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ISRN Applied Mathematics
Volume 2012 (2012), Article ID 952191, 17 pages
http://dx.doi.org/10.5402/2012/952191
Research Article

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.

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