卷 57, 编号 11 (2017)

The problem of approximation of a differentiable function of two variables by partial sums of a double Fourier–Bessel series is considered. Sharp estimates of the rate of convergence of the double Fourier–Bessel series on the class of differentiable functions of two variables characterized by a generalized modulus of continuity are obtained. The proofs of four theorems on this issue, which can be directly applied to solving particular problems of mathematical physics, approximation theory, etc., are presented.

For Laplace transform inversion, a method for constructing quadrature rules of the highest degree of accuracy based on an asymptotic distribution of roots of special orthogonal polynomials on the complex plane is proposed.

The grid-characteristic method on a sequence of embedded hierarchical grids is used to study the reflection and diffraction of elastic seismic waves propagating from an earthquake hypocenter to the Earth’s surface. More specifically, the destruction caused by seismic waves in complex heterogeneous structures, such as multi-story buildings, is analyzed. This study is based on computer modeling with the use of the grid-characteristic method, which provides a detailed description of wave processes in heterogeneous media, takes into account all types of emerging waves, and relies on algorithms that perform well on the boundaries of the integration domain and material interfaces. Applying a sequence of hierarchical grids makes it possible to simulate seismic wave propagation from an earthquake hypocenter to ground facilities of interest—multi-story buildings—and to investigate their seismic resistance.

An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative is considered on the set Ḡ = G ∪ S, where Ḡ = D̅ × [0 ≤ t ≤ T], D̅ = {0 ≤ x ≤ d}, S = S^l ∪ S, and S^l and S₀ are the lateral and lower boundaries. The parameter ε takes arbitrary values from the half-open interval (0,1]. In contrast to the well-known problem for the regular transport equation, for small values of ε, this problem involves a boundary layer of width O(ε) appearing in the neighborhood of S^l; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm. Convergence occurs only if h=dN^-1 ≪ ε and N₀^-1 ≪ 1, where N and N₀ are the numbers of grid intervals in x and t, respectively, and h is the mesh size in x. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in x and uniform in t. On such a grid, a monotone difference scheme for the initial–boundary value problem for the singularly perturbed transport equation converges ε-uniformly in the maximum norm at an Ϭ(N⁻¹ + N₀⁻¹) rate.

A numerical asymptotic model for the breaking of two-dimensional plane relativistic electron oscillations under a small deviation from axial symmetry is developed. The asymptotic theory makes use of the construction of time-uniformly applicable solutions to weakly nonlinear equations. A special finite-difference algorithm on staggered grids is used for numerical simulation. The numerical solutions of axially symmetric one-dimensional relativistic problems yield two-sided estimates for the breaking time. Some of the computations were performed on the “Chebyshev” supercomputer (Moscow State University).

A model kinetic equation is proposed for describing the dynamics of polyatomic gases. The numerical solution of the plane shock structure problem is used to compare it with the R-model. The numerical results are in satisfactory agreement. The model proposed is efficient in the terms of the number of computational operations.

The problem of recognition (classification) by precedents is considered. Issues of improving the recognition ability and the training rate of logical correctors, i.e., the recognition procedures based on the construction of correct sets of elementary classifiers, are studied. The concept of a correct set of generic elementary classifiers is introduced and used to construct and investigate a qualitatively new model of the logical corrector. This model uses a wider class of correcting functions than in the earlier constructed models of logical correctors.