Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki

A method for reconstructing linearly parameterized (in particular, polynomial) functional dependencies from data with interval uncertainty is developed. In many situations, it provides more adequate processing of imprecise measurement and observation results than traditional probability-theoretic approaches. The proposed method uses the mathematical apparatus of interval analysis and is based on the so-called maximum compatibility principle. It allows efficient construction of nonlinear functional dependencies in the form of generalized polynomials from interval data arising in both dependent and independent variables. As a practical example, the processing of real data from an aluminothermic process for industrial waste utilization is considered, where the new method demonstrates significant advantages compared to the traditional least squares method.

A linear integro-differential equation with a singular differential operator in the principal part is studied. For its approximate solution in the space of generalized functions, special generalized versions of the methods of moments and subdomains are proposed and substantiated. Optimality of the methods in order of accuracy is established.

Based on the conjugation of optimal control methods, nonlinear analysis and the theory of ill-posed problems, the regularization of the Lagrange principle (LP) in a non-differential form, in regular and irregular variants, in a nonlinear (non-convex) optimization problem of a general Goursat–Darboux system with a pointwise state equality-constraint is considered. This constraint is understood as an equality in the Hilbert space of square-summable functions and contains a parameter additively included in it, which makes it possible to use a “nonlinear version” of the perturbation method for studying the problem. The main purpose of both variants of the regularized LP is the stable generation of generalized minimizing sequences (GMS) in the problem under consideration, the existence of a solution to which is not assumed a priori. They can be interpreted as GMS-forming (regularizing) operators, which associate with each set of initial data of the problem a subminimal (minimal) of its regular augmented Lagrangian corresponding to this set, the dual variable in which is generated in accordance with the procedures specified in these variants. The construction of the augmented Lagrangian is completely determined by the form of “nonlinear” subdifferentials of a lower semicontinuous and, generally speaking, nonconvex function of values as a function of the problem parameter. The proximal subgradient and the Frechet subdifferential, well known in nonlinear analysis, are used as the latter. In the special case, when the problem is regular, in the sense of the existence of a generalized Kuhn–Tucker vector in it, and its initial data (the integrand of the quality functional and the right-hand side of the controlled system) depend affinely on the control, the limit passage in the relations of the regularized LP leads to classical optimality conditions in the form of the nondifferential Kuhn–Tucker theorem and the Pontryagin maximum principle.

New algorithms for solving the Coulomb two-center problem of discrete and continuous spectra in prolate spheroidal coordinates with separation of independent variables are presented. Energy eigenvalues and separation constants, as well as eigenfunctions of the discrete spectrum, are calculated using the secant method and the finite element method (FEM) on an appropriate grid with a real parameter – the distance between the Coulomb centers. At each step of the secant method, eigen solutions of the discrete spectrum are computed using the KANTBP 5M program implementing FEM in the Maple system. For the continuous spectrum problem (at a fixed energy eigenvalue), it is sufficient to solve the eigenvalue problem for the quasianqular equation with respect to the separation constant and use it when solving the boundary value problem for the quasiradial equation with respect to the unknown phase shift and eigenfunction using the KANTBP 5M program. The results of test calculations agree with reference calculations performed by programs implementing alternative methods in FORTRAN with the required accuracy.

For an incompressible stratified elastic strip, we consider the Poincaré—Steklov operator that maps normal stresses into normal displacements on a part of the boundary. To construct the transfer function (TF) of this operator, a variational formulation of the boundary value problem for displacement transforms is used. A definition is given and the existence and uniqueness are proved for a generalized solution of the variational problem. This problem is approximated by the finite element method. The leading term of the asymptotic expansion of the TF for small and the three-term asymptotic expansion of the TF for large values of the Fourier transform parameter are obtained. Padé approximations of the obtained asymptotic series are constructed. To reduce computational costs a combined approach to calculating the TF has been developed using its asymptotic expansions and Padé approximations.

The H-function, specifically defined for the S-model kinetic equation, is studied for various physical situations. Spatially homogeneous relaxation is considered. A fairly broad class of initial conditions is investigated. It is shown numerically that the H-theorem is valid for them.

Numerical modeling of the separation of a binary gas mixture with similar molecular weights in a thermal micropump based on the radiometric effect is carried out. The simulation method is based on the direct solution of the Boltzmann kinetic equation using a splitting scheme. Relaxation problems are solved using a conservative projection method. Transport equations are solved using first- and second-order schemes. A design of an installation that can be used for separating mixtures of similar masses is proposed. Based on the modeling, an assessment of the efficiency of this device is performed.

This work investigates the applicability of lattice Boltzmann equations based on the Shakhov kinetic equation to modeling rarefied flows under significant external force. As a benchmark problem, a two-dimensional plane Poiseuille flow at different Knudsen numbers and different amplitudes of external force is considered, resulting in characteristic flow velocities corresponding to Mach numbers in the range of 0.4 to 0.6. It is shown that two-dimensional lattice models with 37 velocities can describe nonequilibrium effects beyond the applicability of the continuum medium approximation, in the slip flow regime. Profiles of longitudinal and transverse heat fluxes, as well as velocity and temperature profiles of the rarefied gas, are examined.