# On the Diamond Bessel Heat Kernel

Authors: Wanchak Satsanit
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

We study the heat equation inn 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.


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## Body

## 1. 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)◊k=((∑i=1p∂2∂xi2)2-(∑j=p+1p+q∂2∂xj2)2)k,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:
(1.2)◊k=▵k□k=□k0▵k,
where ▵k is the Laplacian operator iterated k-times defined by
(1.3)▵=∂2∂x12+∂2∂x22+⋯+∂2∂xn2,
and □k is the ultrahyperbolic operator iterated k-times defined by(1.4)□k=(∂2∂x12+∂2∂x22+⋯+∂2∂xp2-∂2∂xp+12-∂2∂xp+22-⋯-∂2∂xp+q2)k.
Kananthai [1, Theorem  1.3] has shown that the convolution (-1)kR2ke(x)*R2kH(x) is an elementary of the operator ◊k. That is(1.5)◊k((-1)kR2ke(x)*R2kH(x))=δ(x),
where R2ke(x) is defined by(1.6)Rαe(x)=|x|α-nWn(α),Wn(α)=πn/22αΓ(α/2)Γ((n-α)/2),α is a complex parameter, n is the dimension of ℝn, and the generalized function RαH(υ) is defined by(1.7)RαH(υ)={υ(α-n)/2Kn(α),forx∈Γ+,0,forx∉Γ+,
and the constant Kn(α) is given by the formula(1.8)Kn(α)=π(n-1)/2Γ((2+α-n)/2)Γ((1-α)/2)Γ(α)Γ((2+α-p)/2)Γ((p-α)/2).
The function RαH(υ) is called the ultrahyperbolic kernel of Marcel Riesz and was introduced by Nozaki (see [2, page  72]).Next, Yildirim et al. (see  [3]) first introduced the Bessel diamond operator ◊Bk iterated k-times, defined by(1.9)◊Bk=((∑i=1pBxi)2-(∑j=p+1p+qBxj)2)k,
where Bxi=∂2/∂xi2+(2υi/xi)(∂/∂xi), 2υi=2αi+1,αi>-1/2,xi>0. The operator ◊Bk can be expressed by ◊Bk=▵Bk□Bk=□Bk▵Bk, where(1.10)▵Bk=(∑i=1pBxi)k,(1.11)□Bk=(∑i=1pBxi-∑j=p+1p+qBxj)k.And Yildirim (see  [4]) have shown that the solution of the convolution form u(x)=(-1)kS2k(x)*R2k(x) is a unique elementary solution of ◊Bk, that is,(1.12)◊Bk((-1)kS2k(x)*R2k(x))=δ,
where S2k(x) is defined by (2.8) with α=2k and R2k(x) is defined by (2.9) with γ=2k.It is well known that for the heat equation(1.13)∂∂tu(x,t)=c2▵u(x,t)
with the initial condition(1.14)u(x,0)=f(x),
where ▵ is the Laplace operator and is defined by (1.3) and (x,t)=(x1,x2,…,xn,t)∈ℝn×(0,∞), we obtain(1.15)u(x,t)=1(4c2πt)n/2∫Rnexp⁡(-|x-y|24c2t)f(y)dy
as the solution of (1.13). Now, (1.15) can be written in the classical form(1.16)u(x,t)=E(x,t)*f(x),
where(1.17)E(x,t)=1(4c2πt)n/2exp⁡(-|x|24c2t).E(x,t) is called the heat kernel, where |x|2=x12+x22+⋯+xn2 and t>0, see [5, pages 208-209]. Moreover, we obtain E(x,t)→δ as t→0, where δ is the Dirac delta distribution.Next, Saglam et al. (see  [6]) have study the following equation,(1.18)∂∂tu(x,t)=c2□Bku(x,t)
with the initial condition(1.19)u(x,0)=f(x),forx∈Rn+,
where the operator □Bk 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}.They obtain the solution in the classical convolution form(1.20)u(x,t)=E(x,t)*f(x),
where the symbol * is the B-convolution in (2.3), as a solution of (1.18), which satisfies (1.19), where(1.21)E(x,t)=Cv∫Ω+ec2t[(y12+⋯+yp2)-(yp+12+⋯+yp+q2)]k∏i=1nJvi-1/2(xi,yi)yi2υidy,
and Ω+⊂Rn+ is the spectrum of E(x,t) for any fixed t>0, and Jvi-1/2(xi,yi) is the normalized Bessel function.Now, the purpose of this work is to study the equation(1.22)∂∂t□Bu(x,t)=c2◊Bu(x,t)
with the initial condition(1.23)□Bu(x,0)=f(x),
where (x,t)=(x1,…,xn,t)∈ℝn×(0,∞),t is a time, c is a positive constant, u(x,t) is an unknown function, and f(x) is a given generalized function for x∈ℝn. We obtain(1.24)u(x,t)=R2H(x)*E(x,t)*f(x)
as a solution of (1.7), where(1.25)E(x,t)=Cv∫Ω+e-c2t(y12+y22+y32+⋯+yn2)∏i=1nJvi-1/2(xi,yi)yi2υidy,
and Ω⊂ℝn is the spectrum of E(x,t) for any fixed t>0, and R2H(x) is defined by (2.6) with α=2. The convolution R2H(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.


## 2. Preliminaries

The shift operator according to the law remarks that this shift operator connected to the Bessel differential operator (see [3, 5, 7, 8]):(2.1)Txyφ(x)=Cv*∫0π⋯∫0πφ(x12+y12-2x1y1cos⁡θ1,…,xn2+yn2-2xnyncos⁡θn)×(∏i=1nsin2vi-1θi)dθ1⋯dθn,
where x,y∈Rn+,Cv*=∏i=1nΓ(vi+1)/Γ(1/2)Γ(vi). We remark that this shift operator is closely connected to the Bessel differential operator (see [3, 5, 7, 8]),
(2.2)d2Udx2+2vxdUdx=d2Udy2+2vydUdy,U(x,0)=f(x),Uy(x,0)=0.
The convolution operator determined by the Txy is as follows:
(2.3)(f*φ)(y)=∫Rn+f(y)Txyφ(x)(∏i=1nyi2vi)dy.
Convolution (2.3) is known as a B-convolution. We note the following properties of the B-convolution and the generalized shift operator.(1)
Txy·1=1,(2)
Tx0·f(x)=f(x),(3)
Iff(x),g(x)∈C(Rn+), g(x) is a bounded function for all x>0, and ∫0∞|f(x)|(∏i=1nxi2vi)dx<∞, then ∫Rn+Txyf(x)g(y)(∏i=1nyi2vi)dy=∫Rn+f(y)Txyg(x)(∏i=1nyi2vi)dy.(4)
From (3), we have the following equality forg(x)=1: ∫Rn+Txyf(x)(∏i=1nyi2vi)dy=∫Rn+f(y)(∏i=1nyi2vi)dy.(5)
(f*g)(x)=(g*f)(x).The Fourier-Bessel transformation and its inverse transformation are defined as follows:(2.4)(FBf)(x)=Cv∫Rn+f(y)(∏i=1nJvi-1/2(xi,yi)yi2υi)dy,(2.5)(FB-1f)(x)=(FBf)(-x),Cv=(∏i=1n2υi-1/2Γ(υi+12))-1,
where Jvi-1/2(xi,yi) 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]):(2.6)FBδ(x)=1,(2.7)FB(f*g)(x)=FBf(x)⋅FBg(x).Definition 2.1.
Letx=(x1,x2,…,xn), ν=(ν1,ν2,…,νn)∈ℝn+. For any complex number α, we define the function Sα(x) by
(2.8)Sα(x)=2n+2|ν|-2αΓ((n+2|ν|-α)/2)|x|α-n-2|ν|∏i=1n2νi-1/2Γ(νi+1/2).Definition 2.2.
Letx=(x1,x2,…,xn),ν=(ν1,ν2,…,νn)∈ℝn+, and denote by V=x12+x22+⋯+xp2-xp+12-xp+22-⋯-xp+q2 the nondegenerated quadratic form. Denote the interior of the forward cone by Γ+={x∈ℝn+:x1>0,x2>0,…,xn>0,V>0}. The function Rβ(x) is defined by
(2.9)Rγ(x)=V(γ-n-2|ν|)/2Kn(γ),
where
(2.10)Kn(γ)=π(n+2|ν|-1)/2Γ((2+γ-n-2|ν|)/2)Γ((1-γ)/2)Γ(γ)Γ((2+γ-p-2|ν|)/2)Γ((p-2|ν|-γ)/2),
and γ is a complex number.
By puttingp=1 in R2k(x) and taking into account Legendre’s duplication formula for Γ(z):
(2.11)Γ(2z)=22z-1π-1/2Γ(z)Γ(z+12),
we obtain
(2.12)Iγ(x)=υ(γ-n-2|ν|)/2Mn(γ)
and υ=x12-x22-⋯+xp2-xp+12-xp+22-⋯-xp+q2, where
(2.13)Mn(2k)=π(n+2|ν|-1)/222k-1Γ(2+2k-n-2|ν|2)Γ(k).Definition 2.3.
The spectrum of the kernelE(x,t) of (1.21) is the bounded support of the Fourier Bessel transform ℱBE(y,t) for any fixed t>0.Definition 2.4.
Letx=(x1,x2,…,xn) be a point in ℝn+, and denote by
(2.14)Γ+={x∈Rn+:x12+x22+⋯+xp2-xp+12-xp+22-⋯-xp+q2>0,ξ1>0}
the set of an interior of the forward cone, and Γ¯+ denotes the closure of Γ+.
LetΩ+ be spectrum of E(x,t) defined by (1.21) for any fixed t>0 and Ω⊂Γ¯+. Let FBE(y,t) be the Fourier Bessel transform of E(x,t), which is defined by
(2.15)FBE(y,t)={e-c2t(y12+y22+y32+⋯+yp2),forξ∈Ω+,0,forξ∉Ω+.Lemma 2.5.
Given the equation▵Bku(x)=δ(x) for x∈ℝn+, where ▵Bk is defined by (1.10). Then,
(2.16)u(x)=(-1)kS2k(x),
where S2k(x) is defined by (2.3), with α=2k. We obtain that (-1)kR2ke(x) is an elementary solution of the operator ▵Bk. That is
(2.17)▵Bk(-1)kS2k(x)=δ(x).Proof.
(See [3, page 379]).Lemma 2.6.
Given the equation□Bku(x)=δ(x) for x∈ℝn+, where □Bk is defined by (1.11). Then,
(2.18)u(x)=R2k(x),
where R2k(x) is defined by (2.4), with γ=2k. We obtain that R2k(x) is an elementary solution of the operator □Bk. That is
(2.19)□BkR2k(x)=δ(x).Proof (see [3, Page  379]).
From (2.8) with k=1, we obtain u(x)=R2H(x) as an elementary solution of the equation
(2.20)□u(x)=δ.
Now, from (2.17),
(2.21)R2H(υ)={υ(2-n-2|ν|)/2Kn(2),forx∈Γ+,0,forx∉Γ+,
We can compute Kn(2) from (2.7) as
(2.22)Kn(2)=π(n+2|ν|-1)/2Γ((4-n-2|ν|)/2)Γ(-1/2)Γ(2)Γ((4-p-2|ν|)/2)Γ((p+2|ν|-2)/2).
By using the formula
(2.23)Γ(z)Γ(1-z)=πsin⁡πz,
we obtain Γ(-1/2)=-2π, Γ(2)=1 and
(2.24)Γ(p+2|ν|-22)Γ(4-p-2|ν|2)=Γ(p+2|ν|-22)Γ(1-p+2|ν|-22)=πsin⁡(π(p+2|ν|-2)/2).
Then, we obtain
(2.25)Kn(2)=π(n+2|ν|-1)/2Γ((4-n)/2)(-2π)sin⁡(π((p+2|ν|-2)/2))π=-2π(n+2|ν|-2)/2Γ(4-n-2|ν|2)sin⁡(π(p+2|ν|-22)).
Thus,
(2.26)R2H(υ)={υ(2-n-2|ν|)/2-2π(n+2|ν|-2)/2Γ((4-n-2|ν|)/2)sin⁡(π((p+2|ν|-2)/2)),forx∈Γ+,0,forx∉Γ+,
where υ=x12+x22+⋯+xp2-xp+12-xp+22-⋯-xp+q2.Lemma 2.7.
LetSα(x) and Rβ(x) be the functions defined by (2.8) and (2.9), respectively. Then
(2.27)Sα(x)*Sβ(x)=Sα+β(x),Rβ(x)*Rα(x)=Rβ+α(x),
where α and β are a positive even number.Proof .
(See [3, pages 171–190]).Lemma 2.8 (Fourier Bessel transform of□Bk operator).
Consider(2.28)FB□Bku(x)=(-1)kV1k(x)FBu(x),
where
(2.29)V1k(x)=(∑i=1pxi2-∑j=p+1p+qxj2)k.Proof.
(See [4]).Lemma 2.9 (Fourier Bessel transform of▵Bk operator).
Consider(2.30)FB▵Bku(x)=(-1)k|x|2kFBu(x),
where
(2.31)|x|2k=(x12+x22+⋯+xn2)k.Proof.
(see [4, 9]).Lemma 2.10.
Fort,v>0, and x,y∈ℝn, we have
(2.32)∫0∞e-c2x2tx2υdx=Γ(υ)2c2υ+1tυ+1/2,∫0∞e-c2x2tJυ-1/2(xy)x2υdx=Γ(υ+1/2)2(c2t)υ+1/2e-y2/4c2t,
where c is a positive constant.Proof.
(See [6, Lemma  3.1.1] and [10]).Lemma 2.11.
Let the operatorL be defined by
(2.33)L=∂∂t-c2▵B,
where ▵B is the Laplace Bessel operator defined by
(2.34)▵B=Bx1+Bx2+Bx3+⋯+Bxn,Bxi=∂2∂xi2+2vixi∂∂xip+q=n is the dimension ℝn+, k is a positive integer, (x1,x2,…+xn)∈ℝn+, and c is a positive constant. Then,
(2.35)E(x,t)=Cv∫Ω+e-c2t(y12+y22+⋯+yn2)∏i=1nJvi-1/2(xi,yi)yi2υidy
is the elementary solution of (2.15) in the spectrum Ω+⊂Rn+ for t>0.Proof.
LetLE(x,t)=δ(x,t), where E(x,t) is the elementary solution of L and δ is the Dirac-delta distribution. Thus,
(2.36)∂∂tE(x,t)-c2(Bx1+Bx2+Bx3+⋯+Bxn)E(x,t)=δ(x)δ(t).
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 ℱBδ(x)=1, we obtain
(2.37)∂∂tFBE(x,t)+c2(x12+x22+x32+⋯+xn2)FBE(x,t)=δ(t).
Thus, we get
(2.38)FBE(x,t)=H(t)e-c2t(x12+x22+x32+⋯+xn2),
where H(t) is the Heaviside function, because H(t)=1 holds for t≥0.
Therefore,(2.39)FBE(x,t)=e-c2t(x12+x22+x32+⋯+xn2),
which has been already given by (2.7). Thus, from (2.5), we have
(2.40)E(x,t)=Cv∫Rn+e-c2t(y12+y22+y32+⋯+yn2)∏i=1nJvi-1/2(xi,yi)yi2υidy,
where Ω+ is the spectrum of E(x,t). Thus, we obtain
(2.41)E(x,t)=Cv∫Ω+e-c2t(y12+y22+y32+⋯+yp2)∏i=1nJvi-1/2(xi,yi)yi2υidy
as an elementary solution of (2.15) in the spectrum Ω+⊂Rn+ for t>0.Definition 2.12.
We can extendE(x,t) to ℝn×ℝ by setting
(2.42)E(x,t)={1(2π)n∫Ωexp⁡(-c2t(ξi2+ξ22+ξ32+⋯+ξn2)),fort>0,0,fort≤0.Lemma 2.13.
Let us consider the equation(2.43)∂∂tu(x,t)-c2▵Bu(x,t)=0
with the initial condition
(2.44)u(x,0)=f(x),
where ▵B is the operator iterated k-times and is defined by
(2.45)▵B=(Bx1+Bx2+Bx3+⋯+B),Bxi=∂2∂xi2+2vixi∂∂xi,p+q=n is the dimension ℝn+,k is a positive integer, u(x,t) is an unknown function for (x,t)=(x1,x2,…,xn,t)∈ℝn+×(0,∞), f(x) is the given generalized function, and c is a positive constant. Then,
(2.46)u(x,t)=E(x,t)*f(x)
is a solution of (2.43), which satisfies (2.44), where E(x,t) is given by (2.35).Proof.
Taking the Fourier Bessel transform, the both sides of (2.43), for x∈ℝn+ and using Lemma 2.9, we obtain
(2.47)∂∂tFBu(x,t)=-c2(x12+x22+x32+⋯+xn2)FBu(x,t).
Thus, we consider the initial condition (2.44), then we have the following equality for (2.47):
(2.48)u(x,t)=f(x)*FB-1e-c2t(x12+x22+x32+⋯+xn2).
Here, if we use (2.4) and (2.5), then we have
(2.49)u(x,t)=f(x)*FB-1e-c2t(y12+y22+y32+⋯+yn2)=∫Rn+FB-1e-c2t(y12+⋯+yn2)Txyf(x)(∏i=1nyi2υi)dy=∫Rn+(Cv∫Rn+e-c2tVk(z)∏i=1nJυi-1/2(yi,zi)zi2υidz)Txyf(x)(∏i=1nyi2υi)dy,
where V(z)=(z12+z22+z32+⋯+zn2). Set
(2.50)E(x,t)=Cv∫Rn+e-c2t(y12+y22+y32+⋯+yn2)∏i=1nJvi-1/2(xi,yi)yi2υidy.
We choose Ω+⊂ℝn+, to be the spectrum of E(x,t), and, by (2.35), we have
(2.51)E(x,t)=Cv∫Rn+e-c2t(y12+y22+y32+⋯+yn2)∏i=1nJvi-1/2(xi,yi)yi2υidy=Cv∫Ω+e-c2t(y12+y22+y32+⋯+yn2)∏i=1nJvi-1/2(xi,yi)yi2υidy.
Thus, (2.49) can be written in the convolution form
(2.52)u(x,t)=E(x,t)*f(x).
Moreover, since E(x,t) exists, we can see that
(2.53)lim⁡t→0E(x,t)=Cv∫Ω+∏i=1nJvi-1/2(xi,yi)yi2vidy=Cv∫Rn+∏i=1nJvi-1/2(xi,yi)yi2vidy=δ(x),forx∈Rn+.
Thus, for the solution u(x,t)=E(x,t)*f(x) of (2.43), we have
(2.54)lim⁡t→0u(x,t)=u(x,0)=δ*f(x)=f(x),
which satisfies (2.44). This completes the proof.


## 3. Main Results

Theorem 3.1.
Given the equation(3.1)∂∂t(□Bu(x,t))=c2◊Bu(x,t)
with the initial condition
(3.2)□Bu(x,0)=f(x),
where f(x) is a given generalized function for ℝn, □B is an ultrahyperbolic Bessel operator, and ◊B is the Diamond Bessel operator defined by (1.11) and (1.9), respectively. Then, we obtain
(3.3)u(x,t)=R2H(x)*E(x,t)*f(x)
as a solution of (3.1), where R2H(x) is given by (2.9) with γ=2 and E(x,t) is given by (2.35).Proof.
Equation (3.1) can be written in the form
(3.4)∂∂t(□Bu(x,t))=c2▵B(□Bu(x,t)).
Let w(x,t)=□Bu(x,t). Thus, the above equation can be written as
(3.5)∂∂t(w(x,t))=c2(▵Bw(x,t)).
We can solve the above equation by applying then-dimensional Fourier Bessel transform to both sides of (3.5). By Lemma 2.7, we obtain
(3.6)w(x,t)=□Bu(x,t)=E(x,t)*f(x).
By convolving both sides of (3.6) by function R2H(x), we obtain
(3.7)R2H(x)*□Bu(x,t)=R2H(x)*E(x,t)*f(x).
By properties of convolution,
(3.8)□R2H(x)*u(x,t)=R2H(x)*E(x,t)*f(x).
By Lemma 2.5, we obtain
(3.9)δ*u(x,t)=R2H(x)*E(x,t)*f(x),
or
(3.10)u(x,t)=R2H(x)*E(x,t)*f(x)
is a solution of (3.1). As shown in (3.5) and by the continuity of convolution,
(3.11)lim⁡t→0□Bu(x,t)=lim⁡t→0(E(x,t)*f(x))=δ*f(x)=f(x).Theorem 3.2 (The properties of the Diamond Bessel Heat KernelR2H(x)*E(x,t)).
(1)R2H(x)*E(x,t) exists and is tempered distribution.
(2)R2H(x)*E(x,t)∈ℂ∞ the space of continuous function and infinitely differentiable.
(3)limt→0(R2H(x)*E(x,t))=R2H(x).
(4)(∂/∂t)□B(R2H(x)*E(x,t))-c2◊B(R2H(x)*E(x,t))=0.Proof.
(1) SinceE(x,t) and R2H(x) are tempered distribution with compact support. Thus, R2H(x)*E(x,t) exists and is a tempered distribution.
(2) We have(3.12)∂n∂xn(R2H(x)*E(x,t))=R2H(x)*∂n∂xnE(x,t),
since E(x,t) is infinitely differentiable and R2H(x)*E(x,t)∈ℂ∞.
(3) By the continuity of the convolution,(3.13)R2H(x)*E(x,t)⟶R2H(x)*δast⟶0.
Thus,
(3.14)lim⁡t→0(R2H(x)*E(x,t))=R2H(x).
(4) Since(3.15)□B(R2H(x)*E(x,t))=□BR2H(x)*E(x,t)=δ*E(x,t)=E(x,t),◊B(R2H(x)*E(x,t))=▵B(□BR2H(x)*E(x,t))=▵B(δ*E(x,t))=▵BE(x,t).
thus
(3.16)∂∂t□B(R2H(x)*E(x,t))-c2◊B(R2H(x)*E(x,t))=∂∂tE(x,t)-c2▵BE(x,t)=(∂∂t-c2▵B)E(x,t)=0,
where E(x,t) is defined by (2.35).



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