On the Diamond Bessel Heat Kernel

Wanchak Satsanit

doi:10.1155/2011/951960

Abstract

We study the heat equation in n dimensional by Diamond Bessel operator. We find the solution by method of convolution and Fourier transform in distribution theory and also obtain an interesting kernel related to the spectrum and the kernel which is called Bessel heat kernel.

Highlights

The operator ♦k has been first introduced by Kananthai 1, is named as the diamond operator iterated k times, and is defined by ♦k ⎛ ⎜ ⎝ p ∂2 i 1 ∂xi2 − pq ⎝ ∂2 j p 1 ∂xj2⎞2⎞k ⎠ ⎟ ⎠, 1.1 p q n, n is the dimension of the space Rn for x x1, x2, . . . , xn ∈ Rn, and k is a nonnegative integer
The operator ♦k can be expressed in the following form:
Is named the Bessel ultrahyperbolic operator iterated k-times and is defined by 1.4, k is a positive integer, u x, t is an unknown function, f x is the given generalized function, and c is a constant, and p q n is the dimension of the Rn {x : x x1, x2, . . . , xn, t, xi > 0, i 1, 2, 3, . . . , n}

Summary

Introduction

The operator ♦k has been first introduced by Kananthai 1 , is named as the diamond operator iterated k times, and is defined by. 1.18 with the initial condition u x, 0 f x , for x ∈ Rn, 1.19 where the operator k B is named the Bessel ultrahyperbolic operator iterated k-times and is defined by 1.4 , k is a positive integer, u x, t is an unknown function, f x is the given generalized function, and c is a constant, and p q n is the dimension of the Rn {x : x x1, x2, .

Preliminaries

1: Rn Txyf x n i

Main Results