Applicable Analysis and Discrete Mathematics SPECTRUM OF THE ...

Applicable Analysis and Discrete Mathematics - Accepted manuscript
Received: December 01, 2011
Revised: Jul 20, 2012
Please cite this article as: Hichem Ben Aouicha and Tahar Moumni: ON THE SPECTRUM OF THE FINITE
LAPLACE TRANSFORM WITH SOME APPLICATIONS,
Appl. Anal. Discrete Math. (to appear)
DOI: 10.2298/AADM120815019B
Applicable Analysis and Discrete Mathematics
available online at http://pefmath.etf.rs
Appl. Anal. Discrete Math. x (xxxx), xxx–xxx.
doi:10.2298/AADMxxxxxxxx
SPECTRUM OF THE FINITE LAPLACE TRANSFORM
AND APPLICATION
Hichem Ben Aouicha and Tahar Moumni
This work is devoted to the computation of the spectrum of the finite Laplace
transform (FLT) and giving some of its applications. For this purpose, we
give two different practical methods. The first one uses a discretization of
the FLT. The second one is based on the Gaussian quadrature method. The
spectrum of the FLT is then used to invert the Laplace transform of time
limited functions as well as the Laplace transform of essentially time limited
functions. Several numerical results are given to illustrate the results of this
work.
1. INTRODUCTION
The finite Laplace transform (FLT) defined as
L 0,b f(x) =
∫ b
0
e −xy f(y)dy, b > 0,
plays an important role in solving boundary value problems for ordinary and partial
differential equations, see [10]. It is used also to solve a weakly singular integral
equation occurring in transfer theory, see [18]. Several applications of this transform
to linear control problems have been studied in [15]. The finite Laplace
transform was studied first by Debnath and Thomas in [11]. They have given some
properties of the FLT. Some other properties can be found in [20]. In [4], the
authors have considered the following integral transform
(1) L a,b f(x) =
∫ b
a
e −xy f(y)dy,
2010 Mathematics Subject Classification. 45P05; 44A10; 45C05; 33C45.
Keywords and Phrases. Integral operator; Laplace transform; Eigenvalue problem, Orthogonal
polynomials.
1

2 Hichem Ben Aouicha and Tahar Moumni
from L 2 [a, b] to L 2 ([0, +∞[) which is also called the finite Laplace transform. Here
0 < a < b are two positive real numbers. They have proved that the differential
operator ˜D given by
˜Df = − ( (x 2 − 1)(α 2 − x 2 )f ′) ′
+ 2
(
x 2 − 1 ) f
commutes with the Stieltjes transform
S α f(x) = (L a,b ) ∗ L a,b f(x) =
∫ α
1
f(y)
x + y dy,
where α = b a
. In [3], the authors have studied the spectral properties of ˜D.
Note here that the parameter a, introduced in (1) cannot be equal to 0. In this
paper, we complete the work given in [3] and we suppose that a = 0 and we study
the spectrum of the operator L 0,b defined from L 2 [0, b] into itself and we give some
of its applications. The eigenfunctions of FLT and their corresponding eigenvalues
are computed by two methods. The first one is based on the discretization of L 0,b
following a suitable set of orthogonal polynomials while the second method is based
on the use of a Gaussian quadrature method. Finally, we use such eigenfunctions
and eigenvalues to invert the finite Laplace transform as well as the Laplace transform
over the set of essentially time limited functions. In the sequel, the Laplace
transform is given by
Lf(x) =
∫ +∞
0
e −xy f(y)dy,
see [10]. Note here that a large number of different methods for the inversion of
the Laplace transform was found in the literature, see for example the extensive
list of papers collected in [16, 17]. A comparison of methods of inversion of the
Laplace transform is given in a survey of Davies and Martin in [5].
The outline of this paper is as follows. In section 2, we provide the reader with two
methods of computation of the eigenfunctions of the finite Laplace transform and
their corresponding eigenvalues. The first method is based on the discretization of
L 0,b . The second method is based on a Gaussian quadrature method. Section 3 is
devoted to some applications of the results given in section 2. The first application
is the inversion of the finite Laplace transform while the second one is the inversion
of the Laplace transform of functions essentially time limited to [0, b]. In section 4,
we illustrate such methods by several numerical results.
2. ON THE COMPUTATION OF THE SPECTRUM OF THE
FINITE LAPLACE TRANSFORM OPERATOR
It is easy to see that L 0,b observed as a map from L 2 [0, b] ti itself is a Hilbert-
Schmidt operator. Moreover, if ∥L 0,b ∥ HS
denotes the Hilbert-Schmidt norm of L 0,b ;
one can check that :
( ∫ b ∫ 1/2 b
∥L 0,b ∥ HS
= e dxdy) −2xy ≤ b.
0
0

SPECTRUM OF THE FINITE LAPLACE TRANSFORM AND APPLICATION 3
Let (µ n ) n≥0 denote the infinite set of the eigenvalues of L 0,b , arranged in the
decreasing order of their magnitude, that is
|µ 0 | > |µ 1 | > .... > |µ n | > ...
In the sequel, we denote by φ n the n th eigenfunction of L 0,b associated with µ n .
That is
(2) L 0,b (φ n )(x) =
∫ b
and we adopt the following normalization of φ n
0
e −xy φ n (y)dy = µ n φ n (x),
( ∫ 1/2 b
∥φ n χ [0,b] ∥ 2 = |φ n (x)| dx) 2 = |µ n |.
0
Note here that both φ n and µ n depend on b, but for simplicity of notation we
have omitted this dependence. In accord to previous statements, we can state the
following result about the eigenfunctions of L 0,b .
Proposition 1. The set B = {φ n ,
n ∈ N} is an orthogonal basis of L 2 [0, b].
The following lemma gives the derivative of µ n with respect to b.
Lemma 1. Let ψ n be the function given by ψ n (x) = φ n (bx). Then
(3)
∂µ n
∂b = (ψ n(1)) 2
µ n
.
Proof. For the proof of the previous lemma, we adopt the techniques used in [19]
to prove a similar result for the eigenvalue of the finite Hankel transform. Let us
introduce the following changes in the integral equation (2):
y = bu, x = bv, ψ n (x) = φ n (bx),
then we obtain the equivalent integral equation
(4) b
∫ 1
0
e −b2 uv ψ n (u)du = µ n ψ n (v).
Let us now differentiate both member of the previous equality with respect to b,
one obtains
(5)
∫ 1
0
e −b2 uv ψ n (u)du − 2b 2 v
∂µ n
∂b ψ ∂ψ n (v)
n(v) + µ n .
∂b
∫ 1
0
ue −b2 uv ψ n (u)du + b
∫ 1
0
e −b2 uv ∂ψ n(u)
du =
∂b

4 Hichem Ben Aouicha and Tahar Moumni
Differentiating (4) with respect to v, one gets
(6) −b 3 ∫ 1
0
ue −b2uv ∂ψ n (v)
ψ n (u)du = µ n .
∂v
Combining (5) and (6) and we use (4) to obtain
(7)
µ n
b ψ n(v) + 2 µ n
b v ∂ψ n(v)
+ b
∂v
∫ 1
0
e −b2 uv ∂ψ n(u)
du = ∂µ n
∂b ∂b ψ ∂ψ n (v)
n(v) + µ n .
∂b
Multiplying both sides of (7) by ψ n (v) and integrate over (0, 1), one finds
µ n
b ∥ψ nχ (0,1) ∥ 2 2 + 2 µ n
b
∫ 1
∫
∂µ 1
n
∂b ∥ψ nχ (0,1) ∥ 2 2 + µ n
0
0
vψ n (v) ∂ψ n(v)
dv + b
∂v
ψ n (v) ∂ψ n(v)
dv.
∂b
∫ 1
0
ψ n (v)
∫ 1
0
e −b2 uv ∂ψ n(u)
dudv =
∂b
By using Fubini’s theorem and (4), the equality (8) can be simply written as follows
(8)
where I =
∫ 1
0
µ n
b ∥ψ nχ (0,1) ∥ 2 2 + 2 µ n
b I = ∂µ n
∂b ∥ψ nχ (0,1) ∥ 2 2,
vψ n (v) ∂ψ n(v)
dv. Integrating I by parts one can easily check that
∂v
(9) I = 1 2
(
(ψn (1)) 2 − ∥ψ n χ (0,1) ∥ 2 )
2 .
Remark also that ∥ψ n χ (0,1) ∥ 2 2 = (µ n) 2
. Thanks to (8) and (9) and the normalization
b
of ψ n one obtains (3).
To proceed further and present our computational methods of (φ n ) n and their
corresponding eigenvalues, we introduce the following results, see for example [9].
Let (P k ) k≥0
be the set of polynomials given by :
√
2k + 1 d k [
(10) P k (x) = √
b
2k+1
k! dx k x k (x − b) k] , k ≥ 0.
The following properties of {P k , k ∈ N} will be used in the sequel:
P 1 : The set {P k , k ∈ N} is an orthonomal basis of L 2 (0, b) .
P 2 :
For all k ∈ N we have
P k+1 (x) = (α k x + β k )P k (x) − γ k P k−1 (x),
where
(11)
α k = 2 √ √
(2k + 3)(2k + 1)
(2k + 3)(2k + 1)
, β k =
b k + 1
k + 1
and γ k = k
√
2k + 3
k + 1 2k − 1 .

SPECTRUM OF THE FINITE LAPLACE TRANSFORM AND APPLICATION 5
Note that from the theory of orthogonal polynomials, one concludes that
∀ n ≥ 0, P n (x) has n distinct zeros inside [0, b]. Moreover, these n different zeros
are simply given as the eigenvalues of a tridiagonal symmetric matrix D of order
n, given by
(12)
D = [d i,j ] 1≤i,j≤n
, d i,i = − β i−1
, d i,i+1 = d i+1,i = −1 , d i,j = 0 if j ≠ i−1, i, i+1,
α i−1 α i−1
where the α i and β i are given by (11).
2.1. MATRIX REPRESENTATION OF L 0,b
In this section, we follow the techniques introduced in [8, 9] to compute the
spectrum of the finite Fourier transform. For the completeness of the paper, we
describe briefly such techniques. Firstly, we introduce the finite moments
M ij =
∫ b
0
x i P j (x)dx, i, j ∈ N,
of the polynomials given by (10). With similar method introduced in [8], we can
easily get :
(m 1 ) For i < j, M ij = 0.
(m 2 ) For j ≤ i,
M ij = bi+1/2√ 2j + 1 (i!) 2
.
(i + j)! (i − j)!
(m 3 ) For j ≤ i,
(13) |M ij | ≤ b2j+1
√ 2j + 1
.
To proceed further, we first expand φ n into its Fourier series following the polynomials
P k
(14) φ n (x) =
where
(15) η n k =
+∞∑
k=0
∫ b
0
η n k P k (x),
φ n (x)P k (x)dx.
x ∈ [0, b],
As in [8], the following lemma gives the decay rate of the coefficients ηk
n
(14).
given in

6 Hichem Ben Aouicha and Tahar Moumni
Lemma 2. Under the previous notations and assumptions, for any positive integer
k ≥ [2eb 2 ], we have the following upper bound of η n k :
(16) |η n k | ≤ beb2 √
2kπ
1
2 k .
Here [x] denotes the integer part of x.
Proof. By combining (15) and (2) together with (13) and Hölder’s inequality, we
obtain
∫ (
|ηk n 1 b ∫ )
b
| =
e −xy φ
∣
n (y)dy P k (x) dx
µ n 0 0
∣
⎛
⎞
∫
1
b ∫ b ∑
=
⎝
(−1) j
|µ n |
x j y j φ n (y)dy⎠ P k (x) dx
∣ 0 0 j!
j≥0
∣
1 ∑ 1
∫ b
∫ b
≤
y j φ
|µ n | j! ∣ n (y)dy
x j P
j≥0
0
∣ ∣ k (x) dx
0
∣
≤
≤
(
1 ∑
∫ 1/2
1 b 2j+1 b
√ y dy) 2j ∥φ n χ [0,b] ∥ 2
|µ n | j! 2j + 1
∑
j≥k
j≥k
b 2j
j!
b2k b2
≤ be
k! .
As a consequence of the following Stirling’s formula, see [1]
one can deduce the inequality (16).
Γ(s + 1) ≥ √ 2πs s+1/2 e −s , ∀ s > 0,
To proceed further, we need the following lemma
Lemma 3. Let k; l ≥ 0 be two integers and let b be a positive real number.
The coefficients a kl = ⟨L 0,b (P k ), P l ⟩ satisfy the following two conditions :
(c 1 ) a kl = ∑ (−1) n
M nl M nk .
n!
n≥ν
0
(c 2 ) |a kl | ≤ 1 ν! |M νlM νk |, where ν = max(k, l).
Proof. Note that the proof of this lemma is based on some techniques similar to
those used in [8]. We have
L 0,b (P k ) (x) =
∫ b
0
e −xy P k (y)dy =
∫ b +∞∑
0
n=0
(−1) n
x n y n P k (y)dy.
n!

SPECTRUM OF THE FINITE LAPLACE TRANSFORM AND APPLICATION 7
It is well known, see [2], that
√ √
b 2k + 1
max | P k(x) |≤ .
x∈[0,b] 2 2
Hence, ∀x ∈ [0, b], the series
Consequently, we have
+∞∑
n=0
(−1) n
x n y n P k (y) converges uniformly in [0, b].
n!
L 0,b (P k ) (x) =
+∞∑
n=0
(−1) n ∫ b
x n y n P k (y)dy =
n!
Since, for j < k, we have M j,k = 0, then, for all k, l ≥ 0,
a kl = ∑ (−1) n
M nl M nk .
n!
n≥ν
0
+∞∑
n=0
(−1) n
x n M nk .
n!
Moreover, we have
|a kl | ≤ 1 ν! |M νlM νk | ,
which completes the proof of the previous Lemma.
As in [8, 9], we can easily check that the spectrum of L 0,b coincides with the
spectrum of A = (a kl ) k,l≥0
, where a kl is given by (c 1 ). Moreover the coefficients
of the Fourier series given by (14) are nothing but the components of the n th
eigenvector of A. In practice, one can get highly accurate approximation ˜µ n of µ n
by using a submatrix of A of order K > 0. The eigenvector (˜η
k n) 0≤k≤K
is taken as
a good approximation in the L 2 -norm of (ηk n) k≥0 . Consequently
(17) ˜φ n (x) =
K∑
˜η k n P k (x), x ∈ [0, b]
k=0
is the approximation of the exact eigenfunction φ n (x) on the interval [0, b].
To approximate the spectrum of the operator L 0,b , by its matrix representation, we
need the following perturbation result on the spectrum of matrices, cited in [12].
Theorem 1. (Weyl’s perturbation theorem)
Let A and B two Hermitian matrices of order n. Let σ(A) = {α 0 ≥ ... ≥ α n−1 }
and σ(B) = {β 0 ≥ ... ≥ β n−1 } denote the spectrum of A and B, respectively. Then,
we have
(18) max
0≤j≤n−1 |α j − β j | ≤ ∥A − B∥ .
Remark 1. The previous theorem stay true with more general hermitian operator
on a Hilbert space.

8 Hichem Ben Aouicha and Tahar Moumni
As a consequence of (c 2 ), we can deduce that the coefficients a kl decays exponentially
to 0 as k + l → +∞. Hence, by using the previous theorem, one can conclude
that, for any positive integer K; one gets highly accurate approximation of the
first K eigenvalues of L 0,b by considering the first K eigenvalues of the appropriate
submatrix of A given by A K = (a kl ) 0≤k,l≤K .
2.2. GAUSSIAN QUADRATURE METHOD
In this paragraph, we use the polynomials (P k ) k and their properties to construct
a quadrature method for the eigenvalue problem (2). Note that the Gaussian
quadrature method of order 2n, associated with the orthogonal P n (x), given by (10)
over [0, b] is given by
∫ 1
0
f(x) x dx ≈
n∑
ω k f(x k ), 1 ≤ k ≤ n,
k=1
where the nodes x k are the eigenvalues of the matrix D given by (12) and the
different quadrature weights (ω k ) 1≤k≤n are simply given by the following practical
formula
ω k = − k n+1 1
k n P n+1 (x k )P n(x ′ k ) , 1 ≤ k ≤ n.
√
2k + 1(2k)!
Here, k n = √ is the highest coefficient of P n (x). For more details on
b
2k+1
(k!) 2
the Gaussian quadrature method, the reader is referred to [7].
The following theorem provides us with a discretization formula for the eigenproblem
(2).
Theorem 2. Let ϵ be an arbitrary { real number satisfying 0 < ϵ < } 1. For a fixed
2 2m b 4m+3/2
positive integer n, let N ε,n = min m ∈ N,
(2m + 1)! ( )
2m 2
< ε |µ n | . Then for any
m
integer N ≥ N ε,n , we have
sup
∣ φ n(x) − 1 ∑
N ω p exp(−xx p )φ n (x p )
µ n ∣ ≤ ε.
x∈[0,b]
p=1
Here, (x p ) 1≤p≤n denote the different zeros of the orthogonal polynomial P n (x) and
φ n (·), µ n are as given by (2).
As a result of the previous theorem, we obtain the following discretization scheme
for the eigenvalue problem (2),
N∑
ω j e −x iyj
˜φ n (y j ) = ˜µ n ˜φ n (x i ), 1 ≤ i, j ≤ N,
j=1

SPECTRUM OF THE FINITE LAPLACE TRANSFORM AND APPLICATION 9
where the x i , y j and the ω i denote the different nodes and weights of our proposed
quadrature method. If A K denotes the square matrix of order K, defined by
A K = ( ω j e −x iy j
)1≤i,j≤K ,
then the set of the eigenvalues of A K is an approximate of a finite subset of the
eigenvalues of the operator L 0,b given by (2). Moreover, for any integer 0 ≤ n ≤ K,
the eigenvector Ũn corresponding to the approximate eigenvalue ˜µ n is given by
Ũ n = ( ˜φ n (x i )) 1≤i≤K
. Finally, to provide approximate values φ n (x) of ˜φ n (x) along
the interval [0, b], we use the following interpolation formula,
(19) ˜φ n (x) = 1 ∑ K
ω j e −xyj ˜φ n (y j ), 0 ≤ x ≤ b.
µ n
j=1
Remark 2. As predicted by the previous theorem, the interpolation formula (19)
is highly accurate. As an example, for b = 5, n = 0 and K = 40, we have found
that
( K
)
1 ∑
1 2
(φ n (x i ) − ˜φ n (x i )) 2 = 2.038073914E − 02,
K
i=0
where x i = ib K .
Now, using similar techniques used in [2, 8, 9], we can assert that the first K
eigenvalues of L 0,b are well approximated by the eigenvalues of A K . This is given
by the following theorem.
Theorem 3. Let σ (L 0,b ) = (µ n ) n≥0 and σ (A N ) = (˜µ n ) 0≤n≤N−1
denote the spectrum
of the finite Laplace operator L 0,b and the matrix A N , where N is a positive
integer larger then N ϵ,n . Then we have
sup |µ n − ˜µ n | ≤ εb √ N.
0≤n≤N−1
3. APPLICATIONS
Our goal here is to invert the Laplace transform of time limited functions
as well as essentially time limited functions by the use of the eigenfunctions φ n of
the finite Laplace transform. Let us suppose that the Laplace transform Lf of an
L 2 −unit norm function f, is known at least on [0, b]. We shall find f.
3.1. INVERSION OF THE LAPLACE TRANSFORM OF TIME
LIMITED FUNCTIONS

10 Hichem Ben Aouicha and Tahar Moumni
We suppose that f is time-limited to [0, b]. That is f(x) = f(x)χ [0,b] (x).
To find the unknown function f, we expand it into its Fourier series following
{φ n , n ∈ N},
(20) f(y) = ∑ k∈N
a k (f)φ k (y), y ∈ [0, b],
where a k (f) = 1 ∫ b
φ k (y)f(y)dy are the unknown Fourier coefficients of f to be
µ k
0
determined in the sequel. Combining (20) and (2), we obtain
(21) L 0,b f(y) = ∑ k∈N
a k (f)µ k φ k (y), y ∈ [0, b].
Multiplying both sides of (21) by φ k (y) and integrate it over [0,b], we conclude that
a k (f) = 1 ∫ b
φ k (y)L 0,b f(y)dy, k ∈ N.
µ k
0
Once (a k (f)) k are known, we use (20) to obtain the unknown data f. Note here
that in practice the series given by (20) is truncated to an order N to obtain the
following function
N∑
f N (y) = a k (f)φ k (y), y ∈ [0, b].
k=0
Then, an error bound of the approximation of f by f N , over [0, b] is given by the
following proposition
Proposition 2. Under the above notations and assumption, we have
∥f − f N ∥ 2 2 =
∑
|a k (f)| 2 µ 2 k ≤ (µ N+1 ) 2 .
k=N+1
As a consequence of the previous proposition, we assert that the f N is a well
approximation of f. This is due to the fast decay of the eigenvalues of L 0,b . This
statement is deduced from the following lemma, see [13, 14].
Lemma 4. Let T be the integral operator given by :
T x(t) =
∫ b
a
k(t, s)x(s)ds,
where k(., .) is symmetric, positive definite and has p th order continuous partial
derivatives. Then, the n th eigenvalue λ n of T satisfies
( ) 1
λ n = o
n p+1 .

SPECTRUM OF THE FINITE LAPLACE TRANSFORM AND APPLICATION 11
3.2. INVERSION OF THE LAPLACE TRANSFORM OF
ESSENTIALLY TIME LIMITED FUNCTIONS
In the sequel, we suppose that f is time-limited to [0, b] at level ϵ b . That is
f ∈ E(ϵ b ) = {f ∈ L 2 ([0, +∞[) , ∥f∥ 2 = 1, ∥fχ ]b,+∞[ ∥ 2 ≤ ϵ b }. Remark that, by
the use the Hölder’s inequality, one gets for x > 0,
(22)
|Lf(x) − L 0,b f(x)| =
≤
≤
∫ +∞
b
(∫ +∞
b
e−2xb
√
2x
.
e −xy f(y)dy
) 1/2 (∫ +∞
) 1/2
e −2xy dy
|f(y)| 2 dy
b
Note here that the right member of (22) decays exponentially to 0 whenever x goes
to +∞. To find the unknown data f, we will use L 0,b f(x) instead of Lf(x). That
is the following equality
(23) L 0,b f(x) =
∫ b
0
e −xy f b (y)dy,
where f b (y) = f(y)χ [0,b] (y). Our aim is to find f b and to show that f b is a well
approximation of f. As it is done previously, we expand f b into its Fourier series
following {φ n , n ∈ N}
N∑
(24) f b (y) = a k (f b )φ k (y),
k=0
y ∈ [0, b].
By using (23), one gets
a k (f b ) = 1 ∫ b
φ k (y)L 0,b f b (y)dy, k ∈ N.
µ k
0
Once the (a k (f b )) k are known, we use (24) to obtain the unknown data f b . Note
here that in practice the series given in (24) is truncated to an order N and the
function f b is approximated by the following one
f b,N (y) =
N∑
a k (f b )φ k (y),
k=0
y ∈ [0, b].
Note that the error of such approximation is given by
∥f b − f b,N ∥ 2 2 =
∑
|a k (f)| 2 µ 2 k ≤ (µ N+1 ) 2 .
k=N+1
The following proposition shows that f b,N is a well approximation f.

12 Hichem Ben Aouicha and Tahar Moumni
Proposition 3. Under the above notations and assumptions, we have
∥f − f b,N ∥ 2 ≤ ϵ b + |µ N+1 |.
Proof. The proof of the previous proposition is based on the triangular inequality
and the fact that the unknown data belongs to E(ϵ b ).
4. NUMERICAL RESULTS
To illustrate the results of section 2, we have considered different values of the
bandwidth b. Also, we have applied the Gaussian quadrature based method for the
computation of the spectrum and the eigenfunctions of the finite Laplace transform
L 0,b with N = 40 quadrature points. Table 1 shows the obtained eigenvalues
µ n with different values of the parameter b. Moreover, we have used (19) with
a maximum truncation order K = N, we obtain accurate approximations to the
normalized φ n (x) along the interval [0, b]. Table 2 lists, for b = 5, the approximate
values ˜φ 0 (x) of φ 0 (x), for different values of x as well as the different approximation
errors in absolute value |φ 0 (x) − ˜φ 0 (x)|. Also, in Figure 1 and 2, we have plotted
the graphs of the first three normalized eigenfunctions of the finite Laplace operator
L 0,b corresponding respectively to the parameter b = 1 and b = 5.
b = 1 b = 5 b = 10
n µ n µ n µ n
0 0.809579221e-00 1.343346838e-00 1.440931952e-00
2 0.216334666e-02 2.168106787e-01 0.386895185e-01
5 0.929136341e-08 3.806005533e-03 0.247988181e-02
10 0.299404378e-18 5.676369190e-07 0.101780894e-04
15 0.798452541e-30 1.167244175e-11 0.176316362e-07
Table 1: Values of the eigenvalues µ n of the finite Laplace transform L 0,b corresponding
to the different values of parameter b = 2, 5, 10.
Acknowledgements. The authors thanks the anonymous referees for suggestions
that improved the style of presentation. This work was supported by a DGRST
research Grant 05/UR/15-02.
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SPECTRUM OF THE FINITE LAPLACE TRANSFORM AND APPLICATION 13
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14 Hichem Ben Aouicha and Tahar Moumni
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Hichem Ben Aouicha
University of Carthage, Department of Mathematics,
Institute of Engineering of Bizerte, Tunisia.
E-mail: hichem.ipeib@gmail.com
Tahar Moumni
University of Carthage,
Higher Institute of Applied Sciences and Technology of Mateur, Tunisia.
E-mail: moumni.tahar1@gmail.com

SPECTRUM OF THE FINITE LAPLACE TRANSFORM AND APPLICATION 15
Figure 1: Graphs of the first three eigenfunctions of the FLT associated to the
parameter b = 1 (φ 0 = solid, φ 1 = dot and φ 2 = dash)
Figure 2: Graphs of the first three eigenfunctions of the FLT associated to the
parameter b = 5 (φ 0 = solid, φ 1 = dot and φ 2 = dash)