Mathematical Problems in Engineering

Volume 2010, Article ID 497676, 12 pages

http://dx.doi.org/10.1155/2010/497676

## On the Operator *⨁* 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 with the initial condition for The operator is the operator iterated k-times and is defined by , where is the dimension of the , , , , , and is a nonnegative integer, is an unknown function for , is a given generalized function, and 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.

#### 1. Introduction

It is well known that for the heat equation
with the initial condition
where is the Laplace operator and , we obtain
as the solution of (1.1). Equation (1.3) can be written as
where
is called * the heat kernel*, where and (see [1, pages 208-209]).

In 2004, Yildirim et al. [2, 3] first introduced the Bessel diamond operator iterated times, defined by where , . The operator can be expressed by , where

And, Yildirim et al. [2, 3] have shown that the solution of the convolution form is a unique elementary solution of that is

Now, the purpose of this work is to study the following equation: with the initial condition where the operator is first introduced by Satsanit and Kananthai [4] and is defined by Let us denote the operator By (1.7) we obtain Thus, (1.13) can be written as where are defined by (1.6) and (1.15), respectively, is the dimension of the , is an unknown function, , is the given generalized function, is a positive integer, and is a constant.

Moreover, Bessel heat kernel has interesting properties and also related to the kernel of an extension of the heat equation. We obtain the solution in the classical convolution form where the symbol is the -convolution in (2.3), as a solution of (1.11), which satisfies (1.12), and and is the spectrum of for any fixed and is the normalized Bessel function.

Before going into details, the following definitions and some important concepts are needed.

#### 2. Preliminaries

The shift operator according to the law remarks that this shift operator connected to the Bessel differential operator (see [2, 3, 5]):
where . We remark that this shift operator is closely connected to the Bessel differential operator (see [4]):
The convolution operator determined by the is as follows:
Convolution (2.3) is known as a *B*-convolution. We note the following properties of the -convolution and the generalized shift operator. (1). (2).(3)If is a bounded function for all , and , then = (4)From (3),we have the following equality fo. (5).

The Fourier-Bessel transformation and its inverse transformation are defined as follows: where is the normalized Bessel function which is the eigenfunction of the Bessel differential operator. The following equalities for Fourier-Bessel transformation are true (see [5–7]):

*Definition 2.1. *The spectrum of the kernel of (1.18) is the bounded support of the Fourier Bessel transform for any fixed .

*Definition 2.2. *Let be a point in and denote by
the set of an interior of the forward cone, and denotes the closure of .

Let be spectrum of defined by (1.18) for any fixed and . Let be the Fourier Bessel transform of which is defined by

Lemma 2.3 (Fourier Bessel transform of operator). *One has
**
where
*

*Proof. *See [8].

Lemma 2.4 (Fourier Bessel transform of operator). *One has
**
where
*

*Proof. *See [8].

Lemma 2.5 (Fourier Bessel transform of operator). *One has
**
where
*

*Proof. *We can use the mathematical induction method; for , we have
where Then, from inverse Fourier transform we obtain
Assume that the statement is true for , that is,
Then, we must prove that is also true for So we have
This completes the proof.

Lemma 2.6. *For and , one has
**
where is a positive constant.*

*Proof. *See [9].

Lemma 2.7. *Let the operator be defined by
**
where is the operator iterated times and is given by
** is the dimension , is a positive integer, and is a positive constant. Then
**
is the elementary solution of (2.21) in the spectrum for *

*Proof. *Let where is the elementary solution of and is the Dirac-delta distribution. Thus
Applying the Fourier Bessel transform, which is defined by (2.4) to the both sides of the above equation and using Lemma 2.5 by considering we obtain
Thus, we get
where is the Heaviside function, because holds for

Therefore,
which has been already defined by (2.7). Thus from (2.5), we have
where is the spectrum of Thus, we obtain
as an elementary solution of (2.21) in the spectrum for

#### 3. Main Results

Theorem 3.1. *Let us consider the equation
**
with the initial condition
**
where is the operator iterated times and is defined by
** is the dimension is a positive integer, is an unknown function for is the given generalized function, and is a positive constant. Then
**
is a solution of (3.1), which satisfies (3.2), where is given by (2.23). In particular, if one puts and in (3.1), then (3.1) reduces to the equation
**
which is related to the Bessel heat equation.*

*Proof. *Taking the Fourier Bessel transform, which is defined by (2.4), of both sides of (3.1) for and using Lemma 2.5, we obtain
Thus, we consider the initial condition (3.2); then we have the following equality for (3.6):
Here, if we use (2.4) and (2.5), then we have
where Set
Since the integral in (3.9) is divergent, therefore we choose to be the spectrum of , and by (2.21), we have
Thus (3.8) can be written in the convolution form
Moreover, since exists, we can see that
hold (see [8]). Thus for the solution of (3.1), then we have
which satisfies (3.2). This completes the proof.

Theorem 3.2. *The kernel defined by (3.10) has the following properties. *(1)*-the space of continuous function for , with infinitely differentiable. *(2)* for all *(3)* for all *

*Proof. *(1) From (3.10) and
we have for

(2) We have since holds. Note here that we use the fact by the Fourier Bessel transformation. Then, we obtain
by direct computation.

(3) This case is obvious by (3.12).

In particular, if we put and in (3.1), then (3.1) reduces to the equation
and we obtain the solution of (3.16) in the convolution form
where is defined by (2.23) with which is related to Bessel heat equation. This completes the proof.

#### Acknowledgments

The authors would like to thank The Thailand Research Fund and Graduate School, Maejo University, Thailand for financial support.

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