Research

18 pages, 328 KiB

Open AccessArticle

Optimizing the Monotonic Properties of Fourth-Order Neutral Differential Equations and Their Applications

by Hend Salah, Osama Moaaz, Sameh S. Askar, Ahmad M. Alshamrani and Elmetwally M. Elabbasy

Symmetry 2023, 15(9), 1744; https://doi.org/10.3390/sym15091744 - 11 Sep 2023

Viewed by 398

We investigate the oscillation of the fourth-order differential equation for a class of functional differential equations of the neutral type. We obtain a new single-oscillation criterion for the oscillation of all the solutions of our equation. We establish new monotonic properties for some [...] Read more.

We investigate the oscillation of the fourth-order differential equation for a class of functional differential equations of the neutral type. We obtain a new single-oscillation criterion for the oscillation of all the solutions of our equation. We establish new monotonic properties for some cases of positive solutions of the studied equation. Moreover, we improve these properties by using an iterative method. This development of monotonic properties contributes to obtaining new and more efficient criteria for verifying the oscillation of the equation. The results obtained extend and improve previous findings in the literature by using an Euler-type equation as an example. The importance of the results was clarified by applying them to some special cases of the studied equation. The fourth-order delay differential equations have great practical importance due to their wide applications in civil, mechanical, and aeronautical engineering. Research on this type of equation is still ongoing due to its remarkable importance in many fields. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

16 pages, 315 KiB

Open AccessArticle

Fourth-Order Emden–Fowler Neutral Differential Equations: Investigating Some Qualitative Properties of Solutions

by Mansour Alatwi, Osama Moaaz, Sameh S. Askar, Ahmad M. Alshamrani and Elmetwally M. Elabbasy

Symmetry 2023, 15(7), 1446; https://doi.org/10.3390/sym15071446 - 19 Jul 2023

Viewed by 580

Abstract

In this article, we investigate some of the qualitative properties of a class of fourth-order neutral differential equations. We start by obtaining new inequalities and relations between the solution and its corresponding function, as well as with its derivatives. The new relations allow [...] Read more.

In this article, we investigate some of the qualitative properties of a class of fourth-order neutral differential equations. We start by obtaining new inequalities and relations between the solution and its corresponding function, as well as with its derivatives. The new relations allow us to improve the monotonic and asymptotic properties of the positive solutions of the studied equation. Then, using an improved approach, we establish new criteria that test the oscillation of all solutions. We also rely on the principle of symmetry between positive and negative solutions to obtain the new criteria. The paper provides illustrative examples that highlight the significance of our findings. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

22 pages, 4668 KiB

Open AccessFeature PaperArticle

Longitudinal–Transverse Vibration of a Functionally Graded Nanobeam Subjected to Mechanical Impact and Electromagnetic Actuation

by Nicolae Herisanu, Bogdan Marinca and Vasile Marinca

Symmetry 2023, 15(7), 1376; https://doi.org/10.3390/sym15071376 - 06 Jul 2023

Cited by 1 | Viewed by 915

Abstract

This study addresses the nonlinear forced vibration of a functionally graded (FG) nanobeam subjected to mechanical impact and electromagnetic actuation. Two symmetrical actuators were present in the mechanical model, and their mechanical behaviors were analyzed considering the symmetry in actuation. The model considered [...] Read more.

This study addresses the nonlinear forced vibration of a functionally graded (FG) nanobeam subjected to mechanical impact and electromagnetic actuation. Two symmetrical actuators were present in the mechanical model, and their mechanical behaviors were analyzed considering the symmetry in actuation. The model considered the longitudinal–transverse vibration of a simple supported Euler–Bernoulli beam, which accounted for von Kármán geometric nonlinearity, including the first-order strain–displacement relationship. The FG nanobeam was made of a mixture of metals and ceramics, while the volume fraction varied in terms of thickness when a power law function was used. The nonlocal Eringen theory of elasticity was used to study the simple supported Euler–Bernoulli nanobeam. The nonlinear governing equations of the FG nanobeam and the associated boundary conditions were gained using Hamilton’s principle. To truncate the system with an infinite degree of freedom, the coupled longitudinal–transverse governing equations were discretized using the Galerkin–Bubnov approach. The resulting nonlinear, ordinary differential equations, which took into account the curvature of the nanobeam, were studied via the Optimal Auxiliary Functions Method (OAFM). For this complex nonlinear problem, an explicit, analytical, approximate solution was proposed near the primary resonance. The simultaneous effects of the following elements were considered in this paper: the presence of a curved nanobeam; the transversal inertia, which is not neglected in this paper; the mechanical impact; and electromagnetic actuation. The present study proposes a highly accurate analytical solution to the abovementioned conditions. Moreover, in these conditions, the study of local stability was developed using two variable expansion methods, the Jacobian matrix and Routh–Hurwitz criteria, and global stability was studied using the Lyapunov function. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

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13 pages, 314 KiB

Open AccessArticle

Time Optimal Feedback Control for 3D Navier–Stokes-Voigt Equations

by Yunxiang Li, Maojun Bin and Cuiyun Shi

Symmetry 2023, 15(5), 1127; https://doi.org/10.3390/sym15051127 - 22 May 2023

Viewed by 881

Abstract

In this article, we discuss a time optimal feedback control for asymmetrical 3D Navier–Stokes–Voigt equations. Firstly, we consider the existence of the admissible trajectories for the asymmetrical 3D Navier–Stokes–Voigt equations by using the well-known Cesari property and the Fillippove’s theorem. Secondly, we study [...] Read more.

In this article, we discuss a time optimal feedback control for asymmetrical 3D Navier–Stokes–Voigt equations. Firstly, we consider the existence of the admissible trajectories for the asymmetrical 3D Navier–Stokes–Voigt equations by using the well-known Cesari property and the Fillippove’s theorem. Secondly, we study the existence result of a time optimal control for the feedback control systems. Lastly, asymmetrical Clarke’s subdifferential inclusions and asymmetrical 3D Navier–Stokes–Voigt differential variational inequalities are given to explain our main results. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

48 pages, 6692 KiB

Open AccessArticle

Testing for Unit Roots in Nonlinear Dynamic Heterogeneous Panels with Logistic Smooth Breaks

by Tolga Omay and Nuri Ucar

Symmetry 2023, 15(3), 747; https://doi.org/10.3390/sym15030747 - 18 Mar 2023

Viewed by 1679

Abstract

In this study, we investigate the validity of the purchasing power parity (PPP) proposition for 34 European and selected global countries. For this purpose, we propose a new unit root test for cross-sectionally dependent heterogeneous panels that allows for gradual structural breaks and [...] Read more.

In this study, we investigate the validity of the purchasing power parity (PPP) proposition for 34 European and selected global countries. For this purpose, we propose a new unit root test for cross-sectionally dependent heterogeneous panels that allows for gradual structural breaks and symmetric nonlinear adjustment toward the equilibrium level. The alternative hypothesis stationary is obtained by symmetric adjustment due to exponential smooth transition autoregression (ESTAR) around a nonlinear trend. Moreover, we provide small sample properties extensively for the newly proposed test. Hence, this alternative hypothesis has been proven to characterize real exchange rate data (REER) correctly. Thus, the newly proposed tests provide an essential basis for modeling the REER series correctly. Finally, we also derive the approximate asymptotic distribution of the proposed tests using new techniques. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

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11 pages, 599 KiB

Open AccessArticle

Temperature Curve of Reflow Furnace Based on Newton’s Law of Cooling

by Bo-yang Li, Shi-you Lin, Li-sha Chen and Ming-yuan Zhao

Symmetry 2023, 15(3), 661; https://doi.org/10.3390/sym15030661 - 06 Mar 2023

Cited by 1 | Viewed by 1113

Abstract

Soldering in a reflow oven is an important and efficient technical means to produce integrated circuit boards. The key to the quality of integrated circuit boards lies in the furnace temperature curve. In this paper, Newton’s law of cooling is used to establish [...] Read more.

Soldering in a reflow oven is an important and efficient technical means to produce integrated circuit boards. The key to the quality of integrated circuit boards lies in the furnace temperature curve. In this paper, Newton’s law of cooling is used to establish the mechanism model of the temperature of each zone of the furnace and the curve of furnace temperature, which can reduce the number of experiments in actual production and obtain a better furnace temperature curve, thus improving production efficiency. Finally, several concrete examples are given to discuss and solve some common problems in the industry. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

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12 pages, 273 KiB

Open AccessArticle

Blow-Up Criterion and Persistence Property to a Generalized Camassa–Holm Equation

by Ying Wang and Yunxi Guo

Symmetry 2023, 15(2), 493; https://doi.org/10.3390/sym15020493 - 13 Feb 2023

Viewed by 716

Abstract

In this paper, a generalized Camassa–Holm equation, which may be used to describe wave motion in the shallow water, is considered. Some dynamic properties are studied for the model. Firstly, a new blow-up criterion for the equation is established. Then, analytical solutions are [...] Read more.

In this paper, a generalized Camassa–Holm equation, which may be used to describe wave motion in the shallow water, is considered. Some dynamic properties are studied for the model. Firstly, a new blow-up criterion for the equation is established. Then, analytical solutions are presented for the first time by using a new method. Finally, we investigate the persistence property for strong solutions. The results we obtain complement earlier results in this direction. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

9 pages, 258 KiB

Open AccessArticle

Atomic Solution for Certain Gardner Equation

by Mohammad Al-Khaleel, Sharifa Al-Sharif and Ameerah AlJarrah

Symmetry 2023, 15(2), 440; https://doi.org/10.3390/sym15020440 - 07 Feb 2023

Cited by 2 | Viewed by 979

Abstract

In this paper, a new technique using a tensor product is presented in order to provide exact solutions to some certain fractional differential equations. Particularly, the well-known third order Gardner’s equation, which is also known in some contexts as KdV-mKdV, of the fractional [...] Read more.

In this paper, a new technique using a tensor product is presented in order to provide exact solutions to some certain fractional differential equations. Particularly, the well-known third order Gardner’s equation, which is also known in some contexts as KdV-mKdV, of the fractional type. This type of equations plays an important role in modeling many symmetric and asymmetric problems. Moreover, the existence of an atomic solution using a tensor product technique for certain second order equations has been proved. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

19 pages, 340 KiB

Open AccessArticle

Nonlocal Pseudo-Parabolic Equation with Memory Term and Conical Singularity: Global Existence and Blowup

by Jiali Yu and Jihong Zhang

Symmetry 2023, 15(1), 122; https://doi.org/10.3390/sym15010122 - 01 Jan 2023

Cited by 1 | Viewed by 1070

Abstract

Considered herein is the initial-boundary value problem for a semilinear parabolic equation with a memory term and non-local source [...] Read more.

Considered herein is the initial-boundary value problem for a semilinear parabolic equation with a memory term and non-local source

w_{t} - Δ_{B} w - Δ_{B} w_{t} + \int_{0}^{t} g (t - τ) Δ_{B} w (τ) d τ = {| w |}^{p - 1} w - \frac{1}{| B |} \int_{B} {| w |}^{p - 1} w \frac{d x_{1}}{x_{1}} d x^{'}

on a manifold with conical singularity, where the Fuchsian type Laplace operator

Δ_{B}

is an asymmetry elliptic operator with conical degeneration on the boundary

x_{1} = 0

. Firstly, we discuss the symmetrical structure of invariant sets with the help of potential well theory. Then, the problem can be decomposed into two symmetric cases: if

w_{0} \in W

and

Π (w_{0}) > 0

, the global existence for the weak solutions will be discussed by a series of energy estimates under some appropriate assumptions on the relaxation function, initial data and the symmetric structure of invariant sets. On the contrary, if

w_{0} \in V

and

Π (w_{0}) < 0

, the nonexistence of global solutions, i.e., the solutions blow up in finite time, is obtained by using the convexity technique. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

23 pages, 865 KiB

Open AccessArticle

Asymptotic Behavior of the Solution to Compressible Navier–Stokes System with Temperature-Dependent Heat Conductivity in an Unbounded Domain

by Wenhuo Su and Jianxin Zhong

Symmetry 2023, 15(1), 112; https://doi.org/10.3390/sym15010112 - 31 Dec 2022

Viewed by 879

Abstract

This paper concerns the one-dimensional compressible Navier–Stokes system with temperature-dependent heat conductivity in

R

with large initial data. We prove that velocity and temperature are uniformly bounded from below and above in time and space when the heat conductivity coefficient takes [...] Read more.

This paper concerns the one-dimensional compressible Navier–Stokes system with temperature-dependent heat conductivity in

R

with large initial data. We prove that velocity and temperature are uniformly bounded from below and above in time and space when the heat conductivity coefficient takes

κ = \bar{κ} (1 + θ^{b})

for all

b > \frac{5}{2}

. In addition, we show that the global solution is asymptotically stable as time tends to infinity. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

14 pages, 304 KiB

Open AccessArticle

Global Existence to Cauchy Problem for 1D Magnetohydrodynamics Equations

by Jianxin Zhong and Xuejun Xie

Symmetry 2023, 15(1), 80; https://doi.org/10.3390/sym15010080 - 28 Dec 2022

Viewed by 859

Abstract

Magnetohydrodynamics are widely used in medicine and biotechnology, such as drug targeting, molecular biology, cell isolation and purification. In this paper, we prove the existence of a global strong solution to the one-dimensional compressible magnetohydrodynamics system with temperature-dependent heat conductivity in unbounded domains [...] Read more.

Magnetohydrodynamics are widely used in medicine and biotechnology, such as drug targeting, molecular biology, cell isolation and purification. In this paper, we prove the existence of a global strong solution to the one-dimensional compressible magnetohydrodynamics system with temperature-dependent heat conductivity in unbounded domains and a large initial value by the Lagrangian symmetry transformation, when the viscosity

μ

is constant and the heat conductivity

κ

, which depends on the temperature, satisfies

κ = \bar{κ} θ^{b} (b > 1)

. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

15 pages, 289 KiB

Open AccessArticle

Double Conformable Sumudu Transform

by Abdallah Al-Rab’a, Sharifa Al-Sharif and Mohammad Al-Khaleel

Symmetry 2022, 14(11), 2249; https://doi.org/10.3390/sym14112249 - 26 Oct 2022

Cited by 1 | Viewed by 1307

Abstract

In this paper, we introduce a new approach to solving fractional initial and boundary value problems involving a heat equation, a wave equation, and a telegraph equation by modifying the double Sumudu transform of the fractional type. We discuss a modified double conformable [...] Read more.

In this paper, we introduce a new approach to solving fractional initial and boundary value problems involving a heat equation, a wave equation, and a telegraph equation by modifying the double Sumudu transform of the fractional type. We discuss a modified double conformable Sumudu transform together with the conditions for its existence. In addition, we prove some more properties of the fractional-type Sumudu transform, including convolution and other properties, which are well known for their use in solving various symmetric and asymmetric problems in applied sciences and engineering. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

14 pages, 321 KiB

Open AccessArticle

Oscillation Results of Third-Order Differential Equations with Symmetrical Distributed Arguments

by Belgees Qaraad, Omar Bazighifan, Ali Hasan Ali, Areej A. Al-Moneef, Awatif Jahman Alqarni and Kamsing Nonlaopon

Symmetry 2022, 14(10), 2038; https://doi.org/10.3390/sym14102038 - 29 Sep 2022

Cited by 7 | Viewed by 1078

Abstract

This paper is concerned with the oscillation and asymptotic behavior of certain third-order nonlinear delay differential equations with distributed deviating arguments. By establishing sufficient conditions for the nonexistence of Kneser solutions and existing oscillation results for the studied equation, we obtain new criteria [...] Read more.

This paper is concerned with the oscillation and asymptotic behavior of certain third-order nonlinear delay differential equations with distributed deviating arguments. By establishing sufficient conditions for the nonexistence of Kneser solutions and existing oscillation results for the studied equation, we obtain new criteria which ensure that every solution oscillates by using the theory of comparison with first-order delay equations and the technique of Riccati transformation. Some examples are presented to illustrate the importance of main results. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

11 pages, 288 KiB

Open AccessArticle

Several Dynamic Properties for the gkCH Equation

by Ying Wang and Yunxi Guo

Symmetry 2022, 14(9), 1772; https://doi.org/10.3390/sym14091772 - 25 Aug 2022

Cited by 1 | Viewed by 874

Abstract

In this paper, we focus on a generalized Camassa–Holm equation (also known as a gkCH equation), which includes both the Camassa–Holm equation and Novikov equation as two special cases. Because of the potential applications in physics, we will further investigate the properties of [...] Read more.

In this paper, we focus on a generalized Camassa–Holm equation (also known as a gkCH equation), which includes both the Camassa–Holm equation and Novikov equation as two special cases. Because of the potential applications in physics, we will further investigate the properties of the equation from a mathematical point of view. More precisely, firstly, we give a new wave-breaking phenomenon. Then, we present the theorem of existence and uniqueness of global weak solutions for the equation, provided that the initial data satisfy certain sign conditions. Finally, we prove the Hölder continuity of a solution map for the equation. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

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