Том 57, № 9 (2017)

A novel statement of the Fourier filtering problem based on the use of matrix Fourier filters instead of conventional multiplier filters is considered. The basic properties of the matrix Fourier filtering for the filters in the Hilbert–Schmidt class are established. It is proved that the solutions with a finite energy to the periodic initial boundary value problem for the quasi-linear functional differential diffusion equation with the matrix Fourier filtering Lipschitz continuously depend on the filter. The problem of optimal matrix Fourier filtering is formulated, and its solvability for various classes of matrix Fourier filters is proved. It is proved that the objective functional is differentiable with respect to the matrix Fourier filter, and the convergence of a version of the gradient projection method is also proved.

Tikhonov’s regularized method of least squares and its generalizations to non-Euclidean norms, including polyhedral, are considered. The regularized method of least squares is reduced to mathematical programming problems obtained by “instrumental” generalizations of the Tikhonov lemma on the minimal (in a certain norm) solution of a system of linear algebraic equations with respect to an unknown matrix. Further studies are needed for problems concerning the development of methods and algorithms for solving reduced mathematical programming problems in which the objective functions and admissible domains are constructed using polyhedral vector norms.

The general optimal control problem subject to irregular constraints is considered for which the factor of the objective functional in Pontryagin’s function may vanish. It turns out that, in the case of p-regular constraints, this drawback can be overcome and a constructive version of the p-order maximum principle can be formulated.

Multicriteria choice methods are developed by applying methods of criteria importance theory with uncertain information on criteria importance and with preferences varying along their scale. Formulas are given for computing importance coefficients and importance scale estimates that are “characteristic” representatives of the feasible set of these parameters. In the discrete case, an alternative with the highest probability of being optimal (for a uniform distribution of parameter value probabilities) can be used as the best one. It is shown how such alternatives can be found using the Monte Carlo method.

A locally one-dimensional scheme for a fractional tracer transport equation of order is considered. An a priori estimate is obtained for the solution of the scheme, and its convergence is proved in the uniform metric.

The asymptotic behavior of the solution with boundary layers in the time-independent mathematical model of reaction–diffusion–advection arising when describing the distribution of greenhouse gases in the surface atmospheric layer is studied. On the basis of the asymptotic method of differential inequalities, the existence of a boundary-layer solution and its asymptotic Lyapunov stability as a steady-state solution of the corresponding parabolic problem is proven. One of the results of this work is the determination of the local domain of the attraction of a boundary-layer solution.

The Richtmyer–Meshkov and Rayleigh–Taylor instabilities in viscoplastic (Bingham) fluids are studied in two-dimensional setting. The evolution of the Richtmyer–Meshkov instability in a Bingham fluid is analyzed as compared with its evolution in a Newtonian fluid. The critical amplitude of the initial perturbation in the velocity field is estimated. Numerical results obtained for Richtmyer–Meshkov and Rayleigh–Taylor instabilities in a Bingham fluid are presented and compared with those obtained for a Newtonian fluid.