Editorial

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33 pages, 370 KiB

Open AccessEditorial

Editorial: S. N. Mergelyan’s Dissertation “Best Approximations in the Complex Domain”

by Sergey N. Mergelyan

Mathematics 2024, 12(7), 939; https://doi.org/10.3390/math12070939 - 22 Mar 2024

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Abstract

Preface by Hovik A [...] Full article

(This article belongs to the Special Issue Differential Equations of Mathematical Physics and Related Problems of Mechanics)

5 pages, 159 KiB

Open AccessEditorial

“Differential Equations of Mathematical Physics and Related Problems of Mechanics”—Editorial 2021–2023

by Hovik A. Matevossian

Mathematics 2024, 12(1), 150; https://doi.org/10.3390/math12010150 - 02 Jan 2024

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Abstract

Based on the published papers in this Special Issue of the elite scientific journal Mathematics, we herein present the Editorial for “Differential Equations of Mathematical Physics and Related Problems of Mechanics”, the main topics of which are fundamental and applied research on [...] Read more.

Based on the published papers in this Special Issue of the elite scientific journal Mathematics, we herein present the Editorial for “Differential Equations of Mathematical Physics and Related Problems of Mechanics”, the main topics of which are fundamental and applied research on differential equations in mathematical physics and mechanics [...] Full article

(This article belongs to the Special Issue Differential Equations of Mathematical Physics and Related Problems of Mechanics)

Research

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12 pages, 430 KiB

Open AccessFeature PaperArticle

Analytical Solutions of Partial Differential Equations Modeling the Mechanical Behavior of Non-Prismatic Slender Continua

by Giovanni Migliaccio

Mathematics 2023, 11(23), 4723; https://doi.org/10.3390/math11234723 - 22 Nov 2023

Viewed by 698

Abstract

Non-prismatic slender continua are the prototypical models of many structural elements used in engineering applications, such as wind turbine blades and towers. Unfortunately, closed-form expressions for stresses and strains in such continua are much more difficult to find than in prismatic ones, e.g., [...] Read more.

Non-prismatic slender continua are the prototypical models of many structural elements used in engineering applications, such as wind turbine blades and towers. Unfortunately, closed-form expressions for stresses and strains in such continua are much more difficult to find than in prismatic ones, e.g., the de Saint-Venant’s cylinder, for which some analytical solutions are known. Starting from a suitable mechanical model of a tapered slender continuum with one dimension much larger than the other tapered two, a variational principle is exploited to derive the field equations, i.e., the set of partial differential equations and boundary conditions that govern its state of stress and strain. The obtained equations can be solved in closed form only in a few cases. Paradigmatic examples in which analytical solutions are obtainable in terms of stresses, strains, or related mechanical quantities of interest in engineering applications are presented and discussed. Full article

(This article belongs to the Special Issue Differential Equations of Mathematical Physics and Related Problems of Mechanics)

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10 pages, 1910 KiB

Open AccessArticle

To the Problem of Discontinuous Solutions in Applied Mathematics

by Valery V. Vasiliev and Sergey A. Lurie

Mathematics 2023, 11(15), 3362; https://doi.org/10.3390/math11153362 - 01 Aug 2023

Viewed by 559

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

(This article belongs to the Special Issue Differential Equations of Mathematical Physics and Related Problems of Mechanics)

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18 pages, 16410 KiB

Open AccessFeature PaperArticle

Transient Wave Propagation in Functionally Graded Viscoelastic Structures

by Sergey Pshenichnov, Radan Ivanov and Maria Datcheva

Mathematics 2022, 10(23), 4505; https://doi.org/10.3390/math10234505 - 29 Nov 2022

Viewed by 1098

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

(This article belongs to the Special Issue Differential Equations of Mathematical Physics and Related Problems of Mechanics)

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26 pages, 691 KiB

Open AccessArticle

Functionally Graded Thin Circular Plates with Different Moduli in Tension and Compression: Improved Föppl–von Kármán Equations and Its Biparametric Perturbation Solution

by Xiao-Ting He, Bo Pang, Jie-Chuan Ai and Jun-Yi Sun

Mathematics 2022, 10(19), 3459; https://doi.org/10.3390/math10193459 - 22 Sep 2022

Cited by 2 | Viewed by 1191

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

(This article belongs to the Special Issue Differential Equations of Mathematical Physics and Related Problems of Mechanics)

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26 pages, 393 KiB

Open AccessFeature PaperArticle

Asymptotic Behavior of Solutions of the Cauchy Problem for a Hyperbolic Equation with Periodic Coefficients (Case: H₀ > 0)

by Hovik A. Matevossian, Maria V. Korovina and Vladimir A. Vestyak

Mathematics 2022, 10(16), 2963; https://doi.org/10.3390/math10162963 - 17 Aug 2022

Cited by 7 | Viewed by 1127

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 \to \infty

, 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:

H_{0} > 0

. Full article

(This article belongs to the Special Issue Differential Equations of Mathematical Physics and Related Problems of Mechanics)

14 pages, 307 KiB

Open AccessArticle

On the Existence and Uniqueness of an R_ν-Generalized Solution to the Stokes Problem with Corner Singularity

by Viktor A. Rukavishnikov and Alexey V. Rukavishnikov

Mathematics 2022, 10(10), 1752; https://doi.org/10.3390/math10101752 - 20 May 2022

Cited by 6 | Viewed by 1098

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

R_{ν}

-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

R_{ν}

-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

R_{ν}

-generalized solution in weighted sets is proved. Full article

(This article belongs to the Special Issue Differential Equations of Mathematical Physics and Related Problems of Mechanics)

11 pages, 276 KiB

Open AccessArticle

Neutral Differential Equations of Second-Order: Iterative Monotonic Properties

by Osama Moaaz, Fahd Masood, Clemente Cesarano, Shami A. M. Alsallami, E. M. Khalil and Mohamed L. Bouazizi

Mathematics 2022, 10(9), 1356; https://doi.org/10.3390/math10091356 - 19 Apr 2022

Cited by 9 | Viewed by 1166

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})}^{'} + \sum_{i = 1}^{n} q_{i} (l) s^{v} (h_{i} (l)) = 0

, where

s \geq s_{0}

. 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

(This article belongs to the Special Issue Differential Equations of Mathematical Physics and Related Problems of Mechanics)

11 pages, 284 KiB

Open AccessArticle

Asymptotic Behavior of Solutions of Integral Equations with Homogeneous Kernels

by Oleg Avsyankin

Mathematics 2022, 10(2), 180; https://doi.org/10.3390/math10020180 - 07 Jan 2022

Cited by 1 | Viewed by 933

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

(This article belongs to the Special Issue Differential Equations of Mathematical Physics and Related Problems of Mechanics)

16 pages, 448 KiB

Open AccessArticle

Analytical Solution of the Three-Dimensional Laplace Equation in Terms of Linear Combinations of Hypergeometric Functions

by Antonella Lupica, Clemente Cesarano, Flavio Crisanti and Artur Ishkhanyan

Mathematics 2021, 9(24), 3316; https://doi.org/10.3390/math9243316 - 20 Dec 2021

Cited by 6 | Viewed by 2577

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

(This article belongs to the Special Issue Differential Equations of Mathematical Physics and Related Problems of Mechanics)

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