Solutions of ?double soliton? type for the multidimensional equation ?u=F(u)

1977 ◽  
Vol 31 (1) ◽  
pp. 293-299 ◽  
Author(s):  
V. S. Buslaev
Keyword(s):  
Multidimensional Equation

scholarly journals A General Formulation of the Transfer Equation. II. Line Formation with General Redistribution

1972 ◽  
Vol 25 (2) ◽  
pp. 177 ◽  
Author(s):  
CJ Cannon
Keyword(s):  
Velocity Field ◽  
Spectral Line ◽  
Transfer Equation ◽  
Functional Form ◽  
System Of Equations ◽  
Line Formation ◽  
Line Radiation ◽  
Multidimensional Equation

The multidimensional equation of transfer for spectral line radiation under a general redistribution law is studied. It is shown that the equation may be rewritten as a system of equations of the "Feautrier" form, which are known to be exceedingly stable and efficient under numerical reduction. It is also shown that the inclusion of a multidimensional differential macroscopic velocity field does not alter the functional form of the equations obtained and therefore may also be treated by the general Feautrier technique.

scholarly journals THE ENVIRONMENT DYNAMICS IDENTIFICATION BASED ON THE MODULAR COMPUTING COMPLEX

World Science ◽  
2021 ◽  
Author(s):  
Borys Moroz ◽  
Gennady Shvachych ◽  
Valentyna Chorna ◽  
Nataliiya Voroshylova
Keyword(s):  
Mathematical Models ◽  
Numerical Models ◽  
Linear Equations ◽  
Industrial Area ◽  
Matrix Computations ◽  
Sustainable Implementation ◽  
Multidimensional Equation ◽  
The Difference ◽  
Dynamics Identification

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.

THE HEUN–SCHRÖDINGER RADIAL EQUATION FOR DH-ATOMS

2005 ◽  
Vol 19 (19n20) ◽  
pp. 981-989 ◽  
Author(s):  
V. F. TARASOV
Keyword(s):  
Coulomb Potential ◽  
Radial Function ◽  
Standard Form ◽  
Electron Cloud ◽  
Quantum Corrections ◽  
Heun Equation ◽  
Radial Equation ◽  
Radial Functions ◽  
Kummer’S Function ◽  
Multidimensional Equation

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.

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