IJPAM: Volume 64, No. 1 (2010)

ON THE OPERATOR $\circledast_B^k$ RELATED TO
THE BESSEL HEAT EQUATION

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


Abstract.In this paper, we study the equation

\begin{displaymath}\frac{\partial}{\partial t} u(x,t)=c^2\circledast^k_B u(x,t)\end{displaymath}

with the initial condition $u(x,0)=f(x)$ for $x\in\mathbb{R}^+_n$, where the operator $\circledast^k_B$ is defined by

\begin{displaymath}\circledast^k_B=\left[\left(B_{x_1}+\cdots+B_{x_p}\right)^3+\left(B_{x_{p+1}}+\cdots+B_{x_{p+q}}\right)^3\right]^k,\end{displaymath}

$p+q=n$ is the dimension of the space $\mathbb{R}^+_n=\{x=(x_1,x_2,\dots,x_n):x_1>0,x_2>0,\dots,x_n>0\}$, $B_{x_i}=\frac{\partial^2}{\partial x_i^2}+
\frac{2v_i}{x_i}\frac{\partial}{\partial x_i}$, $2v_i=2\alpha_i+1$, $\alpha_i>-\frac{1}{2}$, $i=1,2,\dots,n$, $u(x,t)$ is an unknown function for $(x,t)=(x_1,x_2,\dots,x_n,t)\in\mathbb{R}^+_n\times(0,\infty)$, $f(x)$ is a generalized function, $k$ is a positive integer and $c$ is a positive constant. We obtain the solution of such equation which is related to the spectrum and the heat kernel. Moreover, such Bessel heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.

Received: May 12, 2010

AMS Subject Classification: 46F10, 35K05

Key Words and Phrases: heat kernel, Dirac-delta distribution, spectrum, Bessel operator

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 64
Issue: 1