Special Issue "Differential Equations of Mathematical Physics and Related Problems of Mechanics"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Functional Interpolation".

Deadline for manuscript submissions: 31 October 2023 | Viewed by 10113

Special Issue Editors

1. Federal Research Center "Computer Science and Control", Russian Academy of Sciences, Moscow, Russia
2. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia
Interests: partial differential equations & mathematical physics; elasticity system; stokes system; biharmonic (polyharmonic) equation
Special Issues, Collections and Topics in MDPI journals
Department of General Mathematics, Lomonosov Moscow State University, Moscow 11992, Russia
Interests: quasi-classical asymtotics of ODE’s and PDE’s; resurgent analysis; functional analysis
Special Issues, Collections and Topics in MDPI journals
1. Institute of Applied Mechanics, Russian Academy of Sciences, Moscow 125040, Russia
2. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow 119526, Russia
3. Lomonosov Moscow State University, Moscow 11992, Russia
4. Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, Moscow 119333, Russia
Interests: continuum mechanics; gradient theory; micropolar and micro-morphic models

Special Issue Information

Dear Colleagues,

The main topics of this Special Issue are:

  • Mathematical Physics and PDE

Asymtotics of ODEs and PDEs; mathematical physics and PDE including fluid dynamics, wave equation, Boltzmann equation; solvability, regularity, stability and other qualitative properties of linear and nonlinear equations and systems; spectral theory, scattering, inverse problems; variational methods and calculus of variations; fluid dynamics; dynamical systems;

  • Applied Mathematics

Numerical methods of ODEs and PDEs, nonlinear problems, bifurcations, stability, chaos and fractals, fractional calculus;

  • Related Problems of Analysis and Continuum Mechanics

Stochastic models and probabilistic methods including random matrices and stochastic PDE; variational formulations of gradient elasticity theories, micropolar and micromorphic models of solids and fluids; constructive methods for representing solutions in the high-order theories of solids; general representations of solutions; nonlocal effects (interactions) and smooth solutions for bodies with nonsmooth geometry; identification of materials parameters of generalized continuum theories; size-dependent models of thin structures and composite materials; nonclassical dynamic effects in generalized media; gradient theories in multiphysics, including thermodynamic, thermodiffusion, electroelasticity processes in the nano- and microfields of mechanical engineering and related applied problems.

Prof. Dr. Hovik Matevossian
Prof. Dr. Maria Korovina
Prof. Dr. Sergey Lurie
Guest Editors

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Published Papers (9 papers)

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Research

Article
To the Problem of Discontinuous Solutions in Applied Mathematics
Mathematics 2023, 11(15), 3362; https://doi.org/10.3390/math11153362 - 01 Aug 2023
Viewed by 263
Abstract
This paper addresses discontinuities in the solutions of mathematical physics that describe actual processes and are not observed in experiments. The appearance of discontinuities is associated in this paper with the classical differential calculus based on the analysis of infinitesimal quantities. Nonlocal functions [...] Read more.
This paper addresses discontinuities in the solutions of mathematical physics that describe actual processes and are not observed in experiments. The appearance of discontinuities is associated in this paper with the classical differential calculus based on the analysis of infinitesimal quantities. Nonlocal functions and nonlocal derivatives, which are not specified, in contrast to the traditional approach to a point, but are the results of averaging over small but finite intervals of the independent variable are introduced. Classical equations of mathematical physics preserve the traditional form but include nonlocal functions. These equations are supplemented with additional equations that link nonlocal and traditional functions. The proposed approach results in continuous solutions of the classical singular problems of mathematical physics. The problems of a string and a circular membrane loaded with concentrated forces are used to demonstrate the procedure. Analytical results are supported with experimental data. Full article
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Article
Transient Wave Propagation in Functionally Graded Viscoelastic Structures
Mathematics 2022, 10(23), 4505; https://doi.org/10.3390/math10234505 - 29 Nov 2022
Viewed by 743
Abstract
Transient wave processes in viscoelastic structures built from functionally graded material (FGM) still remain almost unexplored. In this article, the problem of the propagation of nonstationary longitudinal waves in an infinite viscoelastic layer of a FGM with plane–parallel boundaries is considered. The physical [...] Read more.
Transient wave processes in viscoelastic structures built from functionally graded material (FGM) still remain almost unexplored. In this article, the problem of the propagation of nonstationary longitudinal waves in an infinite viscoelastic layer of a FGM with plane–parallel boundaries is considered. The physical and mechanical parameters of the FGM depend continuously on the transverse coordinate, while the wave process propagates along the same coordinate. The viscoelastic properties of the material are taken into account employing the linear integral Boltzmann–Volterra relations. The viscoelastic layer of the FGM is replaced by a piecewise-homogeneous layer consisting of a large number of sub-layers (a package of homogeneous layers), thus approximating the continuous inhomogeneity of the FGM. A solution of a non-stationary dynamic problem for a piecewise-homogeneous layer is constructed and, using a specific example, the convergence of the results with an increase in the number of sub-layers in the approximating piecewise-homogeneous layer is shown. Furthermore, the problem of the propagation of nonstationary longitudinal waves in the cross section of an infinitely long viscoelastic hollow FGM cylinder, whose material properties continuously change along the radius, is also considered. The cylinder composed of the FGM is replaced by a piecewise-homogeneous one, consisting of a large number of coaxial layers, for which the solution of the non-stationary dynamic problem is constructed. For both the layer and the cylinder composed of a viscoelastic FGM, the results of calculating the characteristic parameters of the wave processes for the various initial data are presented. The influence of the viscosity and inhomogeneity of the material on the dynamic processes is demonstrated. Full article
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Article
Functionally Graded Thin Circular Plates with Different Moduli in Tension and Compression: Improved Föppl–von Kármán Equations and Its Biparametric Perturbation Solution
Mathematics 2022, 10(19), 3459; https://doi.org/10.3390/math10193459 - 22 Sep 2022
Cited by 1 | Viewed by 805
Abstract
The biparametric perturbation method is applied to solve the improved Föppl–von Kármán equation, in which the improvements of equations come from two different aspects: the first aspect concerns materials, and the other is from deformation. The material considered in this study has bimodular [...] Read more.
The biparametric perturbation method is applied to solve the improved Föppl–von Kármán equation, in which the improvements of equations come from two different aspects: the first aspect concerns materials, and the other is from deformation. The material considered in this study has bimodular functionally graded properties in comparison with the traditional materials commonly used in classical Föppl–von Kármán equations. At the same time, the consideration for deformation deals with not only the large deflection as indicated in classical Föppl–von Kármán equations, but also the larger rotation angle, which is incorporated by adopting the precise curvature formulas but not the simple second-order derivative term of the deflection. To fully demonstrate the effectiveness of the biparametric perturbation method proposed, two sets of parameter combinations, one being a material parameter with central defection and the other being a material parameter with load, are used for the solution of the improved Föppl–von Kármán equations. Results indicate that not only the two sets of solutions from different parameter combinations are consistent, but also they may be reduced to the single-parameter perturbation solution obtained in our previous study. The successful application of the biparametric perturbation method provides new ideas for solving similar nonlinear differential equations. Full article
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Article
Asymptotic Behavior of Solutions of the Cauchy Problem for a Hyperbolic Equation with Periodic Coefficients (Case: H0 > 0)
Mathematics 2022, 10(16), 2963; https://doi.org/10.3390/math10162963 - 17 Aug 2022
Cited by 5 | Viewed by 798
Abstract
The main goal of this article is to study the behavior of solutions of non-stationary problems at large timescales, namely, to obtain an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic [...] Read more.
The main goal of this article is to study the behavior of solutions of non-stationary problems at large timescales, namely, to obtain an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients at large values of the time parameter t. To obtain an asymptotic expansion as t, the basic methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of the Hill operator with periodic coefficients in the case when the operator is positive: H0>0. Full article
Article
On the Existence and Uniqueness of an Rν-Generalized Solution to the Stokes Problem with Corner Singularity
Mathematics 2022, 10(10), 1752; https://doi.org/10.3390/math10101752 - 20 May 2022
Cited by 4 | Viewed by 868
Abstract
We consider the Stokes problem with the homogeneous Dirichlet boundary condition in a polygonal domain with one re-entrant corner on its boundary. We define an Rν-generalized solution of the problem in a nonsymmetric variational formulation. Such defined solution allows us to [...] Read more.
We consider the Stokes problem with the homogeneous Dirichlet boundary condition in a polygonal domain with one re-entrant corner on its boundary. We define an Rν-generalized solution of the problem in a nonsymmetric variational formulation. Such defined solution allows us to construct numerical methods for finding an approximate solution without loss of accuracy. In the paper, the existence and uniqueness of an Rν-generalized solution in weighted sets is proved. Full article
Article
Neutral Differential Equations of Second-Order: Iterative Monotonic Properties
Mathematics 2022, 10(9), 1356; https://doi.org/10.3390/math10091356 - 19 Apr 2022
Cited by 5 | Viewed by 899
Abstract
In this work, we investigate the oscillatory properties of the neutral differential equation [...] Read more.
In this work, we investigate the oscillatory properties of the neutral differential equation (r(l)[(s(l)+p(l)s(g(l)))]v)+i=1nqi(l)sv(hi(l))=0, where ss0. We first present new monotonic properties for the solutions of this equation, and these properties are characterized by an iterative nature. Using these new properties, we obtain new oscillation conditions that guarantee that all solutions are oscillate. Our results are a complement and extension to the relevant results in the literature. We test the significance of the results by applying them to special cases of the studied equation. Full article
Article
Asymptotic Behavior of Solutions of Integral Equations with Homogeneous Kernels
Mathematics 2022, 10(2), 180; https://doi.org/10.3390/math10020180 - 07 Jan 2022
Cited by 1 | Viewed by 687
Abstract
The multidimensional integral equation of second kind with a homogeneous of degree (n) kernel is considered. The special class of continuous functions with a given asymptotic behavior in the neighborhood of zero is defined. It is proved that, if the [...] Read more.
The multidimensional integral equation of second kind with a homogeneous of degree (n) kernel is considered. The special class of continuous functions with a given asymptotic behavior in the neighborhood of zero is defined. It is proved that, if the free term of the integral equation belongs to this class and the equation itself is solvable, then its solution also belongs to this class. To solve this problem, a special research technique is used. The above-mentioned technique is based on the decomposition of both the solution and the free term in spherical harmonics. Full article
Article
Analytical Solution of Stationary Coupled Thermoelasticity Problem for Inhomogeneous Structures
Mathematics 2022, 10(1), 90; https://doi.org/10.3390/math10010090 - 27 Dec 2021
Cited by 2 | Viewed by 1812
Abstract
A mathematical statement for the coupled stationary thermoelasticity is given on the basis of a variational approach and the contact boundary problem is formulated to consider inhomogeneous materials. The structure of general representation of the solution from the set of the auxiliary potentials [...] Read more.
A mathematical statement for the coupled stationary thermoelasticity is given on the basis of a variational approach and the contact boundary problem is formulated to consider inhomogeneous materials. The structure of general representation of the solution from the set of the auxiliary potentials is established. The potentials are analyzed depending on the parameters of the model, taking into account the restrictions associated with additional requirements for the positive definiteness of the potential energy density for the coupled problem in the one-dimensional case. The novelty of this work lies in the fact that it attempts to take into account the effects of higher order coupling between the gradients of the temperature fields and the gradients of the deformation fields. From a mathematical point of view, this leads to a change in the roots of the characteristic equation and affects the structure of the solution. Contact boundary value problems are formulated for modeling inhomogeneous materials and a solution for a layered structure is constructed. The analysis of the influence of the model parameters on the structure of the solution is given. The features of the distribution of mechanical and thermal fields in the region of phase contact with a change in the parameters, which are characteristic only for gradient theories of coupled thermoelasticity and stationary thermal conductivity, are discussed. It is shown, for example, that taking into account the additional parameter of connectivity of gradient fields of deformations and temperatures predicts the appearance of rapidly changing temperature fields and significant localization of heat fluxes in the vicinity of phase contact in inhomogeneous materials. Full article
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Article
Analytical Solution of the Three-Dimensional Laplace Equation in Terms of Linear Combinations of Hypergeometric Functions
Mathematics 2021, 9(24), 3316; https://doi.org/10.3390/math9243316 - 20 Dec 2021
Cited by 4 | Viewed by 2130
Abstract
We present some solutions of the three-dimensional Laplace equation in terms of linear combinations of generalized hyperogeometric functions in prolate elliptic geometry, which simulates the current tokamak shapes. Such solutions are valid for particular parameter values. The derived solutions are compared with the [...] Read more.
We present some solutions of the three-dimensional Laplace equation in terms of linear combinations of generalized hyperogeometric functions in prolate elliptic geometry, which simulates the current tokamak shapes. Such solutions are valid for particular parameter values. The derived solutions are compared with the solutions obtained in the standard toroidal geometry. Full article
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