Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki

A Steklov-type boundary value problem for the Lame operator in a semi-cylinder containing a small cavity is investigated. The case is considered when an elastic, homogeneous isotropic medium filling a region with a small cavity is rigidly coupled to the lateral boundary of a semi-cylinder and the boundary of a small cavity, which corresponds to a homogeneous Dirichlet boundary condition, and the Steklov spectral condition is set on the basis of the semi-cylinder. The main result consists in proving a theorem on the convergence of the eigenelements of such a singularly perturbed boundary value problem to the eigenelements of the limiting problem (in a semi-cylinder without a cavity) with a small parameter ε > 0 tending to zero, characterizing the diameter of the cavity. To prove the theorem, a Hilbert space of infinitely differentiable vector functions with a finite Dirichlet integral over a semicylinder was introduced. In contrast to the situation with a limited domain, in the boundary value problem under study, the condition of finiteness of the Dirichlet integral is essential, since it generally ensures finiteness of the norm in the introduced space. The restriction on the finiteness of the Dirichlet integral made it possible to establish a priori estimates that guarantee the uniqueness of solutions to the limiting and perturbed boundary value problems and to establish the equivalence of norms necessary to prove the existence of a solution to the singularly perturbed boundary value problem under study.

The article is devoted to the analysis of the perturbation coefficients of the nonlinear distortion compensation model in fiber-optic communication lines. The case of long-range signal transmission is considered, for which the effect of signal dispersion is in some sense more significant than nonlinear distortion. This makes it possible to use an approximation of the nonlinear Schrodinger equation based on perturbation theory with respect to a small parameter of nonlinearity to describe the signal propagation process. Using this approximation, analytical expressions are obtained for the coefficients of the first-order model in the case of a Gaussian pulse shape. A number of numerical experiments have been carried out to study the structure of the coefficient matrix. It has been found that this matrix is well approximated by a small rank in the absence of attenuation and amplification. In addition, it was found that when taking into account the effects of signal attenuation and amplification, the rank of the matrix approaching the original matrix with a fixed error is higher than in experiments without attenuation. Research confirms that taking into account the symmetry of the matrix and its approximation with a small rank can reduce the computational complexity of the nonlinear distortion filtering algorithm for a single symbol from O(N²) to O(RN ln N), where N is the size of the matrix and R is its rank.

The analogue of the characteristic function, known from probability theory and mathematical statistics, is used to calculate moments of inertia (inertia integrals) of an arbitrary order. Bodies bounded by a rectangular parallelepiped, an ellipsoid, and an arbitrary tetrahedron are considered as examples.