Copyright © 2010 Wanchak Satsanit and Amnuay Kananthai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Firstly, we studied the solution of the equation ⊗k◊Bku(x)=f(x) where u(x) is an unknown unknown function for x=(x1,x2,…,xn)∈ℝn, f(x) is the generalized function, k is a positive integer. Finally, we have studied the solution of the nonlinear equation ⊗k◊Bku(x)=f(x,□k−1LkΔBk□Bku(x)). It was found that the existence of the solution u(x) of such an equation depends on the condition of f and □k−1LkΔBk□Bku(x). Moreover such solution u(x) is related to the inhomogeneous wave equation depending on the conditions of p, q, and k.