Abstract: Abstract
The unsteady heat equation is considered in thin structures. The asymptotic expansion of the solution constructed earlier is used to evaluate partial derivatives of the solution. The method of partial asymptotic domain decomposition is applied to the unsteady heat equation. It reduces the original 2D model to a hybrid dimension one, partially 2D, partially 1D with some special interface conditions between the 2D and 1D parts. The finite volume method is applied to numerically solve the hybrid dimension model. The error estimate is established. A numerical experiment confirms the theoretical error evaluation. PubDate: 2015-04-01

Abstract: Abstract
A fluid flow along a plate with small irregularities on the surface is considered for large Reynolds numbers. The boundary layer has a double-deck structure, i.e., both a thin boundary layer and the classical Prandtl boundary layer are present. It is proved that the solution of the boundary-value problem thus obtained exists and is unique in the Prandtl boundary layer, and the stability of the solution is investigated at large times. The results of numerical modeling are given. Supported by the Basic Research Program of the National Research University “Higher School of Economics.” PubDate: 2015-04-01

Abstract: Abstract
A periodic system of domains coupled by small windows is considered. In a domain of this kind, we study the band spectrum of a Schrödinger operator subjected to the Neumann condition. We show that, near every isolated eigenvalue of a similar operator in the periodicity cell, there are several nonintersecting bands of the spectrum for the perturbed operator. We also discuss the position of points at which the band functions attain the edges of each band. PubDate: 2015-04-01

Abstract: Abstract
We show that a distribution of the type of the Bose-Einstein distribution describes the van der Waals gas, while the Fermi-Dirac distribution describes the van der Waals liquid. We present the construction of the binodal, the melting curve, and the liquid-to-amorphous-solid transition under negative pressure. The notion of correlation sphere and the two-scale picture on the Hougen-Watson diagram are used. PubDate: 2015-04-01

Abstract: Abstract
All nonequivalent integrable evolution equations of the fifth order of the form
\(u_t = D_x \tfrac{{\delta H}}
{{\delta u}}\)
are found. PubDate: 2015-04-01

Abstract: Abstract
This paper deals with the coarsening operation in dynamical systems where the phase space with a finite invariant measure is partitioned into measurable pieces and the summable function transferred by the phase flow is averaged over these pieces at each instant of time. Letting the time tend to infinity and then refining the partition, we arrive at a modernization of the von Neumann ergodic theorem, which is useful for the purposes of nonequilibrium statistical mechanics. In particular, for fine-grained partitions, we obtain the law of increment of coarse entropy for systems approaching the state of statistical equilibrium. PubDate: 2015-04-01

Abstract: Abstract
Using the Smirnov-Sobolev approach, we reduce the nonstationary problem of diffraction of a plane wave by an impedance wedge to the Hilbert problem on a half-plane. The index of the Hilbert problem is equal to one. The Hilbert problem is “solved in quadratures.” In another way and in another form, a similar diffraction problem was solved earlier by Popandopulos [J. Aust. Math. Soc. 1 (1), 97–106 (1961)]. PubDate: 2015-04-01

Abstract: Abstract
Error estimates for homogenization in L
2- and H
1-norms for an equation with rapidly oscillating quasiperiodic coefficients are studied. PubDate: 2015-04-01

Abstract: Abstract
A singularly perturbed periodic problem for a parabolic reaction-advection-diffusion equation at low advection is studied. The case when there is an internal transition layer under unbalanced nonlinearity is considered. An asymptotic expansion of a solution is constructed. To substantiate the asymptotics thus constructed, the asymptotic method of differential inequalities is used. The Lyapunov asymptotic stability of a periodic solution is studied; the proof uses the Krein-Rutman theorem. PubDate: 2015-04-01

Abstract: Abstract
Exact solutions of the linear water-wave problem describing oblique water waves trapped by a submerged horizontal cylinder of small (but otherwise arbitrary) cross-section are constructed in the form of a convergent series in powers of the small parameter characterizing the “thinness” of the cylinder. The terms of this series are expressed through the solutions of the exterior Neumann problem for the Laplace equation describing the flow of unbounded fluid past the cylinder. PubDate: 2015-04-01

Abstract: Abstract
In this paper, we introduce and investigate a fractional integral operator which contains Fox’s H-function in its kernel. We find solutions to some fractional differential equations by using this operator. The results derived in this paper generalize the results obtained in earlier works by Kilbas et al. [7] and Srivastava and Tomovski [23]. A number of corollaries and consequences of the main results are also considered. Using some of these corollaries, graphical illustrations are presented and it is found that the graphs given here are quite comparable to the physical phenomena of decay processes. PubDate: 2015-01-01

Abstract: Abstract
A modified version of the ASEP model is interpreted as a two-dimensional model of ideal gas. Its properties are studied by simulating its behavior in different situations, using an animation program designed for that purpose. PubDate: 2015-01-01

Abstract: Abstract
In this paper, we consider poly-Bernoulli and higher-order poly-Bernoulli polynomials and derive some new and interesting identities of those polynomials by using umbral calculus. PubDate: 2015-01-01

Abstract: Abstract
By equating to zero all components of the Kröner incompatibility tensor of rank 2n − 4 or of the Riemann tensor dual to the Kröner tensor, n
2(n
2 − 1)/12 independent consistency equations for the stresses in an n-dimensional isotropic elastic medium are derived. The problem concerning the equivalence of the system of these equations to systems following from equating to zero either all n(n + 1)/2 components of the Ricci tensor or only one curvature invariant is investigated. It is shown that the answer to this question depends on the dimension of the space. Three cases are singled out: n = 2 (plane problem of elasticity theory), n = 3 (spatial problem of elasticity theory), and n ⩾ 4. PubDate: 2015-01-01

Abstract: Abstract
The aim of this paper is to construct new generating functions for Hermite base Bernoulli type polynomials, which generalize not only the Milne-Thomson polynomials but also the two-variable Hermite polynomials. We also modify the Milne-Thomson polynomials, which are related to the Bernoulli polynomials and the Hermite polynomials. Moreover, by applying the umbral algebra to these generating functions, we derive new identities for the Bernoulli polynomials of higher order, the Hermite polynomials and numbers of higher order, and the Stirling numbers of the second kind. PubDate: 2015-01-01

Abstract: Abstract
For slow-fast Hamiltonian systems with one fast degree of freedom, we describe the construction of the complete adiabatic invariant and the complete adiabatic term at once in all asymptotic orders by using the small parameter “dynamics” and parallel translations in the phase space. PubDate: 2015-01-01