Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application

Abstract

Quasi-asymptotic behavior of functions as a method has its application in observing many physical phenomena which are expressed by differential equations. The aim of the asymptotic method is to allow one to present the solution of a problem depending on the large (or small) parameter. One application of asymptotic methods in describing physical phenomena is the quasi-asymptotic approximation. The aim of this paper is to look at the quasi-asymptotic properties of multidimensional distributions by extracted variable. Distribution T(x0,x) from S'(Ṝ+1×Rn) has the property of the separability of variables, if it can be represented in form T(x0,x)=∑φi(x0)ψi (x) where distributions, φi(x0) from S'(Ṝ1) and ψi from S(Rn), x0 from Ṝ1+ and x is element Rn different values of do not depend on each other. Distribution T(x0,x) the element S'(Ṝ+1×Rn) is homogeneous and of order α at variable x0 is element Ṝ1+ and x=x1,x2,…,xn from Rn if for k>0 it applies that T(kx0,kx)=kα T(x0,x). The method of separating variables is one of the most widespread methods for solving linear differential equations in mathematical physics. In this paper, the results by V. S Vladimirov are used to present the proof of the basic theorems, regarding the quasi-asymptotic behavior of multidimensional distributions by a singular variable, with the application of quasi-asymptotics to the solution of differential equations.