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$${\oplus }_m^k={[{(\sum _{r=1}^p\frac{{\partial }^2}{\partial x_r^2}+m^2)}^4-{\left(\sum _{j=p+1}^{p+q}\frac{{\partial }^2}{\partial x_j^2}\right)}^4]}^k,$$ where $m$ is a nonnegative real number, $p+q=n$ is the dimension of the Euclidean space ${\mathbb{R}}^n$,$x=(x_1,x_2,\dots ,x_n)\in {\mathbb{R}}^n$, $k$ 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 $x\in {\mathbb{R}}^n$