On the distribution related to the ultra-hyperbolic equations

Abstract

In this paper we consider the distribution e αt □ k δ where α is a constant and α = ( α 1, α 2,…, α n ) ∈ R n the n-dimensional Euclidean space and the variable t = ( t 1, t 2,…, t n ) ∈ R n and □ k is the n-dimensional ultra-hyperbolic operator iterated k-times, δ is the Dirac-delta distribution with □ 0 δ = δ and □ 1 δ = □ δ. At first, all properties of e αt □ k δ are studied and after that we study the application of e αt □ k δ for solving the elementary solution of the equation of the ultra-hyperbolic type by using the convolution method.