Approximate Solution of the Thomas–Fermi Equation for Free Positive Ions

The approximate solution of the nonlinear Thomas–Fermi (TF) equation for ions is found by the Fermi method. The solution is based on the new asymptotic representation of the TF ion size valid for any ionization degree. The two universal functions and their derivatives, introduced by Fermi, are calculated by recent effective algorithms for the Emden–Fowler type equations with the accuracy sufficient for majority of applications. The comparison of our results with those obtained previously shows high accuracy and validity for arbitrary values of ionization degree. This study could potentially be of interest for the statistical TF method applications in physics and chemistry.

Boundary layer expansions for initial value problems with two complex time variables

We study a family of partial differential equations in the complex domain, under the action of a complex perturbation parameter ϵ. We construct inner and outer solutions of the problem and relate them to asymptotic representations via Gevrey asymptotic expansions with respect to ϵ in adequate domains. The asymptotic representation leans on the cohomological approach determined by the Ramis–Sibuya theorem.

ASYMPTOTIC SOLUTION OF THE PROBLEM OF OPTIMAL CONTROL OF SYSTEMS WITH SLOWLY VARIABLE PARAMETERS

Models of non-stationary automatic control systems are differential equations with variable coefficients. Such equations do not integrate in quadratures in the general case. Asymptotic methods are methods of approximate integration of differential equations with variable coefficients. In the article the non-stationary automatic control system with slowly variable parameters is considered. To study this system it is necessary to construct an asymptotic representation of its solution. In the theory of asymptotic integration exist a problem to construction of the asymptotic solution of a system in the presence of a turning point. Special methods have been developed to construct a solution to such systems: Maslov’s canonical operator, the multiphase Kucherenko method, the method of W. Wasow. The purpose of the article is to construct an asymptotic solution of a linear system of differential equations with nonstabilitu spectrum of the main matrix. In this article the asymptotic representation of the solution of the optimal correction problem is constructed. The case of nonstability spectrum of the main matrix and the available of turning points are investigated. Application of the Pontryagin maximum principle to the problem leads to a system with slowly varying coefficients and an nonstable spectrum. Construction of a formal solution of the main system with turning points in the form of a single expression in some cases is possible. The system formed in the process of solving the problem of optimal correction does not allow the mentioned construction. A multiscale method was used to solve this system of equations. Asymptotic estimates for the constructed approximations are given. The studied problem has practical applications in technical and economic systems, in particular in the calculation of the correction of the orbits of artificial satellites. The nonnstability of the spectrum is the cause of the spike phenomenon. Further research may be aimed at finding a unified approach to solving such problems and to ascertain the physical meaning of the turning point in specific systems of automatic control. Keywords: automatic control system, asymptotic solutions, turning point, the problem of optimal correction.

Type classification of extreme quantized characters

The notion of quantized characters was introduced in our previous paper as a natural quantization of characters in the context of asymptotic representation theory for quantum groups. As in the case of ordinary groups, the representation associated with any extreme quantized character generates a von Neumann factor. From the viewpoint of operator algebras (and measurable dynamical systems), it is natural to ask what is the Murray–von Neumann–Connes type of the resulting factor. In this paper, we give a complete solution to this question when the inductive system is of quantum unitary groups $U_{q}(N)$.