THE ENVIRONMENT DYNAMICS IDENTIFICATION BASED ON THE MODULAR COMPUTING COMPLEX

The research aims at covering the mathematical modeling issues of multidimensional applied problems of ecology based on the application of a modular computing complex. The problem of modeling air pollution processes is solved by mathematical models that adequately describe fundamental processes. That reveals issues such as a detailed analysis of the atmosphere of the city or industrial area, short-term forecast of air quality in the region, assessment of long term air purification programs, optimal emission management, transboundary transfer, etc. At the same time, the formulation and methods of solving problems of environmental dynamics identification are considered, which essence is to estimate the input parameters based on the factual information about the modeled system known from the experiment. In these studies, the multidimensional equation of harmful impurities transfer was reduced to a sequence of schemes involving unknown values in a single direction, alternately in the longitudinal, transverse and vertical.The implicit schemes lead to systems of algebraic linear equations with a three-diagonal structure. Thus, the methodological basis of the difference splitting schemes provides the economic and sustainable implementation of numerical models by the scalar runs method. That approach focuses on the fact that the greatest effect of a parallel processor is achieved when it is used to perform matrix computations of linear algebra.In order to analyze the feasibility of mathematical models, a package of applications was developed to compute the transfer of harmful impurities. A solution to several applied problems for the identification of the environmental dynamics is given.

MODIFIED FINITE-DIFFERENCE SCHEME FOR SOLVING ONE CLASS OF CONVECTION-REACTION-DIFFUSION PROBLEMS

The paper reviews approaches to the construction of finite-difference methods for solving time-dependent diffusion equations and transport equations. A modified computational scheme for solving a semilinear multidimensional equation of the «reaction – diffusion – convection» type is presented. The hybrid computational scheme is based on the alternating directions method and the Robert-Weiss scheme.

THE HEUN–SCHRÖDINGER RADIAL EQUATION FOR DH-ATOMS

This article deals with the connection between Schrödinger's multidimensional equation for DH-atoms (D≥1) and the confluent Heun equation with two auxiliary parameters ν and τ, where |1-ν| = o(1) and τ∈ℚ+, which influence the spectrum of eigenvalues, the Coulomb potential and the radial function. The case τ = ν = 1 and D = 3 corresponds to the "standard" form of Schrödinger's equation for a 3H-atom. With the help of parameter ν, e.g., some "quantum corrections" may be considered. The cases 0<τ<1 and τ>1, but â = (n-l-1)τ≥0 is an integer, change the "geometry" of the electron cloud in the atom, i.e. the so-called "exotic" 3H-like atoms arise, where Kummer's function 1F1(-â; c; z) has â zeros and the discrete spectrum depends only on Z/(νn) but not on l and τ. Diagrams of the radial functions [Formula: see text] as n≤3 are given.

Some properties of the functional equationϕ(x)=f(x)+∫0λxg(x,y,ϕ(y))dy

A discussion is given of some of the properties of the functional Volterra Integral equationϕ(x)=f(x)+∫0λxg(x,y,ϕ(y))dy. and of the corresponding multidimensional equation. Sufficient conditions are given for the uniqueness of the solution, and an iterational process is provided for the construction of the solution, together with error estimates. In addition bounds are provided on the solution. The results obtained are illustrated by means of the pantograph equation.