On The Solution-Dimensional of the Composite Operator and Operator

International Scholarly Research Network ISRN Applied Mathematics Volume 2012, Article ID 952191, 17 pages doi:10.5402/2012/952191

Research Article On The Solution n-Dimensional of the Composite ♦k Operator and ♦kB Operator Wanchak Satsanit Department of Mathematics, Faculty of Science, Maejo University, Chiang Mai 50290, Thailand Correspondence should be addressed to Wanchak Satsanit, [email protected] Received 12 June 2012; Accepted 4 September 2012 Academic Editors: C. Lu and G. Mishuris Copyright q 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. Firstly, we study the solution of the equation ♦k ♦kB ux fx, where ♦k ♦kB is the composite of the diamond operator and Bessel diamond operator. Finally, we study of the nonlinear equation ♦k ♦kB ux fx, k−1 k ♦kB . It was found that the existence of the solution ux of such an equation depends on the condition of f and k−1 k ♦kB ux. Moreover, such equation ux is related to the elastic wave equation.

1. Introduction Let k be ultrahyperbolic operator iterated k-times defined by ⎛

⎞k 2 2 2 2 2 2 ∂ ∂ ∂ ∂ ∂ ∂ k ⎝ 2 2 · · · 2 − 2 − 2 − · · · − 2 ⎠ , ∂x1 ∂x2 ∂xp ∂xp1 ∂xp2 ∂xpq

1.1

where p q n, n is the dimension of space Rn and k is a nonnegative integer. Consider the linear differential equation of the form k ux fx,

1.2

where ux and fx are generalized function and x x1 , x2 , . . . , xn ∈ Rn . 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

2

ISRN Applied Mathematics

R2k x which is defined by 2.2 is the unique fundamental solution of 1.2 and Aguirre Tellez 3 also proved that R2k x exists only in case p is odd and n is odd or even and pq n. In 1996, Kananthai 4 has been the first to introduce the operator ♦k which is named as the diamond operator iterated k-times and is defined by ⎛ ⎞2 ⎞k 2 ⎛ pq p 2 2 ∂ ∂ ⎠⎟ ⎜ ♦k ⎝ −⎝ ⎠ , 2 2 ∂x ∂x i1 jp1 i j

1.3

where p q n is the dimension of the space Rn , for x x1 , x2 , . . . , xn ∈ Rn and k is a nonnegative integer. The operator ♦k can be expressed in the form ♦k k k k k ,

1.4

where k is the Laplace operator defined by k

∂2 ∂2 ∂2 · · · ∂x12 ∂x22 ∂xn2

k ,

1.5

and k is the ultrahyperbolic operator iterated k-times and is defined by 1.1. Tellez and x is Kananthai 5, lemma 3.1, page 46 have shown that the convolution −1k Re2k x ∗ RH 2k x are defined by 2.8 and a fundamental solution of the operator ♦k , where Re2k x and RH 2k 2.2, respectively. That is, δx. ♦k −1k Re2k x ∗ RH x 2k

1.6

Furthermore, Yildirim et al. 6 first introduced the Bessel diamond operator ♦kB iterated k-times defined by ⎡ ⎞2 ⎤k 2 ⎛ pq p ⎥ ⎢ Bx i −⎝ Bx j ⎠ ⎦ , ♦kB ⎣ i1

1.7

jp1

where Bxi ∂2 /∂xi2 2υi /xi ∂/∂xi , 2υi 2αi 1, αi > −1/2, xi > 0. The operator ♦kB can be expressed by ♦kB kB kB kB kB , where kB

p i1

k Bx i

,

⎡ ⎤k p pq Bx j ⎦ . kB ⎣ Bxi − i1

jp1

1.8

1.9


3

Yildirim et al. 6 have shown that the solution of the convolution form ux −1k S2k x ∗ R2k x is a unique fundamental solution of the operator ♦kB , that is, ♦kB −1k S2k x ∗ R2k x δ,

1.10

where S2k x and R2k x are defined by 2.11 and 2.15 with α γ 2k, respectively. Now, firstly the purpose of this paper is to study the following equation: ♦k ♦kB ux fx,

1.11

where the operator ♦k defined by 1.3 and ♦kB defined by 1.7 with fx is a generalized function and ux is an unknown function. Finally, we will study the nonlinear of the form ♦k ♦kB ux f x, k−1 k ♦kB ux ,

1.12

with f defined and having continuous first derivative for all x ∈ Ω ∪ ∂Ω, where Ω is an open subset of Rn and ∂Ω denotes the boundary of Ω, and f is bounded on Ω, that is, |f| ≤ N, N is constant. We can find the solution ux of 1.12 which is unique under the boundary condition k−1 k ♦kB ux 0 for x ∈ ∂Ω, 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 x x1 , x2 , . . . , xn be a point of the n-dimensional Euclidean space Rn . Denote by 2 2 2 υ x12 x22 · · · xp2 − xp1 − xp2 − · · · − xpq

2.1

the nondegenerated quadratic form and p q n is the dimension of the space Rn . Let Γ {x ∈ Rn : x1 > 0 and υ > 0} and Γ denote its closure. For any complex number α, define the function ⎧ α−n/2 ⎪ ⎨υ , for x ∈ Γ , H Rα υ Kn α ⎪ ⎩0, for x ∈ / Γ ,

2.2

where the constant Kn α is given by the formula Kn α

π n−1/2 Γ2 α − n/2Γ1 − α/2Γα . Γ 2 α − p /2 Γ p − α /2

2.3

4


The function RH α υ is called the ultrahyperbolic kernel of Marcel Riesz and was introduced by Nozaki 7. It is well known that RH α υ is a function of Reα ≥ n and is a distribution of α if H υ denote the support of RH Reα < n. Let supp RH α α υ and suppose supp Rα υ ⊂ Γ , that is, supp RH α υ, is compact. υ is a fundamental solution of the operator k , that is, From Trione 2, page 11, RH 2k k RH 2k υ δx.

Γz

2.4

By putting p 1 in R2k υ and taking into account Legendre’s duplication formula for 1 , Γ2z 22z−1 π −1/2 ΓzΓ z 2

2.5

then the formula 2.1 reduces to ⎧ α−n/2 ⎪ ⎨u , MαH u Hn α ⎪ ⎩0,

for x ∈ Γ , for x ∈ / Γ ,

2.6

and u x12 − x22 − · · · − xn2 , where Hn α π

n−2/2 α−1

2

α − n 2|ν| α Γ . Γ 2 2

2.7

Mα u is the hyperbolic kernel of Riesz 8, page 31. Definition 2.2. Let x x1 , x2 , . . . , xn be a point of Rn and |x| x12 x22 · · · xn2 the function Reα x denoted by the elliptic kernel of Marcel Riesz which is defined by |x|α−n/2 , Wn α

2.8

π n/2 2α Γα/2 . Γn − α/2

2.9

Reα x

where Wn α

α is a complex parameter and n is the dimension of the space Rn .


5

Let α and β be complex numbers such that α β / n 2r, r 0.1, 2, . . . The function Reα x has the following properties 9: R0 x δx, R−2k x −1k k δx, k {Rα x} −1k Rα−2k x,

2.10

Rα x ∗ Rβ x Rαβ x. Definition 2.3. Let x x1 , x2 , . . . , xn , ν ν1 , ν2 , . . . , νn ∈ Rn . For any complex number α, we define the distribution family Sα x by Sα x

|x|α−n−2|ν| , wn α

2.11

where |x| x12 x22 · · · xn2 , |ν| ν1 ν2 · · · νn and wn α

n

νi −1/2 Γνi 1/2 i1 2 . 2n2|ν|−2α Γn 2|ν| − α/2

2.12

Definition 2.4. Let x x1 , x2 , . . . , xn , ν ν1 , ν2 , . . . , νn ∈ Rn , and denote by 2 2 2 − xp2 − · · · − xpq V x12 x22 · · · xp2 − xp1

2.13

the nondegenerated quadratic form. Denote the interior of the forward cone by Γ {x ∈ Rn : x1 > 0, x2 > 0, . . . , xn > 0, V > 0},

2.14

and Γ denotes its closure. For any complex number γ the distribution family Rγ x is defined by ⎧ γ−n−2|ν|/2 ⎪ ⎨V , Rγ x Kn γ ⎪ ⎩0,

for x ∈ Γ ,

2.15

for x ∈ / Γ ,

where π n2|ν|−1/2 Γ 2 γ − n − 2|ν| /2 Γ 1 − γ /2 Γ γ , Kn γ Γ 2 γ − p − 2|ν| /2 Γ p − γ /2 where γ is a complex number.

2.16

6 Γz

ISRN Applied Mathematics By putting p 1 in R2k x and taking into account Legendre’s duplication formula for 1 , Γ2z 22z−1 π −1/2 ΓzΓ z 2

2.17

we obtain Nγ x

uγ−n−2|ν|/2 , En γ

2.18

and u x12 − x22 − · · · − · · · − xn2 , where 2 γ − n − 2|ν| γ n2|ν|−1/2 γ−1 Γ . En γ π 2 Γ 2 2

2.19

Lemma 2.5. Given the equation kB ux δx for x ∈ Rn , where kB is defined by 1.8, then ux −1k S2k x,

2.20

where S2k x is defined by 2.11, with α 2k. Proof. See 6, page 379. Lemma 2.6. Given the equation kB ux δx for x ∈ Rn , where kB is defined by 1.9. Then ux R2k x,

2.21

where R2k x is defined by 2.15, with γ 2k. Proof. See 6, page 379. Lemma 2.7. Let Sα xand Rβ x be the function defined by 2.11 and 2.15, respectively. Then Sα x ∗ Sβ x Sαβ x, −1k S−2k x ∗ −1k S2k x −12k S−2k2k x S0 x δx, where α and β are a positive even number. Proof. See 10, pages 171–190.

2.22


7

Lemma 2.8. The function R−2k x and −1k S−2k x are the inverse in the convolution algebra of R2k x and −1k S2k x, respectively, that is, R−2k x ∗ R2k x R−2k2k x R0 x δx, Rβ x ∗ Rα x Rβα x.

2.23

Proof. See 6. Lemma 2.9. Given P is a hyper-function, then P δ k P kδk−1 P 0,

2.24

where δk is the Dirac-delta distribution with k derivatives and 2 2 2 P P x x12 x22 · · · xp2 − xp1 − xp2 − · · · − xpq .

2.25

Proof. See 1, page 233. Lemma 2.10. Given the following equation: k ux 0,

2.26

where k is defined by 1.5 and x x1 , x2 , . . . , xn ∈ Rn , then ux δm r 2

2.27

or

ux

−12m−n/21 π n/2 Re x Γm − n/2 24m−n/21 −2m−n/21

2.28

is a homogeneous solution of 2.26 with m n/2−k −1 for k 1, 2, 3, . . .. The function Re−2m−n/21 is defined by 2.8 and α −2m − n/2 1. Proof. We first need to show that the generalized function ux δm r 2 , where r 2 |x|2 x12 x22 · · · xn2 , and ux 0,

2.29

8


where ∂2 /∂x12 ∂2 /∂x22 · · · ∂2 /∂xn2 is a Laplace operator. In fact, ∂ m 2 r 2xi δm1 r 2 , δ ∂xi ∂2 m 2 r 2δm1 r 2 4xi2 δm2 r 2 . δ 2 ∂xi

2.30

Thus n ∂2 m 2 r δ δm r 2 2 i1 ∂xi 2nδm1 r 2 4r 2 δm2 r 2

2nδm1 r 2 − 4m 2δm1 r 2 2n − 4m 2δm1 r 2 , 2 δm r 2 2n − 4m 2 δ m1 r 2 2n − 4m 22n − 4m 3δm2 r 2 ,

2.31

3 δm r 2 2n − 4m 22n − 4m 32n − 4m 4δ m3 r 2 , k δm r 2 2n − 4m 22n − 4m 32n − 4m 4 · · · 2n − 4m k 1δ mk r 2 22

n n − m 2 22 − m 3 22 − m 4 2 2 2 n · · · 22 − m k 1 δmk r 2 . 2

n

Thus n n n −m −2 −m −3 −m −4 k δm r 2 22k 2 2 2 n ··· − m − k 1 δmk r 2 . 2

2.32

Using the following formula:

z − 1z − 2 · · · z − k

−1k Γ−z k 1 , Γm 1

2.33


9

the above expression can be written in the following form: 22k −1k Γn/2 − m − 2 1 δmk r 2 k δm r 2 Γn/2 − m − 2 − k 1 22k −1k Γn/2 − m − 1 mk 2 δ r . Γn/2 − m − k − 1

2.34

If we put m n/2 − k − 1 for k 1, 2, 3, . . . in 2.34, we obtain k δm r 2 0δmk r 2 0.

2.35

ux δm r 2

2.36

It follows that

is homogeneous solution of the equation k ux 0. On the other hand, by Aguirre Tellez 11, we have δm r 2

−1m π n/2 m−n/21 δx Γm − n/2 14m−n/21

n n m− 1 2 −1 π −1 2 e n x. n R m− 1 −2 m− 1 n 2 Γ m− 1 4 2 2 m

2.37

If we put m n/2 − k − 1 in 2.37, we obtain δn/2−k−1 r 2

−1n/2−k−1 π n/2 −1n/2−k−1−n/21 Re x Γn/2 − k − 1 − n/2 14n/2−k−1−n/21 −2n/2−k−1−n/21 −1n/2−k−1 π n/2 −1−k e R2k x Γ−k4−k

2.38

0. By 2.36 and 2.37, we conclude

ux δm r 2

2.39

or ux

−12m−n/21 π n/2 Re x Γm − n/2 24m−n/21 −2m−n/21

is a homogeneous solution of the equation k ux 0. This completes the proof.

2.40

10


Lemma 2.11. Given the following equation: ♦k ♦kB ux 0,

2.41

where ♦k and ♦kB are diamond operator and Bessel diamond operator iterated k-times defined by 1.3 and 1.7, respectively, ux is an unknown function, we obtain k m r2 ux RH 2k u ∗ −1 S2k x ∗ R2k υ ∗ δ

2.42

or k ux RH 2k u ∗ −1 S2k x ∗ R2k υ ∗

−12m−n/21 π n/2 Re x Γm − n/2 24m−n/21 −2m−n/21

2.43

with m n/2 − k − 1 as a homogeneous solution of 2.41. Proof. Since ♦k k k ,

♦kB kB kB .

2.44

Consider the following homogeneous equation: ♦k ♦kB ux 0.

2.45

The above equation can be written as k k kB kB ux 0.

2.46

k kB kB ux δm r 2 .

2.47

By Lemma 2.10, we have

Convolving both sides by RH u ∗ −1k S2k x ∗ T2k x, we obtain 2k k k k k k H m RH r2 . S ux R S ∗ ∗ R ∗ ∗ ∗ R ∗ δ υ −1 x x υ −1 x x 2k 2k 2k 2k B B 2k 2k 2.48 By properties of convolution, we have k k k k H m RH r2 . S R S ∗ ∗ ∗ ux R ∗ ∗ R ∗ δ υ −1 x x υ −1 x x 2k 2k 2k 2k B B 2k 2k 2.49


11

By 2.4, Lemmas 2.5, and 2.6, we obtain k m r2 . S δx ∗ δx ∗ δx ∗ ux RH ∗ ∗ R ∗ δ υ −1 x x 2k 2k 2k

2.50

k m ux RH r2 2k υ ∗ −1 S2k x ∗ R2k x ∗ δ

2.51

Thus

or k ux RH 2k υ ∗ −1 S2k x ∗ R2k x ∗

−12m−n/21 π n/2 Re x Γm − n/2 24m−n/21 −2m−n/21

2.52

is a homogeneous solution of 2.41. Lemma 2.12. Consider the following: ux fx, ux,

2.53

where f is defined and has continuous first derivatives for all x ∈ Ω ∪ ∂Ω, Ω is an open subset of Rn , and ∂Ω is the boundary of Ω. Assume that f is bounded, that is, |fx, u| ≤ N, and the boundary condition ux 0 for x ∈ ∂Ω. Then we obtain ux as a unique solution of 2.53. Proof. We can prove the existence of the solution ux 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: ♦k ♦kB ux fx,

3.1

where ♦k and ♦kB are defined by 1.3 and 1.7, respectively, fx is the generalized function, ux is an unknown function x x1 , x2 , . . . , xn ∈ Rn , and m n/2 − k − 1. We obtain k m r2 S ux RH ∗ ∗ R ∗ δ υ −1 x x 2k 2k 2k k e k RH 2k υ ∗ −1 R2k x ∗ −1 S2k x ∗ R2k x ∗ fx

3.2

12



−12m−n/21 π n/2 Re x Γm − n/2 24m−n/21 −2m−n/21

3.3

k e k RH 2k υ ∗ −1 R2k x ∗ −1 S2k x ∗ R2k x ∗ fx

as a general solution of 3.1. Proof. Consider the following equation: ♦k ♦kB ux fx,

3.4

k k kB ♦kB ux fx.

3.5

or

Convolving both sides of 3.1 by RH υ ∗ −1k Re2k x ∗ −1k S2k x ∗ R2k x, we obtain 2k k e k k k k k RH 2k υ ∗ −1 R2k x ∗ R2k x ∗ −1 S2k x ∗ B B ux k e k RH 2k υ ∗ −1 R2k x ∗ R2k x ∗ −1 S2k x ∗ fx.

3.6

By properties of convolution, we have k e k k k k k RH 2k υ ∗ −1 R2k x ∗ B −1 S2k x ∗ B R2k x ∗ ux k e k RH 2k υ ∗ −1 R2k x ∗ R2k x ∗ −1 S2k x ∗ fx.

3.7

By 2.4, Lemmas 2.5, and 2.6, we obtain k e k 3.8 δx ∗ δx ∗ δx ∗ δx ∗ ux RH 2k υ ∗ −1 R2k x ∗ −1 S2k x ∗ R2k x ∗ fx.

Thus k e k ux RH 2k υ ∗ −1 R2k x ∗ −1 S2k x ∗ R2k x ∗ fx.

3.9

Consider the following homogeneous equation: ♦k ♦kB ux 0.

3.10

By Lemma 2.10, we have a homogeneous solution as k m r2 . ux RH 2k u ∗ −1 S2k x ∗ R2k x ∗ δ

3.11


13

Thus, the general solution of 3.1 is k m r2 ux RH 2k x ∗ −1 S2k x ∗ R2k x ∗ δ

RH 2k υ

∗ −1

k

Re2k x

k

3.12

∗ −1 S2k x ∗ R2k x ∗ fx,


−12m−n/21 π n/2 Re x Γm − n/2 24m−n/21 −2m−n/21

3.13

k e k RH 2k υ ∗ −1 R2k x ∗ −1 S2k x ∗ R2k x ∗ fx.

The proof is complete. Theorem 3.2. Consider the following nonlinear equation: ♦k ♦kB ux f x, k−1 k ♦kB ux ,

3.14

where ♦k , ♦kB , k−1 , and k are defined by 1.3, 1.7, 1.5, and 1.1, respectively. Let f be defined and having continuous first derivatives for all x ∈ Ω ∪ ∂Ω, Ω is an open subset of Rn and ∂Ω denotes the boundary of Ω and n is even with n ≥ 4. Suppose f is bounded, that is, fx, k−1 k ♦kB ux ≤ N,

3.15

and, the boundary condition for all x ∈ ∂Ω let be k−1 k ♦kB ux 0.

3.16

We can assume k−1 k ♦kB ux Ux and Ux is a continuous function for x ∈ ∂Ω, then we obtain k ux −1k−1 Re2k−1 x ∗ RH 2k υ ∗ −1 S2k x ∗ R2k x ∗ Ux

3.17

as a solution of 3.14 with the boundary condition as k ux δ m r 2 ∗ RH 2k υ ∗ −1 S2k x,

3.18

for all x ∈ ∂Ω and m n/2 − k. The function S2k x, R2k x, Re2k−2 x, and RH υ are given by 2k 2.11, 2.15, 2.8, and 2.2, respectively. Moreover, Wx −1k−1 Re21−k x ∗ −1k S−2k x ∗ Ux

3.19

14


is a solution of the following equation: k kB Wx Ux,

3.20

where k , kB are defined by 1.1, 1.9, respectively, and Ux is obtained from 3.11. Furthermore, if we put p k 1, then Wx is reduced to Wx M2H u ∗ N2H u ∗ Ux,

3.21

which is a solution of the following inhomogeneous elastic wave equation:

∂2 ∂2 ∂2 ∂2 − − − · · · − ∂x12 ∂x22 ∂x32 ∂xn2

· Bx1 − Bx2 − Bx3 − · · · − Bxn Wx Ux.

3.22

Proof. We have ♦k ♦kB ux k−1 k ♦kB ux f x, k−1 k ♦kB ux .

3.23

Since ux has continuous derivative up to order 4p for k 1, 2, 3, . . .. thus we can assume k−1 k ♦kB ux Ux,

∀x ∈ Ω.

3.24

♦k ♦kB ux Ux fx, Ux.

3.25

Then 3.17 can be written in the following form:

By 3.2, we have fx, Ux ≤ N,

∀x ∈ Ω,

3.26

k−1 k ♦kB ux 0

∀x ∈ ∂Ω,

3.27

For Ux 0 or

Convolving both sides of 3.24 by k −1k Re2k−1 x ∗ RH 2k x ∗ −1 S2k x ∗ R2k x,

3.28


15

we obtain k k−1 k ♦kB ux S ∗ ∗ R υ −1 x x −1k Re2k−1 x ∗ RH 2k 2k 2k

−1

k

Re2k−1 x

∗

RH 2k x

k

3.29

∗ −1 S2k x ∗ R2k x ∗ Ux.

By properties of convolution, we have

k k k−1 −1k Re2k−1 x ∗ k RH 2k υ ∗ ♦B −1 S2k x ∗ R2k x ∗ ux k ∗ Ux. −1k Re2k−1 x ∗ RH S ∗ ∗ R υ −1 x x 2k 2k 2k

3.30

By Lemma 2.8, we obtain k δ ∗ δ ∗ δ ∗ ux −1k Re2k−1 x ∗ RH ∗ Ux. S ∗ ∗ R υ −1 x x 2k 2k 2k

3.31

Thus k ux −1k Re2k−1 x ∗ RH 2k υ ∗ −1 S2k x ∗ R2k x ∗ Ux,

3.32

as a solution 3.14. Now, considering the boundary condition we have k−1 k ♦kB ux 0.

3.33

k ♦kB ux δm r 2 ,

3.34

By Lemma 2.10, we obtain

with m n/2 − k. Convolving both sides of 3.34 by RH υ ∗ −1k S2k x ∗ R2k x, we obtain 2k k k k k H m r2 RH 2k υ ∗ −1 S2k x ∗ R2k x ♦ ux R2k υ ∗ −1 S2k x ∗ R2k x ∗ δ

3.35

or k mk−2 ux RH 2k υ ∗ −1 S2k x ∗ R2k x ∗ −1

for x ∈ ∂Ω.

−1m π n/2 Re x, Γm − n/2 14m−n/21 −2m−n/21 3.36

16

ISRN Applied Mathematics Lastly, convolving both sides of 3.36 by −1k−1 Re21−k x ∗ −1k S−2k x, we obtain −1k−1 Re21−k x ∗ −1k S−2k x ∗ ux RH 2k x ∗ R2k x ∗ Ux.

3.37

Wx −1k−1 Re21−k x ∗ −1k S−2k x ∗ ux.

3.38

Setting

By Lemmas 2.8 and 2.5, we obtain Wx as a solution of the following equation: k kB Wx Ux.

3.39

H If we put p 1, then RH υ and R2k x are reduced to M2k u and N2k u and are defined 2k by 2.6 and 2.18, respectively. Moreover, if we put p k 1, then the operator k and kB is reduced to

∂2 ∂2 ∂2 ∂2 − 2 − 2 − ··· − 2 , 2 ∂x1 ∂x2 ∂x3 ∂xn

3.40

Bx 1 − B x 2 − B x 3 − · · · − B x n , respectively, and the solution Wx is reduced to Wx M2H u ∗ N2H u ∗ Ux,

3.41

which is solution of the following inhomogeneous elastic wave equation:

∂2 ∂2 ∂2 ∂2 − − − · · · − ∂x12 ∂x22 ∂x32 ∂xn2

· Bx1 − Bx2 − Bx3 − · · · − Bxn Wx Ux.

3.42

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 1 I. M. Gel’fand and G. E. Shilov, Generalized Function, vol. 1, Academic Press, New York, NY, USA, 1964.


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2 S. E. Trione, “On the ultra-hyperbolic kernel,” Trabajos de Mathematica, vol. 116, pp. 1–12, 1987. 3 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. 4 A. Kananthai, “On the solutions of the n-dimensional diamond operator,” Applied Mathematics and Computation, vol. 88, no. 1, pp. 27–37, 1997. 5 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. ¨ urk, 6 H. Yildirim, M. Z. Sarikaya, and S. Ozt “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. 7 Y. Nozaki, “On Riemann-Liouville integral of ultra-hyperbolic type,” K¯odai Mathematical Seminar Reports, vol. 16, pp. 69–87, 1964. 8 M. Riesz, “L’intégrale de Riemann-Liouville et le problème de Cauchy,” Acta Mathematica, vol. 81, pp. 1–223, 1949. 9 S. E. Trione, La Integral de Riemann-Liouville, Courses and Seminarr de Mathatica, Fasciculo 29, Faculad de Ciencias Exactus, Buenos Aires, Argentina. 10 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. 11 M. A. Téllez, “Distributional convolution product between the k—th derivative of Dirac’s delta in |x|2 − m2 ,” Integral Transforms and Special Functions. An International Journal, vol. 10, no. 1, pp. 71–80, 2000. 12 R. Courant and D. Hilbert, Method of Mathematical Physics, vol. 1, Interscience Publishers, New York, NY, USA, 1966.

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