Vol 209 (2022)
Articles
Hyperbolicity of covariant systems of first-order equations for vector and scalar fields
Abstract
We consider a class of first-order systems of quasilinear partial differential equations , that describe time evolution of the pair consisting of a vector field and the set of scalar fields , . The class considered consists of systems that are invariant under time and space translations and covariant under space rotations. We describe the corresponding class of evolution generators, i.e., nonlinear first-order differential operators acting in the functional space . Also, we find conditions under which a pair of operators generates a hyperbolic system.
3-15
On the construction of generalized powers for the Dirac equation of quantum electrodynamics
Abstract
The paper is devoted to applications of the method of generalized powers for constructing a class of solutions of the Dirac equation in the case of a free particle. Possible generalizations of the method are indicated and examples are given.
16-24
On Ulam–Hyers stability of solutions to first-order differential equations with generalized action
Abstract
This paper is devoted to sufficient conditions for the Ulam–Hyers stability of solutions of first-order linear differential equations. We introduce the concept of the Ulam–Hyers stability for equations with unbounded right-hand sides whose solutions are functions of bounded variation and obtain sufficient conditions that guarantee this stability.
25-32
New bifurcation diagram in one model of vortex dynamics
Abstract
We consider a completely Liouville-integrable Hamiltonian system with two degrees of freedom, which includes two limit cases. The first system describes the dynamics of two vortex filaments in a Bose–Einstein condensate enclosed in a harmonic trap. The second system governs the dynamics of point vortices in an ideal fluid in a circular domain. For the case of vortices with arbitrary intensities, we explicitly reduce the problem to a system with one degree of freedom. For intensities of different signs, we detect a new bifurcation diagram, which has not been previously encountered in works on this topic. Also, we obtain a separating curve, which is related to the change of the projections of Liouville tori without changing their number.
33-41
The problem of finding the initial state of a resource network
Abstract
In this paper, we study distributions of resource flows in resource networks. The main problem is to develop methods for finding the initial state (distribution) of resources in a resource network if the state is known at some moment of discrete time. An essential feature is the significant nonlinearity of the resource redistribution process in such networks. We prove that the problem of finding the initial state is solvable and propose approaches for refining the solution and finding the initial state of the resource network in the cases of large and small resources.
42-52
Volterra functional equations and optimization of distributed systems. Special optimal controls
Abstract
The work is a brief review of some results of optimization theory obtained by using Volterra functional equations (VFE). We present the method proposed by the author for using the VFE-description of controlled initial-boundary-value problems for studying special controls on which necessary optimality conditions degenerate. Illustrative examples are given.
53-76
Linear and nonlinear fuzzy averages of systems of fuzzy numbers
Abstract
Linear averages of systems of fuzzy numbers are examined. A class of nonlinear fuzzy average systems of fuzzy numbers is introduced and studied. Fuzzy analogs of numerical inequalities for means are established.
77-87
Systems with dissipation with five degrees of freedom: Analysis and integrability. II. Dynamical systems on tangent bundles
Abstract
The work contains the second and third parts of the survey on the integrability of systems with five degrees of freedom (the first part: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 208, (2022), pp. 91–121). In the first part, the primordial problem from the dynamics of a multidimensional rigid body placed in a nonconservative force field was described in detail. In the second and third parts, we consider more general dynamical systems on tangent bundles to the five-dimensional sphere and other smooth manifolds of a sufficiently wide class. Theorems on sufficient conditions for the integrability of the considered dynamical systems in the class of transcendental functions are proved.
88-107
Some tensor invariants of geodesic, potential, and dissipative systems on the tangent bundles of two-dimensional manifolds
Abstract
In this paper, we construct tensor invariants (differential forms) of homogeneous dynamical systems on the tangent bundles of smooth two-dimensional manifolds. We establish the relationship between the presence of such invariants and the existence of complete sets of first integrals, which are necessary for integrating geodesic, potential, and dissipative systems. Due to force fields, systems considered are dissipative; they are generalizations of systems considered earlier.
108-116
Generalized control problem in diagnostic problems
Abstract
In this paper, we explain such concepts as control sphere, control ellipsoid, and control tube. The solution of the control problem by the method of statistical tests is proposed. The statement of the extended problem of control is formulated and necessary preparations for considering the diagnosis problem are made. This work is the third work of the cycle devoted to control problems.
117-126