Symmetry

Editorial

Jump to: Research

2 pages, 164 KiB

Open AccessEditorial

Special Issue Editorial Asymptotic Methods in the Mechanics and Nonlinear Dynamics

by Igor Andrianov

Symmetry 2022, 14(8), 1647; https://doi.org/10.3390/sym14081647 - 10 Aug 2022

Viewed by 1019

Abstract

The idea of asymptotic approximation is one of the most important and profound in mathematics, especially in the parts of it those are in close contact with physics, mechanics, and engineering [...] Full article

(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)

Research

Jump to: Editorial

11 pages, 274 KiB

Open AccessArticle

Delay Differential Equations of Fourth-Order: Oscillation and Asymptotic Properties of Solutions

by Omar Bazighifan, Maryam Al-Kandari, Khalil S. Al-Ghafri, F. Ghanim, Sameh Askar and Georgia Irina Oros

Symmetry 2021, 13(11), 2015; https://doi.org/10.3390/sym13112015 - 24 Oct 2021

Cited by 4 | Viewed by 1533

Abstract

In this work, by using the comparison method and Riccati transformation, we obtain some oscillation criteria of solutions of delay differential equations of fourth-order in canonical form. These criteria complement those results in the literature. We give two examples to illustrate the main [...] Read more.

In this work, by using the comparison method and Riccati transformation, we obtain some oscillation criteria of solutions of delay differential equations of fourth-order in canonical form. These criteria complement those results in the literature. We give two examples to illustrate the main results. Symmetry plays an essential role in determining the correct methods for solutions to differential equations. Full article

(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)

32 pages, 499 KiB

Open AccessArticle

Padé and Post-Padé Approximations for Critical Phenomena

by Simon Gluzman

Symmetry 2020, 12(10), 1600; https://doi.org/10.3390/sym12101600 - 25 Sep 2020

Cited by 14 | Viewed by 5022

Abstract

We discuss and apply various direct extrapolation methods for calculation of the critical points and indices from the perturbative expansions my means of Padé-techniques and their various post-Padé extensions by means of root and factor approximants. Factor approximants are applied to finding critical [...] Read more.

We discuss and apply various direct extrapolation methods for calculation of the critical points and indices from the perturbative expansions my means of Padé-techniques and their various post-Padé extensions by means of root and factor approximants. Factor approximants are applied to finding critical points. Roots are employed within the context of finding critical index. Additive self-similar approximants are discussed and

D L o g

additive recursive approximants are introduced as their generalization. They are applied to the problem of interpolation. Several examples of interpolation are considered. Full article

(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)

► Show Figures

Figure 1

16 pages, 1040 KiB

Open AccessArticle

Periodic Wave Solutions and Their Asymptotic Property for a Modified Fornberg–Whitham Equation

by Yiren Chen

Symmetry 2020, 12(9), 1517; https://doi.org/10.3390/sym12091517 - 15 Sep 2020

Cited by 1 | Viewed by 1676

Abstract

Recently, periodic traveling waves, which include periodically symmetric traveling waves of nonlinear equations, have received great attention. This article uses some bifurcations of the traveling wave system to investigate the explicit periodic wave solutions with parameter

α

and their asymptotic property for the [...] Read more.

Recently, periodic traveling waves, which include periodically symmetric traveling waves of nonlinear equations, have received great attention. This article uses some bifurcations of the traveling wave system to investigate the explicit periodic wave solutions with parameter

α

and their asymptotic property for the modified Fornberg–Whitham equation. Furthermore, when

α

tends to given parametric values, the elliptic periodic wave solutions become the other three types of nonlinear wave solutions, which include the trigonometric periodic blow-up solution, the hyperbolic smooth solitary wave solution, and the hyperbolic blow-up solution. Full article

(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)

► Show Figures

Figure 1

17 pages, 3124 KiB

Open AccessFeature PaperArticle

Construction of Analytic Solution to Axisymmetric Flow and Heat Transfer on a Moving Cylinder

by Vasile Marinca and Nicolae Herisanu

Symmetry 2020, 12(8), 1335; https://doi.org/10.3390/sym12081335 - 10 Aug 2020

Cited by 14 | Viewed by 2210

Abstract

Based on a new kind of analytical approach, namely the Optimal Auxiliary Functions Method (OAFM), a new analytical procedure is proposed to solve the problem of the annular axisymmetric stagnation flow and heat transfer on a moving cylinder with finite radius. As a [...] Read more.

Based on a new kind of analytical approach, namely the Optimal Auxiliary Functions Method (OAFM), a new analytical procedure is proposed to solve the problem of the annular axisymmetric stagnation flow and heat transfer on a moving cylinder with finite radius. As a novelty, explicit analytical solutions were obtained for the considered complex problem. First, the Navier–Stokes equations were simplified by means of similarity transformations that depended on different parameters and some combinations of these parameters, and the problem under study was reduced to six nonlinear ordinary differential equations with six unknowns. The OAFM proves to be a powerful tool for finding an accurate analytical solution for nonlinear problems, ensuring a fast convergence after the first iteration, even if the small or large parameters are absent, since the determination of the convergence-control parameters is independent of the magnitude of the coefficients that appear in the nonlinear differential equations. Concerning the main novelties of the proposed approach, it is worth mentioning the presence of some auxiliary functions, the involvement of the convergence-control parameters, the construction of the first iteration and much freedom to select the procedure for determining the optimal values of the convergence-control parameters. Full article

(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)

► Show Figures

Figure 1

15 pages, 2233 KiB

Open AccessArticle

Buckling of Corrugated Ring under Uniform External Pressure

by Igor I. Andrianov, Igor V. Andrianov, Alexander A. Diskovsky and Eduard V. Ryzhkov

Symmetry 2020, 12(8), 1250; https://doi.org/10.3390/sym12081250 - 29 Jul 2020

Cited by 4 | Viewed by 2781

Abstract

Stability analysis of a corrugated ring subjected to uniform external pressure is under consideration. Two main approaches to solving this problem are analyzed. The equivalent bending stiffness approach is often used in engineering practice. It is based on some plausible assumptions about the [...] Read more.

Stability analysis of a corrugated ring subjected to uniform external pressure is under consideration. Two main approaches to solving this problem are analyzed. The equivalent bending stiffness approach is often used in engineering practice. It is based on some plausible assumptions about the behavior of a structure. Its advantage is the simplicity of the obtained relations; the disadvantage is the difficulty in estimating the area of applicability. In this paper, we developed an asymptotic homogenization method for calculating the critical pressure for a corrugated ring, which made it possible to mathematically substantiate and refine the equivalent bending stiffness approach. To evaluate the results obtained using the equivalent stiffness approach and asymptotic homogenization method, the imperfection method is used. The influence of the corrugation parameters on buckling pressure is analyzed. Full article

(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)

► Show Figures

Figure 1

11 pages, 5234 KiB

Open AccessArticle

Optimal Perturbation Iteration Method for Solving Fractional Model of Damped Burgers’ Equation

by Sinan Deniz, Ali Konuralp and Mnauel De la Sen

Symmetry 2020, 12(6), 958; https://doi.org/10.3390/sym12060958 - 4 Jun 2020

Cited by 17 | Viewed by 2491

Abstract

The newly constructed optimal perturbation iteration procedure with Laplace transform is applied to obtain the new approximate semi-analytical solutions of the fractional type of damped Burgers’ equation. The classical damped Burgers’ equation is remodeled to fractional differential form via the Atangana–Baleanu fractional derivatives [...] Read more.

The newly constructed optimal perturbation iteration procedure with Laplace transform is applied to obtain the new approximate semi-analytical solutions of the fractional type of damped Burgers’ equation. The classical damped Burgers’ equation is remodeled to fractional differential form via the Atangana–Baleanu fractional derivatives described with the help of the Mittag–Leffler function. To display the efficiency of the proposed optimal perturbation iteration technique, an extended example is deeply analyzed. Full article

(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)

► Show Figures

Figure 1

14 pages, 389 KiB

Open AccessFeature PaperArticle

Waves in Two Coaxial Elastic Cubically Nonlinear Shells with Structural Damping and Viscous Fluid Between Them

by Lev Mogilevich and Sergey Ivanov

Symmetry 2020, 12(3), 335; https://doi.org/10.3390/sym12030335 - 26 Feb 2020

Cited by 8 | Viewed by 1983

Abstract

This article investigates longitudinal deformation waves in physically nonlinear coaxial elastic shells containing a viscous incompressible fluid between them. The presence of a viscous incompressible fluid between the shells, as well as the influence of the inertia of the fluid motion on the [...] Read more.

This article investigates longitudinal deformation waves in physically nonlinear coaxial elastic shells containing a viscous incompressible fluid between them. The presence of a viscous incompressible fluid between the shells, as well as the influence of the inertia of the fluid motion on the amplitude and velocity of the wave, are taken into account. The mathematical model phenomenon is constructed by means of the method of two-scale asymptotic expansion. Structural damping in the shells and surrounding elastic media did not allow discovery of the exact solution of the problem of the deformation waves propagation. This leads to the need for numerical methods. A numerical study of the model constructed in the course of this work is carried out by using a difference scheme for the equation similar to the Crank–Nicholson scheme for the heat equation. In the absence of the structural damping and surrounding media influences, and under the similar initial conditions for both shells, the velocity and amplitude of the wave do not change. The result of the numerical experiment coincides with the exact solution, which is found in the case of the absence of the structural damping and surrounding media influences; therefore, the difference scheme is adequate to the generalized modified Korteweg–de Vries equations system. There is energy is transferred in the presence of the fluid, between the shells. The presence of inertia of the fluid motion leads to a decrease in the velocity of the deformation wave. Full article

(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)

► Show Figures

Figure 1

12 pages, 389 KiB

Open AccessArticle

Exact Solutions and Numerical Simulation of the Discrete Sawada–Kotera Equation

by Aleksandr Zemlyanukhin and Andrey Bochkarev

Symmetry 2020, 12(1), 131; https://doi.org/10.3390/sym12010131 - 9 Jan 2020

Cited by 5 | Viewed by 2362

Abstract

We investigated an integrable five-point differential-difference equation called the discrete Sawada–Kotera equation. On the basis of the geometric series method, a new exact soliton-like solution of the equation is obtained that propagates with positive or negative phase velocity. In terms of the Jacobi [...] Read more.

We investigated an integrable five-point differential-difference equation called the discrete Sawada–Kotera equation. On the basis of the geometric series method, a new exact soliton-like solution of the equation is obtained that propagates with positive or negative phase velocity. In terms of the Jacobi elliptic function, a class of new exact periodic solutions is constructed, in particular stationary ones. Using an exponential generating function for Catalan numbers, Cauchy’s problem with the initial condition in the form of a step is solved. As a result of numerical simulation, the elasticity of the interaction of exact localized solutions is established. Full article

(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)

► Show Figures

Figure 1

15 pages, 4053 KiB

Open AccessArticle

The Verhulst-Like Equations: Integrable OΔE and ODE with Chaotic Behavior

by Igor Andrianov, Galina Starushenko, Sergey Kvitka and Lelya Khajiyeva

Symmetry 2019, 11(12), 1446; https://doi.org/10.3390/sym11121446 - 25 Nov 2019

Cited by 6 | Viewed by 3365

Abstract

In this paper, we study various variants of Verhulst-like ordinary differential equations (ODE) and ordinary difference equations (O

Δ

E). Usually Verhulst ODE serves as an example of a deterministic system and discrete logistic equation is a classic example of a simple system [...] Read more.

In this paper, we study various variants of Verhulst-like ordinary differential equations (ODE) and ordinary difference equations (O

Δ

E). Usually Verhulst ODE serves as an example of a deterministic system and discrete logistic equation is a classic example of a simple system with very complicated (chaotic) behavior. In our paper we present examples of deterministic discretization and chaotic continualization. Continualization procedure is based on Padé approximants. To correctly characterize the dynamics of obtained ODE we measured such characteristic parameters of chaotic dynamical systems as the Lyapunov exponents and the Lyapunov dimensions. Discretization and continualization lead to a change in the symmetry of the mathematical model (i.e., group properties of the original ODE and O

Δ

E). This aspect of the problem is the aim of further research. Full article

(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)

► Show Figures

Figure 1

19 pages, 3465 KiB

Open AccessArticle

Second-Order Sliding Mode Formation Control of Multiple Robots by Extreme Learning Machine

by Dianwei Qian, Guigang Zhang, Jian Wang and Zhimin Wu

Symmetry 2019, 11(12), 1444; https://doi.org/10.3390/sym11121444 - 23 Nov 2019

Cited by 8 | Viewed by 2204

Abstract

This paper addresses a second-order sliding mode control method for the formation problem of multirobot systems. The formation patterns are usually symmetrical. This sliding mode control is based on the super-twisting law. In many real-world applications, the robots suffer from a great diversity [...] Read more.

This paper addresses a second-order sliding mode control method for the formation problem of multirobot systems. The formation patterns are usually symmetrical. This sliding mode control is based on the super-twisting law. In many real-world applications, the robots suffer from a great diversity of uncertainties and disturbances that greatly challenge super-twisting sliding mode formation maneuvers. In particular, such a challenge has adverse effects on the formation performance when the uncertainties and disturbances have an unknown bound. This paper focuses on this issue and utilizes the technique of an extreme learning machine to meet this challenge. Within the leader–follower framework, this paper investigates the integration of the super-twisting sliding mode control method and the extreme learning machine. The output weights of this extreme learning machine are adaptively adjusted so that this integrated formation design has guaranteed closed-loop stability in the sense of Lyaponov. In the end, some simulations are implemented via a multirobot platform, illustrating the superiority and effectiveness of the integrated formation design in spite of uncertainties and disturbances. Full article

(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)

► Show Figures

Figure 1

12 pages, 2402 KiB

Open AccessArticle

Closed Form Solutions for Nonlinear Oscillators Under Discontinuous and Impulsive Periodic Excitations

by Valery Pilipchuk

Symmetry 2019, 11(11), 1420; https://doi.org/10.3390/sym11111420 - 16 Nov 2019

Cited by 1 | Viewed by 2358

Abstract

Periodic responses of linear and nonlinear systems under discontinuous and impulsive excitations are analyzed with non-smooth temporal transformations incorporating temporal symmetries of periodic processes. The related analytical manipulations are illustrated on a strongly nonlinear oscillator whose free vibrations admit an exact description in [...] Read more.

Periodic responses of linear and nonlinear systems under discontinuous and impulsive excitations are analyzed with non-smooth temporal transformations incorporating temporal symmetries of periodic processes. The related analytical manipulations are illustrated on a strongly nonlinear oscillator whose free vibrations admit an exact description in terms of elementary functions. As a result, closed form analytical solutions for the non-autonomous strongly nonlinear case are obtained. Conditions of existence for such solutions are represented as a family of period-amplitude curves. The family is represented by different couples of solutions associated with different numbers of vibration half cycles between any two consecutive pulses. Poincaré sections showed that the oscillator can respond quite chaotically when shifting from the period-amplitude curves. Full article

(This article belongs to the Special Issue Asymptotic Methods in the Mechanics and Nonlinear Dynamics)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Asymptotic Methods in the Mechanics and Nonlinear Dynamics

Share This Special Issue

Special Issue Editor

Special Issue Information

Keywords

Benefits of Publishing in a Special Issue

Published Papers (12 papers)

Editorial

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI