Архив статей

The focus of our study is a dynamic frictional contact model that involves a viscoelastic body and a conductive foundation. We use Coulomb’s law to describe the frictional behavior, while a normal compliance model is employed to simulate the contact. We formulate a variational formulation for the problem, and we establish the existence of its unique weak solution using the Banach fixed point theorem. We propose a fully discrete scheme, using the finite element method for the spatial approximation and the Euler scheme for the discretization of the time derivatives. The errors on the solutions are derived, and the linear convergence is obtained under suitable regularity hypotheses. Some numerical simulations are included to show the performance of method

The paper deals with the concept of basic summability of residue function of interval function, which is a synonym for its differential form. As one comprehensive concept, it includes not only all known concepts of integrability, such as Newton’s, generalized Riemann and generalized Riemann — Stieltjes integrability, but also arithmetic series

We obtain fully constructive results on construction of trigonometric interpolation polynomials with multiple nodes. We construct polynomials interpolating periodic complex– valued functions of a real variable. The polynomials are represented in general form and in the form of expansions over fundamental polynomials. We provide examples and discuss unresolved problems

We show that the inverse scattering transform technique can be applied to obtain the time dependence of scattering data of the Zakharov — Shabat system, which is described by the loaded nonlinear Schr ̈odinger equation in the class of fast decaying functions. In addition we prove that the Cauchy problem for the loaded nonlinear Schr ̈odinger equation is uniquely solvable in the class of rapidly decreasing functions. We provide the explicit expression of a single soliton solution for the loaded nonlinear Schr ̈odinger equation. As an example, we find the soliton solution of the considered problem for an arbitrary non–zero continuous function

The Krause mean process serves as a comprehensive model for the dynamics of opinion exchange within multi–agent system wherein opinions are represented as vectors. In this paper, we propose a framework for opinion exchange dynamics by means of the Krause mean process that is generated by a cubic doubly stochastic matrix with positive influences. The primary objective is to establish a consensus within the multi–agent system

Nowadays, the problem of classification of integrable nonlinear partial differential equations and their discrete analogues in 1+1 dimensions is well–studied. Within the framework of the symmetry approach, there was obtained a complete description of integrable representatives of a number of classes of equations that are interesting from the point of view of application, see [17], [34], [26], [2]. The problem of exhaustive classification of integrable equations containing a large number of independent variables remains less studied due to its extreme complexity. The symmetry approach, which has proven to be the most effective tool for classifying equations of dimension 1+1, is not quite suitable for integrable classification of multidimensional equations. As it is noted in [27], in this problem the symmetry approach loses its efficiency due to problems with nonlocalities involved in higher symmetries

This paper is devoted to studying the reaction–diffusion systems with rapidly oscillating coefficients in the equations and in boundary conditions in domains with locally periodic oscillating boundary; on this boundary a Robin boundary condition is imposed. We consider the supercritical case, when the homogenization changes the Robin boundary condition on the oscillating boundary is to the homogeneous Dirichlet boundary condition in the limit as the small parameter, which characterizes oscillations of the boundary, tends to zero. In this case, we prove that the trajectory attractors of these systems converge in a weak sense to the trajectory attractors of the limit (homogenized) reaction–diffusion systems in the domain independent of the small parameter. For this aim we use the homogenization theory, asymptotic analysis and the approach of V. V. Chepyzhov and M. I. Vishik concerning trajectory attractors of dissipative evolution equations. The homogenization method and asymptotic analysis are used to derive the homogenized reaction–diffusion system and to prove the convergence of solutions. First we define the appropriate auxiliary functional spaces with weak topology, then, we prove the existence of trajectory attractors for these systems and formulate the main Theorem. Finally, we prove the main convergence result with the help of auxiliary lemmas

We obtain a number of spectral and functional inequalities related to Schr ̈odinger operators defined on antisymmetric functions. Among them are Lieb — Thirring and CLR inequalities. Besides, we find new constants for the Sobolev and the Gagliardo — Nirenberg inequalities restricted to antisymmetric functions