Distributed-order calculus: An operator-theoretic interpretation
Within the Bochner-Phillips functional calculus and Hirsch functional calculus, we describe the operators of distributed-order differentiation and integration as functions of the classical operators of differentiation and integration, respectively.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509214296768512 |
|---|---|
| author | Kochubei, A. N. Кочубей, А. Н. |
| author_facet | Kochubei, A. N. Кочубей, А. Н. |
| author_sort | Kochubei, A. N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:47:27Z |
| description | Within the Bochner-Phillips functional calculus and Hirsch functional calculus, we describe the operators of distributed-order differentiation and integration as functions of the classical operators of differentiation and integration, respectively. |
| first_indexed | 2026-03-24T02:37:32Z |
| format | Article |
| fulltext |
UDC 517.9
A. N. Kochubei (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
DISTRIBUTED ORDER CALCULUS:
AN OPERATOR-THEORETIC INTERPRETATION
∗∗∗∗
ÇYSLENNQ ROZPODILENOHO PORQDKU:
TEORETYKO-OPERATORNA INTERPRETACIQ
Within the Bochner – Phillips functional calculus and the Hirsch functional calculus, we describe the
operators of distributed order differentiation and integration as functions of the classical differentiation
and integration operators, respectively.
U meΩax funkcional\nyx çyslen\ Boxnera – Fillipsa ta Xirßa navedeno opys operatoriv
dyferencigvannq ta intehruvannq rozpodilenoho porqdku qk funkcij vid klasyçnyx operatoriv
dyferencigvannq ta intehruvannq.
1. Introduction and preliminaries. In the distributed order calculus [1], used in phy-
sics for modeling ultraslow diffusion and relaxation phenomena, we consider deriva-
tives and integrals of distributed order. The definitions are as follows.
Let µ be a continuous non-negative function on [0, 1]. The distributed order de-
rivative D( )µ of weight µ for a function ϕ on [0, T] is
( )( ) ( )D
µ ϕ t =
0
1
∫ ( )( ) ( ) ( )D
α ϕ µ α αt d (1)
where D( )α is the Caputo – Dzhrbashyan regularized fractional derivative of order α ,
that is
( )( ) ( )D
α ϕ t = 1
1
0
0
Γ( )
( ) ( ) ( )
−
− −
− −∫α
τ ϕ τ τ ϕα αd
dt
t d t
t
, 0 < t < T . (2)
Denote
k ( s ) = s d
−
−∫
α
α
µ α α
Γ( )
( )
1
0
1
, s > 0. (3)
It is obvious that k is a positive decreasing function. The definition (1), (2) can be
rewritten as
( )( ) ( )D
µ ϕ t = d
dt
k t d k t
t
( ) ( ) ( ) ( )− −∫ τ ϕ τ τ ϕ
0
0 . (4)
The right-hand side of (4) makes sense for a continuous function ϕ, for which the de-
rivative d
dt
k t d
t
( ) ( )−∫ τ ϕ τ τ
0
exists.
If a function ϕ is absolutely continuous, then
( )( ) ( )D
µ ϕ t = k t d
t
( ) ( )− ′∫ τ ϕ τ τ
0
. (5)
∗
Partially supported by the Ukrainian Foundation for Fundamental Research (Grant 14.1/003).
© A. N. KOCHUBEI, 2008
478 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
DISTRIBUTED ORDER CALCULUS: AN OPERATOR-THEORETIC INTERPRETATION 479
Below we always assume that µ ∈C3 0 1[ , ], µ ( 1 ) ≠ 0, and either µ ( 0 ) ≠ 0, or
µ ( α ) ∼ a α
ν, a, ν > 0, as α → 0. Under these assumptions (see [1]),
k ( s ) ∼ s s− −1 2 1(log ) ( )µ ,
s → 0,
k ′ ( s ) ∼ – s s− −2 2 1(log ) ( )µ ,
so that k L T∈ 1 0( , ) and k does not belong to any Lp , p > 1. We cannot differenti-
ate under the integral in (4), since k ′ has a non-integrable singularity.
It is instructive to give also the asymptotics of the Laplace transform
K ( z ) = k s e dszs( ) −
∞
∫
0
.
Using (4) we find that
K ( z ) = z dα µ α α−∫ 1
0
1
( ) ,
so that K ( z ) can be extended analytically to an analytic function on C R\ − , R− =
= z z z∈ = ≤{ }C : Im , Re0 0 . If z ∈ −C R\ , z → ∞ , then [1]
K ( z ) =
µ( )
log
log
1 2
z
O z+ ( )( )− ; (6)
see [1] for further properties of K .
The distributed order integral I( )µ is defined as the convolution operator
( )( ) ( )I
µ f t = κ( ) ( )t s f s ds
t
−∫
0
, 0 ≤ t ≤ T, (7)
where κ ( t ) is the inverse Laplace transform of the function z
z z
� 1
K ( )
,
κ ( t ) =
d
dt i
e
z z z
dz
zt
i
i
1
2
1
π
γ
γ
K ( )
− ∞
+ ∞
∫ , γ > 0. (8)
It was proved in [1] that κ ∈ ∞∞C ( , )0 and κ is completely monotone; for small
values of t,
κ ( t ) ≤ C
t
log1, ′κ ( )t ≤ Ct
t
−1 1log . (9)
If f L T∈ 1 0( , ), then D I
( ) ( )µ µ f = f .
The aim of this paper is to clarify the operator-theoretic meaning of the above con-
structions. It is well known that fractional derivatives and integrals can be interpreted
as fractional powers of the differentiation and integration operators in various Banach
spaces; see, for example, [2 – 5].
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
480 A. N. KOCHUBEI
Let A be the differential operator Au = –
du
dx
in L Tp( , )0 , 1 ≤ p < ∞ , with the
boundary condition u ( 0 ) = 0. Its domain D ( A ) consists of absolutely continuous
functions u L Tp∈ ( , )0 , such that u ( 0 ) = 0 and ′ ∈u L Tp( , )0 . We show that on
D ( A ) the distributed order differentiation coincides with the function L ( )− A of the
operator – A, where L ( z ) = z K ( z ) , and the function of an operator is understood in
the sense of the Bochner – Phillips functional calculus (see [6 – 8]).
Moreover, if p = 2 then the distributed order integration operator I
( )µ equals
N ( J ) , where N ( x ) =
1
L( )x
, J is the integration operator, ( Ju ) ( t ) = u d
t
( )τ τ
0∫ . This
result is obtained within Hirsch’s functional calculus [9, 10] giving more detailed re-
sults for a more narrow class of functions. As by-products, we obtain an estimate of
the semigroup generated by – L ( )− A , and an expression for the resolvent of the
operator I( )µ .
2. Functions of the differentiation operator. The semigroup Ut of operators on
the Banach space X = L Tp( , )0 generated by the operator A has the form
( )( )U f xt =
f x t t x T
x t
( ), ,
, ,
− ≤ ≤ <
< <
if
if
0
0 0
x T∈( , )0 , t ≥ 0. This follows from the easily verified formula for the resolvent
R ( λ, A ) = ( )A I− −λ 1 of the operator A :
( R ( λ, A ) u ) ( x ) = – e u y dyx y
x
− −∫ λ( ) ( )
0
; (10)
see [11] for a similar reasoning for operators on Lp( , )0 ∞ . The semigroup Ut is nil-
potent, Ut = 0 for t > T ; compare Sect. 19.4 in [12]. It follows from the expression
(10) and the Young inequality that R A( , )λ ≤ λ−1, λ > 0, so that Ut is a C0-se-
migroup of contractions.
In the Bochner – Phillips functional calculus, for the operator A, as a generator of a
contraction semigroup, and any function f of the form
f ( x ) = ( ) ( )1
0
− + +−
∞
∫ e dt a bxtx σ , a, b ≥ 0, (11)
where σ is a measure on ( 0 , ∞ ) , such that
t
t
dt
1
0
+
∞
∫ σ( ) < ∞ ,
the subordinate C0-semigroup Ut
f is defined by the Bochner integral
Ut
f = ( ) ( )U u dss tσ
0
∞
∫
where the measures σt are defined by their Laplace transforms,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
DISTRIBUTED ORDER CALCULUS: AN OPERATOR-THEORETIC INTERPRETATION 481
e dssx
t
−
∞
∫ σ ( )
0
= e t f x− ( ) .
The class B of functions (11) coincides with the class of Bernstein functions, that
is functions f C C∈ ∞ ∞∞[ , ) ( , )0 0∩ , for which ′f is completely monotone. Below
we show that L ∈ B .
The generator A
f
of the semigroup Ut
f is identified with – f A( )− . On the do-
main D ( A ) ,
A uf = – au bAu U u u dtt+ + −
∞
∫ ( ) ( )σ
0
, u ∈ D ( A ) . (12)
Theorem 1. (i) If u ∈ D ( A ) , then A
L
u = – D
( )µ u .
(ii) The semigroup Ut
L decays at infinity faster than any exponential function:
Ut
L ≤ C er
rt− for any r > 0. (13)
The operator AL has no spectrum.
(iii) The resolvent R A( ),λ − L of the operator – AL has the form
( ( ) ), ( )R A u xλ − L = r x s u s ds
x
λ( ) ( )−∫
0
, u ∈ X , (14)
where
r sλ( ) = 1
λ λ
d
ds
u s( ), (15)
and uλ is the solution of the Cauchy problem
D
( )µ
λu = λ λu , uλ( )0 = 1. (16)
(iv) The inverse ( )− −AL 1 coincides with the distributed order integration ope-
rator I( )µ .
(v) The resolvent of I( )µ has the form
( )( )
I
µ λ− −I u1 = –
1 1
2 1λ λ λu r u− ∗/ , λ ≠ 0. (17)
Proof. Let σ( )dt = – ′k t dt( ) . By (3),
′k t( ) = –
α
α
µ α α
αt
d
− −
−∫
1
0
1
1Γ( )
( ) ,
so that
t
t
dt
1
0
+
∞
∫ σ( ) =
αµ α
α
α σ
α( )
( )
( )
Γ 1 1
0
1
0
− +∫ ∫
−∞
d t
t
dt .
Using the integral formula 2.2.5.25 from [13] we find that
t
t
dt
1
0
+
∞
∫ σ( ) = π αµ α
απ α
α( )
(sin ) ( )Γ 1
0
1
−∫ d < ∞ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
482 A. N. KOCHUBEI
Let us compute the function (11) with a = b = 0. We have
f ( x ) = – ( ) ( )1
0
− ′−
∞
∫ e k t dttx = x e k t dttx−
∞
∫ ( )
0
= x K ( x ) = L ( x ) .
The corresponding expression (12) for A
L
u , u ∈ D ( A ) , is as follows:
( A
L
u ) ( x ) = – [ ]( )( ) ( ) ( )U u x u x k t dtt − ′
∞
∫
0
=
= – [ ]( ) ( ) ( ) ( ) ( )u x t u x k t dt u x k t dt
x
x
− − ′ + ′∫ ∫
∞
0
=
= – k x u x u x t u x k t dt
x
( ) ( ) ( ) ( ) ( )[ ]− − − ′∫
0
.
By (4), we find that A
L
u = – D
( )µ u , u ∈ D ( A ) .
The function L ( z ) is holomorphic for Re z > 0. We will need a detailed infor-
mation (refining (6)) on the behavior of Re L ( σ + i τ ) , σ , τ ∈ R , σ > 0, when τ →
→ ∞ . We have
Re L ( σ + i τ ) = ϕ α σ τ µ α α( , , ) ( )d
0
1
∫
where
ϕ α σ τ( , , ) = ( ) / cos arctanσ τ α τ
σ
α2 2 2+
.
We check directly that ϕ α σ( , , )0 = σα ,
∂
∂
ϕ α σ τ
τ
( , , )
= α σ τ α τ
σ
τ σ α τ
σ
α( ) / cos arctan tan arctan2 2 2 1+
−
− ≥ 0,
and
∂
∂
ϕ α σ τ
τ
( , , )
> 0 for α < 1. This means that the function gσ τ( ) = Re L ( σ + i τ )
(which is even in τ ) is strictly monotone increasing in τ for τ > 0. Its minimal va-
lue is
gσ( )0 = σ µ α αα ( )d
0
1
∫ .
On the other hand,
Re L ( σ + i τ ) ≥ ( ) / cos ( )σ τ απµ α αα2 2 2
0
1
2
+∫ d = 2 22 2
0
2
π
σ τ µ
π
π
π
( ) /
/
cos+
∫ t t t dt =
= 2 2
0
2
π
µ
π
π
e t t dtqt
∫ cos
/
= 2 1 22
0
2
π
µ
π
π
π
e e s s dsq qs/
/
sin− −
∫
where q = 1 2 2
π
σ τlog( )+ . By Watson’s asymptotic lemma (see [14]), since µ ( 1 ) ≠
≠ 0, we have
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
DISTRIBUTED ORDER CALCULUS: AN OPERATOR-THEORETIC INTERPRETATION 483
e s s dsqs− −
∫ µ
π
π
1 2
0
2
sin
/
∼ Cq−2
where C does not depend on σ, τ . Roughening the estimate a little we find that
e
t i− +Re ( )L σ τ ≤ Ce t− −ρ τ ε1 2/
(18)
where 0 < ε < 1 / 2 can be taken arbitrarily, and the positive constants C and ρ do
not depend on σ and τ.
It follows from (18) (see [15]) that for each t > 0 the function x e t x� − L ( ) is
represented by an absolutely convergent Laplace integral. This means that the measure
σt ds( ) has a density m ( t, s ) with respect to the Lebesgue measure. Moreover,
m ( t, s ) =
1
2π
γ
γ
i
e e dzzs t z
i
i
−
− ∞
+ ∞
∫ L ( ) , γ > 0. (19)
Since Ut = 0 for t > T, we have
U ut
f = ( ) ( , )U u m t s dss
T
0
∫ . (20)
The representation (19) yields the expression
m ( t, s ) = e e e d
s
i t i
γ
τ γ τ
π
τ− +
∞
∫ L ( )
0
,
L ( γ + i τ ) = ( ) ( )γ τ µ α αα+∫ i d
0
1
, 0 ≤ τ < ∞ .
We have
m t s( , ) ≤ e e d
s tgγ τ
π
τγ−
∞
∫ ( )
0
.
The above monotonicity property of gγ makes it possible to apply to the last in-
tegral the Laplace asymptotic method [14]. We obtain that, for large values of t,
m t s( , ) ≤ Ct e es tg− −1 0γ γ ( )
.
Changing γ and C we can make the coefficient gγ ( )0 arbitrarily big. By (20), this
leads to the estimate (13).
Due to (13), the resolvent
R ( λ, A
L
) = –
e U dtt
t
T
−∫ λ L
0
, (21)
is an entire function, so that A
L
has no spectrum.
It follows from (21) that
R ( λ, – A
L
) = e U dtt
t
T
λ L
0
∫ ,
and if u ∈ X , Re λ ≤ 0, then
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
484 A. N. KOCHUBEI
( R ( λ, – A
L
) u ) ( x ) = e dt u x s m t s dst
x
λ
0 0
∞
∫ ∫ −( ) ( , ) = r x s u s ds
x
λ( ) ( )−∫
0
where
r sλ( ) = e m t s dttλ ( , )
0
∞
∫ . (22)
For a fixed ω ∈( , )/1 2 1 , let us deform the contour of integration in (19) from the
vertical line to the contour Sγ ω, consisting of the arc
Tγ ω, = z z z∈ = ≤{ }C : , argγ ωπ ,
and two rays
Γγ ω,
± = z z z∈ = ± ≥{ }C : arg ,ωπ γ .
The contour Sγ ω, is oriented in the direction of growth of arg z. By Jordan’s
lemma,
m t s( , ) =
1
2π
γ ω
i
e e dzzs t z
S
−∫ L ( )
,
.
Under this integral, we may integrate in t, as required in (22). We find that
r sλ( ) =
1
2π λ
γ ω
i
e
z
dz
zs
S
L ( )
,
−∫ , s > 0 (23)
(for Re λ > 0, γ should be taken big enough).
If λ = 0, the right-hand side of (23) coincides with that of (8) (see also the
formula (3.4) in [1], and we prove that ( )− −AL 1 = I( )µ .
For λ ≠ 0, we rewrite (23) as
r sλ( ) =
1
2
1
2π λ λ π λ
γ ω γ ω
i
e
z
z
dz
i
e dzzs
S
zs
S
L
L
( )
( )
, ,
−
−∫ ∫ . (24)
For 0 < s < T, we have
e dzzs
Sγ ω,
∫ = – lim
arg
R
zs
z R
z
e dz
→∞
=
< <
∫
ωπ π
,
e dzzs
z R
z
=
< <
∫
ωπ πarg
≤ 2R e dRscosϕ
ωπ
π
ϕ∫ ≤ 2 1R eRsπ ω ωπ( ) cos− → 0,
as R → ∞ .
Thus, the second integral in (24) equals zero, and it remains to compare (24) with
the formula (2.15) of [1] giving an integral representation of the function uλ .
The formula (17) follows from (15) and the general connection between the resol-
vents of an operator and its inverse ([11], Chapter 3, formula (6.18)).
The theorem is proved.
Note that the expression (17) for the resolvent of a distributed order integration
operator is quite similar to the Hille – Tamarkin formula for the resolvent of a
fractional integration operator (see [12], Sect. 23.16). In our case, the function uλ is a
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
DISTRIBUTED ORDER CALCULUS: AN OPERATOR-THEORETIC INTERPRETATION 485
counterpart of the function z E z� α
αλ( ) (for the order α case). However, in our
situation no analog of the entire function Eα (the Mittag-Leffler function) has been
identified so far. Accordingly, our proof of (17) is different from the reasoning in [12].
3. Functions of the integration operator. In this section we assume that p = 2.
Hirsch’s functional calsulus deals with the class R of functions which are conti-
nuous on C \ ( – ∞ , 0 ) , holomorphic on C \ ( – ∞ , 0 ] , transform the upper half-plane
into itself, and transform the semi-axis ( 0 , ∞ ) into itself. The class R is a subclass
of B.
Another important class of functions is the class S of Stieltjes functions
f ( z ) = a
d
z
+
+
∞
∫ ρ λ
λ
( )
0
, z ∈ C \ ( – ∞ , 0 ] ,
where a ≥ 0, ρ is a non-decreasing right-continuous function, such that
d t
t
ρ( )
1
0
+
∞
∫ < ∞ .
If f is a nonzero function from S, then the function
˜( )f z = 1
1f z( )−
also belongs to S.
If f ∈ S , then the function H zf ( ) = f z( )−1 belongs to R . It has the form
H zf ( ) = a z
z
d+
+
∞
∫1
0
λ
ρ λ( ), z ∈ C \ ( – ∞ , 0 ] .
For some classes of linear operators V, the function H Vf ( ) is defined as a closure
of the operator
Wx = ax V I V x d+ + −
∞
∫ ( ) ( )λ ρ λ1
0
, x ∈ D ( V ) .
In particular, this definition makes sense if – V is a generator of a contraction C0-
semigroup, and in this case the above construction is equivalent to the Bochner – Phil-
lips functional calculus [3, 16]. In addition, by Theorem 2 of [9], if ( )− −V 1 is also a
generator of a contraction C0-semigroup, then
[ ]( )H Vf
−1 = H V
f̃
( )−1 . (25)
In order to apply the above theory to our situation, note that [9]
z
α = 1
1 1
0
Γ Γ( ) ( )α α λ
λ λα
− +
∞
−∫ z
z
d , 0 < α < 1,
whence
L ( z ) = z
z
d
1
0
+
∞
∫ λ
β λ λ( )
where
β ( λ ) =
λ µ α
α α
α
α−
−∫ ( )
( ) ( )Γ Γ 1
0
1
d .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
486 A. N. KOCHUBEI
Thus L ( z ) = H zf ( ), with
f ( z ) = 1
0
z
d
+
∞
∫ λ
β λ λ( ) .
It follows from Watson’s lemma [14] that β ( λ ) ≤ C (log )λ −2 for large values of λ .
Therefore
β λ
λ
λ( )
1
0
+
∞
∫ d < ∞ .
Denote N ( z ) = H
f
z˜ ( ) =
1
L ( )z
.
If V = – A, then ( )− −V 1 = – J, where J is the integration operator. It is easy to
check that 〈 + 〉∗( ) ,J J u u ≥ 0 ( 〈⋅ ⋅〉, is the inner product in L T2 0( , )) . Therefore – J
is a generator of a contraction semigroup.
After these preparations, the equality (25) implies the following result.
Theorem 2. The operator I
( )µ of distributed order integration and the integra-
tion operator J are connected by the relation
I
( )µ = N ( J ) .
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Received 19.10.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
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| id | umjimathkievua-article-3170 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:37:32Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/c0/75889998f9bc0064a206f6aedc04edc0.pdf |
| spelling | umjimathkievua-article-31702020-03-18T19:47:27Z Distributed-order calculus: An operator-theoretic interpretation Числення розподіленого порядку: теоретико-операторна інтерпретація Kochubei, A. N. Кочубей, А. Н. Within the Bochner-Phillips functional calculus and Hirsch functional calculus, we describe the operators of distributed-order differentiation and integration as functions of the classical operators of differentiation and integration, respectively. У межах функціональних числень Boxнepa - Філліпса та Хірша наведено опис операторів диференціювання та інтегрування розподіленого порядку як функцій від класичних операторів диференціювання та інтегрування. Institute of Mathematics, NAS of Ukraine 2008-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3170 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 4 (2008); 478–486 Український математичний журнал; Том 60 № 4 (2008); 478–486 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3170/3087 https://umj.imath.kiev.ua/index.php/umj/article/view/3170/3088 Copyright (c) 2008 Kochubei A. N. |
| spellingShingle | Kochubei, A. N. Кочубей, А. Н. Distributed-order calculus: An operator-theoretic interpretation |
| title | Distributed-order calculus: An operator-theoretic interpretation |
| title_alt | Числення розподіленого порядку: теоретико-операторна інтерпретація |
| title_full | Distributed-order calculus: An operator-theoretic interpretation |
| title_fullStr | Distributed-order calculus: An operator-theoretic interpretation |
| title_full_unstemmed | Distributed-order calculus: An operator-theoretic interpretation |
| title_short | Distributed-order calculus: An operator-theoretic interpretation |
| title_sort | distributed-order calculus: an operator-theoretic interpretation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3170 |
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