On the General Solution of the Ultrahyperbolic Bessel Operator
We study the general solution of equation , where is the ultrahyperbolic Bessel operator iterated -times and is defined by , is the dimension of , , , , , is a given generalized function, is an unknown generalized function, is a nonnegative integer, is a positive constant, and .
The -dimensional ultrahyperbolic operator iterated -times is defined by where is the dimension of space , and is a nonnegative integer.
Consider the linear differential equation of the form where and are generalized functions and .
Gel'fand and Shilov  first introduced the fundamental solution of (1.2), which is a complicated form. Later, Trione  has shown that the generalized function , defined by (2.8) with , is a unique fundamental solution of (1.2) and Téllez  also proved that exists only in the case when is odd with odd or even and . A wealth of some effective works on the fundamental solution of the -dimensional classical ultrahyperbolic operator have, presented by Kananthai and Sritanratana [4–9].
In 2004, Yildirim et al.  have introduced the Bessel ultrahyperbolic operator iterated -times with , where , is a nonnegative integer, and is the dimension of . They also have studied the fundamental solution of Bessel ultrahyperbolic operator.
In 2009, Saglam et al.  have developed the operator of (1.3), defined by (1.6), and it is called the ultrahyperbolic Bessel operator iterated -times. They have also studied the product of the ultrahyperbolic Bessel operator related to elastic waves.
Next, Srisombat and Nonlaopon  have studied the weak solution of where and are some generalized functions. They have developed (1.4) into the form which is called the compound ultrahyperbolic Bessel equation. In finding the solution of (1.5), they have used the properties of -convolution for the generalized functions.
The purpose of this study is to find the general solution of equation , where is the ultrahyperbolic Bessel operator iterated -times and is defined by , is the dimension of ,,,, is a given generalized function, is an unknown generalized function, is a nonnegative integer, is a positive constant, and .
Let be the generalized shift operator acting on the function , according to the law [11, 16]: where and . We remark that this shift operator is closely connected to the Bessel differential operator : The convolution operator is determined by the as follows: The convolution (2.3) is known as a -convolution. We note the following properties of the -convolution and the generalized shift operator. (a). (b). (c)If is a bounded function all , and then (d)From (c), we have the following equality for: (e).
Definition 2.1. Let be a point of the -dimensional space . Denote the nondegenerated quadratic form by
where . The interior of the forward cone is defined by , where designates its closure. For any complex number , we define
The function is introduced by [10, 12, 17, 18]. It is well known that is an ordinary function if and is the distribution of if . Let , where denotes the support of .
By putting into (2.7), (2.8), and (2.9), and using the Legendre's duplication of , the formula (2.8) is reduced to where and Note that the function is precisely the Bessel hyperbolic kernel of Marcel Riesz.
The proof of this Lemma is given in .
Lemma 2.3. The -convolutions of tempered distributions. (a), where is any tempered distribution.(b)Let and be defined by (2.8); then exists and is a tempered distribution.(c)Let and be defined by (2.8); then , where and are nonnegative integers.
The proof of this Lemma is given in .
Lemma 2.4. Given that is a hypersurface where is the Dirac-delta distribution with derivatives.
The proof of this Lemma is given in .
Lemma 2.5. Given the equation where is the ultrahyperbolic Bessel operator iterated -times, as defined by (1.6), and , then defined by (2.8) with derivatives, as a solution of (2.15) with and is an even dimension.
Proof. We first show that the generalized function , where , is a solution of
and is defined by (1.6) with and . Now for , we have
Thus, we have
by applying Lemma 2.4 with , where .
Similarly, we have by applying Lemma 2.4 with , where .
Thus, we have by applying Lemma 2.4 with , where .
If , we obtain That is, is a solution of (2.15) with , and is an even dimension. Now can be written in the form From (2.17), we have with , and being an even dimension. By Lemma 2.3(a), we can write (2.24) in the from -convolving both sides of the above equation with the function , we obtain
by Lemma 2.2.
It follows that is a solution of (2.15) with and is an even dimension.
The generalized function 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 where is the ultrahyperbolic Bessel operator iterated -times and is defined by (1.6), is a generalized function, is an unknown generalized function, , and is an even, then (3.1) has the general solution where is a function defined by (2.8) with derivatives.
Proof. -convolving both sides of (3.1) with , we obtain
By Lemma 2.2, we have
So, we obtain that
is the solution of (3.1).
For a homogeneous equation , we have a solution by Lemma 2.5. Thus the general solution of (3.1) is This completes the proof.
By putting , (3.1) becomes the Bessel ultrahyperbolic equation where is the Bessel ultrahyperbolic operator iterated -times, and is defined by (1.3), is a generalized function and is an unknown generalized function. From (3.5) we have that is a solution of (3.8), where defined by (2.8).
From (3.2), we obtain that the general solution of the Bessel ultrahyperbolic equation is Moreover, if we put and (times), then (3.8) is reduced to the Bessel wave equation where is the Bessel wave operator and .
Thus, we obtain as a solution of the Bessel wave equation, since becomes , where is the Bessel ultrahyperbolic kernel of Marcel Riesz, and is defined by (2.11) with . And from (3.2), we obtain the general solution of Bessel wave equation as
where is a solution of Now we put and . By , we obtain that
is the solution of (3.14) with the initial conditions and at and .
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.
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