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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 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

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

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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 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

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

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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

,

,

and

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

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

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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 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 \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)

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 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

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)

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 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})}^{'} + \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)

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 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

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

Open AccessEditor’s ChoiceArticle

Analytical Solution of Stationary Coupled Thermoelasticity Problem for Inhomogeneous Structures

by

Sergey A. Lurie

,

Dmitrii B. Volkov-Bogorodskii

and

Petr A. Belov

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

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

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Open AccessArticle

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

by

,

,

and

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

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

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