On the Operator ⊕k,m Related to the Wave Equation and Laplacian

Abstract

In this article, we study the fundamental solution of the operator $\oplus _{m}^{k}$, iterated $k$-times and is defined by$$\oplus _{m}^{k} = \left[\left(\sum_{r=1}^{p} \frac{\partial^2} {\partial x_r^2}+m^{2}\right)^4 - \left( \sum_{j=p+1}^{p+q} \frac{\partial^2}{\partial x_{j}^2} \right)^4 \right ]^k,$$ where $m$ is a nonnegative real number, $p+q=n$ is the dimension of the Euclidean space $\mathbb{R}^n$,$x=(x_1,x_2,\ldots,x_n)\in\mathbb{R}^n$, $k$ is a nonnegative integer. At first we study the fundamental solution of the operator $\oplus _{m}^{k}$ and after that, we apply such the fundamental solution to solve for the solution of the equation $\oplus _{m}^{k}u(x)= f(x)$, where $f(x)$ is generalized function and $u(x)$ is unknown function for $ x\in \mathbb{R}^{n}$.