Integral Transforms with Non-separated Variables and Discontinuous Coefficients
Multidimensional integral transforms with non-separated variables for discontinuous coefficients problems are constructed in the case where the coefficient discontinuities are on the parallel hyperplanes. Explicit kernel formulas for ideal coupling conditions are obtained. The basic integral transfo...
Gespeichert in:
| Veröffentlicht in: | Журнал математической физики, анализа, геометрии |
|---|---|
| Datum: | 2013 |
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2013
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/106772 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Integral Transforms with Non-separated Variables and Discontinuous Coefficients / O.E. Yaremko // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 4. — С. 594-603. — Бібліогр.: 10 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-106772 |
|---|---|
| record_format |
dspace |
| spelling |
Yaremko, O.E. 2016-10-04T20:06:59Z 2016-10-04T20:06:59Z 2013 Integral Transforms with Non-separated Variables and Discontinuous Coefficients / O.E. Yaremko // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 4. — С. 594-603. — Бібліогр.: 10 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106772 Multidimensional integral transforms with non-separated variables for discontinuous coefficients problems are constructed in the case where the coefficient discontinuities are on the parallel hyperplanes. Explicit kernel formulas for ideal coupling conditions are obtained. The basic integral transform identity is proved. Сконструированы многомерные интегральные преобразования с неразделенными переменными для задач с разрывными коэффициентами в случае, когда разрывы коэффициентов сосредоточены на параллельных гиперплоскостях. Найдены явные формулы для ядер для случая условий идеального сопряжения. Доказано основное тождество интегрального преобразования. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Integral Transforms with Non-separated Variables and Discontinuous Coefficients Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Integral Transforms with Non-separated Variables and Discontinuous Coefficients |
| spellingShingle |
Integral Transforms with Non-separated Variables and Discontinuous Coefficients Yaremko, O.E. |
| title_short |
Integral Transforms with Non-separated Variables and Discontinuous Coefficients |
| title_full |
Integral Transforms with Non-separated Variables and Discontinuous Coefficients |
| title_fullStr |
Integral Transforms with Non-separated Variables and Discontinuous Coefficients |
| title_full_unstemmed |
Integral Transforms with Non-separated Variables and Discontinuous Coefficients |
| title_sort |
integral transforms with non-separated variables and discontinuous coefficients |
| author |
Yaremko, O.E. |
| author_facet |
Yaremko, O.E. |
| publishDate |
2013 |
| language |
English |
| container_title |
Журнал математической физики, анализа, геометрии |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
Multidimensional integral transforms with non-separated variables for discontinuous coefficients problems are constructed in the case where the coefficient discontinuities are on the parallel hyperplanes. Explicit kernel formulas for ideal coupling conditions are obtained. The basic integral transform identity is proved.
Сконструированы многомерные интегральные преобразования с неразделенными переменными для задач с разрывными коэффициентами в случае, когда разрывы коэффициентов сосредоточены на параллельных гиперплоскостях. Найдены явные формулы для ядер для случая условий идеального сопряжения. Доказано основное тождество интегрального преобразования.
|
| issn |
1812-9471 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/106772 |
| citation_txt |
Integral Transforms with Non-separated Variables and Discontinuous Coefficients / O.E. Yaremko // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 4. — С. 594-603. — Бібліогр.: 10 назв. — англ. |
| work_keys_str_mv |
AT yaremkooe integraltransformswithnonseparatedvariablesanddiscontinuouscoefficients |
| first_indexed |
2025-11-25T20:40:19Z |
| last_indexed |
2025-11-25T20:40:19Z |
| _version_ |
1850530674699665408 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2013, vol. 9, No. 4, pp. 594–603
Integral Transforms with Non-separated Variables and
Discontinuous Coefficients
O.E. Yaremko
Penza State University
37, Lermontov Str., Penza 440038, Russia
E-mail: yaremki@mail.ru
Received March 21, 2012, revised January 21, 2013
Multidimensional integral transforms with non-separated variables for
discontinuous coefficients problems are constructed in the case where the
coefficient discontinuities are on the parallel hyperplanes. Explicit kernel
formulas for ideal coupling conditions are obtained. The basic integral trans-
form identity is proved.
Key words: integral transforms, non-separated variables, coupling con-
ditions.
Mathematics Subject Classification 2010: 65R10.
1. Introduction
It is known that the structure of integral transforms, which can be used
to solve the boundary value problems, is determined by the type of differential
equation. A number of transforms have appeared in mathematical literature since
the 70th of the last century in the works by Y.S. Uflyand [1, 2], M.P. Lenuk [3, 4],
L.S. Nayda [4], V.S. Protsenko [5], etc. In particular, the author and I.I. Bavrin
[7] have proposed integral transforms with non-separated variables for solving
multidimensional problems.
Let V ⊂ Rn+1 be the half-space
V =
{
(y1, . . . , yn, x) ∈ Rn+1 : x > 0
}
.
Then the solution of the Dirichlet problem is expressed via the Poisson formula [8],
u(x, y) = Γ
(
n + 1
2
)
π−
n+1
2
∫
y=0
x
[
(y − η)2 + x2
]n+1
2
f(η)dη.
c© O.E. Yaremko, 2013
Integral Transforms with Non-separated Variables
Obviously the Poisson kernel has the form of the Laplace transform
Γ
(
n + 1
2
)
π−
n+1
2
x
[
x2 + (y − η)2
]n+1
2
=
(
1
2π
)n
2
∫ ∞
0
λ
n
2 e−λx
Jn−1
2
(λ |y − η|)
|y − η|n−2
2
dλ,
where Jν is the Bessel function of order ν [8]. Reproducing property of the
Poisson kernel is obtained from the expansion of the function f(y) with respect
to the Laplace operator ∆ eigenfunctions:
f(y) = lim
τ→0
∞∫
0
λ
n
2 e−λτ
1(√
2π
)n
∫
Rn
Jn−2
2
(λ |y − η|)
|y − η|
n−2
2
f (η) dη
dλ.
On the basis of this expansion we can conclude that the integral transforms
with non-separated variables are defined as follows [7]: direct integral Fourier
transform
F [f ] (y, λ) =
1(√
2π
)n
∫
Rn
Jn−2
2
(λ |y − η|)
|y − η|
n−2
2
f (η) dη ≡ f̂ (y, λ) , (1)
inverse Fourier integral transform
F−1[f̂ ](y) = lim
τ→0
∞∫
0
λ
n
2 e−λτ f̂(y;λ)dλ ≡ f(y). (2)
The goal of the paper is to construct multi-dimensional analogues of integral
transforms (1), (2) appropriate for the differential equations with discontinuous
coefficients.
2. One-dimensional Integral Transforms with Discontinuous
Coefficients
In the paper, the integral transforms with discontinuous coefficients are con-
structed in accordance with author’s work [10]. Let ϕ (x, λ) and ϕ∗ (x, λ) be
eigenfunctions of the direct and the dual Sturm–Liouville problems for the Fourier
operator on piecewise-homogeneous axis In,
In =
{
x : x ∈ n+1
U
j=1
(lj−1, lj) , l0 = −∞, ln+1 = ∞, lj < lj+1, j = 1, n
}
.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 595
O.E. Yaremko
The eigenfunction ϕ (x, λ),
ϕ (x, λ) =
n∑
k=2
θ (x− lk−1) θ (lk − x) ϕk (x, λ)
+ θ (l1 − x) ϕ1 (x, λ) + θ (x− ln) ϕn+1 (x, λ) ,
is the solution of the system of separated differential equations
(
a2
m
d2
dx2
+ λ2
)
ϕm (x, λ) = 0, x ∈ (lm, lm+1) ; m = 1, . . . , n + 1,
the coupling conditions
[
αk
m1
d
dx
+ βk
m1
]
ϕk =
[
αk
m2
d
dx
+ βk
m2
]
ϕk+1,
x = lk, k = 1, . . . , n; m = 1, 2,
and the boundary conditions
ϕ1|x=−∞ = 0 , ϕn+1|x=∞ = 0.
Similarly, the eigenfunction ϕ∗ (x, λ),
ϕ∗ (ξ, λ) =
n∑
k=2
θ (ξ − lk−1) θ (lk − ξ) ϕ∗k (ξ, λ)
+ θ (l1 − ξ) ϕ∗1 (ξ, λ) + θ (ξ − ln) ϕ∗n+1 (ξ, λ) ,
is the solution of the system of separated differential equations
(
a2
m
d2
dx2
+ λ2
)
ϕ∗m (x, λ) = 0, x ∈ (lm, lm+1) ; m = 1, . . . , n + 1,
the coupling conditions
1
∆1,k
[
αk
m1
d
dx
+ βk
m1
]
ϕ∗k =
1
∆2,k
[
αk
m2
d
dx
+ βk
m2
]
ϕ∗k+1, x = lk,
where
∆i,k = det
(
αk
1i βk
1i
αk
2i βk
2i
)
k = 1, . . . , n; i,m = 1, 2,
and the boundary conditions
ϕ1|x=−∞ = 0 , ϕn+1|x=∞ = 0.
596 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4
Integral Transforms with Non-separated Variables
We normalize the eigenfunctions as follows:
ϕn+1 (x, λ) = eia−1
n+1xλ. ϕ∗
n+1 (x, λ) = e−ia−1
n+1xλ.
Let Fn and F−1
n be the direct and the inverse Fourier transforms on the
Cartesian axis with n division points defined as (see [10]) :
Fn [f ] (λ) =
n+1∑
m=0
lm∫
lm−1
u∗m (ξ, λ) fm (ξ) dξ ≡ f̂ (λ) , (3)
fk (x) =
1
πi
∞∫
0
uk (x, λ) f̂ (λ) λdλ. (4)
3. The Main Result
We will use the method of delta-like functions [4].
This means that we are looking for the solution of the problem, defined by
the separated matrix systems (n + 1) of parabolic equations
(
∂
∂t
−A2
j
∂2
∂x2
−∆y
)
Uj (t, x, y) = 0, (t, x, y) ∈ D+ ×Rm, j = 1, n + 1 (5)
bounded on the set D ×Rm, D+ = (0,∞)× In, where
In =
{
x : x ∈ n+1
U
j=1
(lj−1, lj) , l0 = −∞, ln+1 = ∞, lj < lj+1, j = 1, n
}
∆y =
∂2
∂y2
1
+ . . . +
∂2
∂y2
m
,
Aj =
(
aj
kl
)
is a positive-definite matrix r × r by the initial conditions
Uj (t, x, y) |t=0 = gj (x, y) , x ∈ In, y ∈ Rm, (6)
by the edge conditions
U1|x=−∞ = 0 , Un+1|x=∞ = 0, (7)
and by the coupling conditions
[
αk
m1
∂
∂x
+ βk
m1
]
Uk =
[
αk
m2
∂
∂x
+ βk
m2
]
Uk+1, (8)
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 597
O.E. Yaremko
x = lk, k = 1, n; m = 1, 2.
Here Uj(t, x, y) is an unknown vector-function, gj(x, y) is a given vector-
function, αk
mi, βk
mi, γ
k
mi, δk
mi are the matrices r × r.
By using the Fourier integral with discontinuous coefficients of Section 2 and
the Fourier integral with non-separated variables (1), (2), we obtain the repre-
sentation for the solution of (3)–(6):
Uk (t, x, y) = − 1
πi
1(√
2π
)m
∫
Rm
n+1∑
j=1
lj∫
lj−1
lim
τ→0
∞∫
0
Jm−2
2
(λ |y − η|)
|y − η|
m−2
2
e−λτλ
m
2 dλ
×
∞∫
−∞
e−β2tϕk (x, β)ϕ∗j (ξ, β) dβ
fj (ξ, η) dξdη, k = 1, n + 1, (9)
where ϕk (x, β) , ϕ∗j (ξ, β) are the eigenfunctions of the direct and the dual Sturm
–Liouville problems, respectively.
We write the integral
∞∫
0
Jm−2
2
(λ |y − η|)
|y − η|
m−2
2
e−λτλ
m
2 dλ
∞∫
−∞
e−β2tϕk (x, β) ϕ∗j (ξ, β) dβ
in the polar coordinates
λ = ρ sinϕ, β = ρ cosϕ, 0 ≤ ρ < ∞, 0 ≤ ϕ ≤ π
to obtain
∞∫
0
ρ
m
2 ρdρ
π∫
0
e−ρ2t cos2 α sin
m
2 α
Jm−2
2
(ρ sinα |y − η|)
|y − η|
m−2
2
e−ρτ sin α
ϕk (x, ρ cosα) ϕ∗j (ξ, ρ cosα) dα.
We carry out the limit τ → 0 in (9) yielding
Uk (t, x, y) = − 1
πi
(
1
2π
)m
2
∫
Rm
n+1∑
j=1
lj∫
lj−1
ρ
m
2 dρ
π∫
0
e−ρ2t cos2 α sin
m
2 α
×
Jm−2
2
(ρ sinα |y − η|)
|y − η|m−2
2
φk (x, ρ cosα) φ∗j (ξ, ρ cosα) dα
)
fj (ξ, η) dξdη.
598 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4
Integral Transforms with Non-separated Variables
If, in addition, we assume that we can carry out the limit t → 0 in the expan-
sion of the eigenfunctions for multidimensional direct Sturm–Liouville problem
fk (x, y) of Sec. 2, we obtain
fk (x, y) = − 1
πi
1(√
2π
)m
∫
Rm
n+1∑
j=1
lj∫
lj−1
∞∫
0
ρ
m
2 ρdρ
π∫
0
si n
m
2 α
Jm−2
2
(ρ sinα |y − η|)
|y − η|
m−2
2
×ϕk(x, ρ cosα)ϕ∗j (ξ, ρ cosα)dα
)
fj (ξ, η) dξdη. (10)
Let us denote
ϕk,j ≡ ϕk,j (ρ, x, ξ, |y − η|) =
π∫
0
sin
m
2 α
Jm−2
2
(ρ sinα |y − η|)
|y − η|
m−2
2
×ϕk (x, ρ cosα) ϕ∗j (ξ, ρ cosα) dα.
It is clear then that formula (10) can be written as
fk (x, y) =
1
π
∞∫
0
ρ
m+1
2 dρ
1(√
2π
)m
∫
Rm
n+1∑
j=1
lj∫
lj−1
ϕk,jfj (ξ, η) dξdη. (11)
Equation (11) allows us to write down the direct and the inverse multidimensional
Fourier transforms with discontinuities on the planes x = lk:
Fn [f ] (x, y, λ) =
1(√
2π
)m
∫
Rm
n+1∑
j=1
lj∫
lj−1
ϕk,j (λ, x, ξ, |y − η|) fj (ξ, η) dξdη, (12)
f(x, y) =
∞∫
0
λ
m
2
+1Fn [f ] (x, y, λ) dλ. (13)
Now we can prove the basic integral identity for differential operator
B = θ (l1 − xt)
(
A2
1
d2
dx2
+ ∆y
)
+
n∑
k=1
θ (x− lk−1) θ (lk − x)
(
A2
k
d2
dx2
+ ∆y
)
+ θ (x− ln)
(
A2
n+1
d2
dx2
+ ∆y
)
.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 599
O.E. Yaremko
Theorem 1. Let
f (x, y) = θ (l1 − x) f1 (x, y) +
n∑
k=2
θ (x− lk−1) θ (lk − x) fk (x, y)
+ θ (x− ln) fn+1 (x, y)
be a twice continuously differentiable on D+ ×Rm vector-function, in which
fn+1(x, y),
∂fn+1(x, y)
∂x
vanishes as x → +∞ and y is fixed,
f1(x, y),
∂f1(x, y)
∂x
vanishes as x → −∞ and y is fixed,
fi(x, y),
∂fi(x, y)
∂yj
vanishes as yj → ±∞ and x, y1, y2, . . . , yj−1, yj+1, . . . , ym are fixed.
Assume also that the coupling conditions (8) are valid.
Then the following holds true:
Fn [B (f)] = −λ2Fn [f ] .
P r o o f. Integrate twice by parts with respect to each of the variables on the
left, taking into account the conditions of the theorem. As a result, the operator
B acts on the kernel:
Fn [B(f)] (x, y, λ) =
1(√
2π
)m
∫
Rm
n+1∑
j=1
lj∫
lj−1
Bj [ϕk,j (λ, x, ξ, |y − η|)] fj (ξ, η) dξdη.
Let us prove the equality Bj [ϕk,j ] = −λ2ϕk,j . We have
Bj [ϕk,j ] =
π∫
0
sin
m
2 α∆η
Jm−2
2
(ρ sinα |y − η|)
|y − η|
m−2
2
×ϕk (x, ρ cosα)
(
ϕ∗j (ξ, ρ cosα)
)
dα
+
π∫
0
sin
m
2 α
Jm−2
2
(ρ sinα |y − η|)
|y − η|
m−2
2
600 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4
Integral Transforms with Non-separated Variables
×ϕk (x, ρ cosα) a2
j
∂2
∂ξ2
(
ϕ∗j (ξ, ρ cosα)
)
dα
= −ρ2 sin2 α
π∫
0
sin
m
2 α
Jm−2
2
(ρ sinα |y − η|)
|y − η|
m−2
2
×ϕk (x, ρ cosα)
(
ϕ∗j (ξ, ρ cosα)
)
dα
−ρ2 cos2 α
π∫
0
sin
m
2 α
Jm−2
2
(ρ sinα |y − η|)
|y − η|
m−2
2
×ϕk (x, ρ cosα)
(
ϕ∗j (ξ, ρ cosα)
)
dα = −ρ2ϕk,j .
We have used above that ϕ∗j (ξ, ρ cosα) are the eigenfunctions of the dual Sturm–
Liouville problems and the relation
∆η
Jm−2
2
(ρ sinα |y − η|)
|y − η|
m−2
2
= −ρ2 sin2 α
Jm−2
2
(ρ sinα |y − η|)
|y − η|
m−2
2
.
By using the basic identity [9], we conclude
ρ
m
2 Jm−2
2
(ρ |y|)
|y|
m−2
2
=
1
(2π)
m
2
∫
Sρ
ei<y,ξ>dSρ.
This completes the proof.
The above formulas for the direct and the inverse Fourier transforms with
non-separated variables are significantly simpler in the case of ideal coupling
conditions on one surface. This case is widely known in engineering practice.
Consider, for the sake of simplicity, the scalar case, assuming that the ideal
coupling conditions are in the plane x = 0,
ϕ1 (x, y) = ϕ2 (x, y) , x = 0, y ∈ Rm;
ϕ′1x (x, y) = νϕ′2x (x, y) , x = 0, y ∈ Rm; ν =
λ2
λ1
.
The one-dimensional components of eigenfunctions are given in [4]:
ϕ1 (x, λ) =
(
cosλ
x
a1
+ i
1√
δ0
sinλ
x
a1
)
(1 + δ0) ;
ϕ2 (x, λ) =
(
cosλ
x
a2
+ i
√
δ0 sinλ
x
a2
)
(1 + δ0) ;
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 601
O.E. Yaremko
ϕ∗k (x, λ) = rkϕk (x, λ), k = 1, 2, r1 =
a2
ν0a2
1
, r2 =
1
a2
, δ0 =
a2
ν0a1
.
The multidimensional components of eigenfunctions with non- separated variables
ϕkj have the form:
ϕ11 =
1 + δ0
a1
Jm−1
2
(
ρ
√
(x−ξ)2
a2
1
+ |y − η|2
)
(
(x−ξ)2
a2
1
+ |y − η|2
)m−1
2
−1− δ0
a1
Jm−1
2
(
ρ
√
(x+ξ)2
a2
1
+ |y − η|2
)
(
(x+ξ)2
a2
1
+ |y − η|2
)m−1
2
,
ϕ12 =
1 + δ0
a2
√
δ0
Jm−1
2
(
ρ
√(
x
a2
− ξ
a1
)2
+ |y − η|2
)
((
x
a2
− ξ
a1
)2
+ |y − η|2
)m−1
2
+
1− δ0
a2
√
δ0
Jm−1
2
(
ρ
√(
x
a2
+ ξ
a1
)2
+ |y − η|2
)
((
x
a2
+ ξ
a1
)2
+ |y − η|2
)m−1
2
,
ϕ21 =
√
δ0
1 + δ0
a1
Jm−1
2
(
ρ
√(
x
a1
− ξ
a2
)2
+ |y − η|2
)
((
x
a1
− ξ
a2
)2
+ |y − η|2
)m−1
2
+
√
δ0
1− δ0
a1
Jm−1
2
(
ρ
√(
x
a1
+ ξ
a2
)2
+ |y − η|2
)
((
x
a1
+ ξ
a2
)2
+ |y − η|2
)m−1
2
,
ϕ22 =
1 + δ0
a2δ0
Jm−1
2
(
ρ
√
(x−ξ)2
a2
2
+ |y − η|2
)
(
(x−ξ)2
a2
2
+ |y − η|2
)m−1
2
−1− δ0
a2δ0
Jm−1
2
(
ρ
√
(x+ξ)2
a2
2
+ |y − η|2
)
(
(x+ξ)2
a2
2
+ |y − η|2
)m−1
2
.
Now the integral transforms given by formulas (12), (13) are constructed.
602 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4
Integral Transforms with Non-separated Variables
4. Conclusion
Let us remark that integral transforms (12), (13) can be used to solve problems
of mathematical physics by using the standard algorithm: find the solution in the
images then return to the originals. An advantage of our formulas is that they
involve just one spectral parameter contained in the final formulas while the
integral transforms with separated variables contain m parameters.
References
[1] Ja.S. Ufljand, Integral Transforms for the Elasticity Theory Problems. Nauka,
Leningrad, 1967. (Russian)
[2] Ja.S. Ufljand, Dual Equations Method in of Mathematical Physics Problems. Nauka,
Leningrad, 1977. (Russian)
[3] M.P. Lenjuk, Integral Hybrid Transforms (Bessel, Legendre). — Ukr. Mat. Zh. 43
(1991), No. 6, 770–779. (Russian)
[4] M.P. Lenjuk, Fourier Integral Transform in Piecewise Homogeneous Half-Line. —
Izv. vuzov. Matematika 4 (1989), 14–18. (Russian)
[5] L.S Najda, Hankel–Legendre Type Integral Hybrid Transforms. — Math. Meth.
Dynamic Systems Anal. Kharkov, 8 (1984), 132–135. (Russian)
[6] V.S. Procenko and A.I. Solov’jov, Some Integral Hybrid Transforms and their Appli-
cations to the Elasticity Theory by Heterogeneous Media. — Prikladnaja mehanika
13 (1982), No. 1, 62–67. (Russian)
[7] I.I. Bavrin and O.E. Yaremko, Integral Fourier Transform on Compact Subset of
RN and their Applications to the Moment Problem. — Pleiades Publishing, Ltd.
Doclady mathematics 374 (2000), No. 2, 177–179.
[8] A.F. Nikiforov and V.B. Uvarov, Special Functions of Mathematical Physics: A Uni-
fied Introduction with Applications. Publisher: Birkhauser, Boton, 1988.
[9] I.N. Sneddon, Fourier Transforms. McGraw-Hill, New York, 1951.
[10] O.E. Yaremko, Matrix Integral Fourier Transform for Problems with Discontinuous
Coefficients and Transformation Operators. — Pleiades Publishing, Ltd. Doclady
mathematics 76 (2007), No. 3, 876–878.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 603
|