Vol 54, No 7 (2018)

We propose an unconditionally stable method for solving the three-dimensional nonstationary Navier–Stokes equations in the velocity–pressure variables. The method is based on a conservative finite-difference scheme and the simultaneous solution of the momentum and continuity equations at each time layer. The velocity and pressure fields are calculated by using a parallel algorithm for solving systems of linear equations by the Gauss method.

In a Hilbert space, we consider an abstract linear parabolic equation defined on an interval with a nonlocal weighted integral condition imposed on the solution. This problem is solved approximately by a projection-difference method with the use of the implicit Euler method in the time variable. The approximation to the problem in the spatial variables is developed with the finite element method in mind. An estimate of the approximate solution is obtained, the convergence of the approximate solutions to the exact solution is proved, and the error estimates, as well as the orders of the rate of convergence, are established.

The problem on a fluid flow in a pulsating vessel is considered in the framework of the quasi-one-dimensional hemodynamic equations. The fluid viscosity is assumed to be anisotropic; i.e., the viscosity coefficient depends on the flow direction. The possible solutions are studied analytically and numerically.

We obtain constructive sufficient conditions for the unique solvability of a periodic boundary value problem for a matrix differential equation that generalizes the Lyapunov and Riccati equations, develop an algorithm for constructing the solution of this equation, estimate the domain where the solution is localized, and study the structural properties of the solution.

We consider two inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative. The inverse problems are reduced to systems of linear Volterra integral equations of the second kind for the unknown functions. These systems are used to prove the existence and uniqueness of the solution of the inverse problems and numerically solve them. The applicability of the methods developed here to the approximate solution of the problem on an unknown source in the heat equation is studied numerically.

We study a linear integro-differential equation with a coefficient that has finite-order zeros. To solve the equation approximately in a distribution space, we suggest and substantiate a generalized collocation method based on special interpolation polynomials.

We consider a linear time-invariant homogeneous system of first-order ordinary differential equations with a noninvertible matrix multiplying the derivative of the unknown vector function and with perturbed coefficients. We introduce a class of perturbations of the coefficient matrices of the system and determine conditions on the perturbations of this class under which they do not affect the internal structure of the system. We obtain sufficient conditions for the robust stability of the system under such perturbations.

We prove that the maximum angle between two arbitrary unstable solution subspaces of an exponentially dichotomous finite-dimensional system on the half-line tends to zero as the time variable tends to infinity. In particular, there exists a positive constant such that the angle between an arbitrary unstable solution subspace and the stable lineal of the system is greater than this constant for all sufficiently large times.