Cauchy problem of the Δ(k) operator related to the Diamond operator and the Laplace operator iterated k times - Bunpog,C. (Eudoxus Press, LLC, 2015)
Given the Laplace operator Δ is defined by the Ultra-hyperbolic operator iterated k times k is defined by where p + q = n is the dimension of the Euclidean space ℝn. In this paper, we study Cauchy problem and fundamental solution of the Δ(k) operator by using Green's identity, In particular, Δ(k) reduces to the Diamond operator if k = 1. Moreover, for q = 0 the ultra-hyperbolic operator 2 reduces to Δ, and Δ(k-1) reduces to the Laplace operator Δk iterated k times. © 2011 by Eudoxus Press,LLC All rights reserved.