IJPAM: Volume 95, No. 3 (2014)

THE PARTIAL DIFFERENTIAL OPERATOR $\diamondsuit _{m,c}^{k}$
RELATED TO THE WAVE EQUATION AND LAPLACIAN

Sudprathai Bupasiri
Faculty of Education
Sakon Nakhon Rajabhat University
Sakon Nakhon, 47000, THAILAND


Abstract. In this article, we study the elementary solution of the operator $\diamondsuit _{m,c}^{k}$ ,iterated $k$-times and is defined by

\begin{displaymath}\diamondsuit _{m,c}^{k}=\left[\left(\frac{1}{c^2}\sum_{i=1}^{... ...}}{\partial x_{j}^{2}} - \frac{m^{2}}{2}\right)^{2}\right]^{k},\end{displaymath}

where $p+q=n, c,m$ are positive real number and $n$ is the dimension of Euclidean space $\mathbb{R}^{n}$, $x\in\mathbb{R}^{n}$ and $k$ is a nonnegative integer. We obtain the elementary solution depending on the conditions of $c, p, q, k$ and $m$.

Received: April 24, 2014

AMS Subject Classification: 46F10

Key Words and Phrases: elementary solution, wave equation, Laplacian

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DOI: 10.12732/ijpam.v95i3.8 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 95
Issue: 3
Pages: 405 - 412


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