# On the General Solution of the Ultrahyperbolic Bessel Operator

Authors: Rattapan Damkengpan; Kamsing Nonlaopon
Journal: Mathematical Problems in Engineering
 (2011)

---

## Abstract

We study the general solution of equation□
B
,
c
k
u
(
x
)
=
f
(
x
), where □
B
,
c
k is the ultrahyperbolic Bessel operator iterated k-times and is defined by □
B
,
c
k
=
[
(
1
/
c
2
)
(
B
x
1
+
B
x
2
+
⋯
+
B
x
p
)
−
(
B
x
p
+
1
+
⋯
+
B
x
p
+
q
)
]
k
,
p
+
q
=
n, n is the dimension of ℝ
n
+
=
{
x
:
x
=
(
x
1
,
x
2
,
…
,
x
n
)
,
x
1
>
0
,
…
,
x
n
>
0
}, B
x
i
=
∂
2
/
∂
x
i
2
+
(
2
v
i
/
x
i
)
(
∂
/
∂
x
i
), 2
v
i
=
2
β
i
+
1, β
i
>
−
1
/
2, x
i
>
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
+.


---

## Body

## 1. Introduction

Then-dimensional ultrahyperbolic operator □
k iterated k-times is defined by (1.1)
□
k
=
(
∂
2
∂
x
1
2
+
∂
2
∂
x
2
2
+
⋯
+
∂
2
∂
x
p
2
-
∂
2
∂
x
p
+
1
2
-
∂
2
∂
x
p
+
2
2
-
⋯
-
∂
2
∂
x
p
+
q
2
)
k
,
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(1.2)
□
k
u
(
x
)
=
f
(
x
)
,
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
2
k
(
x
), defined by (2.8) with |
v
|
=
0, is a unique fundamental solution of (1.2) and Téllez [3] also proved that R
2
k
(
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
},(1.3)
□
B
k
=
(
B
x
1
+
B
x
2
+
⋯
+
B
x
p
-
B
x
p
+
1
-
⋯
-
B
x
p
+
q
)
k
,
where p
+
q
=
n
,
B
x
i
=
∂
2
/
∂
x
i
2
+
(
2
v
i
/
x
i
)
(
∂
/
∂
x
i
)
,
2
v
i
=
2
β
i
+
1
,
β
i
>
-
1
/
2 [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 (1.4)
□
B
,
c
k
u
(
x
)
=
f
(
x
)
,
where u
(
x
) and f
(
x
) are some generalized functions. They have developed (1.4) into the form (1.5)
∑
k
=
0
m
C
k
□
B
,
c
k
u
(
x
)
=
f
(
x
)
,
which is called the compound ultrahyperbolic Bessel equation. In finding the solution of (1.5), they have used the properties of B-convolution for the generalized functions.The purpose of this study is to find the general solution of equation□
B
,
c
k
u
(
x
)
=
f
(
x
), where □
B
,
c
k is the ultrahyperbolic Bessel operator iterated k-times and is defined by (1.6)
□
B
,
c
k
=
[
1
c
2
(
B
x
1
+
B
x
2
+
⋯
+
B
x
p
)
-
(
B
x
p
+
1
+
⋯
+
B
x
p
+
q
)
]
k
p
+
q
=
n, n is the dimension of ℝ
n
+
=
{
x
:
x
=
(
x
1
,
x
2
,
…
,
x
n
)
,
x
1
>
0
,
…
,
x
n
>
0
},B
x
i
=
∂
2
/
∂
x
i
2
+
(
2
v
i
/
x
i
)
(
∂
/
∂
x
i
),2
v
i
=
2
β
i
+
1
,
β
i
>
-
1
/
2,x
i
>
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
+.


## 2. Preliminaries

LetT
x
y be the generalized shift operator acting on the function φ, according to the law [11, 16]: (2.1)
T
x
y
φ
(
x
)
=
C
v
*
∫
0
π
⋯
∫
0
π
φ
(
x
1
2
+
y
1
2
-
2
x
1
y
1
cos
⁡
θ
1
,
…
,
x
n
2
+
y
n
2
-
2
x
n
y
n
cos
⁡
θ
n
)
×
(
∏
i
=
1
n
sin
2
v
i
-
1
θ
i
)
d
θ
1
⋯
d
θ
n
,
where x
,
y
∈
ℝ
n
+ and C
v
*
=
∏
i
=
1
n
(
Γ
(
v
i
+
1
)
/
Γ
(
1
/
2
)
Γ
(
v
i
)
). We remark that this shift operator is closely connected to the Bessel differential operator [11]: (2.2)
d
2
U
d
x
2
+
2
v
x
d
U
d
x
=
d
2
U
d
y
2
+
2
v
y
d
U
d
y
,
U
(
x
,
0
)
=
f
(
x
)
,
U
y
(
x
,
0
)
=
0
.
The convolution operator is determined by the T
x
y as follows: (2.3)
(
f
*
φ
)
(
y
)
=
∫
R
n
+
f
(
y
)
T
x
y
φ
(
x
)
(
∏
i
=
1
n
y
i
2
v
i
)
d
y
.
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)
T
x
y
·
1
=
1.
(b)
T
x
0
·
f
(
x
)
=
f
(
x
).
(c)
Iff
(
x
)
,
g
(
x
)
∈
C
(
ℝ
n
+
)
,
g
(
x
) is a bounded function all x
>
0, and
(2.4)
∫
R
n
+
|
f
(
x
)
|
(
∏
i
=
1
n
x
i
2
v
i
)
d
x
<
∞
,
then
(2.5)
∫
R
n
+
T
x
y
f
(
x
)
g
(
y
)
(
∏
i
=
1
n
y
i
2
v
i
)
d
y
=
∫
R
n
+
f
(
y
)
T
x
y
g
(
x
)
(
∏
i
=
1
n
y
i
2
v
i
)
d
y
.
(d)
From (c), we have the following equality forg
(
x
)
=
1:
(2.6)
∫
R
n
+
T
x
y
f
(
x
)
(
∏
i
=
1
n
y
i
2
v
i
)
d
y
=
∫
R
n
+
f
(
y
)
(
∏
i
=
1
n
y
i
2
v
i
)
d
y
.
(e)
(
f
*
g
)
(
x
)
=
(
g
*
f
)
(
x
).Definition 2.1.
Letx
=
(
x
1
,
x
2
,
…
,
x
n
) be a point of the n-dimensional space ℝ
n
+. Denote the nondegenerated quadratic form by
(2.7)
V
=
c
2
(
x
1
2
+
x
2
2
+
⋯
+
x
p
2
)
-
x
p
+
1
2
-
x
p
+
2
2
-
⋯
-
x
p
+
q
2
,
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
(2.8)
R
α
,
c
H
(
x
)
=
{
V
(
α
-
n
-
2
|
v
|
)
/
2
K
n
(
α
)
,
for
x
∈
Γ
+
,
0
,
for
x
∉
Γ
+
,
where
(2.9)
K
n
(
α
)
=
π
(
n
+
2
|
v
|
-
1
)
/
2
Γ
(
(
2
+
α
-
n
-
2
|
v
|
)
/
2
)
Γ
(
(
1
-
α
)
/
2
)
Γ
(
α
)
Γ
(
(
2
+
α
-
p
-
2
|
v
|
)
/
2
)
Γ
(
(
p
+
2
|
v
|
-
α
)
/
2
)
.
The function R
α
,
c
H
(
x
) is introduced by [10, 12, 17, 18]. It is well known that R
α
,
c
H
(
x
) is an ordinary function if Re
(
α
)
≥
n and is the distribution of α if Re
(
α
)
<
n. Let supp
R
α
,
c
H
(
x
)
⊂
Γ
¯
+, where supp
R
α
,
c
H
(
x
) denotes the support of R
α
,
c
H
(
x
).
By puttingp
=
c
=
1 into (2.7), (2.8), and (2.9), and using the Legendre's duplication of Γ
(
z
),(2.10)
Γ
(
2
z
)
=
2
2
z
-
1
π
-
1
/
2
Γ
(
z
)
Γ
(
z
+
1
2
)
,
the formula (2.8) is reduced to
(2.11)
M
α
H
(
x
)
=
{
V
(
(
α
-
n
-
2
|
v
|
)
/
2
)
H
n
(
α
)
,
for
x
∈
Γ
+
,
0
,
for
x
∉
Γ
+
,
where V
=
x
1
2
-
x
2
2
-
⋯
-
x
n
2 and
(2.12)
H
n
(
α
)
=
π
(
n
+
2
|
v
|
-
1
)
/
2
2
α
-
1
Γ
(
2
+
α
-
n
-
2
|
v
|
2
)
Γ
(
α
2
)
.
Note that the function M
α
H
(
x
) is precisely the Bessel hyperbolic kernel of Marcel Riesz.Lemma 2.2.
Given the equation(2.13)
□
B
,
c
k
u
(
x
)
=
δ
(
x
)
,
where □
B
,
c
k is defined by (1.6) and x
∈
ℝ
n
+, then we obtain u
(
x
)
=
R
2
k
,
c
H
(
x
) as a fundamental solution of (2.13), where R
2
k
,
c
H
(
x
) is defined by (2.8).The proof of this Lemma is given in [14].Lemma 2.3.
TheB-convolutions of tempered distributions. (a)
(
□
B
,
c
k
δ
)
*
u
(
x
)
=
□
B
,
c
k
u
(
x
), where u
(
x
) is any tempered distribution.
(b)
LetR
2
k
,
c
H
(
x
) and R
2
m
,
c
H
(
x
) be defined by (2.8); then R
2
k
,
c
H
(
x
)
*
R
2
m
,
c
H
(
x
) exists and is a tempered distribution.
(c)
LetR
2
k
,
c
H
(
x
) and R
2
m
,
c
H
(
x
) be defined by (2.8); then R
2
k
,
c
H
(
x
)
*
R
2
m
,
c
H
(
x
)
=
R
2
k
+
2
m
,
c
H
(
x
), where k and m are nonnegative integers.The proof of this Lemma is given in [15].Lemma 2.4.
Given thatP is a hypersurface
(2.14)
P
δ
(
m
)
(
P
)
+
m
P
δ
(
m
-
1
)
(
P
)
=
0
,
where δ
(
m
) is the Dirac-delta distribution with m derivatives.The proof of this Lemma is given in [1].Lemma 2.5.
Given the equation(2.15)
□
B
,
c
k
u
(
x
)
=
0
,
where □
B
,
c
k is the ultrahyperbolic Bessel operator iterated k-times, as defined by (1.6), and x
∈
ℝ
n
+, then
(2.16)
u
(
x
)
=
[
R
2
(
k
-
1
)
,
c
H
(
x
)
]
(
m
)
,
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.Proof.
We first show that the generalized functionδ
(
m
)
(
c
2
r
2
-
s
2
), where r
2
=
x
1
2
+
x
2
2
+
⋯
+
x
p
2
,
s
2
=
x
p
+
1
2
+
x
p
+
2
2
+
⋯
+
x
p
+
q
2
,
p
+
q
=
n, is a solution of(2.17)
□
B
,
c
u
(
x
)
=
0
,
and □
B
,
c is defined by (1.6) with k
=
1 and x
∈
ℝ
n
+. Now for 1
≤
i
≤
p, we have
(2.18)
∂
∂
x
i
δ
(
m
)
(
c
2
r
2
-
s
2
)
=
2
c
2
x
i
δ
(
m
+
1
)
(
c
2
r
2
-
s
2
)
,
∂
2
∂
x
i
2
δ
(
m
)
(
c
2
r
2
-
s
2
)
=
2
c
2
δ
(
m
+
1
)
(
c
2
r
2
-
s
2
)
+
4
c
4
x
i
2
δ
(
m
+
2
)
(
c
2
r
2
-
s
2
)
.
Thus, we have
(2.19)
1
c
2
∑
i
=
1
p
[
∂
2
∂
x
i
2
δ
(
m
)
(
c
2
r
2
-
s
2
)
+
2
v
i
x
i
∂
∂
x
i
δ
(
m
)
(
c
2
r
2
-
s
2
)
]
=
2
p
δ
(
m
+
1
)
(
c
2
r
2
-
s
2
)
+
4
c
2
r
2
δ
(
m
+
2
)
(
c
2
r
2
-
s
2
)
+
4
|
v
′
|
δ
(
m
+
1
)
(
c
2
r
2
-
s
2
)
=
(
2
p
+
4
|
v
′
|
)
δ
(
m
+
1
)
(
c
2
r
2
-
s
2
)
+
4
(
c
2
r
2
-
s
2
)
δ
(
m
+
2
)
(
c
2
r
2
-
s
2
)
+
4
s
2
δ
(
m
+
2
)
(
c
2
r
2
-
s
2
)
=
(
2
p
+
4
|
v
′
|
)
δ
(
m
+
1
)
(
c
2
r
2
-
s
2
)
-
4
(
m
+
2
)
δ
(
m
+
1
)
(
c
2
r
2
-
s
2
)
+
4
s
2
δ
(
m
+
2
)
(
c
2
r
2
-
s
2
)
=
[
2
p
+
4
|
v
′
|
-
4
(
m
+
2
)
]
δ
(
m
+
1
)
(
c
2
r
2
-
s
2
)
+
4
s
2
δ
(
m
+
2
)
(
c
2
r
2
-
s
2
)
by applying Lemma 2.4 with P
=
c
2
r
2
-
s
2, where |
v
′
|
=
v
1
+
v
2
+
⋯
+
v
p.
Similarly, we have(2.20)
∑
i
=
p
+
1
p
+
q
[
∂
2
∂
x
i
2
δ
(
m
)
(
c
2
r
2
-
s
2
)
+
2
v
i
x
i
∂
∂
x
i
δ
(
m
)
(
c
2
r
2
-
s
2
)
]
=
[
-
(
2
q
+
4
|
v
′′
|
)
+
4
(
m
+
2
)
]
δ
(
m
+
1
)
(
c
2
r
2
-
s
2
)
+
4
c
2
r
2
δ
(
m
+
2
)
(
c
2
r
2
-
s
2
)
by applying Lemma 2.4 with P
=
c
2
r
2
-
s
2, where |
v
′′
|
=
v
p
+
1
+
v
p
+
2
+
⋯
+
v
p
+
q.
Thus, we have(2.21)
□
B
,
c
δ
(
m
)
(
c
2
r
2
-
s
2
)
=
1
c
2
∑
i
=
1
p
[
∂
2
∂
x
i
2
+
2
v
i
x
i
∂
∂
x
i
]
δ
(
m
)
(
c
2
r
2
-
s
2
)
-
∑
i
=
p
+
1
p
+
q
[
∂
2
∂
x
i
2
+
2
v
i
x
i
∂
∂
x
i
]
δ
(
m
)
(
c
2
r
2
-
s
2
)
=
[
2
(
p
+
q
+
2
|
v
|
)
-
8
(
m
+
2
)
]
δ
(
m
+
1
)
(
c
2
r
2
-
s
2
)
-
4
(
c
2
r
2
-
s
2
)
δ
(
m
+
2
)
(
c
2
r
2
-
s
2
)
=
[
2
(
n
+
2
|
v
|
)
-
8
(
m
+
2
)
]
δ
(
m
+
1
)
(
c
2
r
2
-
s
2
)
+
4
(
m
+
2
)
δ
(
m
+
1
)
(
c
2
r
2
-
s
2
)
=
[
2
(
n
+
2
|
v
|
)
-
4
(
m
+
2
)
]
δ
(
m
+
1
)
(
c
2
r
2
-
s
2
)
by applying Lemma 2.4 with P
=
c
2
r
2
-
s
2, where |
v
|
=
|
v
′
|
+
|
v
′′
|.
If[
2
(
n
+
2
|
v
|
)
-
4
(
m
+
2
)
]
=
0, we obtain(2.22)
□
B
,
c
δ
(
m
)
(
c
2
r
2
-
s
2
)
=
0
.
That is, u
(
x
)
=
δ
(
m
)
(
c
2
r
2
-
s
2
) is a solution of (2.15) with m
=
(
n
+
2
|
v
|
-
4
)
/
2
,
n
+
2
|
v
|
≥
4, and n is an even dimension. Now □
B
,
c
k
u
(
x
)
=
0 can be written in the form
(2.23)
□
B
,
c
(
□
B
,
c
k
-
1
u
(
x
)
)
=
0
.
From (2.17), we have
(2.24)
□
B
,
c
k
-
1
u
(
x
)
=
δ
(
m
)
(
c
2
r
2
-
s
2
)
with m
=
(
n
+
2
|
v
|
-
4
)
/
2
,
n
+
2
|
v
|
≥
4, and n being an even dimension. By Lemma 2.3(a), we can write (2.24) in the from
(2.25)
□
B
,
c
k
-
1
δ
*
u
(
x
)
=
δ
(
m
)
(
c
2
r
2
-
s
2
)
.
B-convolving both sides of the above equation with the function R
2
(
k
-
1
)
,
c
H
(
x
), we obtain
(2.26)
R
2
(
k
-
1
)
,
c
H
(
x
)
*
□
B
,
c
k
-
1
δ
*
u
(
x
)
=
R
2
(
k
-
1
)
,
c
H
(
x
)
*
δ
(
m
)
(
c
2
r
2
-
s
2
)
,
□
B
,
c
k
-
1
[
R
2
(
k
-
1
)
,
c
H
(
x
)
]
*
u
(
x
)
=
[
R
2
(
k
-
1
)
,
c
H
(
x
)
]
(
m
)
,
δ
*
u
(
x
)
=
u
(
x
)
=
[
R
2
(
k
-
1
)
,
c
H
(
x
)
]
(
m
)
,
by Lemma2.2.
It follows thatu
(
x
)
=
[
R
2
(
k
-
1
)
,
c
H
(
x
)
]
(
m
) is a solution of (2.15) with m
=
(
n
+
2
|
v
|
-
4
)
/
2
,
n
+
2
|
v
|
≥
4 and n is an even dimension.
The generalized functionδ
(
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].


## 3. Main Result

Theorem 3.1.
Given the equation(3.1)
□
B
,
c
k
u
(
x
)
=
f
(
x
)
,
where □
B
,
c
k 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
(3.2)
u
(
x
)
=
[
R
2
(
k
-
1
)
,
c
H
(
x
)
]
(
m
)
+
R
2
k
,
c
H
(
x
)
*
f
(
x
)
,
where [
R
2
k
,
c
H
(
x
)
]
(
m
) is a function defined by (2.8) with m derivatives.Proof.
B-convolving both sides of (3.1) with R
2
k
,
c
H
(
x
), we obtain(3.3)
R
2
k
,
c
H
(
x
)
*
(
□
B
,
c
k
u
(
x
)
)
=
R
2
k
,
c
H
(
x
)
*
f
(
x
)
.
By Lemma 2.2, we have
(3.4)
□
B
,
c
k
(
R
2
k
,
c
H
(
x
)
)
*
u
(
x
)
=
δ
*
u
(
x
)
=
R
2
k
,
c
H
(
x
)
*
f
(
x
)
.
So, we obtain that
(3.5)
u
(
x
)
=
R
2
k
,
c
H
(
x
)
*
f
(
x
)
is the solution of (3.1).
For a homogeneous equation□
B
,
c
k
u
(
x
)
=
0, we have a solution(3.6)
u
(
x
)
=
[
R
2
(
k
-
1
)
,
c
H
(
x
)
]
(
m
)
by Lemma 2.5. Thus the general solution of (3.1) is
(3.7)
u
(
x
)
=
[
R
2
(
k
-
1
)
,
c
H
(
x
)
]
(
m
)
+
R
2
k
,
c
H
(
x
)
*
f
(
x
)
.
This completes the proof.By puttingc
=
1, (3.1) becomes the Bessel ultrahyperbolic equation(3.8)
□
B
k
w
(
x
)
=
f
(
x
)
,
where □
B
k 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 (3.9)
w
(
x
)
=
R
2
k
H
(
x
)
*
f
(
x
)
is a solution of (3.8), where R
2
k
H
(
x
)
=
R
2
k
,
1
H
(
x
) defined by (2.8).From (3.2), we obtain that the general solution of the Bessel ultrahyperbolic equation is(3.10)
w
(
x
)
=
[
R
2
(
k
-
1
)
H
(
x
)
]
(
m
)
+
R
2
k
H
(
x
)
*
f
(
x
)
.
Moreover, if we put k
=
1
,
p
=
1 and x
1
=
t(times), then (3.8) is reduced to the Bessel wave equation (3.11)
□
B
w
(
x
)
=
(
B
t
-
∑
i
=
2
n
B
x
i
)
w
(
x
)
=
f
(
x
)
,
where (3.12)
□
B
=
B
t
-
∑
i
=
2
n
B
x
i
is the Bessel wave operator and B
x
i
=
∂
2
/
∂
x
i
2
+
(
2
v
i
/
x
i
)
(
∂
/
∂
x
i
).Thus, we obtainw
(
x
)
=
M
2
(
x
)
*
f
(
x
) as a solution of the Bessel wave equation, since R
2
H
(
x
) becomes M
2
H
(
x
), where M
2
H
(
x
) is the Bessel ultrahyperbolic kernel of Marcel Riesz, and is defined by (2.11) with α
=
2. And from (3.2), we obtain the general solution of Bessel wave equation as (3.13)
w
(
x
)
=
δ
(
m
)
(
x
)
+
M
2
H
(
x
)
*
f
(
x
)
,whereδ
(
m
)
(
x
) is a solution of (3.14)
(
B
t
-
∑
i
=
2
n
B
x
i
)
w
(
x
)
=
0
.
Now we put V
=
t
2
-
x
2
2
-
x
3
2
-
⋯
-
x
n
2 and s
2
=
x
2
2
+
x
3
2
+
⋯
+
x
n
2. By [24], we obtain that (3.15)
w
(
x
,
t
)
=
δ
(
m
)
(
t
2
-
s
2
)is the solution of (3.14) with the initial conditions w
(
x
,
0
)
=
0 and ∂
w
(
x
,
0
)
/
∂
t
=
(
-
1
)
m
2
π
m
+
1
δ
(
x
) at t
=
0 and x
=
(
x
2
,
x
3
,
…
,
x
n
)
∈
ℝ
n
-
1
+.



---

*Source: 579645-2011-08-10.xml*