Mathematical Models and Methods in Applied Sciences

In this paper we consider a particle moving in a random distribution of obstacles. Each obstacle generates an inverse power law potential ε^s/|x|^s, where ε is a small parameter and s>2. Such a rescaled potential is truncated at distance ε^γ-1, where γ∈] 0, 1[ is suitably large. We also assume that the scatterer density is diverging as ε^-d+1, where d is the dimension of the physical space.

We prove that, as ε→0 (the Boltzmann–Grad limit), the probability density of a test particle converges to a solution of the linear (uncutoff) Boltzmann equation with the cross-section relative to the potential V(x)=|x|^-s.

We prove that under a small perturbation (in the Euclidean metric) of the boundary of a Lipschitz domain there corresponds a small variation (in the uniform norm) of the elastic far-field pattern. The corresponding estimate is of Hölder type. This is done under the assumptions that the Lipschitz character of the perturbation is bounded and that the frequency of the elastic waves is either sufficiently small or has a strictly positive imaginary part.

In this paper we introduce and analyze an a posteriori error estimator for the approximation of the eigenvalues and eigenvectors of a second-order elliptic problem obtained by the mixed finite element method of Raviart–Thomas of the lowest order. We define an error estimator of the residual type which can be computed locally from the approximate eigenvector and prove that the estimator is equivalent to the norm of the error in the approximation of the eigenvector up to higher order terms. The constants involved in this equivalence depend on the corresponding eigenvalue but are independent of the mesh size, provided the meshes satisfy the usual minimum angle condition. Moreover, the square root of the error in the approximation of the eigenvalue is also bounded by a constant times the estimator.

A mixed-hybrid formulation of the elasticity problem with a nonconforming symmetric approximation of the stress–tensor is considered. Based on such a formulation, a new finite element of low order with minimal number of degrees of freedom is constructed. Optimal error estimates are derived. Moreover all estimates are valid uniformly with respect to compressibility and apply for the Stokes problem. Finally, an equivalence between this finite element and the piecewise quadratic nonconforming approximation of the elasticity problem is established.

This work is devoted to diffusion stochastic processes (DSPs) with nonlinear coefficients in n-dimensional Euclidean space at high n (n is much greater than a few units). It deals with expectation and variance of a nonstationary process whereas our accompanying work deals with covariance and spectral density of a stationary process. Combined, analytical-numerical approach is a reasonable and perhaps the only way to treat high-dimensional DSPs in practice. Each of the above works develops the corresponding parts of the analytical basis for this combined treatment. The present work proposes approximate analytical expressions for the expectation and variance in the form of two ordinary differential equation (ODE) systems. They are derived within DSP theory without any techniques directly related to stochastic differential equations. Both ODE systems allow for space nonhomogeneities of the diffusion and damping matrixes and thereby do take nonlinearities of the DSP coefficients into account. Some related topics like invariant processes and the aspects of practical implementation of the above expressions are discussed as well. A proper attention is paid to formulation of some features important in applications of DSPs to the real-world problems. The results of this work can equally be used in various engineering fields.

This work is devoted to diffusion stochastic processes (DSPs) with nonlinear coefficients in n-dimensional Euclidean space at high n (n is much greater than a few units). It deals with covariance and spectral density of a stationary process whereas our accompanying work deals with expectation and variance of a nonstationary process. Combined, analytical–numerical approach is a reasonable and perhaps the only way to treat high-dimensional DSPs in practice. Each of the above works develops the corresponding parts of the analytical basis for this combined treatment. The present work proposes approximate analytical expressions for the covariance in the efficient integral form. The integral formula is based on the obtained analytical results for the covariance, allows for nonlinearities of the drift function, and is derived within DSP theory. No techniques directly related to stochastic differential equations are involved. The approximate expression for the covariance is also applied to evaluate the approximation for the spectral density. Some aspects of practical implementation of these approximations are discussed. The results of this work can equally be used in various engineering fields.