On the General Solution of the Ultrahyperbolic Bessel Operator

We study the general solution of equation kB,cu x f x , where kB,c is the ultrahyperbolic Bessel operator iterated k-times and is defined by kB,c 1/c2 Bx1 Bx2 · · · Bxp − Bxp 1 · · · Bxp q , p q n, n is the dimension of R n {x : x x1, x2, . . . , xn , x > 0, . . . , xn > 0}, Bxi ∂2/∂x2 i 2vi/xi ∂/∂xi , 2vi 2βi 1, βi > −1/2, xi > 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.


Introduction
The n-dimensional ultrahyperbolic operator k iterated k-times is defined by 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 where u x and f x are generalized functions and x x 1 , x 2 , . . ., 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 x 1 , x 2 , . . ., x n , x 1 > 0, . . ., x n > 0}, 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 .

Preliminaries
Let T y x be the generalized shift operator acting on the function ϕ, according to the law 11, 16 : We remark that this shift operator is closely connected to the Bessel differential operator 11 :

2.2
The convolution operator is determined by the T y x as follows: 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 x is a bounded function all x > 0, and d From c , we have the following equality for g x 1:

Mathematical Problems in Engineering
Definition 2.1.Let x x 1 , x 2 , . . ., x n be a point of the n-dimensional space R n .Denote the nondegenerated quadratic form by where p q n.The interior of the forward cone is defined by Γ {x x 1 , . . ., x n ∈ R n : x i > 0, i 1, . . ., n and V > 0}, where Γ designates its closure.For any complex number α, we define where 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 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 , the formula 2.8 is reduced to Proof.We first show that the generalized function δ m c 2 r 2 − s 2 , where and B,c is defined by 1.6 with k 1 and x ∈ R n .Now for 1 ≤ i ≤ p, we have

6
Mathematical Problems in Engineering Thus, we have Similarly, we have 2.20 by applying Lemma 2.4 with P c 2 r 2 − s 2 , where Thus, we have  The generalized function δ m c 2 r 2 −s 2 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 .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

Theorem 3 . 1 .
Given 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 12Note that the function M H α x is precisely the Bessel hyperbolic kernel of Marcel Riesz.The proof of this Lemma is given in 15 .
m is the Dirac-delta distribution with m derivatives.The proof of this Lemma is given in 1 .c is the ultrahyperbolic Bessel operator iterated k-times, as defined by 1.6 , and x ∈ R n , then 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.
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 m is a function defined by 2.8 with m derivatives.mR H 2k x * f x .3.10