Journal ArticleDOI

# A class of Fredholm equations and systems of equations related to the Kontorovich-Lebedev and the Fourier integral transforms

01 May 2020-Turkish Journal of Mathematics (The Scientific and Technological Research Council of Turkey)-Vol. 44, Iss: 3, pp 643-655
TL;DR: In this article, the Kontorovich-Lebedev, Fourier sine, and Fourier cosine integral transforms are solved in closed form for a class of Fredholm integral equations with non-degenerate kernels.
Abstract: In this article, we solve in closed form a class of Fredholm integral equations and systems of Fredholm integral equations with nondegenerate kernels by using techniques of convolutions and generalized convolutions related to the Kontorovich-Lebedev, Fourier sine, and Fourier cosine integral transforms.

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Turk J Math
(2020) 44: 643 655
doi:10.3906/mat-2001-24
Turkish Journal of Mathematics
http://journals.tubitak.gov.tr/math/
Research Article
A class of Fredholm equations and systems of equations related to the
Kontorovich-Lebedev and the Fourier integral transforms
Trinh TUAN
1,
, Nguyen Thanh HONG
2
1
Department of Mathematics, Electric Power University, Hanoi, Vietnam
2
School for Gifted Students, Hanoi National University of Education, Hanoi, Vietnam
Received: 09.01.2020 Accepted/Published Online: 04.03.2020 Final Version: 08.05.2020
Abstract: In this article, we solve in closed form a class of Fredholm integral equations and systems of Fredholm
integral equations with nondegenerate kernels by using techniques of convolutions and generalized convolutions related
to the Kontorovich-Lebedev, Fourier sine, and Fourier cosine integral transforms.
Key words: Convolution, Fourier sine, Fourier cosine, Kontorovich–Lebedev, transform, Fredholm equation
1. Introduction
Fredholm integral equation is of the form (see [4])
α(x)f(x) + λ
Z
Γ
k(x, τ)f (τ ) = g(x), (1.1)
where α(x), g(x) are given functions, f is unknown function, λ C , k(x, τ ) is a kernel, and the integral is on
the curve Γ, or in general case Γ = Γ
0
+ Γ
1
+ ... + Γ
n
, Γ
i
is collection of curve.
The Fredholm integral equation was studied rst by I. Fredholm, and further developed by Riesz [3].
In the last two decades, the theory of abstract Volterra and Fredholm integral equation has undergone rapid
development. To a large extent this was due to the applications of this theory to problems in mathematical
physics, such as viscoelasticity, heat conduction in materials with memory, electrodynamics with memory, and
to the need of tools to tackle the problems arising in these elds. Many interesting phenomena are not found
with dierential equations but observed in specic examples of integral equations (see [4, 5]).
However, the equation (1.1) is only solved by a method for approximating solutions. The equation (1.1)
was only solved in closed form for some classes of degenerate kernels. The solution in closed form of the equation
(1.1) in general case is still open.
In recent years, the equation (1.1) with Γ = [0, T ] and the kernel is a periodic function that has been
studied by many authors (see [1] and references therein). In [811, 1315], the authors studied the equation
(1.1) in case Γ = (0, +) and kernel is of the Toeplitz plus Hankel type.
In this paper, we attempt to solve in closed form a class of Fredholm integral equations and systems of
Fredholm integral equations with nondegenerate kernels and Γ = (0, +) by using techniques of convolutions
Correspondence: tuantrinhpsac@yahoo.com
2010 AMS Mathematics Subject Classication: 33C10, 44A35, 45E10, 45J05.
643

TUAN and HONG/Turk J Math
and generalized convolutions related to Kontorovich–Lebedev and Fourier sine, Fourier cosine integral transform.
These convolutions and inverse formulas were studied in [6, 12, 1517].
The main results of the article are presented in Sections 3 and 4. In Section 3, we obtain the closed
form of solutions of the equation (1.1) in the case of Γ = (0, +), α(x) = 0, λ = 1, and the kernel
k(x, τ) = k
1
(x, τ ) + k
2
(x, τ ) is certain nondegenerate kernel (3.1). In Section 4, we solve in closed form a
class systems of Fredholm integral equations (4.1). The results obtained in Theorems 3.1, 3.2, 4.1, and 4.2 give
explicit formulas of solutions which contain of integral transformations such as Kontorovich–Lebedev, Fourier
sine, Fourier cosine. Each equation of system (4.1) is of the form (1.1), with Γ = (0, +), α(x) = 1, λ = 1, and
the kernels k
3
, k
4
are nondegenerated.
The key tool in proofs of results in papers [810, 14, 15] is Wiener–Levy theorem. However, we prove
Theorems 3.1, 3.2, 4.1 and 4.2 by a method without using Wiener–Levy theorem. Tuan et al. [13] studied the
equation (1.1) with a kernel of the generalized convolution. It is harder to make function spaces of solution for
the equation in Theorem 3.1, 3.2 if compared with results in [13].
2. Related integral transforms and function spaces
In this section, we recall several integral transform and function spaces which are usefull throughout this article.
The Fourier cosine transform (F
c
) and its inverse formula (F
1
c
) in L
1
(R
+
) are dened by (see [2])
(F
c
f)(y) =
r
2
π
Z
+
0
f(x) cos(xy)dx, y > 0, (2.1)
and
f(x) =
r
2
π
Z
+
0
(F
c
f)(y) cos(xy)dy, x > 0. (2.2)
The Fourier sine transform (F
s
) and its inverse formula (F
1
s
) in L
1
(R
+
) are dened by (see [2])
(F
s
f)(y) =
r
2
π
Z
+
0
f(x) sin(xy)dx, y > 0, (2.3)
and
f(x) =
r
2
π
Z
+
0
(F
s
f)(y) sin(xy)dy, x > 0. (2.4)
The Kontorovich–Lebedev integral transform (K ) in L
1
(R
+
) is of the form (see [12])
(Kf)(y) =
Z
+
0
K
iy
(x)f(x)dx, y > 0, (2.5)
here, K
iy
(x) is the Macdonald function (see [12])
K
iy
(x) =
Z
+
0
e
x cosh u
cos(yu)dy, y 0, x > 0.
The inverse Kontorovich–Lebedev transform (K
1
) is of the form
(K
1
f)(x) = f (x) =
2
π
2
x
Z
+
0
y sinh(πy)K
iy
(x)(Kf)(y)dx, y > 0. (2.6)
644

TUAN and HONG/Turk J Math
In the space L
2
(R
+
), the integral transforms (F
c
), (F
s
) are of the form
F
{
c
s
}
f
(y) = lim
N→∞
Z
N
0
f(x)
cos xy
sin xy
dx, y > 0. (2.7)
The inverse transforms of the (F
c
) và (F
s
) are dened by formulas (2.2) and (2.4) respectively.
In the space L
2
(R
+
, x
α
), α R , the Kontorovich–Lebedev integral transform (K) is of the form
(Kf)(y) = lim
N→∞
Z
N
1
N
K
iy
(x)f(x)dx, y > 0. (2.8)
The inverse transform is dened by formula (2.6).
The function space L
p
(R
+
, γ) with the weight function γ is dened as follow (see [2])
L
p
(R
+
, γ) =
(
f :
Z
+
0
γ(x)|f (x)|
p
dx
1
p
<
)
, 1 p < .
Note that
L
p
(R
+
) L
p
(R
+
, γ).
The function space L
α,β
p
is introduced in a research of Yakubovich and Britvina [16]
L
α,β
p
L
p
(R
+
, K
0
(βx)x
α
), α R, 0 < β 1.
This function spaces is equived the norm
||f||
L
α,β
p
=
Z
+
0
|f(x)|
p
K
0
(βx)x
α
dx
1
p
< +.
Note that L
p
(R
+
) L
α,β
p
, L
α,1
1
L
1
(R
+
, K
0
(x)x
α
), and
K
0
(x) =
Z
+
0
e
x cosh u
du, x > 0.
3. A class of Fredholm’s type integral equations
In this section, we solve in close form a class of integral equations of Fredholm’s type with nondegenerate kernel
k(x, τ) = k
1
(x, τ ) + k
2
(x, τ ) with the help of convolutions and generalized convolutions techniques. Namely, we
use the convolutions and generalized convolutions related to the Kontorovich–Lebedev (K ), Fourier sine (F
s
),
Fourier cosine (F
c
) transforms (see [12, 16, 17]) and their inverses.
Consider the following problem:
Z
+
0
[k
1
(x, τ ) + k
2
(x, τ )]f(τ ) = g(x), x > 0, (3.1)
645

TUAN and HONG/Turk J Math
where
k
1
(x, τ ) =
1
2π
Z
+
0
1
x
[e
x cosh(τ +θ)
e
x cosh(τ θ)
]h(θ)dθ, x, τ > 0.
k
2
(x, τ ) =
1
2π
2
Z
+
0
[sinh(τ + θ)e
x cosh(τ +θ)
+ sinh(τ θ)e
x cosh(τ θ)
]φ(θ)dθ, x, τ > 0,
(3.2)
where h, φ are given functions in L
2
(R
+
), g L
2
(R
+
, x
α
) (α R ) is given function, and f is unknown
function.
Theorem 3.1 Suppose that
π
2
(F
s
h)(y) + y(F
c
φ) = 0, y > 0 such that
y sinh(πy)(Kg)(y)
π
2
(F
s
h)(y) + y(F
c
φ)(y)
L
2
(R
+
),
then the equation (3.1) has the solution in L
2
(R
+
), which is of the form
f(x) =
r
2
π
Z
+
0
y sinh(πy)(Kg)(y)
π
2
(F
s
h)(y) + y(F
c
φ)(y)
sin(xy)dy, x > 0.
Proof In order to prove this theorem, we use the generalized convolution for the Kontorovich–Lebedev, Fourier
sine transforms which is dened by (see [17]):
(f
γ
1
1
h)(x) :=
1
2π
Z
R
+
2
1
x
[e
x cosh(τ +θ)
e
x cosh(τ θ)
]f(τ )h(θ) , x > 0, (3.3)
where γ
1
(y) =
1
y sinh(πy)
.
For f, h L
2
(R
+
), the generalized convolution (f
γ
1
1
h) L
2
(R
+
, x
α
) and satises the following
factorization equality:
K(f
γ
1
1
h)(y) =
π
2
γ
1
(y)(F
s
f)(y)(F
s
h)(y), y > 0. (3.4)
The generalized convolution for the Kontorovich–Lebedev, Fourier transforms is dened by (see [12])
(f
γ
2
2
h)(x) :=
1
2π
2
Z
R
+
2
[sinh(τ + θ)e
x cosh(τ +θ)
+ sinh(τ θ)e
x cosh(τ θ)
]f(τ )h(θ) , x > 0, (3.5)
where γ
2
(y) =
1
sinh(πy)
.
For f, h L
2
(R
+
), the generalized convolution (f
γ
2
2
h) L
2
(R
+
, x
α
) , α R and satises the following
factorization equality
K(f
γ
2
2
h)(y) = γ
2
(y)(F
c
f)(y)(F
s
h)(y), y > 0. (3.6)
646

TUAN and HONG/Turk J Math
When the kernels k
1
, k
2
are dened by formula (3.2), the equation (3.1) can be rewritten in the dual convolutions
equation as follow:
(f
γ
1
1
h)(x) + (φ
γ
2
2
f)(x) = g(x), x > 0, (3.7)
where f is unknown function, γ
1
(y) =
1
y sinh(πy)
, γ
2
(y) =
1
sinh(πy)
, h, φ are given functions in L
2
(R
+
), g is
given function in L
2
(R
+
, x
α
), α R.
Applying the (K) transform on both sides of equation (3.7), with the help of factorization equalities
(3.4), (3.6), we have
K(f
γ
1
1
h)(y) + K(φ
γ
2
2
f)(y) = (Kg)(y), y > 0.
This implies that
π
2
1
y sinh(πy)
(F
s
f)(y)(F
s
h)(y) +
1
sinh(πy)
(F
c
φ)(y)(F
s
f)(y) = (Kg)(y), y > 0,
or equivalent,
(F
s
f)(y) =
y sinh(πy)(Kg)(y)
π
2
(F
s
h)(y) + y(F
c
φ)(y)
L
2
(R
+
).
Thanks to the inverse formular of Fourier sine transform (2.4), we have the solution in L
2
(R
+
) as follows
f(x) =
r
2
π
Z
+
0
y sinh(πy)(Kg)(y)
π
2
(F
s
h)(y) + y(F
c
φ)(y)
sin(xy)dy, x > 0. (3.8)
2
Example 1. We use Theorem 3.1 to solve in closed form in the case
φ(x) = h(x) =
r
2
π
e
x
L
2
(R
+
),
g(x) =
e
x
π
2
x
r
fπx
2
πxe
2x
erfe(
2x)
!
L
2
(R
+
, x
α
), α R,
then,
(F
s
h)(y) =
2
π
y
1 + y
2
, (F
c
φ)(y) =
1
1 + y
2
.
Therefore,
π
2
(
F
s
h
)(
y
) +
y
(
F
c
φ
)(
y
) = (
2
π
+ 1)
y
1 + y
2
= 0
, y >
0
.
Thanks to formular (2.16.48.8) in [7], we have (Kg)(y) =
y
sinh(πy) cosh(πy)
. Thus,
y sinh(πy)(Kg)(y)
π
2
(F
s
h)(y) + y(F
c
φ)(y)
=
y(1 + y
2
)
(
2
π
+ 1) cosh(πy)
L
2
(R
+
), y > 0.
647

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29 Sep 2004
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• ...1) with Γ = [0, T ] and the kernel is a periodic function that has been studied by many authors (see [1] and references therein)....

[...]

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### "A class of Fredholm equations and s..." refers background or methods in this paper

• ...Proof In order to prove this theorem, we use the generalized convolution for the Kontorovich–Lebedev, Fourier sine transforms which is defined by (see [17]):...

[...]

• ...Namely, we use the convolutions and generalized convolutions related to the Kontorovich–Lebedev (K ), Fourier sine (Fs ), Fourier cosine (Fc ) transforms (see [12, 16, 17]) and their inverses....

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• ...1), we will use the generalized convolution with the weight function γ1(y) for the Kontorovich–Lebedev and the Fourier sine transform ( f γ1 ∗ 1 h ) (see [17]) defined by (3....

[...]

• ...These convolutions, generalized convolution were studied in [6, 15, 17]....

[...]

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• ...[17] Yakubovich SB, Britvina LE....

[...]

• ...[16] Yakubovich SB, Britvina LE....

[...]

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• ...The function space Lα,βp is introduced in a research of Yakubovich and Britvina [16] Lα,βp ≡ Lp(R+,K0(βx)xα), α ∈ R, 0 β ≤ 1....

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