Vol 210 (2022)
Статьи
6-11
On the solvability of boundary-value problems for third-order equations of parabolic-hyperbolic type with lower terms
Abstract
In this paper, boundary-value problems for a third-order mixed differential equation of the parabolic-hyperbolic type with a fractional Gerasimov–Caputo operator are examined. Classes of functions that ensure the unique solvability of the boundary-value problem are found. The existence and uniqueness of a solution of the boundary-value problem are proved.
12-23
On the solvability of a boundary-value problem for a third-order differential equation with multiple characteristics
Abstract
In this paper, we discuss the unique solvability of a boundary-value problem for an inhomogeneous third-order partial differential equation with multiple characteristics. Using the Green function, we is construct a solution of this boundary-value problem in the explicit form.
24-34
On the theory of periodic solutions of systems of hyperbolic equations in the plane
Abstract
A periodic problem on the plane for a system of second-order hyperbolic equations with mixed derivatives is considered. The existence of a unique classical solution of the problem is examined and methods of constructing it are discussed.
35-48
Enumeration of labeled thorn graphs
Abstract
A thorn graph is a connected graph that becomes smooth after a single removal of end points together with their incident edges. An explicit formula is obtained for the number of labeled thorn graphs with given numbers of vertices and edges, and the corresponding asymptotics is found for the number of such graphs with a large number of vertices. It is proved that with a uniform probability distribution, almost all labeled connected sparse graphs are not thorn graphs.
49-54
Nonlocal problem for a fractional-order mixed-type equation with involution
Abstract
In this paper, we examine the unique solvability of a nonlocal problem for a nonlocal analog of a mixed parabolic-hyperbolic equation with a generalized Riemann-Liouville operator and involution with respect to the space variable. A criterion for the uniqueness of the solution is established and sufficient conditions for the unique solvability of the problem are determined. By the method of separation of variables, a solution is constructed in the form of an absolutely and uniformly convergent series with respect to eigenfunctions of the corresponding one-dimensional spectral problem. The stability of the solution of the problem under consideration under a nonlocal condition is established.
55-65
Boundary-value problem with an integral conjugation condition for a partial differential equation with the fractional Riemann-Liouville derivative that describes gas flows in a channel surrounded by a porous medium
Abstract
A boundary-value problem with an integral conjugation condition for a mixed equation with a fractional integro-differential operator was examined. The main result of the work is the proof of the unique solvability of the boundary-value problem with an integral conjugation condition for the equation consisting of two partial differential equations with the fractional Riemann-Liouville derivative in a rectangular domain. The problem is reduced to a Volterra integral equation of the second kind. The special role of the conjugation condition in the solvability of the problem is shown.
66-76
Integrable homogeneous dynamical systems with dissipation on the tangent bundles of four-dimensional manifolds. I. Equations of geodesic lines
Abstract
In many problems of dynamics, systems arise whose position spaces are four-dimensional manifolds. Naturally, the phase spaces of such systems are the tangent bundles of the corresponding manifolds. Dynamical systems considered have variable dissipation, and the complete list of first integrals consists of transcendental functions expressed in terms of finite combinations of elementary functions. In this paper, we prove the integrability of more general classes of homogeneous dynamical systems with variable dissipation on tangent bundles of four-dimensional manifolds.
77-95
Some tensor invariants of geodesic, potential, and dissipative systems on the tangent bundles of three-dimensional manifolds
Abstract
In this paper, we present tensor invariants (differential forms) for homogeneous dynamical systems on the tangent bundles of smooth three-dimensional manifolds and demonstrate the connection between the presence of these invariants and the existence of a complete set of first integrals, which is necessary for integrating geodesic, potential, and dissipative systems.
96-105
Dynamical systems and classification of malfunctions in problems of differential diagnostics
Abstract
In this paper, we discuss a universal approach to the study of control (not always smooth) dynamical systems and possible malfunctions in such dynamical systems. The universal concepts of reference malfunctions and their neighborhoods are introduced.
106-116
Optimal control of inverse thermal processes in a parabolic equation with nonlinear deviations in time
Abstract
In this paper, we examine the weakly generalized solvability of a nonlinear inverse problem in the nonlinear optimal control of thermal processes for one type of parabolic differential equation with nonlinear deviations. We formulate necessary optimality conditions for nonlinear control and obtain formulas for approximate calculating the state functions of the controlled process, the restoration function, and the optimal control function.
117-135