On the ultra-hyperbolic wave operator

Abstract

In this paper, we study the generalized wave equation of the form ∂2 /∂t2|u(x,t) +c2(□) k u(x,t)=0 with the initial conditions u(x, 0) = f(x), ∂/∂t(x, 0) = g(x), where u(x, t) ⊂ Rℝn × 0, ∞), Rℝn is the n-dimensional Euclidean space, k is the ultra-hyperbolic operator iterated K-times defined by □k=(∂/∂x 2+∂/∂x22+...+∂2/ ∂x2p-∂2/∂x2p/∂/ ∂x2p+1-∂2/∂x2p-2-...-∂2/∂x2p+q) k p + q = n, c is a positive constant, k is a nonnegative integer, f and g are continuous and absolutely integrable functions. We obtain u(x,t) as a solution for such equation. Moreover, by ε-approximation we also obtain the asymptotic solution u(x,t) = 0(ε-n/k). In particularly, if we put n = 1, k = 2 and q = 0, the u(x, t) reduces to the solution of the beam equation ∂2/∂t2u(x, t) +c2∂4/∂x4u(x,t)=0. © 2009 Academic Publications.