Special Issue "Asymptotic Methods in the Mechanics and Nonlinear Dynamics"

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics and Symmetry".

Deadline for manuscript submissions: closed (31 August 2020).

Special Issue Editor

Prof. Dr. Igor Andrianov
Website
Guest Editor
Karl-Marx-Allee 4, D-50769, Koeln, Germany.
Interests: asymptotology; mechanics of solids; nonlinear dynamics
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Special Issue Information

Dear Colleagues,

This Special Issue of Symmetry is dedicated to asymptotic modelling. Galileo’s concept of idealization is the cornerstone of contemporary science. Idealization is based on increasing the symmetry of the original system, and its tool in applied mathematics is asymptotic analysis. It is no exaggeration to say that the basic models of applied mathematics, physics, and mechanics are asymptotic. There are many methods for constructing asymptotic models, and their development, generalization, and application are of fundamental importance both for theorists and for engineers. We hope this Special Issue will help theorists to find new tasks and areas of application for their knowledge, and engineers to find new methods for their practice.

This Special Issue invites research and review papers on various fields of theoretical physics and applied mathematics, including classical and quantum mechanics, mechanics of fluids and solids, and asymptotology.

Prof. Dr. Igor Andrianov
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Asymptotology
  • asymptotic modelling
  • regular and singular perturbation problems
  • Padé approximants and others summation and interpolation procedures
  • homogenization
  • kontinualization

Published Papers (10 papers)

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Research

Open AccessArticle
Padé and Post-Padé Approximations for Critical Phenomena
Symmetry 2020, 12(10), 1600; https://doi.org/10.3390/sym12101600 - 25 Sep 2020
Cited by 1 | Viewed by 290
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 DLog 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)
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Open AccessArticle
Periodic Wave Solutions and Their Asymptotic Property for a Modified Fornberg–Whitham Equation
Symmetry 2020, 12(9), 1517; https://doi.org/10.3390/sym12091517 - 15 Sep 2020
Viewed by 288
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)
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Open AccessFeature PaperArticle
Construction of Analytic Solution to Axisymmetric Flow and Heat Transfer on a Moving Cylinder
Symmetry 2020, 12(8), 1335; https://doi.org/10.3390/sym12081335 - 10 Aug 2020
Viewed by 349
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)
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Open AccessArticle
Buckling of Corrugated Ring under Uniform External Pressure
Symmetry 2020, 12(8), 1250; https://doi.org/10.3390/sym12081250 - 29 Jul 2020
Viewed by 461
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)
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Open AccessArticle
Optimal Perturbation Iteration Method for Solving Fractional Model of Damped Burgers’ Equation
Symmetry 2020, 12(6), 958; https://doi.org/10.3390/sym12060958 - 04 Jun 2020
Cited by 4 | Viewed by 437
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)
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Open AccessFeature PaperArticle
Waves in Two Coaxial Elastic Cubically Nonlinear Shells with Structural Damping and Viscous Fluid Between Them
Symmetry 2020, 12(3), 335; https://doi.org/10.3390/sym12030335 - 26 Feb 2020
Cited by 1 | Viewed by 471
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)
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Open AccessArticle
Exact Solutions and Numerical Simulation of the Discrete Sawada–Kotera Equation
Symmetry 2020, 12(1), 131; https://doi.org/10.3390/sym12010131 - 09 Jan 2020
Cited by 1 | Viewed by 507
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)
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Open AccessArticle
The Verhulst-Like Equations: Integrable OΔE and ODE with Chaotic Behavior
Symmetry 2019, 11(12), 1446; https://doi.org/10.3390/sym11121446 - 25 Nov 2019
Cited by 2 | Viewed by 633
Abstract
In this paper, we study various variants of Verhulst-like ordinary differential equations (ODE) and ordinary difference equations (OΔE). 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)
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Open AccessArticle
Second-Order Sliding Mode Formation Control of Multiple Robots by Extreme Learning Machine
Symmetry 2019, 11(12), 1444; https://doi.org/10.3390/sym11121444 - 23 Nov 2019
Cited by 1 | Viewed by 656
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)
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Open AccessArticle
Closed Form Solutions for Nonlinear Oscillators Under Discontinuous and Impulsive Periodic Excitations
Symmetry 2019, 11(11), 1420; https://doi.org/10.3390/sym11111420 - 16 Nov 2019
Viewed by 666
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)
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