We consider spectral features and an iterative scheme of finding a spectrum of the product of two non-commutative partially symmetric operators in the Hilbert space H. In this case, it is assumed that one of the operators is compact, the second one not necessarily being compact and even bounded in H. Numerical implementation of the iterative scheme for finding the operator spectrum of the problem of eigen-oscillations of the Rayleigh beam is presented.

A numerical model for studying the dynamic mixing of sea and river waters in estuary areas is proposed. The model is based on two-dimensional longitudinal vertical stratified fluid mechanics equations and an equation of salt transport. The model is used to simulate local density currents in the estuaries of branched deltas of Siberian rivers. The results of numerical experiments are given. The flow dynamic structure and the salinity profile are compared with observational data.

This paper presents an algorithm based on the spectral Laguerre method for approximation of time derivatives as applied to a problem of seismic wave propagation in porous media with energy dissipation. The initial system of equations is written as a first-order hyperbolic system in terms of velocities, stresses, and pore pressure. To numerically solve the problem, a combination of an analytical Laguerre transform and a finite-difference method is used. The method proposed in the paper is an analog of a well-known spectral method based on the Fourier transform. However, unlike the Fourier transform, the integral Laguerre transform with respect to time reduces the initial problem to a system of equations in which the expansion parameter is present only in the right-hand side of the equations as a recurrence relation. As compared to the finite-difference method, with an analytical transform in the spectral method it is possible to reduce the original problem to a system of differential equations having only derivatives with respect to the spatial coordinates. This allows using the known stable difference scheme for the recurrence solutions to similar systems. Such an approach is effective when solving dynamic problems for porous media. Because of the presence of a second longitudinal wave with low velocity, the use of difference schemes in all the coordinates to obtain stable solutions requires a small step consistent both with respect to time and space, which inevitably increases the execution time.

In this paper, an implicit method of decomposition of cubic Hermite splines using a new type of multiwavelets with supercompact supports is investigated. A splitting algorithm of wavelet-transforms of solving two three-diagonal systems of linear equations with strict diagonal dominance in parallel is justified. Results of some numerical experiments are presented.

Convergence of a splitting method scheme for the nonlinear Boltzmann equation is considered. Using the splitting method scheme, boundedness of the positive solutions in a space of continuous functions is obtained. By means of the solution boundedness and some a priori estimates, convergence of the splitting method scheme and uniqueness of the limiting element are proved. The limiting element satisfies an equivalent integral Boltzmann equation. Thereby global in time solvability of the nonlinear Boltzmann equation is shown.

It is well known that the problem of maximization of any difference of convex functions can be turned into a convex maximization problem; here, the aim is a piecewise convex maximization problem instead. Although this may seem harder, sometimes the dimension may be reduced by 1, and the local search may be improved by using extreme points of the closure of the convex hull of better points. We show that it is always the case for both binary and permutation problems and give, as such instances, piecewise convex formulations for the maximum clique problem and the quadratic assignment problem.

Implicit methods applied to the numerical solution of systems of ordinary differential equations (ODEs) with an identically singular matrix multiplying the derivative of the sought-for vector-function are considered. The effects produced by losing L-stability of a classical implicit Euler scheme when solving such stiff systems are discussed.

A problem of unconstrained minimization of a semicoercive nondifferentiable functional corresponding to a model friction problem is reduced to a problem of constrained minimization of a differentiable functional. An algorithm based on iterative proximal regularization of a modified Lagrangian functional is used for solving the problem thus obtained. Convergence of a finite-element solution is investigated. Results of numerical calculations are presented.