Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2010, Article ID 497676, 12 pages
doi:10.1155/2010/497676
Research Article
On the Operator kB Related to Bessel
Heat Equation
Wanchak Satsanit
Department of Mathematics, Faculty of Science, Maejo University, Chiang Mai 50290, Thailand
Correspondence should be addressed to Wanchak Satsanit, aunphue@live.com
Received 12 April 2010; Accepted 27 June 2010
Academic Editor: Carlo Cattani
Copyright q 2010 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.
We study the equation ∂/∂tux, t c2 ⊕kB ux, t with the initial condition ux, 0 fx for
p
x ∈ Rn . The operator ⊕kB is the operator iterated k-times and is defined by ⊕kB i1 Bxi 4 −
pq
jp1 Bxi 4 k , where p q n is the dimension of the Rn , Bxi ∂2 /∂xi2 2vi /xi ∂/∂xi , 2vi 2αi 1, αi > −1/2, i 1, 2, 3, . . . , n, and k is a nonnegative integer, ux, t is an unknown function
for x, t x1 , x2 , . . . , xn , t ∈ Rn × 0, ∞, fx is a given generalized function, and c is a positive
constant. We obtain the solution of such equation, which is related to the spectrum and the kernel,
which is so called Bessel heat kernel. Moreover, such Bessel heat kernel has interesting properties
and also related to the kernel of an extension of the heat equation.
1. Introduction
It is well known that for the heat equation
∂
ux, t c2 Δux, t
∂t
1.1
ux, 0 fx,
1.2
with the initial condition
where Δ we obtain
n
i1
∂2 /∂xi2 is the Laplace operator and x, t x1 , x2 , . . . , xn , t ∈ Rn × 0, ∞,
ux, t 1
n/2
4c2 πt
x − y2
f y dy
exp −
2
4c t
Rn
1.3
2
Mathematical Problems in Engineering
as the solution of 1.1. Equation 1.3 can be written as
ux, t Ex, t ∗ fx,
1.4
where
Ex, t 1
4c2 πt
n/2
|x|2
exp − 2
4c t
.
1.5
Ex, t is called the heat kernel, where |x|2 x12 x22 · · · xn2 and t > 0 see 1, pages 208-209.
In 2004, Yildirim et al. 2, 3 first introduced the Bessel diamond operator ♦kB iterated
k-times, defined by
⎛
p
⎜ ♦kB ⎝
Bx i
⎛
2
pq
−⎝
i1
⎞2 ⎞k
⎟
Bx j ⎠ ⎠ ,
1.6
jp1
where Bxi ∂2 /∂xi2 2υi /xi ∂/∂xi , 2υi 2αi 1, αi > −1/2, xi > 0. The operator ♦kB can be
expressed by ♦kB ΔkB kB kB ΔkB , where
ΔkB
p
k
Bx i
,
1.7
i1
kB
⎛
⎞k
p
pq
⎝ Bx i −
Bx j ⎠ .
i1
1.8
jp1
And, Yildirim et al. 2, 3 have shown that the solution of the convolution form
ux −1k S2k x ∗ R2k x
1.9
is a unique elementary solution of ♦kB that is
♦kB −1k S2k x ∗ R2k x δ.
1.10
Now, the purpose of this work is to study the following equation:
k
∂
ux, t c2
ux, t
∂t
B
1.11
with the initial condition
ux, 0 fx,
for x ∈ Rn ,
1.12
Mathematical Problems in Engineering
3
where the operator ⊕kB is first introduced by Satsanit and Kananthai 4 and is defined by
k
⎛
p
⎜ ⎝
Bx i
⎛
4
−⎝
i1
B
⎛
p
⎜ ⎝
Bx i
pq
⎞4 ⎞k
⎟
Bx i ⎠ ⎠
jp1
2
⎞2 ⎞k ⎛
⎞2 ⎞k
2 ⎛ pq
pq
p
⎟ ⎜
⎟
−⎝
Bx i ⎠ ⎠ ⎝
Bx i
⎝
Bx i ⎠ ⎠ .
⎛
i1
jp1
i1
1.13
jp1
Let us denote the operator
⎛
⎜
kB ⎝
⎛
2
p
pq
⎝
Bx i
i1
⎞2 ⎞k
⎟
Bx j ⎠ ⎠ .
1.14
jp1
By 1.7 we obtain
⎡
⎞2 ⎤k
2 ⎛ pq
p
⎥
⎢
Bxi2
⎝
Bxj2 ⎠ ⎦
kB ⎣
i1
jp1
ΔB B
2
Δ2B 2B
2
2
ΔB − B
2
2 k
1.15
k
.
Thus, 1.13 can be written as
k
♦kB kB ,
1.16
B
where ♦kB and kB are defined by 1.6 and 1.15, respectively, p q n is the dimension
of the Rn x : x x1 , x2 , . . . , xn , t, xi > 0, i 1, 2, 3, . . . , n, ux, t is an unknown function,
Bxi ∂2 /∂xi2 2υi /xi ∂/∂xi , 2υi 2αi 1, αi > −1/2, fx is the given generalized function,
k is a positive integer, and c is a constant.
Moreover, Bessel heat kernel has interesting properties and also related to the kernel
of an extension of the heat equation. We obtain the solution in the classical convolution form
ux, t Ex, t ∗ fx,
1.17
4
Mathematical Problems in Engineering
where the symbol ∗ is the B-convolution in 2.3, as a solution of 1.11, which satisfies 1.12,
and
Ex, t Cv
4 k
4
Ω
e
2
2
c2 ty12 ···yp2 −yp1
···ypq
n
Jvi −1/2 xi , yi yi2υi dy,
1.18
i1
and Ω ⊂ Rn is the spectrum of Ex, t for any fixed t > 0 and Jvi −1/2 xi , yi is the normalized
Bessel function.
Before going into details, the following definitions and some important concepts are
needed.
2. Preliminaries
The shift operator according to the law remarks that this shift operator connected to the Bessel
differential operator see 2, 3, 5:
y
Tx ϕx Cv∗
π
0
···
π ϕ
x12 y12 −2x1 y1 cos θ1 , . . . , xn2 yn2 −2xn yn cos θn
0
×
n
2.1
sin
2vi −1
θi dθ1 · · · dθn ,
i1
where x, y ∈ Rn , Cv∗ ni1 Γvi 1/Γ1/2Γvi . We remark that this shift operator is closely
connected to the Bessel differential operator see 4:
d2 U 2v dU d2 U 2v dU
,
x dx
y dy
dx2
dy2
Ux, 0 fx,
2.2
Uy x, 0 0.
y
The convolution operator determined by the Tx is as follows:
f ∗ϕ y Rn
n
2v
y
i
dy.
f y Tx ϕx
yi
2.3
i1
Convolution 2.3 is known as a B-convolution. We note the following properties of the Bconvolution and the generalized shift operator.
y
1 Tx · 1 1.
2 Tx0 · fx fx.
!∞
3 If fx, gx ∈ CRn , gx is a bounded function for all x > 0, and 0 |fx|
!
!
y
y
ni1 xi2vi dx < ∞, then Rn Tx fxgy ni1 yi2vi dy Rn fyTx gx ni1 yi2vi dy.
!
y
4 From 3,we have the following equality for gx 1 : Rn Tx fx ni1 yi2vi dy !
fy ni1 yi2vi dy.
Rn
5 f ∗ gx g ∗ fx.
Mathematical Problems in Engineering
5
The Fourier-Bessel transformation and its inverse transformation are defined as follows:
FB−1 f
FB f x Cv
Rn
n
2υi
dy,
f y
Jvi −1/2 xi , yi yi
2.4
i1
Cv x FB f −x,
n
υi −1/2
2
i1
−1
1
Γ υi ,
2
2.5
where Jvi −1/2 xi , yi is the normalized Bessel function which is the eigenfunction of the Bessel
differential operator. The following equalities for Fourier-Bessel transformation are true see
5–7:
FB δx 1,
2.6
FB f ∗ g x FB fx · FB gx.
2.7
Definition 2.1. The spectrum of the kernel Ex, t of 1.18 is the bounded support of the
Fourier Bessel transform FB Ey, t for any fixed t > 0.
Definition 2.2. Let x x1 , x2 , . . . , xn be a point in Rn and denote by
"
#
2
2
2
Γ x ∈ Rn : x12 x22 · · · xp2 − xp1
− xp2
− · · · − xpq
> 0, ξ1 > 0
2.8
the set of an interior of the forward cone, and Γ denotes the closure of Γ .
Let Ω be spectrum of Ex, t defined by 1.18 for any fixed t > 0 and Ω ⊂ Γ . Let
FB Ey, t be the Fourier Bessel transform of Ex, t, which is defined by
FB E y, t ⎧
4
4 k
2
2
⎪
···ypq
⎨ec2 ty12 ···yp2 −yp1
for ξ ∈ Ω ,
⎪
⎩0
for ξ /
∈ Ω .
2.9
Lemma 2.3 Fourier Bessel transform of kB operator. One has
FB kB ux −1k V1k xFB ux,
2.10
⎛
⎞k
p
pq
V1k x ⎝ xi2 −
xj2 ⎠ .
2.11
where
i1
Proof. See 8.
jp1
6
Mathematical Problems in Engineering
Lemma 2.4 Fourier Bessel transform of ΔkB operator. One has
FB ΔkB ux −1k |x|2k FB ux,
2.12
k
|x|2k x12 x22 · · · xn2 .
2.13
where
Proof. See 8.
Lemma 2.5 Fourier Bessel transform of ⊕kB operator. One has
FB
k
ux V k xFB ux,
2.14
B
where
⎛
p
⎜ 2
V k x ⎝
xi
i1
4
⎛
−⎝
pq
⎞4 ⎞k
⎟
xj2 ⎠ ⎠ .
2.15
jp1
Proof. We can use the mathematical induction method; for k 1, we have
FB
n
2υi
dy
u x Cv
u y
Jvi −1/2 xi , yi yi
Rn
B
Cv
B
i1
n
Jvi −1/2 xi , yi yi2υi
♦B B u y
Rn
i1
dy
n
Δ2B 2B 2υi
Cv
dy, g y ♦B u y
g y
Jvi −1/2 xi , yi yi
2
Rn
i1
2
Δ g 2B g
FB B
x
2
2 2
2
2
− · · · − xpq
−12 x12 · · · xn2 −12 x12 · · · xp2 − xp1
FB gx
2
2 2 2
2
2
2
2
x1 x2 · · · xp xp1 · · · xpq
FB ♦B ux
Mathematical Problems in Engineering
⎞ ⎛
⎞
⎛
p 2 ⎛ pq ⎞2
p 2 ⎛ pq ⎞2
⎟ ⎜
⎟
⎜
xi2
⎝
xj2 ⎠ ⎠ · ⎝
xi2
−⎝
xj2 ⎠ ⎠FB ux
⎝
i1
jp1
x12 x22 · · · xp2
4
i1
7
jp1
4 2
2
− xp1
· · · xpq
FB ux
V xFB ux,
2.16
4
4
2
2
where V x x12 x22 · · · xp2 − xp1
· · · xpq
. Then, from inverse Fourier transform
we obtain
ux FB−1 V xFB ux.
2.17
B
Assume that the statement is true for k − 1, that is,
k−1
ux FB−1 V k−1 xFB ux.
2.18
B
Then, we must prove that is also true for k ∈ N. So we have
k
ux B
k−1
B
ux
B
2.19
FB−1 V xFB FB−1 V k−1 xFB ux
FB−1 V k xFB ux.
This completes the proof.
Lemma 2.6. For t, v > 0 and x, y ∈ Rn , one has
∞
e−c
2 2
x dx 0
∞
e
−c2 x2 t
Γυ
x t 2υ
2c2υ1 tυ1/2
Jυ−1/2 xy x dx 0
2υ
,
Γυ 1/2
2c2 tυ1/2
2.20
e
−y2 /4c2 t
,
where c is a positive constant.
Proof. See 9.
Lemma 2.7. Let the operator L be defined by
L
k
∂
− c2
,
∂t
B
2.21
8
Mathematical Problems in Engineering
where ⊕kB is the operator iterated k-times and is given by
k
⎛
p
⎜ ⎝
Bx i
i1
B
4
⎛
−⎝
⎞4 ⎞k
⎟
Bx i ⎠ ⎠ ,
pq
jp1
2.22
∂2
2vi ∂
,
2
xi ∂xi
∂xi
Bx i p q n is the dimension Rn , k is a positive integer, x1 , x2 , . . . , xn ∈ Rn , and c is a positive
constant. Then
Ex, t Cv
4 k
4
Ω
e
2
2
c2 ty12 ···yp2 −yp1
···ypq
n
Jvi −1/2 xi , yi yi2υi dy
2.23
i1
is the elementary solution of 2.21 in the spectrum Ω ⊂ Rn for t > 0.
Proof. Let LEx, t δx, t, where Ex, t is the elementary solution of L and δ is the Diracdelta distribution. Thus
k
∂
Ex, t − c2
Ex, t δxδt.
∂t
B
2.24
Applying the Fourier Bessel transform, which is defined by 2.4 to the both sides of the above
equation and using Lemma 2.5 by considering FB δx 1, we obtain
(
4 4 )k
∂
2
2
FB Ex, t − c2 x12 x22 · · · xp2 − xp1
· · · xpq
FB Ex, t δt.
∂t
2.25
Thus, we get
FB Ex, t Hte
4 k
4
2
2
c2 tx12 x22 ···xp2 −xp1
···xpq
,
2.26
where Ht is the Heaviside function, because Ht 1 holds for t ≥ 0.
Therefore,
FB Ex, t e
4 k
4
2
2
c2 tx12 x22 ···xp2 −xp1
···xpq
,
2.27
which has been already defined by 2.7. Thus from 2.5, we have
Ex, t Cv
4
Rn
e
4 k
2
2
c2 ty12 ···yp2 −yp1
···ypq
n
i1
Jvi −1/2 xi , yi yi2υi dy,
2.28
Mathematical Problems in Engineering
9
where Ω is the spectrum of Ex, t. Thus, we obtain
Ex, t Cv
Ω
n
Jvi −1/2 xi , yi yi2υi dy
4 k
4
e
2
2
c2 ty12 ···yp2 −yp1
···ypq
2.29
i1
as an elementary solution of 2.21 in the spectrum Ω ⊂ Rn for t > 0.
3. Main Results
Theorem 3.1. Let us consider the equation
k
∂
ux, t − c2
ux, t 0
∂t
B
3.1
ux, 0 fx,
3.2
with the initial condition
where ⊕kB is the operator iterated k-times and is defined by
k
B
⎛
p
⎜ ⎝
Bx i
i1
Bx i 4
⎛
−⎝
pq
⎞4 ⎞k
⎟
Bx i ⎠ ⎠ ,
jp1
3.3
∂2
2vi ∂
,
2
xi ∂xi
∂xi
p q n is the dimension Rn , k is a positive integer, ux, t is an unknown function for x, t x1 , x2 , . . . , xn , t ∈ Rn × 0, ∞, fx is the given generalized function, and c is a positive constant.
Then
ux, t Ex, t ∗ fx
3.4
is a solution of 3.1, which satisfies 3.2, where Ex, t is given by 2.23. In particular, if one puts
k 1 and q 0 in 3.1, then 3.1 reduces to the equation
∂
ux, t − c2 Δ4B ux, t 0,
∂t
which is related to the Bessel heat equation.
3.5
10
Mathematical Problems in Engineering
Proof. Taking the Fourier Bessel transform, which is defined by 2.4, of both sides of 3.1 for
x ∈ Rn and using Lemma 2.5, we obtain
4 4 k
∂
2
2
2
2
2
FB ux, t c
FB ux, t.
x1 · · · xp − xp1 · · · xpq
∂t
3.6
Thus, we consider the initial condition 3.2; then we have the following equality for 3.6:
ux, t fx ∗ FB−1 e
4 k
4
2
2
c2 tx12 ···xp2 −xp1
···xpq
.
3.7
dy
3.8
Here, if we use 2.4 and 2.5, then we have
ux, t fx ∗ FB−1 e
Rn
Rn
4 k
4
2
2
c2 ty12 ···yp2 −yp1
···ypq
FB−1 e
y
Tx fx
n
yi2υi
i1
4 k
4
2
2
c2 ty12 ···yp2 −yp1
···ypq
Cv
e
Rn
c2 tV k z
n
n
2υ
2υi
y
i
dy,
Jυi −1/2 yi , zi zi dz Tx fx
yi
i1
i1
4
4
where V z z21 z22 · · · z2p − z2p1 z2p2 · · · z2pq . Set
Ex, t Cv
4
Rn
e
4 k
2
2
c2 ty12 ···yp2 −yp1
···ypq
n
Jvi −1/2 xi , yi yi2υi dy.
3.9
i1
Since the integral in 3.9 is divergent, therefore we choose Ω ⊂ Rn to be the spectrum of
Ex, t, and by 2.21, we have
Ex, t Cv
4
Rn
e
4
Ω
n
Jvi −1/2 xi , yi yi2υi dy
i1
Cv
4 k
2
2
c2 ty12 ···yp2 −yp1
···ypq
e
4 k
2
2
c2 ty12 ···yp2 −yp1
···ypq
n
3.10
Jvi −1/2 xi , yi yi2υi dy.
i1
Thus 3.8 can be written in the convolution form
ux, t Ex, t ∗ fx.
3.11
Mathematical Problems in Engineering
11
Moreover, since Ex, t exists, we can see that
lim
t→0
Ex, t Cv
n
Ω i1
Cv
n
Rn
δx,
Jvi −1/2 xi , yi yi2vi dy
Jvi −1/2 xi , yi yi2vi dy
3.12
i1
for x ∈ Rn ,
hold see 8. Thus for the solution ux, t Ex, t ∗ fx of 3.1, then we have
lim
t→0
ux, t ux, 0 δ ∗ fx fx,
3.13
which satisfies 3.2. This completes the proof.
Theorem 3.2. The kernel Ex, t defined by 3.10 has the following properties.
1 Ex, t ∈ C∞ -the space of continuous function for x ∈ Rn , t > 0 with infinitely
differentiable.
2 ∂/∂t − c2 ⊕kB Ex, t 0 for all x ∈ Rn , t > 0.
3 lim
t→0
Ex, t δ for all x ∈ Rn .
Proof. 1 From 3.10 and
∂n
Ex, t Cv
∂tn
Rn
n
4 k
2
2
∂n c2 ty12 ···yp2 4 −yp1
···ypq
e
Jvi −1/2 xi , yi yi2υi dy.
n
∂t
i1
3.14
we have Ex, t ∈ C∞ for x ∈ Rn , t > 0.
2 We have ux, t Ex, t since ux, t Ex, t ∗ fx holds. Note here that we use
the fact fx δx by the Fourier Bessel transformation. Then, we obtain
k
∂
2
Ex, t 0
−c
∂t
B
3.15
by direct computation.
3 This case is obvious by 3.12.
In particular, if we put k 1 and q 0 in 3.1, then 3.1 reduces to the equation
∂
ux, t − c2 Δ4B ux, t 0,
∂t
3.16
and we obtain the solution of 3.16 in the convolution form
ux, t Ex, t ∗ fx,
3.17
12
Mathematical Problems in Engineering
where Ex, t is defined by 2.23 with k 1 which is related to Bessel heat equation. This
completes the proof.
Acknowledgments
The authors would like to thank The Thailand Research Fund and Graduate School, Maejo
University, Thailand for financial support.
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