Vol 232 (2024)

The work is devoted to constructing approximate solutions of a parabolic differential equation with the Bessel operator. Solutions are sought in the form of a linear combination of piecewise continuous, compactly supported basis functions. The construction of the solution is performed in two stages. Initially, the approximation in a spatial variable is performed by using the Bubnov–Galerkin projection-grid method. Then the approximation in t is carried out by using the finite-difference method. The resulting system of equations is solved by the tridiagonal matrix algorithm.

In this work, we construct new mathematical models of cooling and freezing of living biological tissue with a flat, long ruler applicator located on its surface. The models are two-dimensional boundary-value problems (including Stefan-type problems) and have applications in cryosurgery. The method of numerical study of these problems is based on smoothing discontinuous functions and applying locally one-dimensional difference schemes to “smoothed” problems without explicitly identifying the boundaries of the influence of cold and the boundaries of the phase transition.

Using the well-known Riemann method and a new method for compensating the boundary regime with the right-hand side of the equation, we obtain the Riemann formulas for the unique and stable classical solution of the first mixed problem for a linear general inhomogeneous telegraph equation with variable coefficients in the first quadrant. From the formulation of the mixed problem, the definition of classical solutions, and the established criterion for the smoothness of the right-hand side of the equation, we obtain a criterion of the well-posedness in the Hadamard sense. This criterion consists of smoothness requirements and three conditions for matching the right-hand side of the equation and the boundary and initial data. The validity of the Riemann formulas and the well-posedness criterion is confirmed by their coincidence with the well-known formulas of the classical solution and the well-posedness criterion for the model telegraph equation.

For a control system with partial derivatives, a criterion for the complete controllability is derived by using the cascade decomposition method based on the transition from the original system to reduced systems in subspaces. We obtain a function, which belongs to a subspace of minimal dimension and determines the type of solution of the program control problem, i.e., the state and control functions in the analytical form. Necessary and sufficient conditions for the existence of the determining function are established and a scheme of its construction is given. Necessary and sufficient conditions for the existence of a determining function in the polynomial, exponential, and fractional-rational forms are found; formulas for constructing such functions are proposed. For the original system, a solution of the program control problem is constructed.

We study the initial-boundary-value problem in a half-strip for a second-order inhomogeneous hyperbolic equation with constant coefficients and a nonzero potential containing a mixed derivative. The equation considered is the equation of transverse vibrations of a moving finite string. The problems with general initial conditions (nonzero string profile and nonzero initial velocity of string points) and fixed ends (Dirichlet conditions) are examined. Theorems on the existence and uniqueness of a solution are formulated and formulas for the solution are obtained.

The problem of detecting anomalous behavior in large software systems can be reduced to the problem of detecting anomalies in text data streams. In this paper, we propose an approach based on a combination of deep learning (an autoencoder using convolutional neural networks and a single-layer fully connected decoder) and approaches based on the fuzzy clustering method. The solution proposed allows one to construct vector representations of groups of sequential events and identify outliers in the data using a developed layer based on fuzzy clustering and radial basis functions methods.