Vol 56, No 12 (2016)

We present an application of the p-regularity theory to the analysis of non-regular (irregular, degenerate) nonlinear optimization problems. The p-regularity theory, also known as the p-factor analysis of nonlinear mappings, was developed during last thirty years. The p-factor analysis is based on the construction of the p-factor operator which allows us to analyze optimization problems in the degenerate case. We investigate reducibility of a non-regular optimization problem to a regular system of equations which do not depend on the objective function. As an illustration we consider applications of our results to non-regular complementarity problems of mathematical programming and to linear programming problems.

The solvability of the boundary value and extremum problems for the convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of substances is proven. The role of the control in the extremum problem is played by the boundary function in the Dirichlet condition. For a particular reaction coefficient in the extremum problem, the optimality system and estimates of the local stability of its solution to small perturbations of the quality functional and one of specified functions is established.

For a strongly elliptic system of second-order equations of a special form, formulas for the Poisson integral and Green’s function in a circle and an ellipse are obtained. The operator under consideration is represented by the sum of the Laplacian and a residual part with a small parameter, and the solution to the Dirichlet problem is found in the form of a series in powers of this parameter. The Poisson formula is obtained by the summation of this series.

The problem of excitation of an anisotropic media-filled waveguide at critical frequencies is considered. An example of a dispersion curve with two rather than one or three singular points is presented. The possibility of excitation of back waves is studied. The character of the increase in the field upon resonance excitation of a waveguide is considered.

A finite-difference TVD scheme is presented for problems in nonequilibrium wave dynamics of heterogeneous media with different velocities and temperatures but with identical pressures of the phases. A nonlinear form of artificial viscosity depending on the phase relaxation time is proposed. The computed solutions are compared with exact self-similar ones for an equilibrium heterogeneous medium. The performance of the scheme is demonstrated by numerical simulation with varying particle diameters, grid sizes, and particle concentrations. It is shown that the scheme is efficient in terms of Fletcher’s criterion as applied to stiff problems.

A model of measuring the level of a viscous incompressible liquid in a tank as based on the liquid level in a measuring tube is investigated. The tank is in the field of gravity, and the tank liquid level varies according to some law. As a result, a Dirichlet boundary value problem for a nonlinear integrodifferential equation of parabolic type is obtained. A global existence and uniqueness theorem is proved for a weak solution of the problem. In the case of a tank level decreasing linearly with time, it is shown numerically that the liquid level in the measuring tube oscillates with a decaying amplitude with respect to the tank level.