IJPAM: Volume 56, No. 4 (2009)

THE OPERATOR $\circledast^{k}$ RELATED
TO TRIHARMONIC WAVE EQUATION

Wanchak Satsanit$^1$, Amnuay Kananthai$^2$
$^{1,2}$Department of Mathematics
Regina Coeli College
166, Charoenprathet Road Changkran, Chiang Mai, 50100, THAILAND
$^1$e-mail: aunphue@live.com


Abstract.In this paper, we study the solution of equation $\circledast^{k}u(x)=f(x)$, where $\circledast^{k}$ is the operator iterated $k$ times and is defined by

\begin{eqnarray*}
\circledast^{k}&=&\left(\left(\sum^{p}_{i=1}\frac{\partial^2}...
..._{j=p+1}\frac{\partial^2}{\partial
x^2_j}\right)^{3}\right)^{k},
\end{eqnarray*}

where $p+q=n$ is the dimension of the Euclidean space $\mathbb{R}^n$, $u(x)$ is an unknown function for $x=(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n$, $f(x)$ is the generalized function, $k$ is a positive integer and $u(x)$ is an unknown function.

It is found that the solution $u(x)$ depends on the conditions of $p$ and $q$ and moreover such a solution is related to the solution of the Laplace equation. In particularly if we put $k=1$ and we obtain solution of the triharmonic wave equation.

Received: August 21, 2009

AMS Subject Classification: 47F05

Key Words and Phrases: generalization function, temper distribution, diamond operator, triharmonic wave equation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 56
Issue: 4