1585

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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

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17 pages, 1717 KiB

Open AccessArticle

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 13 | Viewed by 1938

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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

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14 pages, 449 KiB

Open AccessArticle

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 1447

Abstract

We perform a Lie analysis of

(2 k + 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

(2 k + 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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

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10 pages, 274 KiB

Open AccessArticle

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 2011

Abstract

We investigate the boundary value problem for biharmonic operators on the Heisenberg group. The inherent features of

H^{n}

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

H^{n}

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

H^{n} .

Full article

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

18 pages, 447 KiB

Open AccessArticle

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 9 | Viewed by 1891

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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

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10 pages, 287 KiB

Open AccessArticle

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 6 | Viewed by 1655

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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

16 pages, 366 KiB

Open AccessFeature PaperArticle

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 8 | Viewed by 2323

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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

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11 pages, 1755 KiB

Open AccessArticle

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 2220

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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

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10 pages, 230 KiB

Open AccessFeature PaperArticle

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 2566

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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

20 pages, 10063 KiB

Open AccessArticle

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 - 2 Nov 2020

Cited by 7 | Viewed by 2745

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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

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12 pages, 288 KiB

Open AccessArticle

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 1929

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

ϕ_{t t} (t, ω) - Δ ϕ (t, ω) + ϕ_{t} (t, ω) = \frac{1}{Γ (1 - ρ)} \int_{0}^{t} {(t - σ)}^{- ρ} {| ϕ (σ, ω) |}^{q} d σ + μ (ω), t > 0

,

ω \in R^{N}

imposing the condition

(ϕ (0, ω), ϕ_{t} (0, ω)) = (ϕ_{0} (ω), ϕ_{1} (ω)) in R^{N},

where

N \geq 1

,

q > 1

,

0 < ρ < 1

,

ϕ_{i} \in L_{l o c}^{1} (R^{N})

,

i = 0, 1

,

μ \in L_{l o c}^{1} (R^{N})

and

μ ≢ 0

. Namely, it is shown that, if

ϕ_{0}, ϕ_{1} \geq 0

,

μ \in L^{1} (R^{N})

and

\int_{R^{N}} μ (ω) d ω > 0

, then for all

q > 1

, the considered problem has no global weak solution. Full article

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

12 pages, 287 KiB

Open AccessArticle

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 2165

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} \partial_{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) \in (0, \infty) \times R^{N}

, where

N \geq 1

,

ξ \in 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} \in L_{l o c}^{1} ([0, \infty), R)

, and provide two illustrative examples. Full article

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

16 pages, 323 KiB

Open AccessArticle

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 - 6 May 2020

Cited by 6 | Viewed by 3653

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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

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9 pages, 246 KiB

Open AccessArticle

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 - 4 Mar 2020

Cited by 1 | Viewed by 2290

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) \in (0, \infty) \times D^{c}

under the nonhomegeneous Neumann boundary condition

\frac{\partial u}{\partial ν} (t, x) = λ (x), (t, x) \in (0, \infty) \times \partial D,

where

L : = i \partial_{t} + Δ

is the Schrödinger operator,

D = B (0, 1)

is the open unit ball in

R^{N}

,

N \geq 2

,

D^{c} = R^{N} ∖ D

,

p > 1

,

κ \in C

,

κ \neq 0

,

λ \in L^{1} (\partial D, C)

is a nontrivial complex valued function, and

\partial ν

is the outward unit normal vector on

\partial D

, relative to

D^{c}

. Namely, under a certain condition imposed on

(κ, λ)

, we show that if

N \geq 3

and

p < p_{c}

, where

p_{c} = \frac{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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

13 pages, 284 KiB

Open AccessArticle

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 34 | Viewed by 4286

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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

19 pages, 345 KiB

Open AccessArticle

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 4 | Viewed by 2335

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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

Other

Jump to: Editorial, Research

7 pages, 232 KiB

Open AccessComment

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 - 1 Jun 2020

Cited by 3 | Viewed by 2582

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

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

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