On the Diamond Bessel Heat Kernel

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.


Introduction
The operator ♦ k has been first introduced by Kananthai 1 , is named as the diamond operator iterated k times, and is defined by 1.1 p q n, n is the dimension of the space R n for x x 1 , x 2 , . . ., x n ∈ R n , and k is a nonnegative integer.The operator ♦ k can be expressed in the following form: where k is the Laplacian operator iterated k-times defined by and k is the ultrahyperbolic operator iterated k-times defined by 1.4 Kananthai 1, Theorem 1.3 has shown that the convolution −1 k R e 2k x * R H 2k x is an elementary of the operator ♦ k .That is where R e 2k x is defined by α is a complex parameter, n is the dimension of R n , and the generalized function R H α υ is defined by and the constant K n α is given by the formula 1.9 where

1.11
And Yildirim see 4 have shown that the solution of the convolution form where S 2k x is defined by 2.8 with α 2k and R 2k x is defined by 2.9 with γ 2k.It is well known that for the heat equation with the initial condition where is the Laplace operator and is defined by 1.3 and x, t 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 R n {x : x x 1 , x 2 , . . ., x n , t , x i > 0, i 1, 2, 3, . . ., n}.
They obtain the solution in the classical convolution form where the symbol * is the B-convolution in 2.3 , as a solution of 1.18 , which satisfies 1.19 , where and Ω ⊂ R n is the spectrum of E x, t for any fixed t > 0, and J v i −1/2 x i , y i is the normalized Bessel function.Now, the purpose of this work is to study the equation with the initial condition and f x is a given generalized function for x ∈ R n .We obtain as a solution of 1.7 , where and Ω ⊂ R n is the spectrum of E x, t for any fixed t > 0, and R H 2 x is defined by 2.6 with α 2. The convolution R H 2 x * E x, t is called the Diamond Bessel Heat Kernel, and all properties will be studied in details.Before proceeding, the following definitions and concepts are needed.

Preliminaries
The shift operator according to the law remarks that this shift operator connected to the Bessel differential operator see 3, 5, 7, 8 : We remark that this shift operator is closely connected to the Bessel differential operator see 3, 5, 7, 8 ,

2.2
The convolution operator determined by the T y x is as follows: Convolution 2.3 is known as a B-convolution.We note the following properties of the Bconvolution and the generalized shift operator. 1 x is a bounded function for all x > 0, and 4 From 3 , we have the following equality for g x The Fourier-Bessel transformation and its inverse transformation are defined as follows: where J v i −1/2 x i , y i is the normalized Bessel function which is the eigenfunction of the Bessel differential operator.The following equalities for Fourier-Bessel transformation are true see 3, 5, 7, 8 : For any complex number α, we define the function S α x by p q the nondegenerated quadratic form.Denote the interior of the forward cone by Γ {x ∈ R n : where and γ is a complex number.By putting p 1 in R 2k x and taking into account Legendre's duplication formula for Γ z : the set of an interior of the forward cone, and Γ denotes the closure of Γ .
Let Ω be spectrum of E x, t defined by 1.21 for any fixed t > 0 and Ω ⊂ Γ .Let F B E y, t be the Fourier Bessel transform of E x, t , which is defined by

2.21
We can compute K n 2 from 2.7 as

2.22
By using the formula

2.24
Then, we obtain

2.25
Thus, Lemma 2.7.Let S α x and R β x be the functions defined by 2.8 and 2.9 , respectively.Then

2.27
where α and β are a positive even number.
Lemma 2.8 Fourier Bessel transform of k B operator .Consider where
Lemma 2.9 Fourier Bessel transform of k B operator .Consider where

2.32
where c is a positive constant.
Lemma 2.11.Let the operator L be defined by where B is the Laplace Bessel operator defined by is the elementary solution of 2.15 in the spectrum Ω ⊂ R n for t > 0.
Proof.Let LE x, t δ x, t , where E x, t is the elementary solution of L and δ is the Diracdelta distribution.Thus,

2.36
Applying the Fourier Bessel transform, which is defined by 2.4 to the both sides of the above equation and using Lemma 2.7 by considering F B δ x 1, we obtain

2.37
Thus, we get where H t is the Heaviside function, because H t 1 holds for t ≥ 0. Therefore, which has been already given by 2.7 .Thus, from 2.5 , we have where Ω is the spectrum of E x, t .Thus, we obtain as an elementary solution of 2.15 in the spectrum Ω ⊂ R n for t > 0.
Definition 2.12.We can extend E x, t to R n × R by setting Proof.Taking the Fourier Bessel transform, the both sides of 2.43 , for x ∈ R n and using Lemma 2.9, we obtain

2.47
Thus, we consider the initial condition 2.44 , then we have the following equality for 2.47 :

12 Journal of Applied Mathematics
Here, if we use 2.4 and 2.5 , then we have x i , y i y 2υ i i dy.

2.50
We choose Ω ⊂ R n , to be the spectrum of E x, t , and, by 2.35 , we have

2.51
Thus, 2.49 can be written in the convolution form u x, t E x, t * f x .

2.52
Moreover, since E x, t exists, we can see that

13
Definition 2.3.The spectrum of the kernel E x, t of 1.21 is the bounded support of the Fourier Bessel transform F B E y, t for any fixed t > 0.
Definition 2.4.Let x x 1 , x 2 , . . ., x n be a point in R n , and denote by is the dimension R n , k is a positive integer, u x, t is an unknown function for x, t x 1 , x 2 , . . ., x n , t ∈ R n × 0, ∞ , f x is the given generalized function, and c is a positive constant.
The properties of the Diamond Bessel Heat Kernel R H 2 x * E x, t .(1)RH 2 x * E x, t exists and is tempered distribution.(2)RH 2 x * E x, t ∈ C ∞ the space of continuous function and infinitely differentiable.(3)limt → 0 R H 2 x * E x, t R H 2 x .(4)∂/∂t B R H 2 x * E x, t − c 2 ♦ B R H 2 x * E x, t 0.Proof. 1 Since E x, t and R H 2 x are tempered distribution with compact support.Thus, R H 2 x * E x, t exists and is a tempered distribution.is infinitely differentiable and R H 2 x * E x, t ∈ C ∞ .3 By the continuity of the convolution, where E x, t is defined by 2.35 .