IJPAM: Volume 52, No. 1 (2009)

ON THE ULTRA-HYPERBOLIC WAVE OPERATOR

Wanchak Satsanit$^1$, Amnuay Kananthai$^2$
$^{1,2}$Department of Mathematics
Faculty of Science
Chiang Mai University
Chiang Mai, 50200, THAILAND
$^2$e-mail: malamnka@science.cmu.ac.th


Abstract.In this paper, we study the generalized wave equation of the form

\begin{displaymath}\frac{\partial^2}{\partial t^2}u(x,t)+c^2(\Box)^ku(x,t)=0\end{displaymath}

with the initial conditions

\begin{displaymath}u(x,0)=f(x),~~ \frac{\partial}{\partial t}u(x,0)=g(x)\,,\end{displaymath}

where $u(x,t)\in \mathbb{R}^n\times [0,\infty)$, $\mathbb{R}^n$ is the $n$-dimensional Euclidean space, $\Box^k$ is the ultra-hyperbolic operator iterated $k-$times defined by

\begin{displaymath}\Box^k=\left(\displaystyle \displaystyle \frac{\partial^2}{\... ...p+2}^2}-\cdots-\frac{\partial^2}{\partial x_{p+q}^2}\right)^k,\end{displaymath}

$p+q=n$, $c$ is a positive constant, $k$ is a nonnegative integer, $f$ and $g$ are continuous and absolutely integrable functions. We obtain $u(x,t)$ as a solution for such equation. Moreover, by $\epsilon$-approximation we also obtain the asymptotic solution $u(x,t)=O(\epsilon^{-n/k})$. In particularly, if we put $n=1,$ $k=2$ and $q=0$, the $u(x,t)$ reduces to the solution of the beam equation

\begin{displaymath}\frac{\partial^2}{\partial t^2}u(x,t)+c^2\frac{\partial^4}{\partial x^4}u(x,t)=0.\end{displaymath}



Received: March 12, 2009

AMS Subject Classification: 35L05

Key Words and Phrases: generalized wave equation, beam equation, tempered distribution

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 52
Issue: 1