Vol 231 (2024)
Статьи
Periodic solutions of a differential equation with relay nonlinearity with delay
Abstract
For one class of second-order differential equations with relay nonlinearity and delay, orbitally stable periodic solutions are found by means of the recurrence operator, which is a suspension over some one-dimensional mapping. The analysis of this one-dimensional mapping shows that there exist domains of parameters for which exponentially orbitally stable periodic solutions exist.
3-12
Analytical estimates of the accuracy of wind profile reconstruction from lidar scanning data
Abstract
We consider the problem of reconstructing three components of wind velocity from measurement data of the radial component along directions uniformly located on the surface of a vertical cone using the least squares method. Estimates are obtained for the maximum error in the reconstruction of each component of the wind speed vector and for the mean square errors in the asymptotic approximation. Estimates are obtained taking into account the completeness of measurement data.
13-26
Construction of regularized asymptotics for the solution of a singularly perturbed mixed problem on the half-axis for the inhomogeneous Schrödinger-type equation with the potential V(x) = x
Abstract
In this paper, we propose a method for constructing an asymptotic solution to a singularly perturbed mixed problem on the half-axis for a nonstationary inhomogeneous Schrödinger-type equation in the coordinate representation in the case of violation of the stability conditions for the spectrum of the limit operator. The chosen profile of the potential energy leads to a spectral singularity of the limit operator, which, within the framework of S. A. Lomov’s regularization method, is usually called a strong turning point.
27-43
Maximum flow in parallel networks with connected arcs
Abstract
The well-known problem of finding maximum flows in classical networks has many solution algorithms that have a polynomial computational complexity depending on the size of the network. In general, the problem of finding maximum flows for networks with connected arcs is NP-complete. However, among the previously studied networks with connected arcs, there are networks for which the calculation of maximum flows is feasible in a time polynomially depending on the size of the network. This work is devoted to determining the influence of the topology of networks with connected arcs on the possibility of finding maximum flows in polynomial time. In this paper, for a class of parallel networks with connected arcs, we propose a fast polynomial algorithm for finding maximum flows.
44-52
Moment functions for a solution of a stochastic system of partial differential equations
Abstract
In this paper, we consider the Cauchy problem for a linear inhomogeneous system of first-order partial differential equations with two random coefficients and a random inhomogeneity. Explicit formulas for the moment functions of the solution are obtained: mathematical expectation, mixed moment functions, and the second moment function. As applications, explicit formulas for mixed moment functions and the second moment function for solutions of an equation with independent Gaussian random coefficients are obtained.
53-67
Green’s formulas for the Kipriyanov ΔB - operator in the weighted linear form
Abstract
For the Kipriyanov singular differential operator in the Euclidean n - half-space, the general Green formula and two its particular cases corresponding to special values of the parameter are obtained.
68-73
Treatment of symmetry in the Ritz method for the Schrödinger equation in crystals with a basis
Abstract
This paper is devoted to treatment of symmetry in the Schrödinger equation with a periodic potential for crystals with a basis. We present a general group-theoretical approach, which yields the matrix elements of the Hamiltonian in the tight-binding approximation, using the spatial symmetry of the problem, time reversal symmetry, and the Hermitian property of the Hamiltonian. The developed mathematical theory generalizes the well-known result for crystals with two atoms in the unit cell to the case of crystals with several atoms in the unit cell.
74-82
On the asymptotic stability of one equation with a discrete retarded argument
Abstract
In this paper, we consider a functional differential equation with a discrete retarded argument, a constant delay, and a term without delay. The problem of the asymptotic stability of this equation is reduced to the problem of studying the spectrum of the operator of shift along trajectories. Simple coefficient necessary conditions for the asymptotic stability of this equation are obtained.
83-88
On the reconstruction of solutions of the Cauchy problem for the singular heat equation
Abstract
The problem of reconstructing solutions of the singular heat equation on the positive part of the real axis at a certain moment of time is solved by inaccurate measurements of this solution at other previous moments of time. Explicit expressions for the optimal reconstruction method and its errors are obtained.
89-99
On the construction of solutions of the inhomogeneous biharmonic equation in problems of mechanics of thin isotropic plates
Abstract
In this paper, we propose a method for constructing a solution of the inhomogeneous biharmonic equation as applied to problems in the mechanics of thin isotropic plates. The method is based on the Chebyshev polynomial approximation of the eighth-order mixed partial derivative of the unknown function. Chebyshev polynomials of the first kind were used as basis functions. The proposed method is used to simulate the bending of an elastic isotropic rectangular plate under the action of a transverse load. The results obtained by the collocation method are analyzed; the roots of Chebyshev polynomials of the first kind are used as collocation points.
100-106
l - Problem of moments in problems of optimal control and state estimation for multidimensional fractional linear systems
Abstract
In this paper, we consider multidimensional dynamical systems whose states are described by systems of linear fractional differential equations of different order. We examine problems of optimal control and optimal state estimation for systems with the Caputo and Riemann–Liouville fractional differentiation operators. We prove that under certain conditions both problems can be reduced to the l -problem of moments. For the resulting problem, the solvability conditions are verified and, in a number of cases, exact solutions are constructed.
107-114
Linear conjugation problem for the Cauchy–Riemann equation with a strong singularity in the lowest coefficient in a domain with piecewise smooth boundary
Abstract
In this work, a general solution of the Cauchy–Riemann equation with strong singularities in the lowest coefficient is constructed and the boundary-value problem of linear conjugation in a domain with a piecewise smooth boundary is examined.
115-123
124-133