Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 951960, 15 pages
doi:10.1155/2011/951960
Research Article
On the Diamond Bessel Heat Kernel
Wanchak Satsanit
Department of Mathematics, Faculty of Science, Maejo University, Chiangmai 50290, Thailand
Correspondence should be addressed to Wanchak Satsanit, aunphue@live.com
Received 19 July 2011; Accepted 2 November 2011
Academic Editor: Tak-Wah Lam
Copyright q 2011 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 heat equation in n dimensional by Diamond Bessel operator. We find the solution by
method of convolution and Fourier transform in distribution theory and also obtain an interesting
kernel related to the spectrum and the kernel which is called Bessel heat kernel.
1. Introduction
The operator ♦k has been first introduced by Kananthai 1, is named as the diamond operator
iterated k times, and is defined by
⎛
⎜
♦k ⎝
p
2
∂
2
i1 ∂xi
2
⎞2 ⎞k
∂ ⎠ ⎟
−⎝
⎠ ,
2
jp1 ∂xj
⎛
pq
2
1.1
pq n, 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 following form:
♦k k k k 0k ,
1.2
where k is the Laplacian operator iterated k-times defined by
∂2
∂2
∂2
·
·
·
,
∂x12 ∂x22
∂xn2
1.3
2
Journal of Applied Mathematics
and k is the ultrahyperbolic operator iterated k-times defined by
⎞k
2
2
2
2
2
2
∂
∂
∂
∂
∂
∂
⎝ 2 2 ··· 2 − 2 − 2 − ··· − 2 ⎠ .
∂x1 ∂x2
∂xp ∂xp1 ∂xp2
∂xpq
⎛
k
1.4
x is an
Kananthai 1, Theorem 1.3 has shown that the convolution −1k Re2k x ∗ RH
2k
elementary of the operator ♦k . That is
♦k −1k Re2k x ∗ RH
2k x δx,
1.5
where Re2k x is defined by
Reα x |x|α−n
,
Wn α
π n/2 2α Γα/2
,
Wn α Γn − α/2
1.6
α is a complex parameter, n is the dimension of Rn , and the generalized function RH
α υ is
defined by
⎧
α−n/2
⎪
⎨υ
, for x ∈ Γ ,
Kn α
RH
α υ ⎪
⎩0,
for x ∈
/ Γ ,
1.7
and 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
1.8
The function RH
α υ is called the ultrahyperbolic kernel of Marcel Riesz and was introduced
by Nozaki see 2, page 72.
Next, Yildirim et al. see 3 first introduced the Bessel diamond operator ♦kB iterated
k-times, defined by
⎛
⎞2 ⎞k
2 ⎛ pq
p
⎟
⎜
Bx i
−⎝
Bx j ⎠ ⎠ ,
♦kB ⎝
i1
jp1
1.9
Journal of Applied Mathematics
3
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
p
k
Bx i ,
kB
1.10
i1
kB
⎞k
⎛
p
pq
⎝ Bx i −
Bx j ⎠ .
i1
1.11
jp1
And Yildirim see 4 have shown that the solution of the convolution form ux −1k S2k x ∗ R2k x is a unique elementary solution of ♦kB , that is,
♦kB −1k S2k x ∗ R2k x δ,
1.12
where S2k x is defined by 2.8 with α 2k and R2k x is defined by 2.9 with γ 2k.
It is well known that for the heat equation
∂
ux, t c2 ux, t
∂t
1.13
ux, 0 fx,
1.14
with the initial condition
where is the Laplace operator and is defined by 1.3 and x, t x1 , x2 , . . . , xn , t ∈ Rn ×
0, ∞, we obtain
ux, t 1
n/2
4c2 πt
x − y2
f y dy
exp −
2
4c t
Rn
1.15
as the solution of 1.13. Now, 1.15 can be written in the classical form
ux, t Ex, t ∗ fx,
1.16
where
Ex, t 1
4c2 πt
n/2
|x|2
exp − 2
4c t
.
1.17
Ex, t is called the heat kernel, where |x|2 x12 x22 · · · xn2 and t > 0, see 5, pages 208-209.
Moreover, we obtain Ex, t → δ as t → 0, where δ is the Dirac delta distribution.
4
Journal of Applied Mathematics
Next, Saglam et al. see 6 have study the following equation,
∂
ux, t c2 kB ux, t
∂t
1.18
with the initial condition
ux, 0 fx,
for x ∈ Rn ,
1.19
where the operator kB is named the Bessel ultrahyperbolic operator iterated k-times and is
defined by 1.4, k is a positive integer, ux, t is an unknown function, fx is the given
generalized function, and c is a constant, and p q n is the dimension of the Rn {x : x x1 , x2 , . . . , xn , t, xi > 0, i 1, 2, 3, . . . , n}.
They obtain the solution in the classical convolution form
ux, t Ex, t ∗ fx,
1.20
where the symbol ∗ is the B-convolution in 2.3, as a solution of 1.18, which satisfies 1.19,
where
Ex, t Cv
Ω
e
2
2
c2 ty12 ···yp2 −yp1
···ypq
k
n
Jvi −1/2 xi , yi yi2υi dy,
1.21
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.
Now, the purpose of this work is to study the equation
∂
B ux, t c2 ♦B ux, t
∂t
1.22
B ux, 0 fx,
1.23
with the initial condition
where x, t x1 , . . . , xn , t ∈ Rn × 0, ∞, t is a time, c is a positive constant, ux, t is an
unknown function, and fx is a given generalized function for x ∈ Rn . We obtain
ux, t RH
2 x ∗ Ex, t ∗ fx
1.24
as a solution of 1.7, where
Ex, t Cv
Ω
e−c
2
ty12 y22 y32 ···yn2 n
Jvi −1/2 xi , yi yi2υi dy,
i1
1.25
Journal of Applied Mathematics
5
and Ω ⊂ Rn is the spectrum of Ex, t for any fixed t > 0, and RH
2 x is defined by 2.6
x
∗
Ex,
t
is
called
the
Diamond
Bessel Heat Kernel, and
with α 2. The convolution RH
2
all properties will be studied in details. Before proceeding, the following definitions and
concepts are needed.
2. Preliminaries
The shift operator according to the law remarks that this shift operator connected to the Bessel
differential operator see 3, 5, 7, 8:
y
Tx ϕx Cv∗
×
π
···
0
n
π ϕ
x12 y12 − 2x1 y1 cos θ1 , . . . , xn2 yn2 − 2xn yn cos θn
0
2.1
sin2vi −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 3, 5, 7, 8,
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
y
2vi
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,
for all x >
3 If fx,
∈ CRn , gx is a bounded
function
∞ gx y
<
∞, then Rn Tx fxgy ni1 yi2vi dy
and 0 |fx| ni1 xi2vi dx
y
fyTx gx ni1 yi2vi dy.
Rn
4 From 3, we have the following equality for gx 1:
fy ni1 yi2vi dy.
Rn
5 f ∗ gx g ∗ fx.
Rn
y
Tx fx
n
2vi
i1 yi dy
0,
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Journal of Applied Mathematics
The Fourier-Bessel transformation and its inverse transformation are defined as follows:
FB−1 f
FB f x Cv
Rn
n
2υi
f y
Jvi −1/2 xi , yi yi
dy,
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
3, 5, 7, 8:
FB δx 1,
2.6
FB f ∗ g x FB fx · FB gx.
2.7
Definition 2.1. Let x x1 , x2 , . . . , xn , ν ν1 , ν2 , . . . , νn ∈ Rn . For any complex number α,
we define the function Sα x by
Sα x 2n2|ν|−2α Γn 2|ν| − α/2|x|α−n−2|ν|
.
n ν −1/2
i
Γνi 1/2
i1 2
2.8
Definition 2.2. Let x x1 , x2 , . . . , xn , ν ν1 , ν2 , . . . , νn ∈ Rn , and denote by V x12 x22 2
2
2
− xp2
− · · · − xpq
the nondegenerated quadratic form. Denote the interior of
· · · xp2 − xp1
the forward cone by Γ {x ∈ Rn : x1 > 0, x2 > 0, . . . , xn > 0, V > 0}. The function Rβ x is
defined by
Rγ x V γ−n−2|ν|/2
,
Kn γ
2.9
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 − 2|ν| − γ /2
2.10
and γ is a complex number.
By putting p 1 in R2k x and taking into account Legendre’s duplication formula for
Γz:
Γ2z 2
2z−1
π
−1/2
1
,
ΓzΓ z 2
2.11
Journal of Applied Mathematics
7
we obtain
Iγ x υγ−n−2|ν|/2
Mn γ
2.12
2
2
2
− xp2
− · · · − xpq
, where
and υ x12 − x22 − · · · xp2 − xp1
Mn 2k π
n2|ν|−1/2 2k−1
2
2 2k − n − 2|ν|
Γk.
Γ
2
2.13
Definition 2.3. The spectrum of the kernel Ex, t of 1.21 is the bounded support of the
Fourier Bessel transform FB Ey, t for any fixed t > 0.
Definition 2.4. Let x x1 , x2 , . . . , xn be a point in Rn , and denote by
2
2
2
− xp2
− · · · − xpq
> 0, ξ1 > 0
Γ x ∈ Rn : x12 x22 · · · xp2 − xp1
2.14
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 1.21 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 ⎧ 2 2 2 2
2
⎨e−c ty1 y2 y3 ···yp ,
for ξ ∈ Ω ,
⎩0,
for ξ ∈
/ Ω .
2.15
Lemma 2.5. Given the equation kB ux δx for x ∈ Rn , where kB is defined by 1.10. Then,
ux −1k S2k x,
2.16
where S2k x is defined by 2.3, with α 2k. We obtain that −1k Re2k x is an elementary solution
of the operator kB . That is
kB −1k S2k x δx.
2.17
Proof. See 3, page 379.
Lemma 2.6. Given the equation kB ux δx for x ∈ Rn , where kB is defined by 1.11. Then,
ux R2k x,
2.18
where R2k x is defined by 2.4, with γ 2k. We obtain that R2k x is an elementary solution of the
operator kB . That is
kB R2k x δx.
2.19
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Journal of Applied Mathematics
Proof (see [3, Page 379]). From 2.8 with k 1, we obtain ux RH
2 x as an elementary
solution of the equation
ux δ.
2.20
⎧
2−n−2|ν|/2
⎪
⎨υ
, for x ∈ Γ ,
H
Kn 2
R2 υ ⎪
⎩0,
for x ∈
/ Γ ,
2.21
Now, from 2.17,
We can compute Kn 2 from 2.7 as
Kn 2 π n2|ν|−1/2 Γ4 − n − 2|ν|/2Γ−1/2Γ2
.
Γ 4 − p − 2|ν| /2 Γ p 2|ν| − 2 /2
2.22
By using the formula
ΓzΓ1 − z π
,
sin πz
2.23
√
we obtain Γ−1/2 −2 π, Γ2 1 and
4 − p − 2|ν|
p 2|ν| − 2
p 2|ν| − 2
p 2|ν| − 2
Γ
Γ
Γ 1−
Γ
2
2
2
2
π
.
sin π p 2|ν| − 2 /2
2.24
Then, we obtain
√ π n2|ν|−1/2 Γ4 − n/2 −2 π sin π p 2|ν| − 2 /2
Kn 2 π
p 2|ν| − 2
4 − n − 2|ν|
n2|ν|−2/2
−2π
Γ
sin π
.
2
2
2.25
Thus,
RH
2 υ
⎧
⎪
⎨
−2π n2|ν|−2/2 Γ4
⎪
⎩0,
υ2−n−2|ν|/2
,
− n − 2|ν|/2 sin π p 2|ν| − 2 /2
2
2
2
− xp2
− · · · − xpq
.
where υ x12 x22 · · · xp2 − xp1
for x ∈ Γ ,
for x ∈
/ Γ ,
2.26
Journal of Applied Mathematics
9
Lemma 2.7. Let Sα x and Rβ x be the functions defined by 2.8 and 2.9, respectively. Then
Sα x ∗ Sβ x Sαβ x,
2.27
Rβ x ∗ Rα x Rβα x,
where α and β are a positive even number.
Proof . See 3, pages 171–190.
Lemma 2.8 Fourier Bessel transform of kB operator. Consider
FB kB ux −1k V1k xFB ux,
2.28
⎞k
⎛
p
pq
V1k x ⎝ xi2 −
xj2 ⎠ .
2.29
where
i1
jp1
Proof. See 4.
Lemma 2.9 Fourier Bessel transform of kB operator. Consider
FB kB ux −1k |x|2k FB ux,
2.30
k
|x|2k x12 x22 · · · xn2 .
2.31
where
Proof. see 4, 9.
Lemma 2.10. For t, v > 0, and x, y ∈ Rn , we have
∞
e−c
2 2
x dx x t 2υ
0
∞
e
−c2 x2 t
Γυ
,
2c2υ1 tυ1/2
Jυ−1/2 xy x dx 0
2υ
Γυ 1/2
2c2 tυ1/2
2.32
e
−y2 /4c2 t
,
where c is a positive constant.
Proof. See 6, Lemma 3.1.1 and 10.
Lemma 2.11. Let the operator L be defined by
L
∂
− c 2 B ,
∂t
2.33
10
Journal of Applied Mathematics
where B is the Laplace Bessel operator defined by
B Bx 1 Bx 2 Bx 3 · · · Bx n ,
Bx i 2.34
∂2
2vi ∂
2
xi ∂xi
∂xi
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
Ω
e−c
2
ty12 y22 ···yn2 n
Jvi −1/2 xi , yi yi2υi dy
2.35
i1
is the elementary solution of 2.15 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,
∂
Ex, t − c2 Bx1 Bx2 Bx3 · · · Bxn Ex, t δxδt.
∂t
2.36
Applying the Fourier Bessel transform, which is defined by 2.4 to the both sides of the above
equation and using Lemma 2.7 by considering FB δx 1, we obtain
∂
FB Ex, t c2 x12 x22 x32 · · · xn2 FB Ex, t δt.
∂t
2.37
Thus, we get
FB Ex, t Hte−c
2
tx12 x22 x32 ···xn2 ,
2.38
where Ht is the Heaviside function, because Ht 1 holds for t ≥ 0.
Therefore,
FB Ex, t e−c
2
tx12 x22 x32 ···xn2 ,
2.39
which has been already given by 2.7. Thus, from 2.5, we have
Ex, t Cv
Rn
e−c
2
ty12 y22 y32 ···yn2 n
Jvi −1/2 xi , yi yi2υi dy,
2.40
i1
where Ω is the spectrum of Ex, t. Thus, we obtain
Ex, t Cv
Ω
e−c
2
ty12 y22 y32 ···yp2 n
Jvi −1/2 xi , yi yi2υi dy
i1
2.41
Journal of Applied Mathematics
11
as an elementary solution of 2.15 in the spectrum Ω ⊂ Rn for t > 0.
Definition 2.12. We can extend Ex, t to Rn × R by setting
⎧
⎪
⎨
1
n
Ex, t 2π
⎪
⎩0,
Ω
exp −c2 t ξi2 ξ22 ξ32 · · · ξn2 , for t > 0,
for t ≤ 0.
2.42
Lemma 2.13. Let us consider the equation
∂
ux, t − c2 B ux, t 0
∂t
2.43
ux, 0 fx,
2.44
with the initial condition
where B is the operator iterated k-times and is defined by
B Bx1 Bx2 Bx3 · · · B,
Bx i 2.45
∂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
2.46
is a solution of 2.43, which satisfies 2.44, where Ex, t is given by 2.35.
Proof. Taking the Fourier Bessel transform, the both sides of 2.43, for x ∈ Rn and using
Lemma 2.9, we obtain
∂
FB ux, t −c2 x12 x22 x32 · · · xn2 FB ux, t.
∂t
2.47
Thus, we consider the initial condition 2.44, then we have the following equality for 2.47:
ux, t fx ∗ FB−1 e−c
2
tx12 x22 x32 ···xn2 .
2.48
12
Journal of Applied Mathematics
Here, if we use 2.4 and 2.5, then we have
ux, t fx ∗ FB−1 e−c
Rn
Rn
ty12 y22 y32 ···yn2 2
2
2
y
FB−1 e−c ty1 ···yn Tx fx
n
yi2υi
dy
i1
2
Cv
Rn
e−c
2
tV k z
2.49
n
n
i
y
Jυi −1/2 yi , zi z2υ
dz
T
fx
yi2υi
x
i
i1
dy,
i1
where V z z21 z22 z23 · · · z2n . Set
Ex, t Cv
Rn
e−c
n
Jvi −1/2 xi , yi yi2υi dy.
ty12 y22 y32 ···yn2 2
2.50
i1
We choose Ω ⊂ Rn , to be the spectrum of Ex, t, and, by 2.35, we have
Ex, t Cv
Rn
Cv
Ω
e−c
2
ty12 y22 y32 ···yn2 n
Jvi −1/2 xi , yi yi2υi dy
i1
e−c
2
ty12 y22 y32 ···yn2 n
Jvi −1/2 xi , yi yi2υi dy.
2.51
i1
Thus, 2.49 can be written in the convolution form
ux, t Ex, t ∗ fx.
2.52
Moreover, since Ex, t exists, we can see that
lim Ex, t Cv
t→0
n
Ω i1
Cv
n
Rn
δx,
Jvi −1/2 xi , yi yi2vi dy
Jvi −1/2 xi , yi yi2vi dy
2.53
i1
for x ∈ Rn .
Thus, for the solution ux, t Ex, t ∗ fx of 2.43, we have
limux, t ux, 0 δ ∗ fx fx,
t→0
which satisfies 2.44. This completes the proof.
2.54
Journal of Applied Mathematics
13
3. Main Results
Theorem 3.1. Given the equation
∂
B ux, t c2 ♦B ux, t
∂t
3.1
B ux, 0 fx,
3.2
with the initial condition
where fx is a given generalized function for Rn , B is an ultrahyperbolic Bessel operator, and ♦B is
the Diamond Bessel operator defined by 1.11 and 1.9, respectively. Then, we obtain
ux, t RH
2 x ∗ Ex, t ∗ fx
3.3
as a solution of 3.1, where RH
2 x is given by 2.9 with γ 2 and Ex, t is given by 2.35.
Proof. Equation 3.1 can be written in the form
∂
B ux, t c2 B B ux, t.
∂t
3.4
Let wx, t B ux, t. Thus, the above equation can be written as
∂
wx, t c2 B wx, t.
∂t
3.5
We can solve the above equation by applying the n-dimensional Fourier Bessel
transform to both sides of 3.5. By Lemma 2.7, we obtain
wx, t B ux, t Ex, t ∗ fx.
3.6
By convolving both sides of 3.6 by function RH
2 x, we obtain
H
RH
2 x ∗ B ux, t R2 x ∗ Ex, t ∗ fx.
3.7
By properties of convolution,
H
RH
2 x ∗ ux, t R2 x ∗ Ex, t ∗ fx.
3.8
δ ∗ ux, t RH
2 x ∗ Ex, t ∗ fx,
3.9
By Lemma 2.5, we obtain
14
Journal of Applied Mathematics
or
ux, t RH
2 x ∗ Ex, t ∗ fx
3.10
is a solution of 3.1. As shown in 3.5 and by the continuity of convolution,
limB ux, t lim Ex, t ∗ fx δ ∗ fx fx.
t→0
t→0
3.11
H
Theorem 3.2 The properties of the Diamond Bessel Heat Kernel RH
2 x ∗ Ex, t. (1) R2 x ∗
Ex, t exists and is tempered distribution.
∞
(2) RH
2 x ∗ Ex, t ∈ C the space of continuous function and infinitely differentiable.
H
(3) limt → 0 R2 x ∗ Ex, t RH
2 x.
x
∗
Ex,
t
−
c2 ♦B RH
(4) ∂/∂tB RH
2
2 x ∗ Ex, t 0.
Proof. 1 Since Ex, t and RH
2 x are tempered distribution with compact support. Thus,
x
∗
Ex,
t
exists
and
is
a
tempered distribution.
RH
2
2 We have
∂n H
∂n
R2 x ∗ Ex, t RH
Ex, t,
2 x ∗
n
∂x
∂xn
3.12
∞
since Ex, t is infinitely differentiable and RH
2 x ∗ Ex, t ∈ C .
3 By the continuity of the convolution,
H
RH
2 x ∗ Ex, t −→ R2 x ∗ δ
as t −→ 0.
3.13
Thus,
H
lim RH
2 x ∗ Ex, t R2 x.
t→0
3.14
4 Since
B RH
B RH
∗
Ex,
t
x
2
2 x ∗ Ex, t δ ∗ Ex, t Ex, t,
♦B RH
2 x
∗ Ex, t B B RH
2 x
3.15
∗ Ex, t B δ ∗ Ex, t B Ex, t.
thus
∂
2
H
−
c
B RH
R
♦
∗
Ex,
t
∗
Ex,
t
x
x
B
2
2
∂t
∂
∂
Ex, t − c2 B Ex, t − c2 B Ex, t 0,
∂t
∂t
where Ex, t is defined by 2.35.
3.16
Journal of Applied Mathematics
15
Acknowledgments
The authors would like to thank The Thailand Research Fund and Office of the Higher
Education Commission and Maejo University, Chiang Mai, Thailand, for financial support
and also Professor Amnuay Kananthai, Department of Mathematics, Chiang Mai University,
for the helpful of discussion.
References
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6 A. Saglam, H. Yildirim, and M. Z. Sarikaya, “On the Bessel ultra-hyperbolic heat equation,” Thai
Journal of Mathematics, vol. 8, no. 1, pp. 149–159, 2010.
7 I. A. Kiprijanov, “Fourier-Bessel transforms and imbedding theorems for weight classes,” Trudy Matematicheskogo Instituta imeni V. A. Steklova, vol. 89, pp. 130–213, 1967.
8 B. M. Levitan, “Expansion in Fourier series and integrals with Bessel functions,” Uspekhi Matematicheskikh Nauk, vol. 6, pp. 102–143, 1951 Russian.
9 H. Yildirim and A. Sağlam, “On the generalized Bessel heat equation related to the generalized Bessel
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