On the diamond wave operator

Abstract

In this paper, we study the solution of the Diamond-wave operator L which is defined by where is the Diamond operator, x ∈ R{double-struck}n-the n dimensional Euclidean space, t ≥ 0, and p + q = n is the dimension of R{double-struck}n. By considering the equation Lu(x, t) = 0 with the suitable initial conditions. We obtained the unique solution u(x, t) of such equation. Moreover, we obtained the boundedness of u(x, t) subject to the suitable initial conditions. In particular, if we put n = 1, p = 1 and q = 0 we also obtained the solution of the beam equation.