Том 59, № 9 (2019)

A feature of the general Lagrange multiplier method as applied to variational problems is studied in the case when a part of the boundary of the domain of the state variables and control functions is a characteristic of the system of governing partial differential equations. It is shown that, if the state variables and the control functions are not related on this boundary part and do not influence the objective functional value, then the compatibility conditions for the system of governing equations do not need to be taken into account in varying the functional. Otherwise, the compatibility conditions along the characteristic have to be included in the generalized Lagrange functional with their own multipliers.

A second-order semilinear differential equation with slowly varying parameters is considered. With frozen parameters, the corresponding autonomous equation has fixed points: a saddle point and stable nodes. Upon deformation of the parameters, the saddle–node pair merges. An asymptotic solution near such a dynamic bifurcation is constructed. It is found that, in a narrow transition layer, the principal terms of the asymptotics are described by the Riccati and Kolmogorov–Petrovsky–Piskunov equations. An important result is finding the dragging out of the stability: the moment of disruption significantly shifts from the moment of bifurcation. The exact assertions are illustrated by the results of numerical experiments.

Many applied problems lead to the necessity of solving inverse problems for partial differential equations. In particular, much attention is paid to the problem of identifying coefficients of equations using some additional information. The problem of determining the time dependence of the right-hand side of a multidimensional hyperbolic equation using information about the solution at an interior point of the computational domain is considered. An approximate solution is obtained using a standard finite element spatial approximation and implicit schemes for time approximations. The computational algorithm is based on a special decomposition of the solution of the inverse problem when the transition to a new time level is ensured by solving standard elliptic problems. Numerical results for a model two-dimensional problem are given, which demonstrate the potentialities of the computational algorithms proposed to approximately solve inverse problems.

A mathematical model of the development of a tropical cyclone is considered. It consists of a family of equations obtained by transforming the equation of inviscid non-heat conductive gas (air) motion to the form of equations on wind trajectories in an axially symmetric cylindrical domain. The numerical solution of these equations shows the increase of the wind velocity in accordance with the steam condensation and air warming; later, the velocity becomes stable as the liquid or small pieces of ice accumulate in the air and the friction of water against air decelerates the air updraft.

For a singularly perturbed parabolic equation \({{\epsilon }^{2}}\left( {{{a}^{2}}\frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}} - \frac{{\partial u}}{{\partial t}}} \right) = F(u,x,t,\epsilon )\) in a rectangle, a problem with boundary conditions of the first kind is considered. At the corner points of the rectangle, the function \(F\) is assumed to be quadratic and nonmonotonic with respect to the variable \(u\) on the interval from the root of the degenerate equation to the boundary value. The main attention is paid to constructing the main term of the corner part of the asymptotics of the solution as \(\epsilon \to 0\).

The importance of this study is caused by the existence of applied machine learning problems that cannot be adequately solved in the classical statement of the logical data analysis. Based on a generalization of basic concepts, a scheme for synthesizing correct supervised classification procedures is proposed. These procedures are focused on specifying partial order relations on sets of feature values. It is shown that the construction of classification procedures requires a key intractable discrete problem to be solved. This is the dualization problem over products of partially ordered sets. The matrix formulation of this problem is given. The effectiveness of the proposed approach to the supervised classification problem is illustrated on model data.

Within the formalism of the multicommodity flow model, changes in the functional characteristics of a multiuser network after a damage are studied. To analyze the original limiting capabilities of the system, the maximum flow is calculated for each pair of vertices independently of other pairs. All edges of the corresponding minimum cut are removed, and the maximum possible flows for all origin–destination pairs are found in the damaged network and compared with their initial values. The detriment is evaluated for various cuts (structural damages). The influence of structural damages on the attainable values of the flow for each pair of vertices is investigated. This makes it possible to rank the origin–destination pairs by their susceptibility to the influence of damages from a given class. Ranking is performed on the basis of a bi-criteria detriment evaluation model. This approach is used for analyzing the vulnerability of large territory distributed systems, including telecommunication systems, communication and control systems.