On the General Solution of the Ultrahyperbolic Bessel Operator

Abstract

We study the general solution of equation □^𝑘_𝐵,𝑐𝑢(𝑥)=𝑓(𝑥), where □^𝑘_𝐵,𝑐 is the ultrahyperbolic Bessel operator iterated 𝑘-times and is defined by □^𝑘_𝐵,𝑐=[(1/𝑐²)(𝐵_𝑥1+𝐵_𝑥2+⋯+𝐵_𝑥𝑝)−(𝐵_𝑥𝑝+1+⋯+𝐵_{𝑥𝑝+𝑞})]^𝑘,𝑝+𝑞=𝑛, 𝑛 is the dimension of ℝ⁺_𝑛={𝑥∶𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛),_𝑥>0,…,𝑥_𝑛>0}, 𝐵_𝑥𝑖=𝜕²/𝜕𝑥²_𝑖+(2𝑣_𝑖/𝑥_𝑖)(𝜕/𝜕𝑥_𝑖), 2𝑣_𝑖=2𝛽_𝑖+1, 𝛽_𝑖>−1/2, 𝑥_𝑖>0(𝑖=1,2,…,𝑛), 𝑓(𝑥) is a given generalized function, 𝑢(𝑥) is an unknown generalized function, 𝑘 is a nonnegative integer, 𝑐 is a positive constant, and 𝑥∈ℝ⁺_𝑛.

1. Introduction

The 𝑛-dimensional ultrahyperbolic operator □^𝑘 iterated 𝑘-times is defined by □^𝑘=𝜕²𝜕𝑥²₁+𝜕²𝜕𝑥²₂𝜕+⋯+²𝜕𝑥²_𝑝−𝜕²𝜕𝑥²_𝑝+1−𝜕²𝜕𝑥²_𝑝+2𝜕−⋯−²𝜕𝑥²_𝑝+𝑞^𝑘, (1.1) where 𝑝+𝑞=𝑛,𝑛 is the dimension of space ℝ^𝑛, and 𝑘 is a nonnegative integer.

Consider the linear differential equation of the form □^𝑘𝑢(𝑥)=𝑓(𝑥), (1.2) where 𝑢(𝑥) and 𝑓(𝑥) are generalized functions and 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛.

Gel'fand and Shilov [1] first introduced the fundamental solution of (1.2), which is a complicated form. Later, Trione [2] has shown that the generalized function 𝑅_2𝑘(𝑥), defined by (2.8) with |𝑣|=0, is a unique fundamental solution of (1.2) and Téllez [3] also proved that 𝑅_2𝑘(𝑥) exists only in the case when 𝑝 is odd with 𝑛 odd or even and 𝑝+𝑞=𝑛. A wealth of some effective works on the fundamental solution of the 𝑛-dimensional classical ultrahyperbolic operator have, presented by Kananthai and Sritanratana [4–9].

In 2004, Yildirim et al. [10] have introduced the Bessel ultrahyperbolic operator iterated 𝑘-times with 𝑥∈ℝ⁺_𝑛={𝑥∶𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛),𝑥₁>0,…,𝑥_𝑛>0},□^𝑘_𝐵=𝐵_𝑥1+𝐵_𝑥2+⋯+𝐵_𝑥𝑝−𝐵_𝑥𝑝+1−⋯−𝐵_{𝑥𝑝+𝑞}^𝑘, (1.3) where 𝑝+𝑞=𝑛,𝐵_𝑥𝑖=𝜕²/𝜕𝑥²_𝑖+(2𝑣_𝑖/𝑥_𝑖)(𝜕/𝜕𝑥_𝑖),2𝑣_𝑖=2𝛽_𝑖+1,𝛽_𝑖>−1/2 [11], 𝑘 is a nonnegative integer, and 𝑛 is the dimension of ℝ⁺_𝑛. They also have studied the fundamental solution of Bessel ultrahyperbolic operator.

In 2007, Sarikaya and Yildirim [12] have studied the weak solution of the compound Bessel ultrahyperbolic equation and also studied the Bessel ultrahyperbolic heat equation [13].

In 2009, Saglam et al. [14] have developed the operator of (1.3), defined by (1.6), and it is called the ultrahyperbolic Bessel operator iterated 𝑘-times. They have also studied the product of the ultrahyperbolic Bessel operator related to elastic waves.

Next, Srisombat and Nonlaopon [15] have studied the weak solution of □^𝑘_𝐵,𝑐𝑢(𝑥)=𝑓(𝑥), (1.4) where 𝑢(𝑥) and 𝑓(𝑥) are some generalized functions. They have developed (1.4) into the form ^𝑚_𝑘=0𝐶_𝑘□^𝑘_𝐵,𝑐𝑢(𝑥)=𝑓(𝑥), (1.5) which is called the compound ultrahyperbolic Bessel equation. In finding the solution of (1.5), they have used the properties of 𝐵-convolution for the generalized functions.

The purpose of this study is to find the general solution of equation □^𝑘_𝐵,𝑐𝑢(𝑥)=𝑓(𝑥), where □^𝑘_𝐵,𝑐 is the ultrahyperbolic Bessel operator iterated 𝑘-times and is defined by □^𝑘_𝐵,𝑐=1𝑐²𝐵_𝑥1+𝐵_𝑥2+⋯+𝐵_𝑥𝑝−𝐵_𝑥𝑝+1+⋯+𝐵_{𝑥𝑝+𝑞}^𝑘 (1.6)𝑝+𝑞=𝑛, 𝑛 is the dimension of ℝ⁺_𝑛={𝑥∶𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛),𝑥₁>0,…,𝑥_𝑛>0},𝐵_𝑥𝑖=𝜕²/𝜕𝑥²_𝑖+(2𝑣_𝑖/𝑥_𝑖)(𝜕/𝜕𝑥_𝑖),2𝑣_𝑖=2𝛽_𝑖+1,𝛽_𝑖>−1/2,𝑥_𝑖>0(𝑖=1,2,…,𝑛),𝑓(𝑥) is a given generalized function, 𝑢(𝑥) is an unknown generalized function, 𝑘 is a nonnegative integer, 𝑐 is a positive constant, and 𝑥∈ℝ⁺_𝑛.

2. Preliminaries

Let 𝑇^𝑦_𝑥 be the generalized shift operator acting on the function 𝜑, according to the law [11, 16]: 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 and 𝐶^∗_𝑣=∏^𝑛_𝑖=1(Γ(𝑣_𝑖+1)/Γ(1/2)Γ(𝑣_𝑖)). We remark that this shift operator is closely connected to the Bessel differential operator [11]: 𝑑²𝑈𝑑𝑥²+2𝑣𝑥𝑑𝑈=𝑑𝑑𝑥²𝑈𝑑𝑦²+2𝑣𝑦𝑑𝑈,𝑈()(𝑥),𝑈𝑑𝑦𝑥,0=𝑓_𝑦(𝑥,0)=0. (2.2) The convolution operator is determined by the 𝑇^𝑦_𝑥 as follows: 𝑓∗𝜑𝑦=_ℝ+𝑛𝑓𝑦𝑇^𝑦_𝑥𝜑(𝑥)^𝑛_𝑖=1𝑦^2𝑣𝑖_𝑖𝑑𝑦. (2.3) The convolution (2.3) is known as a 𝐵-convolution. We note the following properties of the 𝐵-convolution and the generalized shift operator. (a)𝑇^𝑦_𝑥⋅1=1. (b)𝑇⁰_𝑥⋅𝑓(𝑥)=𝑓(𝑥). (c)If 𝑓(𝑥),𝑔(𝑥)∈𝐶(ℝ⁺_𝑛),𝑔(𝑥) is a bounded function all 𝑥>0, and _ℝ+𝑛𝑓(𝑥)^𝑛_𝑖=1𝑥^2𝑣𝑖_𝑖𝑑𝑥<∞, (2.4) then _ℝ+𝑛𝑇^𝑦_𝑥𝑓(𝑥)𝑔𝑦^𝑛_𝑖=1𝑦^2𝑣𝑖_𝑖𝑑𝑦=_ℝ+𝑛𝑓𝑦𝑇^𝑦_𝑥𝑔(𝑥)^𝑛_𝑖=1𝑦^2𝑣𝑖_𝑖𝑑𝑦. (2.5)(d)From (c), we have the following equality for𝑔(𝑥)=1: _ℝ+𝑛𝑇^𝑦_𝑥𝑓(𝑥)^𝑛_𝑖=1𝑦^2𝑣𝑖_𝑖𝑑𝑦=_ℝ+𝑛𝑓𝑦^𝑛_𝑖=1𝑦^2𝑣𝑖_𝑖𝑑𝑦. (2.6)(e)(𝑓∗𝑔)(𝑥)=(𝑔∗𝑓)(𝑥).

Definition 2.1. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) be a point of the 𝑛-dimensional space ℝ⁺_𝑛. Denote the nondegenerated quadratic form by 𝑉=𝑐²𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝−𝑥²_𝑝+1−𝑥²_𝑝+2−⋯−𝑥²_𝑝+𝑞, (2.7) where 𝑝+𝑞=𝑛. The interior of the forward cone is defined by Γ₊={𝑥=(𝑥₁,…,𝑥_𝑛)∈ℝ⁺_𝑛∶𝑥_𝑖>0,𝑖=1,…,𝑛and𝑉>0}, where Γ₊ designates its closure. For any complex number 𝛼, we define 𝑅^𝐻_𝛼,𝑐(𝑥)=𝑉^{(𝛼−𝑛−2|𝑣|)/2}𝐾_𝑛(𝛼),for𝑥∈Γ₊,0,for𝑥∉Γ₊, (2.8) where 𝐾_𝑛(𝛼)=𝜋^{(𝑛+2|𝑣|−1)/2}Γ((2+𝛼−𝑛−2|𝑣|)/2)Γ((1−𝛼)/2)Γ(𝛼)Γ2+𝛼−𝑝−2|𝑣|/2Γ𝑝+2|𝑣|−𝛼/2. (2.9) The function 𝑅^𝐻_𝛼,𝑐(𝑥) is introduced by [10, 12, 17, 18]. It is well known that 𝑅^𝐻_𝛼,𝑐(𝑥) is an ordinary function if Re(𝛼)≥𝑛 and is the distribution of 𝛼 if Re(𝛼)<𝑛. Let supp𝑅^𝐻_𝛼,𝑐(𝑥)⊂Γ₊, where supp𝑅^𝐻_𝛼,𝑐(𝑥) denotes the support of 𝑅^𝐻_𝛼,𝑐(𝑥).
By putting 𝑝=𝑐=1 into (2.7), (2.8), and (2.9), and using the Legendre's duplication of Γ(𝑧),Γ(2𝑧)=2^2𝑧−1𝜋^−1/2Γ(𝑧)Γ1𝑧+2, (2.10) the formula (2.8) is reduced to 𝑀^𝐻_𝛼(𝑥)=𝑉^{((𝛼−𝑛−2|𝑣|)/2)}𝐻_𝑛(𝛼),for𝑥∈Γ₊,0,for𝑥∉Γ₊, (2.11) where 𝑉=𝑥²₁−𝑥²₂−⋯−𝑥²_𝑛 and 𝐻_𝑛(𝛼)=𝜋^{(𝑛+2|𝑣|−1)/2}2^𝛼−1Γ2+𝛼−𝑛−2|𝑣|2Γ𝛼2. (2.12) Note that the function 𝑀^𝐻_𝛼(𝑥) is precisely the Bessel hyperbolic kernel of Marcel Riesz.

Lemma 2.2. Given the equation □^𝑘_𝐵,𝑐𝑢(𝑥)=𝛿(𝑥), (2.13) where □^𝑘_𝐵,𝑐 is defined by (1.6) and 𝑥∈ℝ⁺_𝑛, then we obtain 𝑢(𝑥)=𝑅^𝐻_2𝑘,𝑐(𝑥) as a fundamental solution of (2.13), where 𝑅^𝐻_2𝑘,𝑐(𝑥) is defined by (2.8).

The proof of this Lemma is given in [14].

Lemma 2.3. The 𝐵-convolutions of tempered distributions. (a)(□^𝑘_𝐵,𝑐𝛿)∗𝑢(𝑥)=□^𝑘_𝐵,𝑐𝑢(𝑥), where 𝑢(𝑥) is any tempered distribution.(b)Let 𝑅^𝐻_2𝑘,𝑐(𝑥) and 𝑅^𝐻_2𝑚,𝑐(𝑥) be defined by (2.8); then 𝑅^𝐻_2𝑘,𝑐(𝑥)∗𝑅^𝐻_2𝑚,𝑐(𝑥) exists and is a tempered distribution.(c)Let 𝑅^𝐻_2𝑘,𝑐(𝑥) and 𝑅^𝐻_2𝑚,𝑐(𝑥) be defined by (2.8); then 𝑅^𝐻_2𝑘,𝑐(𝑥)∗𝑅^𝐻_2𝑚,𝑐(𝑥)=𝑅^𝐻_{2𝑘+2𝑚,𝑐}(𝑥), where 𝑘 and 𝑚 are nonnegative integers.

The proof of this Lemma is given in [15].

Lemma 2.4. Given that 𝑃 is a hypersurface 𝑃𝛿^(𝑚)(𝑃)+𝑚𝑃𝛿^(𝑚−1)(𝑃)=0, (2.14) where 𝛿^(𝑚) is the Dirac-delta distribution with 𝑚 derivatives.

The proof of this Lemma is given in [1].

Lemma 2.5. Given the equation □^𝑘_𝐵,𝑐𝑢(𝑥)=0, (2.15) where □^𝑘_𝐵,𝑐 is the ultrahyperbolic Bessel operator iterated 𝑘-times, as defined by (1.6), and 𝑥∈ℝ⁺_𝑛, then 𝑢(𝑥)=𝑅^𝐻_{2()𝑘−1,𝑐}(𝑥)^(𝑚), (2.16) defined by (2.8) with 𝑚 derivatives, as a solution of (2.15) with 𝑚=((𝑛+2|𝑣|−4)/2),𝑛+2|𝑣|≥4 and 𝑛 is an even dimension.

Proof. We first show that the generalized function 𝛿^(𝑚)(𝑐²𝑟²−𝑠²), where 𝑟²=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝,𝑠²=𝑥²_𝑝+1+𝑥²_𝑝+2+⋯+𝑥²_𝑝+𝑞,𝑝+𝑞=𝑛, is a solution of□_𝐵,𝑐𝑢(𝑥)=0, (2.17) and □_𝐵,𝑐 is defined by (1.6) with 𝑘=1 and 𝑥∈ℝ⁺_𝑛. Now for 1≤𝑖≤𝑝, we have 𝜕𝜕𝑥_𝑖𝛿^(𝑚)𝑐²𝑟²−𝑠²=2𝑐²𝑥_𝑖𝛿^(𝑚+1)𝑐²𝑟²−𝑠²,𝜕²𝜕𝑥²_𝑖𝛿^(𝑚)𝑐²𝑟²−𝑠²=2𝑐²𝛿^(𝑚+1)𝑐²𝑟²−𝑠²+4𝑐⁴𝑥²_𝑖𝛿^(𝑚+2)𝑐²𝑟²−𝑠². (2.18) Thus, we have 1𝑐^2𝑝_𝑖=1𝜕²𝜕𝑥²_𝑖𝛿^(𝑚)𝑐²𝑟²−𝑠²+2𝑣_𝑖𝑥_𝑖𝜕𝜕𝑥_𝑖𝛿^(𝑚)𝑐²𝑟²−𝑠²=2𝑝𝛿^(𝑚+1)𝑐²𝑟²−𝑠²+4𝑐²𝑟²𝛿^(𝑚+2)𝑐²𝑟²−𝑠²+4𝑣^′𝛿^(𝑚+1)𝑐²𝑟²−𝑠²=2𝑝+4𝑣^′𝛿^(𝑚+1)𝑐²𝑟²−𝑠²+4𝑐²𝑟²−𝑠²𝛿^(𝑚+2)𝑐²𝑟²−𝑠²+4𝑠²𝛿^(𝑚+2)𝑐²𝑟²−𝑠²=2𝑝+4𝑣^′𝛿^(𝑚+1)𝑐²𝑟²−𝑠²−4(𝑚+2)𝛿^(𝑚+1)𝑐²𝑟²−𝑠²+4𝑠²𝛿^(𝑚+2)𝑐²𝑟²−𝑠²=2𝑝+4𝑣^′−4(𝑚+2)𝛿^(𝑚+1)𝑐²𝑟²−𝑠²+4𝑠²𝛿^(𝑚+2)𝑐²𝑟²−𝑠² (2.19) by applying Lemma 2.4 with 𝑃=𝑐²𝑟²−𝑠², where |𝑣^′|=𝑣₁+𝑣₂+⋯+𝑣_𝑝.
Similarly, we have^𝑝+𝑞_𝑖=𝑝+1𝜕²𝜕𝑥²_𝑖𝛿^(𝑚)𝑐²𝑟²−𝑠²+2𝑣_𝑖𝑥_𝑖𝜕𝜕𝑥_𝑖𝛿^(𝑚)𝑐²𝑟²−𝑠²=−2𝑞+4𝑣^′′+4(𝑚+2)𝛿^(𝑚+1)𝑐²𝑟²−𝑠²+4𝑐²𝑟²𝛿^(𝑚+2)𝑐²𝑟²−𝑠² (2.20) by applying Lemma 2.4 with 𝑃=𝑐²𝑟²−𝑠², where |𝑣^′′|=𝑣_𝑝+1+𝑣_𝑝+2+⋯+𝑣_𝑝+𝑞.
Thus, we have□_𝐵,𝑐𝛿^(𝑚)𝑐²𝑟²−𝑠²=1𝑐^2𝑝_𝑖=1𝜕²𝜕𝑥²_𝑖+2𝑣_𝑖𝑥_𝑖𝜕𝜕𝑥_𝑖𝛿^(𝑚)𝑐²𝑟²−𝑠²−^𝑝+𝑞_𝑖=𝑝+1𝜕²𝜕𝑥²_𝑖+2𝑣_𝑖𝑥_𝑖𝜕𝜕𝑥_𝑖𝛿^(𝑚)𝑐²𝑟²−𝑠²=2𝑝+𝑞+2|𝑣|−8(𝑚+2)𝛿^(𝑚+1)𝑐²𝑟²−𝑠²−4𝑐²𝑟²−𝑠²𝛿^(𝑚+2)𝑐²𝑟²−𝑠²=[2(𝑛+2|𝑣|)−8(𝑚+2)]𝛿^(𝑚+1)𝑐²𝑟²−𝑠²+4(𝑚+2)𝛿^(𝑚+1)𝑐²𝑟²−𝑠²=[2(𝑛+2|𝑣|)−4(𝑚+2)]𝛿^(𝑚+1)𝑐²𝑟²−𝑠² (2.21) by applying Lemma 2.4 with 𝑃=𝑐²𝑟²−𝑠², where |𝑣|=|𝑣^′|+|𝑣^′′|.
If [2(𝑛+2|𝑣|)−4(𝑚+2)]=0, we obtain□_𝐵,𝑐𝛿^(𝑚)𝑐²𝑟²−𝑠²=0. (2.22) That is, 𝑢(𝑥)=𝛿^(𝑚)(𝑐²𝑟²−𝑠²) is a solution of (2.15) with 𝑚=(𝑛+2|𝑣|−4)/2,𝑛+2|𝑣|≥4, and 𝑛 is an even dimension. Now □^𝑘_𝐵,𝑐𝑢(𝑥)=0 can be written in the form □_𝐵,𝑐□^𝑘−1_𝐵,𝑐𝑢(𝑥)=0. (2.23) From (2.17), we have □^𝑘−1_𝐵,𝑐𝑢(𝑥)=𝛿^(𝑚)𝑐²𝑟²−𝑠² (2.24) with 𝑚=(𝑛+2|𝑣|−4)/2,𝑛+2|𝑣|≥4, and 𝑛 being an even dimension. By Lemma 2.3(a), we can write (2.24) in the from □^𝑘−1_𝐵,𝑐𝛿∗𝑢(𝑥)=𝛿^(𝑚)𝑐²𝑟²−𝑠². (2.25)𝐵-convolving both sides of the above equation with the function 𝑅^𝐻_{2(𝑘−1),𝑐}(𝑥), we obtain 𝑅^𝐻_{2()𝑘−1,𝑐}(𝑥)∗□^𝑘−1_𝐵,𝑐𝛿∗𝑢(𝑥)=𝑅^𝐻_{2()𝑘−1,𝑐}(𝑥)∗𝛿^(𝑚)𝑐²𝑟²−𝑠²,□^𝑘−1_𝐵,𝑐𝑅^𝐻_{2()𝑘−1,𝑐}(𝑥)∗𝑢(𝑥)=𝑅^𝐻_{2()𝑘−1,𝑐}(𝑥)^(𝑚),𝛿∗𝑢(𝑥)=𝑢(𝑥)=𝑅^𝐻_{2()𝑘−1,𝑐}(𝑥)^(𝑚), (2.26)
by Lemma 2.2.
It follows that 𝑢(𝑥)=[𝑅^𝐻_{2(𝑘−1),𝑐}(𝑥)]^(𝑚) is a solution of (2.15) with 𝑚=(𝑛+2|𝑣|−4)/2,𝑛+2|𝑣|≥4 and 𝑛 is an even dimension.
The generalized function 𝛿^(𝑚)(𝑐²𝑟²−𝑠²) mentioned in Lemma 2.5 has been also studied on the aspect of multiplicative product, distributional product and applications, for more details, see [19–23].

3. Main Result

Theorem 3.1. Given the equation □^𝑘_𝐵,𝑐𝑢(𝑥)=𝑓(𝑥), (3.1) where □^𝑘_𝐵,𝑐 is the ultrahyperbolic Bessel operator iterated 𝑘-times and is defined by (1.6), 𝑓(𝑥) is a generalized function, 𝑢(𝑥) is an unknown generalized function, 𝑥∈ℝ⁺_𝑛, and 𝑛 is an even, then (3.1) has the general solution 𝑢(𝑥)=𝑅^𝐻_{2()𝑘−1,𝑐}(𝑥)^(𝑚)+𝑅^𝐻_2𝑘,𝑐(𝑥)∗𝑓(𝑥), (3.2) where [𝑅^𝐻_2𝑘,𝑐(𝑥)]^(𝑚) is a function defined by (2.8) with 𝑚 derivatives.

Proof. 𝐵-convolving both sides of (3.1) with 𝑅^𝐻_2𝑘,𝑐(𝑥), we obtain𝑅^𝐻_2𝑘,𝑐(𝑥)∗□^𝑘_𝐵,𝑐𝑢(𝑥)=𝑅^𝐻_2𝑘,𝑐(𝑥)∗𝑓(𝑥). (3.3) By Lemma 2.2, we have □^𝑘_𝐵,𝑐𝑅^𝐻_2𝑘,𝑐(𝑥)∗𝑢(𝑥)=𝛿∗𝑢(𝑥)=𝑅^𝐻_2𝑘,𝑐(𝑥)∗𝑓(𝑥). (3.4) So, we obtain that 𝑢(𝑥)=𝑅^𝐻_2𝑘,𝑐(𝑥)∗𝑓(𝑥) (3.5) is the solution of (3.1).
For a homogeneous equation □^𝑘_𝐵,𝑐𝑢(𝑥)=0, we have a solution𝑢(𝑥)=𝑅^𝐻_{2()𝑘−1,𝑐}(𝑥)^(𝑚) (3.6) by Lemma 2.5. Thus the general solution of (3.1) is 𝑢(𝑥)=𝑅^𝐻_{2()𝑘−1,𝑐}(𝑥)^(𝑚)+𝑅^𝐻_2𝑘,𝑐(𝑥)∗𝑓(𝑥). (3.7) This completes the proof.

By putting 𝑐=1, (3.1) becomes the Bessel ultrahyperbolic equation□^𝑘_𝐵𝑤(𝑥)=𝑓(𝑥), (3.8) where □^𝑘_𝐵 is the Bessel ultrahyperbolic operator iterated 𝑘-times, and is defined by (1.3), 𝑓(𝑥) is a generalized function and 𝑤(𝑥) is an unknown generalized function. From (3.5) we have that 𝑤(𝑥)=𝑅^𝐻_2𝑘(𝑥)∗𝑓(𝑥) (3.9) is a solution of (3.8), where 𝑅^𝐻_2𝑘(𝑥)=𝑅^𝐻_2𝑘,1(𝑥) defined by (2.8).

From (3.2), we obtain that the general solution of the Bessel ultrahyperbolic equation is𝑤(𝑥)=𝑅^𝐻_2()𝑘−1(𝑥)^(𝑚)+𝑅^𝐻_2𝑘(𝑥)∗𝑓(𝑥). (3.10) Moreover, if we put 𝑘=1,𝑝=1 and 𝑥₁=𝑡(times), then (3.8) is reduced to the Bessel wave equation □_𝐵𝑤(𝑥)=𝐵_𝑡−^𝑛_𝑖=2𝐵_𝑥𝑖𝑤(𝑥)=𝑓(𝑥), (3.11) where □_𝐵=𝐵_𝑡−^𝑛_𝑖=2𝐵_𝑥𝑖 (3.12) is the Bessel wave operator and 𝐵_𝑥𝑖=𝜕²/𝜕𝑥²_𝑖+(2𝑣_𝑖/𝑥_𝑖)(𝜕/𝜕𝑥_𝑖).

Thus, we obtain 𝑤(𝑥)=𝑀₂(𝑥)∗𝑓(𝑥) as a solution of the Bessel wave equation, since 𝑅^𝐻₂(𝑥) becomes 𝑀^𝐻₂(𝑥), where 𝑀^𝐻₂(𝑥) is the Bessel ultrahyperbolic kernel of Marcel Riesz, and is defined by (2.11) with 𝛼=2. And from (3.2), we obtain the general solution of Bessel wave equation as 𝑤(𝑥)=𝛿^(𝑚)(𝑥)+𝑀^𝐻₂(𝑥)∗𝑓(𝑥), (3.13)

where 𝛿^(𝑚)(𝑥) is a solution of 𝐵_𝑡−^𝑛_𝑖=2𝐵_𝑥𝑖𝑤(𝑥)=0. (3.14) Now we put 𝑉=𝑡²−𝑥²₂−𝑥²₃−⋯−𝑥²_𝑛 and 𝑠²=𝑥²₂+𝑥²₃+⋯+𝑥²_𝑛. By [24], we obtain that 𝑤(𝑥,𝑡)=𝛿^(𝑚)𝑡²−𝑠² (3.15)

is the solution of (3.14) with the initial conditions 𝑤(𝑥,0)=0 and 𝜕𝑤(𝑥,0)/𝜕𝑡=(−1)^𝑚2𝜋^𝑚+1𝛿(𝑥) at 𝑡=0 and 𝑥=(𝑥₂,𝑥₃,…,𝑥_𝑛)∈ℝ⁺_𝑛−1.

Acknowledgments

The authors would like to thank an anonymous referee who provided very useful comments and suggestions. This work is supported by the Commission on Higher Education, the Thailand Research Fund, and Khon Kaen University (contract number MRG5380118), and the Centre of Excellence in Mathematics, Thailand.