ISRN Applied Mathematics

Volume 2012, Article ID 952191, 17 pages

http://dx.doi.org/10.5402/2012/952191

## On The Solution -Dimensional of the Composite Operator and Operator

Department of Mathematics, Faculty of Science, Maejo University, Chiang Mai 50290, Thailand

Received 12 June 2012; Accepted 4 September 2012

Academic Editors: C. Lu and G. Mishuris

Copyright © 2012 Wanchak Satsanit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Firstly, we study the solution of the equation , where is the composite of the diamond operator and Bessel diamond operator. Finally, we study of the nonlinear equation . It was found that the existence of the solution of such an equation depends on the condition of and . Moreover, such equation is related to the elastic wave equation.

#### 1. Introduction

Let be ultrahyperbolic operator iterated -times 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 function and .

Gel’fand and Shilov [1, pages 279–282] first introduced the fundamental solution of (1.2) which is complicated form. Later Trione [2] has shown that the generalized function which is defined by (2.2) is the unique fundamental solution of (1.2) and Aguirre Tellez [3] also proved that exists only in case is odd and is odd or even and .

In 1996, Kananthai [4] has been the first to introduce the operator which is named as the diamond operator iterated -times and is defined by where is the dimension of the space , for and is a nonnegative integer. The operator can be expressed in the form where is the Laplace operator defined by and is the ultrahyperbolic operator iterated -times and is defined by (1.1). Tellez and Kananthai [5, lemma 3.1, page 46] have shown that the convolution is a fundamental solution of the operator , where and are defined by (2.8) and (2.2), respectively. That is,

Furthermore, Yildirim et al. [6] first introduced the Bessel diamond operator iterated -times defined by where , . The operator can be expressed by , where

Yildirim et al. [6] have shown that the solution of the convolution form is a unique fundamental solution of the operator , that is, where and are defined by (2.11) and (2.15) with , respectively.

Now, firstly the purpose of this paper is to study the following equation: where the operator defined by (1.3) and defined by (1.7) with is a generalized function and is an unknown function.

Finally, we will study the nonlinear of the form with defined and having continuous first derivative for all , where is an open subset of and denotes the boundary of , and is bounded on , that is, is constant. We can find the solution of (1.12) which is unique under the boundary condition for , and we obtain the solution related to the elastic wave equation.

Before going to that point, the following definitions and some concepts are needed.

#### 2. Preliminaries

*Definition 2.1. *Let be a point of the -dimensional Euclidean space . Denote by
the nondegenerated quadratic form and is the dimension of the space . Let and and denote its closure. For any complex number , define the function
where the constant is given by the formula
The function is called the ultrahyperbolic kernel of Marcel Riesz and was introduced by Nozaki [7].

It is well known that is a function of and is a distribution of if . Let supp denote the support of and suppose supp , that is, supp , is compact.

From Trione [2, page 11], is a fundamental solution of the operator , that is,

By putting in and taking into account Legendre's duplication formula for

then the formula (2.1) reduces to
and , where

is the hyperbolic kernel of Riesz [8, page 31].

*Definition 2.2. * Let be a point of and the function denoted by the elliptic kernel of Marcel Riesz which is defined by
where
is a complex parameter and is the dimension of the space .

Let and be complex numbers such that The function has the following properties [9]:

*Definition 2.3. *Let . For any complex number , we define the distribution family by
where and

*Definition 2.4. *Let , and denote by
the nondegenerated quadratic form. Denote the interior of the forward cone by
and denotes its closure. For any complex number the distribution family is defined by
where
where is a complex number.

By putting in and taking into account Legendre's duplication formula for
we obtain
and , where

Lemma 2.5. *Given the equation for , where is defined by (1.8), then
**
where is defined by (2.11), with .*

*Proof. *See [6, page 379].

Lemma 2.6. *Given the equation for , where is defined by (1.9). Then
**
where is defined by (2.15), with . *

* Proof. *See [6, page 379].

Lemma 2.7. *Let and be the function defined by (2.11) and (2.15), respectively. Then
**
where and are a positive even number.*

*Proof. *See [10, pages 171–190].

Lemma 2.8. *The function and are the inverse in the convolution algebra of and , respectively, that is,
*

*Proof. *See [6].

Lemma 2.9. *Given is a hyper-function, then
**
where is the Dirac-delta distribution with derivatives and
*

*Proof. *See [1, page 233].

Lemma 2.10. *Given the following equation:
**
where is defined by (1.5) and , then
**
or
**
is a homogeneous solution of (2.26) with for . The function is defined by (2.8) and .*

* Proof. *We first need to show that the generalized function , where , and
where is a Laplace operator. In fact,
Thus
Thus
Using the following formula:
the above expression can be written in the following form:
If we put for in (2.34), we obtain
It follows that
is homogeneous solution of the equation . On the other hand, by Aguirre Tellez [11], we have
If we put in (2.37), we obtain
By (2.36) and (2.37), we conclude
or
is a homogeneous solution of the equation . This completes the proof.

Lemma 2.11. *Given the following equation:
**
where and are diamond operator and Bessel diamond operator iterated -times defined by (1.3) and (1.7), respectively, is an unknown function, we obtain
**
or
**
with as a homogeneous solution of (2.41).*

*Proof. *Since
Consider the following homogeneous equation:

The above equation can be written as

By Lemma 2.10, we have

Convolving both sides by , we obtain
By properties of convolution, we have
By (2.4), Lemmas 2.5, and 2.6, we obtain
Thus
or
is a homogeneous solution of (2.41).

Lemma 2.12. *Consider the following:
**
where is defined and has continuous first derivatives for all , is an open subset of , and is the boundary of . Assume that is bounded, that is, , and the boundary condition for . Then we obtain as a unique solution of (2.53).*

*Proof. *We can prove the existence of the solution of (2.53) by the method of iterations and the Schuder's estimates. The details of the proof are given by Courant and Hilbert, [12, pages 369–372].

#### 3. Main Results

Theorem 3.1. *Given the following equation:
**
where and are defined by (1.3) and (1.7), respectively, is the generalized function, is an unknown function , and . We obtain
**
or
**
as a general solution of (3.1).*

*Proof. *Consider the following equation:
or
Convolving both sides of (3.1) by , we obtain
By properties of convolution, we have
By (2.4), Lemmas 2.5, and 2.6, we obtain

Thus
Consider the following homogeneous equation:
By Lemma 2.10, we have a homogeneous solution as
Thus, the general solution of (3.1) is
or
The proof is complete.

Theorem 3.2. *Consider the following nonlinear equation:
**
where , and are defined by (1.3), (1.7), (1.5), and (1.1), respectively. Let be defined and having continuous first derivatives for all is an open subset of and denotes the boundary of and is even with . Suppose is bounded, that is,
**
and, the boundary condition for all let be
**
We can assume and is a continuous function for , then we obtain
**
as a solution of (3.14) with the boundary condition as
**
for all and . The function , and are given by (2.11), (2.15), (2.8), and (2.2), respectively. Moreover,
**
is a solution of the following equation:
**
where are defined by (1.1), (1.9), respectively, and is obtained from (3.11). Furthermore, if we put , then is reduced to
**
which is a solution of the following inhomogeneous elastic wave equation:
*

*Proof. * We have
Since has continuous derivative up to order for . thus we can assume
Then (3.17) can be written in the following form:
By (3.2), we have
For or
Convolving both sides of (3.24) by
we obtain
By properties of convolution, we have
By Lemma 2.8, we obtain
Thus
as a solution (3.14).

Now, considering the boundary condition we have
By Lemma 2.10, we obtain
with . Convolving both sides of (3.34) by , we obtain
or
for .

Lastly, convolving both sides of (3.36) by , we obtain
Setting
By Lemmas 2.8 and 2.5, we obtain as a solution of the following equation:
If we put , then and are reduced to and and are defined by (2.6) and (2.18), respectively. Moreover, if we put , then the operator and is reduced to
respectively, and the solution is reduced to
which is solution of the following inhomogeneous elastic wave equation:
The proof is complete.

#### Acknowledgment

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

#### References

- I. M. Gel'fand and G. E. Shilov,
*Generalized Function*, vol. 1, Academic Press, New York, NY, USA, 1964. - S. E. Trione, “On the ultra-hyperbolic kernel,”
*Trabajos de Mathematica*, vol. 116, pp. 1–12, 1987. View at Google Scholar - M. A. Tellez, “The distributional Hankel transform of Marcel Riesz's ultrahyperbolic kernel,”
*Studies in Applied Mathematics*, vol. 93, no. 2, pp. 133–162, 1994. View at Google Scholar · View at Zentralblatt MATH - A. Kananthai, “On the solutions of the
*n*-dimensional diamond operator,”*Applied Mathematics and Computation*, vol. 88, no. 1, pp. 27–37, 1997. View at Publisher · View at Google Scholar - M. A. Tellez and A. Kananthai, “On the convolution product of the distributional families related to the diamond operator,”
*Le Matematiche*, vol. 57, no. 1, pp. 39–48, 2002. View at Google Scholar · View at Zentralblatt MATH - H. Yildirim, M. Z. Sarikaya, and S. Öztürk, “The solutions of the
*n*-dimensional Bessel diamond operator and the Fourier-Bessel transform of their convolution,”*Proceedings of the Indian Academy of Sciences*, vol. 114, no. 4, pp. 375–387, 2004. View at Publisher · View at Google Scholar - Y. Nozaki, “On Riemann-Liouville integral of ultra-hyperbolic type,”
*Kōdai Mathematical Seminar Reports*, vol. 16, pp. 69–87, 1964. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Riesz, “L'intégrale de Riemann-Liouville et le problème de Cauchy,”
*Acta Mathematica*, vol. 81, pp. 1–223, 1949. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. E. Trione, La Integral de Riemann-Liouville, Courses and Seminarr de Mathatica, Fasciculo 29, Faculad de Ciencias Exactus, Buenos Aires, Argentina.
- A. M. T. Aguirre, “Some properties of Bessel elliptic kernel and Bessel ultrahyperbolic kernel,”
*Thai Journal of Mathematics*, vol. 6, no. 1, pp. 171–190, 2008. View at Google Scholar · View at Zentralblatt MATH - M. A. Téllez, “Distributional convolution product between the
*k*—th derivative of Dirac's delta in ${|x|}^{2}-{m}^{2}$,”*Integral Transforms and Special Functions. An International Journal*, vol. 10, no. 1, pp. 71–80, 2000. View at Publisher · View at Google Scholar - R. Courant and D. Hilbert,
*Method of Mathematical Physics*, vol. 1, Interscience Publishers, New York, NY, USA, 1966.