Fundamental solution for natural powers of the fractional Laplace and Dirac operators in the Riemann-Liouville sense

TLDR: In this paper, the fundamental solution of natural powers of the n-parameter fractional Laplace and Dirac operators defined via Riemann-Liouville fractional derivatives is studied.

Abstract: In this paper, we study the fundamental solution of natural powers of the n-parameter fractional Laplace and Dirac operators defined via Riemann–Liouville fractional derivatives. To do this we use iteration through the fractional Poisson equation starting from the fundamental solutions of the fractional Laplace $$\Delta _{a^+}^\alpha $$ and Dirac $$D_{a^+}^\alpha $$ operators, admitting a summable fractional derivative. The family of fundamental solutions of the corresponding natural powers of fractional Laplace and Dirac operators are expressed in operator form using the Mittag–Leffler function.

Full text

Fundamental solution for natural powers of the fractional
Laplace and Dirac operators in the Riemann-Liouville
sense
∗
A. Di Teodoro
]
, M. Ferreira
†,‡
, N. Vieira
‡
]
Departamento de Matemáticas, Colegio de Ciencias e Ingenierías
Universidad San Francisco de Quito-Ecuador, Diego de Robles y vía Interoceánica.
E-mail: nditeodoro@usfq.edu.ec
†
School of Technology and Management
Polytechnic Institute of Leiria
P-2411-901, Leiria, Portugal.
E-mail: milton.ferreira@ipleiria.pt
‡
CIDMA - Center for Research and Development in Mathematics and Applications
Department of Mathematics, University of Aveiro
Campus Universitário de Santiago, 3810-193 Aveiro, Portugal.
E-mails: mferreira@ua.pt, nloureirovieira@gmail.com
Abstract
In this paper, we study the fundamental solution for natural powers of the
n
-parameter fractional Laplace
and Dirac operators dened via Riemann-Liouville fractional derivatives. To do this we use iteration through
the fractional Poisson equation starting from the fundamental solutions of the fractional Laplace
∆
α
a
+
and
Dirac
D
α
a
+
operators, admitting a summable fractional derivative. The family of fundamental solutions of
the corresponding natural powers of fractional Laplace and Dirac operators are expressed in operator form
using the Mittag-Leer function.
Keywords:
Fractional Cliord Analysis; Fractional derivatives; Fundamental solution; Poisson's equa-
tion; Laplace transform.
MSC 2010:
30G35; 26A33; 35J05; 31A30; 44A10.
1 Introduction
During the last decades, the study of the so-called fractional Laplace operator has received the attention of
several authors (see for example [1,14] and references therein indicated). This operator is dened as a singular
integral operator or as a Fourier multiplier in Fourier domain and has the purpose of extending the harmonic
function theory of the Laplace operator by taking into account the long-range interactions that occur in a
number of applications. Motivated by fractional calculus and fractional derivatives it appeared recently new
denitions for fractional Laplace operators (see [6,7]).
In this paper we consider a
n
-parameter fractional Laplace operator dened in
n
-dimensional space and the
associated
n
-parameter fractional Dirac operator over a Cliord algebra, both dened via Riemann-Liouville
fractional derivatives with dierent fractional order of dierentiation for each direction. Previous approaches
∗
The nal version is published in
Advances in Applied Cliord Algebras
,
30
-No.1, (2020), 118. It as available via the website
https://doi.org/10.1007/s00006-019-1029-1
1

for this type of operators can be found in [6,7]. There the authors studied eigenfunctions and fundamental solu-
tions for the three-parameter fractional Laplace operator dened with Caputo and Riemann-Liouville fractional
derivatives, and derived also fundamental solutions for the corresponding fractional Dirac operator which fac-
torizes the fractional Laplace operator. In both cases, the authors applied an operational approach via Laplace
transform to construct general families of fundamental solutions.
The aim of this paper is to present an expression for the family of fundamental solutions for the natural
powers of the
n
-parameter fractional Laplace operator, as well as a family of fundamental solutions for the
natural powers of the fractional Dirac operator. To do this, we use the fundamental solution of the Laplace
operator
∆
α
a
+
and the fundamental solution of the Dirac operator
D
α
a
+
and the iteration process using the
fractional Poisson equation in order to get the families of fundamental solutions expressed in operator form
using the Mittag-Leer function.
We explain now how this paper is organized. In Section 2 we recall some basic knowledge about fractional
calculus and Cliord analysis. In Section 3 we solve the Poisson equation for the
n
-parameter fractional Laplace
operator, being this the key for the paper, because it connects the fundamental solution of previous order of the
powers of the operator with the next order. In Section 4 are presented the fundamental solutions for natural
powers of the
n
-parameter fractional Laplace operator together with a detailed discussion for the integer case.
To nish, in Section 5 we present the fundamental solutions for natural powers of the
n
-parameter fractional
Dirac operator.
2 Preliminaries
2.1 Fractional Calculus
Let

D
α
a
+
f

(x)
denote the fractional Riemann-Liouville derivative of order
α > 0
(see [11])
(D
α
a
+
f) (x) =

d
dx

m
1
Γ(m − α)
Z
x
a
f(t)
(x − t)
α−m+1
dt, m = [α] + 1, x > a,
(1)
where
[α]
means the integer part of
α.
When
0 < α < 1
then (1) takes the form
(D
α
a
+
f) (x) =
d
dx
1
Γ(1 − α)
Z
x
a
f(t)
(x − t)
α
dt.
(2)
The Riemann-Liouville fractional integral of order
α > 0
is given by (see [11])
(I
α
a
+
f) (x) =
1
Γ(α)
Z
x
a
f(t)
(x − t)
1−α
dt, x > a.
(3)
We denote by
I
α
a
+
(L
1
)
the class of functions
f
represented by the fractional integral (3) of a summable function,
that is
f = I
α
a
+
ϕ, ϕ ∈ L
1
(a, b).
A description of this class of functions was given in [13].
Theorem 2.1
A function
f ∈ I
α
a
+
(L
1
), α > 0
if and only if
I
m−α
a
+
f ∈ AC
m
([a, b])
,
m = [α]+1
and
(I
m−α
a
+
f)
(k)
(a) =
0, k = 0, . . . , m − 1.
In Theorem 2.1
AC
m
([a, b])
denotes the class of functions
f
, which are continuously dierentiable on the segment
[a, b]
up to order
m − 1
and
f
(m−1)
is absolutely continuous on
[a, b]
. Removing the last condition in Theorem
2.1 we obtain the class of functions that admits a summable fractional derivative.
Denition 2.2
(see [13]) A function
f ∈ L
1
(a, b)
has a summable fractional derivative

D
α
a
+
f

(x)
if

I
m−α
a
+

(x) ∈
AC
m
([a, b])
, where
m = [α] + 1.
If a function
f
admits a summable fractional derivative, then the composition of (1) and (3) can be written in
the form (see [13, Thm. 2.4])
(I
α
a
+
D
α
a
+
f) (x) = f(x) −
m−1
X
k=0
(x − a)
α−k−1
Γ(α − k)

I
m−α
a
+
f

(m−k−1)
(a), m = [α] + 1.
(4)
2

We remark that if
f ∈ I
α
a
+
(L
1
)
then (4) reduces to

I
α
a
+
D
α
a
+
f

(x) = f(x)
. Nevertheless we note that
D
α
a
+
I
α
a
+
f =
f
in both cases. This is a particular case of a more general property (cf. [12, (2.114)])
D
α
a
+

I
γ
a
+
f

= D
α−γ
a
+
f, α ≥ γ > 0.
(5)
It is important to remark that the semigroup property for the composition of fractional derivatives does not
hold in general (see [12, Sect. 2.3.6]). In fact, the property
D
β
a
+
(D
α
a
+
f) = D
β+α
a
+
f
(6)
holds whenever
f
(j)
(a
+
) = 0, j = 0, 1, . . . , m − 1,
(7)
and
f ∈ AC
m−1
([a, b])
,
f
(m)
∈ L
1
(a, b)
and
m = [α] + 1
. There are other sucient conditions that ensure the
semigroup property (see [7]).
One important function used in this paper is the two-parameter Mittag-Leer function
E
µ,ν
(z)
[9], which
is dened in terms of the power series by
E
µ,ν
(z) =
∞
X
k=0
z
k
Γ(µk + ν)
, µ > 0, ν > 0, z ∈ C.
(8)
In particular, the function
E
µ,ν
(z)
is entire of order
ρ =
1
µ
and type
σ = 1
. Two important fractional integral
and dierential formulae involving the two-parametric Mittag-Leer function are the following (see [9, pp.
87-88]
I
α
a
+

(x − a)
ν−1
E
µ,ν
(k(x − a)
µ
)

= (x − a)
α+ν−1
E
µ,ν+α
(k(x − a)
µ
)
(9)
for all
α > 0, k ∈ C, x > a, µ > 0, ν > 0,
and
D
α
a
+

(x − a)
ν−1
E
µ,ν
(k(x − a)
µ
)

= (x − a)
ν−α−1
E
µ,ν−α
(k(x − a)
µ
)
(10)
for all
α > 0, k ∈ C, x > a, µ > 0, ν > 0, ν 6= α − p,
with
p = 0, . . . , m − 1,
and
m = [α] + 1.
Remark 2.3
For
ν = α − p,
with
p = 0, . . . , m − 1
and
m = [α] + 1
we have that
D
α
a
+
((x − a)
α−p−1
) = 0,
which implies that the rst term in the series expansion of
(x − a)
ν−1
E
µ,ν
(k(x − a)
µ
)
vanishes. Therefore, the
derivation rule (10) must be replaced in these cases by the following derivation rule:
D
α
a
+

(x − a)
α−p−1
E
µ,α−p
(k(x − a)
µ
)

= (x − a)
µ−p−1
k E
µ,µ−p
(k(x − a)
µ
) , p = 0, . . . , n − 1.
(11)
The approach presented in this paper is based on the Laplace transform and leads to the solution of a linear
Abel integral equation of the second kind.
Theorem 2.4
( [9, Thm. 4.2]) Let
f ∈ L
1
[a, b], α > 0
and
λ ∈ C.
Then the integral equation
u(x) = f(x) +
λ
Γ(α)
Z
x
a
(x − t)
α−1
u(t) dt, x ∈ [a, b]
has a unique solution
u(x) = f(x) + λ
Z
x
a
(x − t)
α−1
E
α,α
(λ(x − t)
α
)f(t) dt.
(12)
2.2 Cliord analysis
Let
{e
1
, · · · , e
n
}
be the standard basis of the Euclidean vector space in
R
n
. The associated Cliord algebra
R
0,n
is the free algebra generated by
R
n
modulo
x
2
= −||x||
2
e
0
, where
x ∈ R
n
and
e
0
is the neutral element
with respect to the multiplication operation in the Cliord algebra
R
0,n
. The dening relation induces the
multiplication rules
e
i
e
j
+ e
j
e
i
= −2δ
ij
,
(13)
3

where
δ
ij
denotes the Kronecker's delta. In particular,
e
2
i
= −1
for all
i = 1, . . . , n
. The standard basis vectors
thus operate as imaginary units. A vector space basis for
R
0,n
is given by the set
{e
A
: A ⊆ {1, . . . , n}}
with
e
A
= e
l
1
e
l
2
. . . e
l
r
, where
1 ≤ l
1
< . . . < l
r
≤ n
,
0 ≤ r ≤ n, e
∅
:= e
0
:= 1
. Each
a ∈ R
0,n
can be written in the
form
a =
P
A
a
A
e
A
, with
a
A
∈ R
. The conjugation in the Cliord algebra
R
0,n
is dened by
a =
P
A
a
A
e
A
,
where
e
A
= e
l
r
e
l
r−1
. . . e
l
1
, and
e
j
= −e
j
for
j = 1, . . . , n, e
0
= e
0
= 1
.
Cliord analysis can be regarded as a higher-dimensional generalization of complex function theory in the
sense of the Riemann approach. An
R
0,n
−
valued function
f
over
Ω ⊂ R
n
1
has the representation
f =
P
A
e
A
f
A
,
with components
f
A
: Ω → R
0,n
. Properties such as continuity or dierentiability have to be understood com-
ponentwise. Next, we recall the Euclidean Dirac operator
D =
P
n
j=1
e
j
∂
x
j
, which factorizes the
n
-dimensional
Euclidean Laplace, i.e.,
D
2
= −∆ = −
P
n
j=1
∂x
2
j
. An
R
0,n
-valued function
f
is called
left-monogenic
if it
satises
Du = 0
on
Ω
(resp.
right-monogenic
if it satises
uD = 0
on
Ω
).
For more details about Cliord algebras and basic concepts of its associated function theory we refer the
interested reader for example to [3,8]. Connections between Cliord analysis and fractional calculus were studied
in [6,7,10,15].
3 The Poisson problem
Let
Ω =
Q
n
j=1
[a
j
, b
j
]
be any bounded open rectangular domain, let
α = (α
1
, . . . , α
n
)
, with
α
i
∈]0, 1]
,
i = 1, . . . , n
,
and let us consider the
n
-parameter fractional Laplace operator
∆
α
a
+
dened over
Ω
by means of the Riemann-
Liouville fractional derivative given by
∆
α
a
+
=
n
X
j=1
∂
1+α
j
x
+
j
.
(14)
The previous fractional operator is associated to the corresponding fractional Dirac operator dened by
D
α
a
+
=
n
X
j=1
e
j
∂
1+α
j
2
x
+
j
.
(15)
For
j = 1, . . . , n
the partial derivatives
∂
1+α
j
x
+
j
and
∂
1+α
j
2
x
+
j
are the Riemann-Liouville fractional derivatives (2) of
orders
1 + α
j
and
1+α
j
2
, with respect to the variable
x
j
∈ [a
j
, b
j
]
. Like in the three-dimensional case (see [7]),
under certain conditions we have
∆
α
a
+
= −D
α
a
+
D
α
a
+
. Due to the nature of the fundamental solutions of these
operators we additionally need to consider the variable
bx = (x
2
, . , x
n
) ∈
b
Ω =
Q
n
j=2
[a
j
, b
j
]
, and the fractional
Laplace and Dirac operators acting on
bx
dened by:
b
∆
α
a
+
=
n
X
j=2
∂
1+α
j
x
+
j
,
b
D
α
a
+
=
n
X
j=2
∂
1+α
j
2
x
+
j
.
(16)
Consider the following Poisson problem
∆
α
a
+
v(x) = u(x),
(17)
where we suppose that
v(x) = v(x
1
, . . . , x
n
)
admits summable fractional derivative
∂
1+α
1
x
+
1
v(x)
in the variable
x
1
and belongs to
I
1+α
j
a
+
j
(L
1
)
in the variables
x
j
, for
j = 2, . . . , n
. Starting to apply the fractional integral
I
1+α
1
a
+
1
to both sides of the previous equation and taking into account (4) we get
v(x) −
(x
1
− a
1
)
α
1
Γ(α
1
+ 1)

∂
α
1
x
+
1
v

(a
1
, bx) −
(x
1
− a
1
)
α
1
−1
Γ(α
1
)

I
1−α
1
a
+
1
v

(a
1
, bx) +
n
X
k=2

I
1+α
1
a
+
1
∂
1+α
k
x
+
k
v

(x) =

I
1+α
1
a
+
1
u

(x).
Applying successively the fractional integrals
I
1+α
j
a
+
j
, with
j = 2, . . . , n
, to both sides of the previous equation,
recalling that we supposed that
v
belongs to
I
1+α
j
a
+
j
(L
1
)
in the variables
x
j
, applying Fubini's theorem, and
4

rearranging the terms, we obtain:




I
1+α
1
a
+
1
n
X
k=2
n
Y
j=2
j6=k
I
1+α
j
a
+
j
v




(x) +


n
Y
j=2
I
1+α
j
a
+
j
v


(x) −


n
Y
j=1
I
1+α
j
a
+
j
u


(x)
=
(x
1
− a
1
)
α
1
Γ(α
1
+ 1)


n
Y
j=2
I
1+α
j
a
+
j
f
1


(bx) +
(x
1
− a
1
)
α
1
−1
Γ(α
1
)


n
Y
j=2
I
1+α
j
a
+
j
f
0


(bx),
(18)
where
f
0
and
f
1
are fractional initial conditions given by
f
0
(bx) =

I
1−α
1
a
+
1
v

(a
1
, bx), f
1
(bx) =

∂
α
1
x
+
1
v

(a
1
, bx).
(19)
We observe that the fractional integrals in (18) are Laplace-transformable functions. Therefore, we may apply
the n-dimensional Laplace transform with respect to
x
2
, . . . , x
n
, which we dene by
F(bs) = F(s
2
, . . . , s
n
) = L{f}(s
2
, . . . , s
n
) =
Z
+∞
a
2
. . .
Z
+∞
a
n
exp
−
n
X
p=2
s
p
x
p
!
f(x
2
, . . . , x
n
) dx
n
· · · dx
2
.
Taking into account its convolution and operational properties (see [4,11]), we obtain the following relations for
each term in (18):
L







I
1+α
1
a
+
1
n
X
k=2
n
Y
j=2
j6=k
I
1+α
j
a
+
j
v







(x
1
, bs) =
n
X
k=2
n
Y
p=2
p6=k
s
−1−α
p
p

I
1+α
1
a
+
1
V

(x
1
, bs), k = 2, . . . , n,
L



n
Y
j=2
I
1+α
j
a
+
j
v



(x
1
, bs) =
n
Y
p=2
s
−1−α
p
p
V(x
1
, bs),
L



n
Y
j=1
I
1+α
j
a
+
j
u



(x
1
, bs) =
n
Y
p=2
s
−1−α
p
p

I
1+α
1
a
+
1
U

(x
1
, bs),
L



(x
1
− a
1
)
α
1
−1
Γ(α
1
)


n
Y
j=2
I
1+α
j
a
+
j
f
0





(x
1
, bs) =
(x
1
− a
1
)
α
1
−1
Γ(α
1
)
n
Y
p=2
s
−1−α
p
p
F
0
(bs),
L



(x
1
− a
1
)
α
1
Γ(α
1
+ 1)


n
Y
j=2
I
1+α
j
a
+
j
f
1





(x
1
, bs) =
(x
1
− a
1
)
α
1
Γ(α
1
+ 1)
n
Y
p=2
s
−1−α
p
p
F
1
(bs).
Combining all the resulting terms and multiplying by
Q
n
p=2
s
1+α
p
p
we obtain the following second kind homo-
geneous integral equation of Volterra type:
V(x
1
, bs) +
1
Γ(α
1
+ 1)
n
X
p=2
s
1+α
p
p
Z
x
1
a
1
(x
1
− t)
α
1
V(t, bs) dt =

F (x
1
, bs) +

I
1+α
1
a
+
1
U

(x
1
, bs)

,
(20)
where
F (x
1
, bs) =
(x
1
− a
1
)
α
1
−1
Γ(α
1
)
F
0
(bs) +
(x
1
− a
1
)
α
1
Γ(α
1
+ 1)
F
1
(bs)
and
F
k
(bs) = L {f } (s),
with
k = 0, 1.
Using (12), we have that the unique solution of (20) in the class of
summable functions is:
V(x
1
, bs) = F (x
1
, bs) +

I
1+α
1
a
+
1
U

(x
1
, bs)
−
n
X
p=2
s
1+α
p
p
Z
x
1
a
1
(x
1
− t)
α
E
1+α
1
,1+α
1
−(x
1
− t)
α
1
+1
n
X
p=2
s
1+α
p
p
!

F (t, bs) +

I
1+α
1
a
+
1
U

(x
1
, bs)

dt,
(21)
5

Frequently asked questions

Q1. What have the authors contributed in "Fundamental solution for natural powers of the fractional laplace and dirac operators in the riemann-liouville sense∗" ?

In this paper, the authors study the fundamental solution for natural powers of the n-parameter fractional Laplace and Dirac operators de ned via Riemann-Liouville fractional derivatives.