Том 59, № 7 (2023)

We consider the Dirichlet problem for a second-order differential–difference equation in divergence form with variable coefficients on a finite interval Q = (0, d). Conditions on the right-hand side of the equation ensuring the smoothness of the generalized solution on the entire interval are studied. It is proved that the generalized solution of the problem belongs to the Sobolev space W22(Q)  if the right-hand side is orthogonal in the space L2(Q)  to finitely many linearly independent functions

We consider the first boundary value problem for uniformly parabolic systems of the second order with one spatial variable in bounded and semibounded domains with nonsmooth lateral boundaries. The coefficients of the system satisfy the Hölder condition and do not depend on the time variable. For continuous initial and boundary functions, the existence and uniqueness of the classical solution of this problem is established

We introduce the concepts of a learning algorithm, an objective function, a recognition system, a class of patterns, a training set, a reward algorithm, a finitely convergent algorithm, an adaptive control system, a control goal, control tactics, adaptation time, etc., related to the problem of artificial intelligence in the processes of learning and adaptation. The general problem of self-learning (unsupervised learning)—about the separation of sets—in terms of the classical calculus of variations is posed. The generality of the problem is due to the introduction of an additional time variable into the analysis. The problem is solved by determining extremal conditions under which the minimization of the overall average risk functional is achieved. Problems corresponding to nonfixed and fixed time intervals are considered. For these two cases, expressions are found for calculating variations in cost functionals. Necessary conditions are indicated for determining the extremal values of the self-learning process (separation of classes of a set of patterns) in time.

Domain decomposition methods are used for the approximate solution of boundary value problems for partial differential equations on parallel computing systems. The specifics of nonstationary problems is most completely taken into account when using noniterative domain decomposition schemes. Regionally additive schemes are constructed on the basis of various classes of splitting schemes. A new class of domain decomposition schemes with an additive representation of the solution on a new time level is distinguished that is based on splitting the domain into subdomains based on a partition of unity. An example of the Cauchy problem for first-order evolution equations with a positive self-adjoint operator in a finite-dimensional Hilbert space is considered. Unconditionally stable two- and three-level splitting schemes are constructed for the corresponding system of equations.

Issues related to the lack of convergence in the application of formally path-conservative difference schemes for solving nonconservative hyperbolic systems of equations are numerically investigated. This problem is central in constructing well-posed difference schemes for solving this class of problems. The basic concepts of the theory of nonconservative hyperbolic systems of equations and the corresponding problems of constructing difference schemes for their solution are outlined. A variant of the HLL method is proposed that allows using an arbitrary explicitly specified path. For a model system of Burgers equations, the shock adiabates and paths corresponding to the viscous regularization of a system of a given form are explicitly calculated. The reasons for the lack of convergence of numerical solutions of exact ones in the case of incorrect application of the corresponding algorithms are analyzed. It is shown that, at least in the particular case considered, a variant of the HLL method that is formally conservative along the way gives the correct solution of the problem.

A multidimensional system of ordinary differential equations with relay hysteresis is considered. The system parameters are assumed to be such that there exists a family of continuous operators each of which maps some connected compact set into itself. In this case, the operator corresponds to a periodic orbit with an even number of switching points in the phase space of the system. For the operator family, a necessary and sufficient condition for the existence of a single fixed point is obtained.