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`Mathematical Problems in EngineeringVolume 2010 (2010), Article ID 497676, 12 pagesdoi:10.1155/2010/497676`
Research Article

## On the Operator ⨁Bk Related to Bessel Heat Equation

Department of Mathematics, Faculty of Science, Maejo University, Chiang Mai 50290, Thailand

Received 12 April 2010; Accepted 27 June 2010

Academic Editor: Carlo Cattani

Copyright © 2010 Wanchak Satsanit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the equation (/t)u(x,t)=c2Bku(x,t) with the initial condition u(x,0)=f(x) for xRn+. The operator Bk is the operator iterated k-times and is defined by Bk=((i=1pBxi)4-(j=p+1p+qBxi)4)k, where p+q=n is the dimension of the Rn+, Bxi=2/xi2+(2vi/xi)(/xi), 2vi=2αi+1, αi>-1/2, i=1,2,3,,n, and k is a nonnegative integer, u(x,t) is an unknown function for (x,t)=(x1,x2,,xn,t)Rn+×(0,), f(x) is a given generalized function, and c is a positive constant. We obtain the solution of such equation, which is related to the spectrum and the kernel, which is so called Bessel heat kernel. Moreover, such Bessel heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.