On the Diamond Bessel Heat Kernel

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.

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 ◊^𝑘=^𝑝_𝑖=1𝜕²𝜕𝑥²_𝑖²−^𝑝+𝑞_𝑗=𝑝+1𝜕²𝜕𝑥²_𝑗^2𝑘, (1.1)𝑝+𝑞=𝑛, 𝑛 is the dimension of the space ℝ^𝑛 for 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛, and 𝑘 is a nonnegative integer. The operator ◊^𝑘 can be expressed in the following form: ◊^𝑘=▵^𝑘□^𝑘=□^𝑘0▵^𝑘, (1.2) where ▵^𝑘 is the Laplacian operator iterated 𝑘-times defined by 𝜕▵=²𝜕𝑥²₁+𝜕²𝜕𝑥²₂𝜕+⋯+²𝜕𝑥²_𝑛, (1.3) and □^𝑘 is the ultrahyperbolic operator iterated 𝑘-times defined by□^𝑘=𝜕²𝜕𝑥²₁+𝜕²𝜕𝑥²₂𝜕+⋯+²𝜕𝑥²_𝑝−𝜕²𝜕𝑥²_𝑝+1−𝜕²𝜕𝑥²_𝑝+2𝜕−⋯−²𝜕𝑥²_𝑝+𝑞^𝑘. (1.4) Kananthai [1, Theorem  1.3] has shown that the convolution (−1)^𝑘𝑅^𝑒_2𝑘(𝑥)∗𝑅^𝐻_2𝑘(𝑥) is an elementary of the operator ◊^𝑘. That is◊^𝑘(−1)^𝑘𝑅^𝑒_2𝑘(𝑥)∗𝑅^𝐻_2𝑘(𝑥)=𝛿(𝑥), (1.5) where 𝑅^𝑒_2𝑘(𝑥) is defined by𝑅^𝑒_𝛼(𝑥)=|𝑥|^𝛼−𝑛𝑊_𝑛(𝛼),𝑊_𝑛(𝛼)=𝜋^𝑛/22^𝛼Γ(𝛼/2)Γ)),((𝑛−𝛼/2 (1.6)𝛼 is a complex parameter, 𝑛 is the dimension of ℝ^𝑛, and the generalized function 𝑅^𝐻_𝛼(𝜐) is defined by𝑅^𝐻_𝛼(𝜐)=𝜐^{(𝛼−𝑛)/2}𝐾_𝑛(𝛼),for𝑥∈Γ₊,0,for𝑥∉Γ₊, (1.7) and the constant 𝐾_𝑛(𝛼) is given by the formula𝐾_𝑛(𝛼)=𝜋^(𝑛−1)/2Γ((2+𝛼−𝑛)/2)Γ((1−𝛼)/2)Γ(𝛼)ΓΓ.2+𝛼−𝑝/2𝑝−𝛼/2 (1.8) The function 𝑅^𝐻_𝛼(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 ◊^𝑘_𝐵 iterated 𝑘-times, defined by◊^𝑘_𝐵=^𝑝_𝑖=1𝐵_𝑥𝑖²−^𝑝+𝑞_𝑗=𝑝+1𝐵_𝑥𝑗^2𝑘, (1.9) where 𝐵_𝑥𝑖=𝜕²/𝜕𝑥²_𝑖+(2𝜐_𝑖/𝑥_𝑖)(𝜕/𝜕𝑥_𝑖), 2𝜐_𝑖=2𝛼_𝑖+1,𝛼_𝑖>−1/2,𝑥_𝑖>0. The operator ◊^𝑘_𝐵 can be expressed by ◊^𝑘_𝐵=▵^𝑘_𝐵□^𝑘_𝐵=□^𝑘_𝐵▵^𝑘_𝐵, where▵^𝑘_𝐵=^𝑝_𝑖=1𝐵_𝑥𝑖^𝑘□_𝑘𝐵,=^𝑝_𝑖=1𝐵_𝑥𝑖−^𝑝+𝑞_𝑗=𝑝+1𝐵_𝑥𝑗^𝑘. (1.10)(1.11)

And Yildirim (see  [4]) have shown that the solution of the convolution form 𝑢(𝑥)=(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥) is a unique elementary solution of ◊^𝑘_𝐵, that is,◊^𝑘_𝐵(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)=𝛿, (1.12) 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𝜕𝑢()𝜕𝑡𝑥,𝑡=𝑐²▵𝑢(𝑥,𝑡) (1.13) with the initial condition𝑢(𝑥,0)=𝑓(𝑥), (1.14) where ▵ is the Laplace operator and is defined by (1.3) and (𝑥,𝑡)=(𝑥₁,𝑥₂,…,𝑥_𝑛,𝑡)∈ℝ^𝑛×(0,∞), we obtain𝑢()=1𝑥,𝑡4𝑐²𝜋𝑡^{𝑛/2ℝ𝑛}exp−𝑥−𝑦²4𝑐²𝑡𝑓𝑦𝑑𝑦 (1.15) as the solution of (1.13). Now, (1.15) can be written in the classical form𝑢(𝑥,𝑡)=𝐸(𝑥,𝑡)∗𝑓(𝑥), (1.16) where𝐸()=1𝑥,𝑡4𝑐²𝜋𝑡^𝑛/2−|𝑥|exp²4𝑐²𝑡. (1.17)𝐸(𝑥,𝑡) is called the heat kernel, where |𝑥|²=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑛 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,𝜕𝑢()𝜕𝑡𝑥,𝑡=𝑐²□^𝑘_𝐵𝑢(𝑥,𝑡) (1.18) with the initial condition𝑢(𝑥,0)=𝑓(𝑥),for𝑥∈𝑅⁺_𝑛, (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 𝑅⁺_𝑛={𝑥∶𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛,𝑡),𝑥_𝑖>0,𝑖=1,2,3,…,𝑛}.

They obtain the solution in the classical convolution form𝑢(𝑥,𝑡)=𝐸(𝑥,𝑡)∗𝑓(𝑥), (1.20) where the symbol ∗ is the 𝐵-convolution in (2.3), as a solution of (1.18), which satisfies (1.19), where𝐸(𝑥,𝑡)=𝐶_𝑣Ω+𝑒^{𝑐2𝑡[(𝑦21+⋯+𝑦2𝑝)−(𝑦2𝑝+1+⋯+𝑦2𝑝+𝑞)]𝑘𝑛}_𝑖=1𝐽_{𝑣𝑖−1/2}𝑥_𝑖,𝑦_𝑖𝑦^2𝜐𝑖_𝑖𝑑𝑦, (1.21) and Ω⁺⊂𝑅⁺_𝑛 is the spectrum of 𝐸(𝑥,𝑡) for any fixed 𝑡>0, and 𝐽_{𝑣𝑖−1/2}(𝑥_𝑖,𝑦_𝑖) is the normalized Bessel function.

Now, the purpose of this work is to study the equation𝜕□𝜕𝑡^𝐵𝑢(𝑥,𝑡)=𝑐²◊_𝐵𝑢(𝑥,𝑡) (1.22) with the initial condition□_𝐵𝑢(𝑥,0)=𝑓(𝑥), (1.23) where (𝑥,𝑡)=(𝑥₁,…,𝑥_𝑛,𝑡)∈ℝ^𝑛×(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∈ℝ^𝑛. We obtain𝑢(𝑥,𝑡)=𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)∗𝑓(𝑥) (1.24) as a solution of (1.7), where𝐸(𝑥,𝑡)=𝐶_𝑣Ω+𝑒^{−𝑐2𝑡(𝑦21+𝑦22+𝑦23+⋯+𝑦2𝑛)𝑛}_𝑖=1𝐽_{𝑣𝑖−1/2}𝑥_𝑖,𝑦_𝑖𝑦^2𝜐𝑖_𝑖𝑑𝑦, (1.25) and Ω⊂ℝ^𝑛 is the spectrum of 𝐸(𝑥,𝑡) for any fixed 𝑡>0, and 𝑅^𝐻₂(𝑥) is defined by (2.6) with 𝛼=2. The convolution 𝑅^𝐻₂(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]):𝑇^𝑦_𝑥𝜑(𝑥)=𝐶^∗_𝑣^𝜋₀⋯^𝜋₀𝜑𝑥²₁+𝑦²₁−2𝑥₁𝑦₁cos𝜃₁,…,𝑥²_𝑛+𝑦²_𝑛−2𝑥_𝑛𝑦_𝑛cos𝜃_𝑛×^𝑛_𝑖=1sin^{2𝑣𝑖−1}𝜃_𝑖𝑑𝜃₁⋯𝑑𝜃_𝑛, (2.1) where 𝑥,𝑦∈𝑅⁺_𝑛,𝐶^∗_𝑣=∏^𝑛_𝑖=1Γ(𝑣_𝑖+1)/Γ(1/2)Γ(𝑣_𝑖). We remark that this shift operator is closely connected to the Bessel differential operator (see [3, 5, 7, 8]), 𝑑²𝑈𝑑𝑥²+2𝑣𝑥𝑑𝑈=𝑑𝑑𝑥²𝑈𝑑𝑦²+2𝑣𝑦𝑑𝑈,𝑈()(𝑥)𝑑𝑦𝑥,0=𝑓,𝑈_𝑦(𝑥,0)=0. (2.2) The convolution operator determined by the 𝑇^𝑦_𝑥 is as follows: 𝑓∗𝜑𝑦=_𝑅+𝑛𝑓𝑦𝑇^𝑦_𝑥𝜑(𝑥)^𝑛_𝑖=1𝑦^2𝑣𝑖_𝑖𝑑𝑦. (2.3) Convolution (2.3) is known as a 𝐵-convolution. We note the following properties of the 𝐵-convolution and the generalized shift operator.(1)𝑇^𝑦_𝑥⋅1=1,(2)𝑇⁰_𝑥⋅𝑓(𝑥)=𝑓(𝑥),(3) If 𝑓(𝑥),𝑔(𝑥)∈𝐶(𝑅⁺_𝑛), 𝑔(𝑥) is a bounded function for all 𝑥>0, and ∫^∞₀|𝑓(𝑥)|(∏^𝑛_𝑖=1𝑥^2𝑣𝑖_𝑖)𝑑𝑥<∞, then ∫_𝑅+𝑛𝑇^𝑦_𝑥𝑓(𝑥)𝑔(𝑦)(∏^𝑛_𝑖=1𝑦^2𝑣𝑖_𝑖)𝑑𝑦=∫_𝑅+𝑛𝑓(𝑦)𝑇^𝑦_𝑥𝑔(𝑥)(∏^𝑛_𝑖=1𝑦^2𝑣𝑖_𝑖)𝑑𝑦.(4) From (3), we have the following equality for 𝑔(𝑥)=1: ∫_𝑅+𝑛𝑇^𝑦_𝑥𝑓(𝑥)(∏^𝑛_𝑖=1𝑦^2𝑣𝑖_𝑖)𝑑𝑦=∫_𝑅+𝑛𝑓(𝑦)(∏^𝑛_𝑖=1𝑦^2𝑣𝑖_𝑖)𝑑𝑦.(5)(𝑓∗𝑔)(𝑥)=(𝑔∗𝑓)(𝑥).

The Fourier-Bessel transformation and its inverse transformation are defined as follows:ℱ_𝐵𝑓(𝑥)=𝐶_{𝑣𝑅+𝑛}𝑓𝑦^𝑛_𝑖=1𝐽_{𝑣𝑖−1/2}𝑥_𝑖,𝑦_𝑖𝑦^2𝜐𝑖_𝑖ℱ𝑑𝑦,_𝐵−1𝑓(𝑥)=ℱ_𝐵𝑓(−𝑥),𝐶_𝑣=^𝑛_𝑖=12^{𝜐𝑖−1/2}Γ𝜐_𝑖+12⁻¹, (2.4)(2.5) where 𝐽_{𝑣𝑖−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]):ℱ_𝐵𝛿(𝑥)ℱ=1,_𝐵𝑓∗𝑔(𝑥)=ℱ_𝐵𝑓(𝑥)⋅ℱ_𝐵𝑔(𝑥). (2.6)(2.7)

Definition 2.1. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛), 𝜈=(𝜈₁,𝜈₂,…,𝜈_𝑛)∈ℝ⁺_𝑛. For any complex number 𝛼, we define the function 𝑆_𝛼(𝑥) by 𝑆_𝛼(𝑥)=2^{𝑛+2|𝜈|−2𝛼}Γ((𝑛+2|𝜈|−𝛼)/2)|𝑥|^{𝛼−𝑛−2|𝜈|}∏^𝑛_𝑖=12^{𝜈𝑖−1/2}Γ𝜈_𝑖+1/2. (2.8)

Definition 2.2. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛),𝜈=(𝜈₁,𝜈₂,…,𝜈_𝑛)∈ℝ⁺_𝑛, and denote by 𝑉=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝−𝑥²_𝑝+1−𝑥²_𝑝+2−⋯−𝑥²_𝑝+𝑞 the nondegenerated quadratic form. Denote the interior of the forward cone by Γ₊={𝑥∈ℝ⁺_𝑛∶𝑥₁>0,𝑥₂>0,…,𝑥_𝑛>0,𝑉>0}. The function 𝑅_𝛽(𝑥) is defined by 𝑅_𝛾(𝑥)=𝑉^{(𝛾−𝑛−2|𝜈|)/2}𝐾_𝑛𝛾, (2.9) where 𝐾_𝑛𝛾=𝜋^{(𝑛+2|𝜈|−1)/2}Γ2+𝛾−𝑛−2|𝜈|/2Γ1−𝛾/2Γ𝛾Γ|𝜈|Γ|𝜈|,2+𝛾−𝑝−2/2𝑝−2−𝛾/2 (2.10) 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): Γ(2𝑧)=2^2𝑧−1𝜋^−1/2Γ(𝑧)Γ1𝑧+2, (2.11) we obtain 𝐼_𝛾(𝑥)=𝜐^{(𝛾−𝑛−2|𝜈|)/2}𝑀_𝑛𝛾 (2.12) and 𝜐=𝑥²₁−𝑥²₂−⋯+𝑥²_𝑝−𝑥²_𝑝+1−𝑥²_𝑝+2−⋯−𝑥²_𝑝+𝑞, where 𝑀_𝑛(2𝑘)=𝜋^{(𝑛+2|𝜈|−1)/2}2^2𝑘−1Γ2+2𝑘−𝑛−2|𝜈|2Γ(𝑘). (2.13)

Definition 2.3. The spectrum of the kernel 𝐸(𝑥,𝑡) of (1.21) is the bounded support of the Fourier Bessel transform ℱ_𝐵𝐸(𝑦,𝑡) for any fixed 𝑡>0.

Definition 2.4. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) be a point in ℝ⁺_𝑛, and denote by Γ₊=𝑥∈ℝ⁺_𝑛∶𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝−𝑥²_𝑝+1−𝑥²_𝑝+2−⋯−𝑥²_𝑝+𝑞>0,𝜉₁>0 (2.14) 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_𝐵𝐸(𝑦,𝑡) be the Fourier Bessel transform of 𝐸(𝑥,𝑡), which is defined by ℱ_𝐵𝐸𝑦,𝑡=𝑒^{−𝑐2𝑡(𝑦21+𝑦22+𝑦23+⋯+𝑦2𝑝),}for𝜉∈Ω₊,0,for𝜉∉Ω₊. (2.15)

Lemma 2.5. Given the equation ▵^𝑘_𝐵𝑢(𝑥)=𝛿(𝑥) for 𝑥∈ℝ⁺_𝑛, where ▵^𝑘_𝐵 is defined by (1.10). Then, 𝑢(𝑥)=(−1)^𝑘𝑆_2𝑘(𝑥), (2.16) where 𝑆_2𝑘(𝑥) is defined by (2.3), with 𝛼=2𝑘. We obtain that (−1)^𝑘𝑅^𝑒_2𝑘(𝑥) is an elementary solution of the operator ▵^𝑘_𝐵. That is ▵^𝑘_𝐵(−1)^𝑘𝑆_2𝑘(𝑥)=𝛿(𝑥). (2.17)

Proof. (See [3, page 379]).

Lemma 2.6. Given the equation □^𝑘_𝐵𝑢(𝑥)=𝛿(𝑥) for 𝑥∈ℝ⁺_𝑛, where □^𝑘_𝐵 is defined by (1.11). Then, 𝑢(𝑥)=𝑅_2𝑘(𝑥), (2.18) where 𝑅_2𝑘(𝑥) is defined by (2.4), with 𝛾=2𝑘. We obtain that 𝑅_2𝑘(𝑥) is an elementary solution of the operator □^𝑘_𝐵. That is □^𝑘_𝐵𝑅_2𝑘(𝑥)=𝛿(𝑥). (2.19)

Proof (see [3, Page  379]). From (2.8) with 𝑘=1, we obtain 𝑢(𝑥)=𝑅^𝐻₂(𝑥) as an elementary solution of the equation □𝑢(𝑥)=𝛿. (2.20) Now, from (2.17), 𝑅^𝐻₂(𝜐)=𝜐^{(2−𝑛−2|𝜈|)/2}𝐾_𝑛(2),for𝑥∈Γ₊,0,for𝑥∉Γ₊, (2.21) We can compute 𝐾_𝑛(2) from (2.7) as 𝐾_𝑛(2)=𝜋^{(𝑛+2|𝜈|−1)/2}Γ((4−𝑛−2|𝜈|)/2)Γ(−1/2)Γ(2)Γ|𝜈|Γ|𝜈|.4−𝑝−2/2𝑝+2−2/2 (2.22) By using the formula Γ(𝑧)Γ()=𝜋1−𝑧,sin𝜋𝑧 (2.23) we obtain Γ(−1/2)=−2√𝜋, Γ(2)=1 and Γ𝑝+2|𝜈|−22Γ4−𝑝−2|𝜈|2|𝜈|=Γ𝑝+2−22Γ|𝜈|1−𝑝+2−22=𝜋𝜋|𝜈|.sin𝑝+2−2/2 (2.24) Then, we obtain 𝐾_𝑛(2)=𝜋^{(𝑛+2|𝜈|−1)/2}Γ((4−𝑛)/2)−2√𝜋sin𝜋𝑝+2|𝜈|−2/2𝜋=−2𝜋^{(𝑛+2|𝜈|−2)/2}Γ4−𝑛−2|𝜈|2𝜋|𝜈|sin𝑝+2−22. (2.25) Thus, 𝑅^𝐻₂(𝜐)=𝜐^{(2−𝑛−2|𝜈|)/2}−2𝜋^{(𝑛+2|𝜈|−2)/2}Γ((4−𝑛−2|𝜈|)/2)sin𝜋𝑝+2|𝜈|−2/2,for𝑥∈Γ₊,0,for𝑥∉Γ₊, (2.26) where 𝜐=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝−𝑥²_𝑝+1−𝑥²_𝑝+2−⋯−𝑥²_𝑝+𝑞.

Lemma 2.7. Let 𝑆_𝛼(𝑥) and 𝑅_𝛽(𝑥) be the functions defined by (2.8) and (2.9), respectively. Then 𝑆_𝛼(𝑥)∗𝑆_𝛽(𝑥)=𝑆_𝛼+𝛽(𝑥),𝑅_𝛽(𝑥)∗𝑅_𝛼(𝑥)=𝑅_𝛽+𝛼(𝑥), (2.27) where 𝛼 and 𝛽 are a positive even number.

Proof . (See [3, pages 171–190]).

Lemma 2.8 (Fourier Bessel transform of □^𝑘_𝐵 operator). Consider ℱ_𝐵□^𝑘_𝐵𝑢(𝑥)=(−1)^𝑘𝑉^𝑘₁(𝑥)ℱ_𝐵𝑢(𝑥), (2.28) where 𝑉^𝑘₁(𝑥)=^𝑝_𝑖=1𝑥²_𝑖−^𝑝+𝑞_𝑗=𝑝+1𝑥²_𝑗^𝑘. (2.29)

Proof. (See [4]).

Lemma 2.9 (Fourier Bessel transform of ▵^𝑘_𝐵 operator). Consider ℱ_𝐵▵^𝑘_𝐵𝑢(𝑥)=(−1)^𝑘|𝑥|^2𝑘ℱ_𝐵𝑢(𝑥), (2.30) where |𝑥|^2𝑘=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑛^𝑘. (2.31)

Proof. (see [4, 9]).

Lemma 2.10. For 𝑡,𝑣>0, and 𝑥,𝑦∈ℝ^𝑛, we have ^∞₀𝑒^{−𝑐2𝑥2𝑡}𝑥^2𝜐Γ(𝜐)𝑑𝑥=2𝑐^2𝜐+1𝑡^𝜐+1/2_∞0,𝑒^{−𝑐2𝑥2𝑡}𝐽_𝜐−1/2𝑥𝑦𝑥^2𝜐Γ()𝑑𝑥=𝜐+1/22𝑐²𝑡^𝜐+1/2𝑒^{−𝑦2/4𝑐2𝑡}, (2.32) where 𝑐 is a positive constant.

Proof. (See [6, Lemma  3.1.1] and [10]).

Lemma 2.11. Let the operator 𝐿 be defined by 𝜕𝐿=𝜕𝑡−𝑐²▵_𝐵, (2.33) where ▵_𝐵 is the Laplace Bessel operator defined by ▵_𝐵=𝐵_𝑥1+𝐵_𝑥2+𝐵_𝑥3+⋯+𝐵_𝑥𝑛,𝐵_𝑥𝑖=𝜕²𝜕𝑥²_𝑖+2𝑣_𝑖𝑥_𝑖𝜕𝜕𝑥_𝑖 (2.34)𝑝+𝑞=𝑛 is the dimension ℝ⁺_𝑛, 𝑘 is a positive integer, (𝑥₁,𝑥₂,…+𝑥_𝑛)∈ℝ⁺_𝑛, and 𝑐 is a positive constant. Then, 𝐸(𝑥,𝑡)=𝐶_𝑣Ω+𝑒^{−𝑐2𝑡(𝑦21+𝑦22+⋯+𝑦2𝑛)𝑛}_𝑖=1𝐽_{𝑣𝑖−1/2}𝑥_𝑖,𝑦_𝑖𝑦^2𝜐𝑖_𝑖𝑑𝑦 (2.35) 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 Dirac-delta distribution. Thus, 𝜕𝐸()𝜕𝑡𝑥,𝑡−𝑐²𝐵_𝑥1+𝐵_𝑥2+𝐵_𝑥3+⋯+𝐵_𝑥𝑛𝐸(𝑥,𝑡)=𝛿(𝑥)𝛿(𝑡). (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_𝐵𝛿(𝑥)=1, we obtain 𝜕ℱ𝜕𝑡^𝐵𝐸(𝑥,𝑡)+𝑐²𝑥²₁+𝑥²₂+𝑥²₃+⋯+𝑥²_𝑛ℱ_𝐵𝐸(𝑥,𝑡)=𝛿(𝑡). (2.37) Thus, we get ℱ_𝐵𝐸(𝑥,𝑡)=𝐻(𝑡)𝑒^{−𝑐2𝑡(𝑥21+𝑥22+𝑥23+⋯+𝑥2𝑛)}, (2.38) where 𝐻(𝑡) is the Heaviside function, because 𝐻(𝑡)=1 holds for 𝑡≥0.
Therefore, ℱ_𝐵𝐸(𝑥,𝑡)=𝑒^{−𝑐2𝑡(𝑥21+𝑥22+𝑥23+⋯+𝑥2𝑛)}, (2.39) which has been already given by (2.7). Thus, from (2.5), we have 𝐸(𝑥,𝑡)=𝐶_𝑣ℝ+𝑛𝑒^{−𝑐2𝑡(𝑦21+𝑦22+𝑦23+⋯+𝑦2𝑛)𝑛}_𝑖=1𝐽_{𝑣𝑖−1/2}𝑥_𝑖,𝑦_𝑖𝑦^2𝜐𝑖_𝑖𝑑𝑦, (2.40) where Ω⁺ is the spectrum of 𝐸(𝑥,𝑡). Thus, we obtain 𝐸(𝑥,𝑡)=𝐶_𝑣Ω+𝑒^{−𝑐2𝑡(𝑦21+𝑦22+𝑦23+⋯+𝑦2𝑝)𝑛}_𝑖=1𝐽_{𝑣𝑖−1/2}𝑥_𝑖,𝑦_𝑖𝑦^2𝜐𝑖_𝑖𝑑𝑦 (2.41) 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 𝐸()=1𝑥,𝑡(2𝜋)^𝑛_Ωexp−𝑐²𝑡𝜉²_𝑖+𝜉²₂+𝜉²₃+⋯+𝜉²_𝑛,for𝑡>0,0,for𝑡≤0. (2.42)

Lemma 2.13. Let us consider the equation 𝜕𝑢()𝜕𝑡𝑥,𝑡−𝑐²▵_𝐵𝑢(𝑥,𝑡)=0 (2.43) with the initial condition 𝑢(𝑥,0)=𝑓(𝑥), (2.44) where ▵_𝐵 is the operator iterated 𝑘-times and is defined by ▵_𝐵=𝐵_𝑥1+𝐵_𝑥2+𝐵_𝑥3,𝐵+⋯+𝐵_𝑥𝑖=𝜕²𝜕𝑥²_𝑖+2𝑣_𝑖𝑥_𝑖𝜕𝜕𝑥_𝑖, (2.45)𝑝+𝑞=𝑛 is the dimension ℝ⁺_𝑛,𝑘 is a positive integer, 𝑢(𝑥,𝑡) is an unknown function for (𝑥,𝑡)=(𝑥₁,𝑥₂,…,𝑥_𝑛,𝑡)∈ℝ⁺_𝑛×(0,∞), 𝑓(𝑥) is the given generalized function, and 𝑐 is a positive constant. Then, 𝑢(𝑥,𝑡)=𝐸(𝑥,𝑡)∗𝑓(𝑥) (2.46) 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∈ℝ⁺_𝑛 and using Lemma 2.9, we obtain 𝜕ℱ𝜕𝑡^𝐵𝑢(𝑥,𝑡)=−𝑐²𝑥²₁+𝑥²₂+𝑥²₃+⋯+𝑥²_𝑛ℱ_𝐵𝑢(𝑥,𝑡). (2.47) Thus, we consider the initial condition (2.44), then we have the following equality for (2.47): 𝑢(𝑥,𝑡)=𝑓(𝑥)∗ℱ_𝐵⁻¹𝑒^{−𝑐2𝑡(𝑥21+𝑥22+𝑥23+⋯+𝑥2𝑛)}. (2.48) Here, if we use (2.4) and (2.5), then we have 𝑢(𝑥,𝑡)=𝑓(𝑥)∗ℱ_𝐵⁻¹𝑒^{−𝑐2𝑡(𝑦21+𝑦22+𝑦23+⋯+𝑦2𝑛)}=_ℝ+𝑛ℱ_𝐵⁻¹𝑒^{−𝑐2𝑡(𝑦21+⋯+𝑦2𝑛)}𝑇^𝑦_𝑥𝑓(𝑥)^𝑛_𝑖=1𝑦^2𝜐𝑖_𝑖=𝑑𝑦_ℝ+𝑛𝐶_𝑣ℝ+𝑛𝑒^{−𝑐2𝑡𝑉𝑘𝑛(𝑧)}_𝑖=1𝐽_{𝜐𝑖−1/2}𝑦_𝑖,𝑧_𝑖𝑧^2𝜐𝑖_𝑖𝑑𝑧𝑇^𝑦_𝑥𝑓(𝑥)^𝑛_𝑖=1𝑦^2𝜐𝑖_𝑖𝑑𝑦, (2.49) where 𝑉(𝑧)=(𝑧²₁+𝑧²₂+𝑧²₃+⋯+𝑧²_𝑛). Set 𝐸(𝑥,𝑡)=𝐶_𝑣ℝ+𝑛𝑒^{−𝑐2𝑡(𝑦21+𝑦22+𝑦23+⋯+𝑦2𝑛)𝑛}_𝑖=1𝐽_{𝑣𝑖−1/2}𝑥_𝑖,𝑦_𝑖𝑦^2𝜐𝑖_𝑖𝑑𝑦. (2.50) We choose Ω⁺⊂ℝ⁺_𝑛, to be the spectrum of 𝐸(𝑥,𝑡), and, by (2.35), we have 𝐸(𝑥,𝑡)=𝐶_𝑣ℝ+𝑛𝑒^{−𝑐2𝑡(𝑦21+𝑦22+𝑦23+⋯+𝑦2𝑛)𝑛}_𝑖=1𝐽_{𝑣𝑖−1/2}𝑥_𝑖,𝑦_𝑖𝑦^2𝜐𝑖_𝑖𝑑𝑦=𝐶_𝑣Ω+𝑒^{−𝑐2𝑡(𝑦21+𝑦22+𝑦23+⋯+𝑦2𝑛)𝑛}_𝑖=1𝐽_{𝑣𝑖−1/2}𝑥_𝑖,𝑦_𝑖𝑦^2𝜐𝑖_𝑖𝑑𝑦. (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 lim_𝑡→0𝐸(𝑥,𝑡)=𝐶_𝑣Ω+^𝑛_𝑖=1𝐽_{𝑣𝑖−1/2}𝑥_𝑖,𝑦_𝑖𝑦^2𝑣𝑖_𝑖𝑑𝑦=𝐶_𝑣ℝ+𝑛^𝑛_𝑖=1𝐽_{𝑣𝑖−1/2}𝑥_𝑖,𝑦_𝑖𝑦^2𝑣𝑖_𝑖(𝑥)𝑑𝑦=𝛿,for𝑥∈ℝ⁺_𝑛. (2.53) Thus, for the solution 𝑢(𝑥,𝑡)=𝐸(𝑥,𝑡)∗𝑓(𝑥) of (2.43), we have lim_𝑡→0𝑢(𝑥,𝑡)=𝑢(𝑥,0)=𝛿∗𝑓(𝑥)=𝑓(𝑥), (2.54) which satisfies (2.44). This completes the proof.

3. Main Results

Theorem 3.1. Given the equation 𝜕□𝜕𝑡^𝐵𝑢(𝑥,𝑡)=𝑐²◊_𝐵𝑢(𝑥,𝑡) (3.1) with the initial condition □_𝐵𝑢(𝑥,0)=𝑓(𝑥), (3.2) where 𝑓(𝑥) is a given generalized function for ℝ^𝑛, □_𝐵 is an ultrahyperbolic Bessel operator, and ◊_𝐵 is the Diamond Bessel operator defined by (1.11) and (1.9), respectively. Then, we obtain 𝑢(𝑥,𝑡)=𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)∗𝑓(𝑥) (3.3) as a solution of (3.1), where 𝑅^𝐻₂(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 𝜕□𝜕𝑡^𝐵𝑢(𝑥,𝑡)=𝑐²▵_𝐵□_𝐵𝑢(𝑥,𝑡). (3.4) Let 𝑤(𝑥,𝑡)=□_𝐵𝑢(𝑥,𝑡). Thus, the above equation can be written as 𝜕(𝑤(𝜕𝑡𝑥,𝑡))=𝑐²▵_𝐵𝑤(𝑥,𝑡). (3.5)
We can solve the above equation by applying the 𝑛-dimensional Fourier Bessel transform to both sides of (3.5). By Lemma 2.7, we obtain 𝑤(𝑥,𝑡)=□_𝐵𝑢(𝑥,𝑡)=𝐸(𝑥,𝑡)∗𝑓(𝑥). (3.6) By convolving both sides of (3.6) by function 𝑅^𝐻₂(𝑥), we obtain 𝑅^𝐻₂(𝑥)∗□_𝐵𝑢(𝑥,𝑡)=𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)∗𝑓(𝑥). (3.7) By properties of convolution, □𝑅^𝐻₂(𝑥)∗𝑢(𝑥,𝑡)=𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)∗𝑓(𝑥). (3.8) By Lemma 2.5, we obtain 𝛿∗𝑢(𝑥,𝑡)=𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)∗𝑓(𝑥), (3.9) or 𝑢(𝑥,𝑡)=𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)∗𝑓(𝑥) (3.10) is a solution of (3.1). As shown in (3.5) and by the continuity of convolution, lim_𝑡→0□_𝐵𝑢(𝑥,𝑡)=lim_𝑡→0𝐸(𝑥,𝑡)∗𝑓(𝑥)=𝛿∗𝑓(𝑥)=𝑓(𝑥). (3.11)

Theorem 3.2 (The properties of the Diamond Bessel Heat Kernel 𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)). (1) 𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡) exists and is tempered distribution.
(2) 𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)∈ℂ^∞ the space of continuous function and infinitely differentiable.
(3) lim_𝑡→0(𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡))=𝑅^𝐻₂(𝑥).
(4) (𝜕/𝜕𝑡)□_𝐵(𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡))−𝑐²◊_𝐵(𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡))=0.

Proof. (1) Since 𝐸(𝑥,𝑡) and 𝑅^𝐻₂(𝑥) are tempered distribution with compact support. Thus, 𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡) exists and is a tempered distribution.
(2) We have 𝜕^𝑛𝜕𝑥^𝑛𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)=𝑅^𝐻₂(𝑥)∗𝜕^𝑛𝜕𝑥^𝑛𝐸(𝑥,𝑡), (3.12) since 𝐸(𝑥,𝑡) is infinitely differentiable and 𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)∈ℂ^∞.
(3) By the continuity of the convolution, 𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)⟶𝑅^𝐻₂(𝑥)∗𝛿as𝑡⟶0. (3.13) Thus, lim_𝑡→0𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)=𝑅^𝐻₂(𝑥). (3.14)
(4) Since □_𝐵𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)=□_𝐵𝑅^𝐻₂(𝑥)()()(),◊∗𝐸𝑥,𝑡=𝛿∗𝐸𝑥,𝑡=𝐸𝑥,𝑡_𝐵𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)=▵_𝐵□_𝐵𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)=▵_𝐵(𝛿∗𝐸(𝑥,𝑡))=▵_𝐵𝐸(𝑥,𝑡). (3.15) thus 𝜕□𝜕𝑡^𝐵𝑅^𝐻₂(𝑥)∗𝐸(𝑥,𝑡)−𝑐²◊_𝐵𝑅^𝐻₂(𝑥)()=𝜕∗𝐸𝑥,𝑡𝐸()𝜕𝑡𝑥,𝑡−𝑐²▵_𝐵𝐸()=𝜕𝑥,𝑡𝜕𝑡−𝑐²▵_𝐵𝐸(𝑥,𝑡)=0, (3.16) where 𝐸(𝑥,𝑡) is defined by (2.35).

Acknowledgments

The authors would like to thank The Thailand Research Fund and Office of the Higher Education Commission and Maejo University, Chiang Mai, Thailand, for financial support and also Professor Amnuay Kananthai, Department of Mathematics, Chiang Mai University, for the helpful of discussion.