Vol 213 (2022)
Статьи
Singlet linear equation for one-particle distribution function in statistical physics of surface phenomena in liquids
Abstract
In this work we suggest the algorithm to solve the linear Fredholm integral equation of the second kind for the one-particle distribution function of simple liquid near the hard surface. The core and the right part of the equation are guessed using Percus-Yevick approximation defined on finite interval for spatial macroscopic liquid. We suggest an approach to solve the equation analytically for each interval where function is defined.
3–9
Lie algebras of projective motions of five-dimensional pseudo-riemannian spaces. II. Integration of the Eisenhart equations
Abstract
This work is devoted to the problem of studying multidimensional pseudo-Riemannian manifolds that admit Lie algebras of infinitesimal projective (in particular, affine) transformations, wider than Lie algebras of infinitesimal homotheties. Such manifolds have numerous geometric and physical applications. This paper is the second part of the work. The first part: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2022. — 212. — P. 10-29. Continuation will be published in future issues.
10–37
An inverse problem for a class of degenerate evolution multi-term equations with Gerasimov-Caputo derivatives
Abstract
Issues of well-posedness of linear inverse problems for equations with several Gerasimov- Caputo fractional derivatives in Banach spaces are investigated. The inverse coefficient problem is considered for an equation solved with respect to the highest fractional derivative containing bounded operators at lower order derivatives. The criterion of well-posedness of such a problem is proved. A similar inverse problem for an equation with a degenerate operator at the highest derivative, assuming the relative 0-boundedness of a pair of operators at two higher derivatives, is reduced to two problems on subspaces for equations solved with respect to the highest derivative. The obtained well-posedness criteria allowed us to investigate one class of inverse problems for equations with polynomials from an elliptic differential operator with respect to spatial variables and with several Gerasimov-Caputo time derivatives.
38–46
Operator forms and methods of the maximum principle in optimal control problems with constraints
Abstract
New constructive forms of well-known optimality conditions for constrained controlled systems in the form of fixed point problems in the control space are considered. Optimality conditions proposed allows one to apply the theory and methods of fixed points to develop new iterative algorithms for finding extremal controls in the class of constrained optimal control problems.
47–53
Construction of solutions to a degenerate reaction-diffusion system with a general nonlinearity in the cases of cylindrical and spherical symmetry
Abstract
We consider a reaction-diffusion system with a general nonlinearity with cylindrical or spherical symmetry. For this system, we find a solution of the diffusion-wave type propagating over a zero background with a finite velocity. The solution is constructed as a Taylor series with recurrent coefficients whose convergence is proved by the majorant method and the Cauchy-Kovalevskaya theorem. The research is supplemented by numerical calculations based on the expansion in radial basis functions. This paper continues a series of our publications devoted to the study of wave-type solutions in the class of analytical functions.
54–62
On the solvability of control synthesis problems for nonlinear oscillatory optimization processes described by integro-differential equations
Abstract
The solvability of synthesis problems for distributed and boundary controls in minimizing problems for piecewise linear functionals for oscillatory processes described by partial integrodifferential equations with Fredholm integral operators are examined. For the Bellman functional, a specific integro-differential equation is obtained. An algorithm for constructing a solution of the control synthesis problem of distributed and boundary controls is described. A procedure for determining controls as functions (functionals) of the state of the controlled process is constructed.
63–71
Inverse problem for the Boussinesq-Love equation
Abstract
For an abstract, high-order, incomplete Sobolev-type equation, an inverse problem with final redefinition is considered. Conditions for the unique solvability of the problem are found. Some special cases are considered. The main result contains necessary and sufficient conditions for the existence and uniqueness of a solution of the inverse problem for high-order, Sobolev-type equations.
This technique is applied to the study of the inverse problem for the Boussinesq-Love equation.
72–79
On the well-posedness of an inverse problem for a degenerate evolutionary equation with the dzhrbashyan-nersesyan fractional derivative
Abstract
In this paper, we find necessary and sufficient conditions for the well-posedness of linear inverse coefficient problems for degenerate evolutionary equations with the Dzhrbashyan-Nersesyan fractional derivative in Banach spaces. We examine an inverse problem with a constant unknown coefficient under the generalized Showalter-Sidorov conditions and the condition of p-boundedness of a pair of operators in it. The general result is applied to the inverse problem for the system of dynamics of a viscoelastic Kelvin-Voigt fluid with the Dzhrbashyan-Nersesyan fractional derivative in time.
80–88
On one approach to the optimization of state-linear controlled systems with terminal constraints
Abstract
In the class of state-linear optimal control problems with terminal constraints, we consider the problem of nonlocal improvement of an admissible control preserving all terminal constraints. We apply an approach based on solving a special system of functional equations. The corresponding system is interpreted as a fixed-point problem; to the solution of this problem we apply the theory of fixed points.
89–95
Systems with dissipation with a finite number of degrees of freedom: analysis and integrability. Iii. Systems on the tangent bundles of smooth n - dimensional manifolds
Abstract
This paper is the third part of a survey on the integrability of systems with a large number n of degrees of freedom (the first part: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 211 (2022), pp. 41-74; the second part: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 212 (2022), pp. 139-148). The review consists of three parts. In the first part, the primordial problem from the dynamics of a multidimensional rigid body placed in a nonconservative force field is described in detail. The second part is devoted to more general dynamical systems on the tangent bundles to the n-dimensional sphere. In this third part, we discuss dynamical systems on the tangent bundles to 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.
96-109
Polynomial automorphisms, quantization, and Jacobian conjecture related problems. I. Introduction
Abstract
The purpose of this review is the collection and systematization of results concerning the quantization approach to the some classical aspects of non-commutative algebras, especially to the Jacobian conjecture. We start with quantization proof of Bergman centralizing theorem, then discourse authomorphisms of INd-schemes authomorphisms, then go to aproximation issues. Last chapter dedicated to relations between P I-theory Burnside type theorems and Jacobian Conjecture (Jagzev approach). This issue contains the first part of the work; continuation will be published in future issues.
110–144