Hilfer-Prabhakar Derivatives and Some Applications

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by  Roberto Garra

Hilfer–Prabhakar Derivatives and Some Applications
Roberto Garra1
1Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Universita` di Roma.
Rudolf Gorenflo2
2Department of Mathematics and Informatics, Free University of Berlin.
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1 Federico Polito3
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3Dipartimento di Matematica “G. Peano”, Universit`a degli Studi di Torino.
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y Zˇivorad Tomovski4
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M 4Department of Mathematics, Sts. Cyril and Methodius University, Skopje.

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Abstract
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We present a generalization of Hilfer derivatives in which Riemann–Liouville integrals are replaced by
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more general Prabhakar integrals. We analyze and discuss its properties. Furthermore, we show some
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a applications of these generalized Hilfer–Prabhakar derivatives in classical equations of mathematical
m physics such as the heat and the free electron laser equations, and in difference-differential equations
[ governing the dynamics of generalized renewal stochastic processes.

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Keywords: Hilfer–Prabhakar derivatives, Prabhakar Integrals, Mittag–Leffler functions, Generalized
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Poisson Processes
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6 1. Introduction
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Intherecentyearsfractionalcalculushasgainedmuchinterestmainlythankstotheincreasingpresence
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4 of research works in the applied sciences considering models based on fractional operators. Beside
1 that, the mathematical study of fractional calculus has proceeded, leading to intersections with other
: mathematical fields such as probability and the study of stochastic processes.
v
i In the literature, several different definitions of fractional integrals and derivatives are present. Some
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of them such as the Riemann–Liouville integral, the Caputo and the Riemann–Liouville derivatives
r are thoroughly studied and actually used in applied models. Other less-known definitions such as the
a
Hadamard and Marchaud derivatives are mainly subject of mathematical investigation (the reader
interested in fractional calculus in general can consult one of the classical reference texts such as
[38, 20, 33]).
InthispaperweintroduceanovelgeneralizationofderivativesofbothRiemann–LiouvilleandCaputo
types and show the effect of using it in equations of mathematical physics or related to probability. In
order to do so, we start from the definition of generalized fractional derivatives given by R. Hilfer [16].
Theso-calledHilferfractionalderivativeisinfactaveryconvenientwaytogeneralizebothdefinitionsof
derivativesasitactuallyinterpolatesthembyintroducingonlyoneadditionalrealparameterν ∈[0,1].
The further generalization that we are going to discuss in this paper is given by replacing Riemann–
Liouville fractional integrals with Prabhakar integrals in the definition of Hilfer derivatives. We recall
Preprint submitted to Elsevier May 13, 2014
that the Prabhakar integral [36] is obtained by modifying the Riemann–Liouville integral operator by
extendingitskernelwithathree-parameterMittag–Lefflerfunction,afunctionwhichextendsthewell-
known two-parameter Mittag–Leffler function. This latter function was used by J.D. Tamarkin [44]
in 1930 and later gained importance in treating problems of fractional relaxation and oscillation, see
e.g. [25] for a grand survey. The Hilfer–Prabhakar derivative (which contains the Hilfer derivative
as a specific case) interpolates the Prabhakar derivative, first introduced in [18] and its Caputo-
like regularized counterpart. In Section 4, we study some of its properties and, in Section 5, we
discuss some related applications of interest in mathematical physics and probability. We commence
by analyzing the time-fractional heat equation involving Hilfer–Prabhakar derivatives. We discuss
the main differences between the solution of the Cauchy problems involving the non-regularized and
the regularized operators. Another integro-differential equation of interest for applications is the free
electron laser (FEL) integral equation [6]. This equation arises in the description of the unsatured
behavior of the free electron laser. Several generalizations of this equation involving other fractional
operators have been studied in literature (see e.g. [19]). Taking inspiration from these works, we
study a FEL-type integro-differential equation involving Hilfer–Prabhakar fractional derivatives. A
further application that we study in Section 5.3 regards the derivation of a renewal point process
which is in fact a direct generalization of the classical homogeneous Poisson process and the time-
fractional Poisson process. The connection with Hilfer–Prabhakar derivatives comes from the fact
that the state probabilities are governed by time-fractional difference-differential equations involving
Hilfer–Prabhakar derivatives. We give a complete discussion of the main properties of this process,
providing also the explicit form of the probability generating function, a subordination representation
in terms of a time-changed Poisson process, and its renewal structure.
Before introducing the definition of the Hilfer–Prabhakar derivatives, in Section 2, in order to make
the paper self-contained, we recall some basic definitions and results of fractional calculus. Section 3
presents instead the definition of Hilfer derivatives along with some of their properties.
2. Preliminaries on fractional calculus
Before introducing the non-regularized and regularized Hilfer–Prabhakar differential operators, for
the reader’s convenience, in this section we recall some definitions of classical fractional operators.
In particular, the classical Riemann–Liouville derivative and its regularized operator (the so-called
Caputoderivative)willbedescribed. However, inordertogainmoreinsightonfractionalcalculusthe
reader can consult the classical reference books [38, 33, 20, 7].
Definition 2.1 (Riemann–Liouville integral). Let f ∈ L1 [a,b], where −∞ ≤ a < t < b ≤ ∞, be a
loc
locally integrable real-valued function. The Riemann–Liouville integral is defined as
1 (cid:90) t f(u)
Iα f(t)= du=(f ∗K )(t), α>0, (1)
a+ Γ(α) (t−u)1−α α
a
where K (t)=tα−1/Γ(α).
α
Definition 2.2 (Riemann–Liouville derivative). Let f ∈ L1[a,b], −∞ ≤ a < t < b ≤ ∞, and
f ∗K ∈Wm,1[a,b], m=(cid:100)α(cid:101), α>0, where Wm,1[a,b] is the Sobolev space defined as
m−α
(cid:26) dm (cid:27)
Wm,1[a,b]= f ∈L1[a,b]: f ∈L1[a,b] . (2)
dtm
The Riemann–Liouville derivative of order α>0 is defined as
dm 1 dm (cid:90) t
Dα f(t)= Im−αf(t)= (t−s)m−1−αf(s)ds. (3)
a+ dtm a+ Γ(m−α)dtm
a
2
For n ∈ N, we denote by ACn[a,b] the space of real-valued functions f(t) which have continuous
derivativesuptoordern−1on[a,b]suchthatf(n−1)(t)belongstothespaceofabsolutelycontinuous
functions AC[a,b]:
(cid:26) dn−1 (cid:27)
ACn[a,b]= f :[a,b]→R: f(x)∈AC[a,b] . (4)
dxn−1
Definition 2.3 (Caputo derivative). Let α>0, m=(cid:100)α(cid:101), and f ∈ACm[a,b]. The Caputo derivative
of order α>0 is defined as
dm 1 (cid:90) t dm
CDα f(t)=Im−α f(t)= (t−s)m−1−α f(s)ds. (5)
a+ a+ dtm Γ(m−α) dsm
a
InthespaceofthefunctionsbelongingtoACm[a,b]thefollowingrelationbetweenRiemann–Liouville
and Caputo derivatives holds [17].
Theorem 2.1. For f ∈ACm[a,b], m=(cid:100)α(cid:101), α∈R+\N, the Riemann–Liouville derivative of order α
of f exists almost everywhere and it can be written as
m(cid:88)−1 (x−a)k−α
Dα f(t)=CDα f(t)+ f(k)(a+). (6)
a+ a+ Γ(k−α+1)
k=0
The above theorem gives the set of functions where the Riemann–Liouville derivative can be regular-
ized. Moreover, if f(t)∈ACm[a,b], we have (see e.g. [1])
dk
lim Im−αf(t)=0, ∀ 0≤k ≤m−1. (7)
t→a+ dtk a+
Indeed, taking the Laplace transform of both sides of (6) the equality holds if (7) is true.
See also Gorenflo and Mainardi [11] for more information. In that paper (page 228) the Caputo
fractional derivative was baptized so, ad late it gained rising popularity by Podlubny’s book [33] on
fractional differential equations.
3. Hilfer derivatives
Inaseriesofworks(see[17]andthereferencestherein), R.Hilferstudiedapplicationsofageneralized
fractionaloperatorhavingtheRiemann–LiouvilleandtheCaputoderivativesasspecificcases(seealso
[45, 43]).
Definition 3.1 (Hilfer derivative). Let µ ∈ (0,1), ν ∈ [0,1], f ∈ L1[a,b], −∞ ≤ a < t < b ≤ ∞,
f ∗K ∈AC1[a,b]. The Hilfer derivative is defined as
(1−ν)(1−µ)
(cid:18) (cid:19)
d
Dµ,νf(t)= Iν(1−µ) (I(1−ν)(1−µ)f) (t), (8)
a+ a+ dt a+
Hereafterandwithoutlossofgeneralityweseta=0. Thegeneralization(8), forν =0, coincideswith
the Riemann–Liouville derivative (3) and for ν = 1 with the Caputo derivative (5). A relevant point
in the following discussion regards the initial conditions that should be considered in order to solve
fractional Cauchy problems involving Hilfer derivatives. Indeed, in view of the Laplace transform of
the Hilfer derivative ([45], formula (1.6))
L[Dµ,νf](s)=sµL[f](s)−sν(µ−1)(I(1−ν)(1−µ)f)(0+), (9)
0+ 0+
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itisclearthattheinitialconditionsthatmustbeconsideredareoftheform(I(1−ν)(1−µ)f)(0+),i.e.on
0+
theinitialvalueofthefractionalintegraloforder(1−ν)(1−µ). Theseinitialconditionsdonothavea
clearphysicalmeaningunlessν =1. InordertoobtainaregularizedversionoftheHilferderivative,we
must restrict ourselves to the set of absolutely continuous functions AC1[0,b] and therefore applying
Theorem 2.1 we obtain, for µ∈(0,1),
(cid:18) (cid:19)
d
Dµ,νf(t)= Iν(1−µ) (I(1−ν)(1−µ)f) (t) (10)
0+ 0+ dt 0+
(cid:18) d (cid:19) tνµ−ν−µf(0+)
= Iν(1−µ)I(1−ν)(1−µ) f (t)+Iν(1−µ)
0+ 0+ dt 0+ Γ(1−ν−µ+νµ)
d t−µf(0+) t−µf(0+)
=I1−µ f(t)+ =CDµ f(t)+ ,
0+ dt Γ(1−µ) 0+ Γ(1−µ)
where we used the well-known semi-group property of Riemann–Liouville integrals and where CDµ
0+
is the Caputo derivative (5). From (10) it follows that in the space AC1[0,b] the Hilfer derivative (8)
coincides with the Riemann–Liouville derivative of order µ, and the regularized Hilfer derivative can
be written as
t−µf(0+)
Dµ,νf(t)− , (11)
0+ Γ(1−µ)
which coincides with CDµ and which in fact does not depend on the parameter ν.
0+
4. Hilfer–Prabhakar derivatives
We introduce a generalization of Hilfer derivatives by substituting in (8) the Riemann–Liouville inte-
grals with a more general integral operator with kernel
eγ (t)=tµ−1Eγ (ωtρ), t∈R, ρ,µ,ω,γ ∈C, (cid:60)(ρ),(cid:60)(µ)>0, (12)
ρ,µ,ω ρ,µ
where
(cid:88)∞ Γ(γ+k) xk
Eγ (x)= , (13)
ρ,µ Γ(γ)Γ(ρk+µ) k!
k=0
is the generalized Mittag–Leffler function first investigated in [36]. The so-called Prabhakar integral is
defined as follows [36, 18].
Definition 4.1 (Prabhakar integral). Let f ∈L1[0,b], 0<t<b≤∞. The Prabhakar integral can be
written as
(cid:90) t
Eγ f(t)= (t−y)µ−1Eγ [ω(t−y)ρ]f(y)dy =(f ∗eγ )(t), (14)
ρ,µ,ω,0+ ρ,µ ρ,µ,ω
0
where ρ,µ,ω,γ ∈C, with (cid:60)(ρ),(cid:60)(µ)>0.
We also recall that the left-inverse to the operator (14), the Prabhakar derivative, was introduced in
[18]. We define it below in a slightly different form.
Definition 4.2 (Prabhakar derivative). Let f ∈ L1[0,b], 0 < x < b ≤ ∞, and f ∗e−γ (·) ∈
ρ,m−µ,ω
Wm,1[0,b], m=(cid:100)µ(cid:101). The Prabhakar derivative is defined as
dm
Dγ f(x)= E−γ f(x) (15)
ρ,µ,ω,0+ dxm ρ,m−µ,ω,0+
where µ,ω,γ,ρ∈C, (cid:60)(µ),(cid:60)(ρ)>0.
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Observing now that the Riemann–Liouville integrals in (3) can be expressed in terms of Prabhakar
integrals as
Im−(µ+θ)f(x)=E0 f(x), (16)
0+ ρ,m−(µ+θ),ω,0+
we have that,
dm
Dγ f(x)= E−γ f(x) (17)
ρ,µ,ω,0+ dxm ρ,m−µ,ω,0+
dm
= Im−(µ+θ)E−γ f(x)
dxm ρ,θ,ω,0+
=Dµ+θE−γ f(x), θ ∈C, (cid:60)(θ)>0,
0+ ρ,θ,ω,0+
where we used the fact that (see [18], Theorem 8)
Eγ Eσ f(x)=Eγ+σ f(x). (18)
ρ,µ,ω,0+ ρ,ν,ω,0+ ρ,µ+ν,ω,0+
Note that formula (17) concides with the definition given by [18]. As expected, the inverse operator
(17) of the Prabhakar integral generalizes the Riemann–Liouville derivative. Its regularized Caputo
counterpart is given, for functions f ∈ACm[0,b], 0<x<b≤∞, by
dm
CDγ f(x)=E−γ f(x) (19)
ρ,µ,ω,0+ ρ,m−µ,ω,0+dxm
m−1
(cid:88)
=Dγ f(x)− xk−µE−γ (ωxρ)f(k)(0+).
ρ,µ,ω,0+ ρ,k−µ+1
k=0
Proposition 4.1. Let µ>0 and f ∈ACm[0,b], 0<x<b≤∞. Then
(cid:34) m(cid:88)−1xk (cid:35)
CDγ f(x)=Dγ f(x)− f(k)(0+) . (20)
ρ,µ,ω,0+ ρ,µ,ω,0+ k!
k=0
Proof. It easily follows from (17) and Corollary 2.3 of [18].
We are now ready to define the Hilfer–Prabhakar derivative, interpolating (19) and (17).
Definition 4.3 (Hilfer–Prabhakar derivative). Let µ ∈ (0,1), ν ∈ [0,1], and let f ∈ L1[a,b], 0 < t <
b≤∞, f ∗e−γ(1−ν) (·)∈AC1[0,b]. The Hilfer–Prabhakar derivative is defined by
ρ,(1−ν)(1−µ),ω
(cid:18) (cid:19)
d
Dγ,µ,ν f(t)= E−γν (E−γ(1−ν) f) (t), (21)
ρ,ω,0+ ρ,ν(1−µ),ω,0+dt ρ,(1−ν)(1−µ),ω,0+
where γ,ω ∈R, ρ>0, and where E0 f =f.
ρ,0,ω,0+
We observe that (21) reduces to the Hilfer derivative for γ = 0. Moreover, for ν = 1 and ν = 0 it
coincides with (19) and (17), respectively (note that m=1).
Lemma 4.1. The Laplace transform of (21) is given by
(cid:18) (cid:19)
d
L E−γν (E−γ(1−ν) f) (s) (22)
ρ,ν(1−µ),ω,0+dt ρ,(1−ν)(1−µ),ω,0+
(cid:104) (cid:105)
=sµ[1−ωs−ρ]γL[f](s)−s−ν(1−µ)[1−ωs−ρ]γν E−γ(1−ν) f(t) .
ρ,(1−ν)(1−µ),ω,0+
t=0+
5
Proof. By recurring to formula (2.19) of [18], i.e.
L(cid:2)tµ−1E−γ(ωtρ)(cid:3)(s)=s−µ(1−ωs−ρ)γ, γ,ω,ρ,µ∈C, (cid:60)(µ)>0, (23)
ρ,µ
with s∈C, (cid:60)(s)>0, |ωs−ρ|<1. We can write
(cid:18) (cid:19)
d
L E−γν (E−γ(1−ν) f) (s) (24)
ρ,ν(1−µ),ω,0+dt ρ,(1−ν)(1−µ),ω,0+
(cid:104) (cid:105) (cid:20)d (cid:21)
=L tν(1−µ)−1E−γν (ωtρ) (s)·L (E−γ(1−ν) f) (s)
ρ,ν(1−µ) dt ρ,(1−ν)(1−µ),ω,0+
(cid:104) (cid:105)
=s−ν(1−µ)[1−ωs−ρ]γνsL t(1−ν)(1−µ)−1E−γ(1−ν) (ωtρ) (s)L[f](s)
ρ,(1−ν)(1−µ)
(cid:104) (cid:105)
−s−ν(1−µ)[1−ωs−ρ]γν E−γ(1−ν) f(t)
ρ,(1−ν)(1−µ),ω,0+
t=0+
(cid:104) (cid:105)
=sµ[1−ωs−ρ]γL[f](s)−s−ν(1−µ)[1−ωs−ρ]γν E−γ(1−ν) f(t) .
ρ,(1−ν)(1−µ),ω,0+
t=0+
In order to consider Cauchy problems involving initial conditions depending only on the function and
its integer-order derivatives we use the regularized version of (21), that is, for f ∈AC1[0,b], we have
(cid:18) (cid:19) (cid:18) (cid:19)
d d
CDγ,µ f(t)= E−γν E−γ(1−ν) f (t)= E−γ f (t), (25)
ρ,ω,0+ ρ,ν(1−µ),ω,0+ ρ,(1−ν)(1−µ),ω,0+dt ρ,1−µ,ω,0+dt
We remark that, in the regularized version of the Hilfer–Prabhakar derivative (as well as in the regu-
larized Hilfer derivative—see (10)), there is no dependence on the interpolating parameter ν.
Lemma 4.2. The Laplace transform of the operator (25) is given by
L[CDγ,µ f](s)=sµ[1−ωs−ρ]γL[f](s)−sµ−1[1−ωs−ρ]γf(0+). (26)
ρ,ω,0+
Proof. It ensues from similar calculations to those in Lemma 4.1.
From Lemmas 4.1 and 4.2 we have that the relation between the two operators (21) and (25) is given
by
CDγ,µ f(t)=Dγ,µ,ν f(t)−t−µE−γ (ωtρ)f(0+), (27)
ρ,ω,0+ ρ,ω,0+ ρ,1−µ
observing that, for absolutely continuous functions f ∈AC1[0,b],
(cid:104) (cid:105)
E−γ(1−ν) f(t) =0, (28)
ρ,(1−ν)(1−µ),ω,0+
t=0+
and
L−1[sµ−1[1−ωs−ρ]γ](t)f(0+)=t−µE−γ (ωtρ)f(0+). (29)
ρ,1−µ
5. Applications
We show below some applications of Hilfer–Prabhakar derivatives in equations of interest for mathe-
matical physics and probability.
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5.1. Time-fractional heat equation
In the recent years more and more papers have been devoted to the mathematical analysis of versions
of the time-fractional heat equation and to the study of its applications in mathematical physics and
probability theory (see for example [29, 23, 24, 39, 46, 35] and the references therein).
Herewestudyageneralizationofthetime-fractionalheatequationinvolvingHilfer–Prabhakarderiva-
tives. We present analytical results for the time-fractional heat equation involving both regularized
and non regularized Hilfer–Prabhakar derivatives in order to highlight the main differences between
the two cases.
We start by considering the fractional heat equation involving the non-regularized operator Dγ,µ,ν .
ρ,ω,0+
Theorem 5.1. The solution to the Cauchy problem

D(cid:104)ργ,,ωµ,,0ν+u(x,t)=K∂∂x22u(x(cid:105),t), t>0, x∈R,
E−γ(1−ν) u(x,t) =g(x), (30)
ρ,(1−ν)(1−µ),ω,0+
lim u(x,t)=0, t=0+
x→±∞
with µ∈(0,1), ν ∈[0,1], ω ∈R, K,ρ>0, γ ≥0, is given by
u(x,t)=(cid:90) +∞dke−ikxgˆ(k) 1 (cid:88)∞ (−K)ntµ(n+1)−ν(µ−1)−1Eγ(n+1−ν) (ωtρ)k2n. (31)
2π ρ,µ(n+1)−ν(µ−1)
−∞ n=0
Proof. We denote with u˜(x,s) = L(u)(x,s) the Laplace transform with respect to the time variable
t and uˆ(k,t) = F(u)(k,t) the Fourier transform with respect to the space variable x. Taking the
Fourier–Laplace transform of (30), by formula (22), we have
sµ(1−ωs−ρ)γuˆ˜(k,s)−sν(µ−1)(1−ωs−ρ)γνgˆ(k)=−Kk2uˆ˜(k,s), (32)
so that
sν(µ−1)(1−ωs−ρ)γνgˆ(k)
uˆ˜(k,s)= (33)
sµ(1−ωs−ρ)γ +Kk2
(cid:18) Kk2 (cid:19)−1
=s−µ+ν(µ−1)(1−ωs−ρ)−γ(1−ν)gˆ(k) 1+
sµ(1−ωs−ρ)γ
= (cid:88)∞ (cid:0)−Kk2(cid:1)ns−µ(n+1)+ν(µ−1)(1−ωs−ρ)−γ(n+1−ν)gˆ(k), (cid:12)(cid:12)(cid:12) Kk2 (cid:12)(cid:12)(cid:12)<1.
(cid:12)sµ(1−ωs−ρ)γ(cid:12)
n=0
Inverting first the Laplace transform it yields
∞
uˆ(k,t)= (cid:88)(−K)ntµ(n+1)−ν(µ−1)−1Eγ(n+1−ν) (ωtρ)k2ngˆ(k). (34)
ρ,µ(n+1)−ν(µ−1)
n=0
Note that the inversion term by term of the Laplace transform is possible in view of Theorem 30.1 by
Doetsch[8]providedtochooseasufficientlylargeabscissafortheinverseintegralandbyrecallingthat
the generalized Mittag Leffler function is defined as an absolutely convergent series. The convergence
of (34)andingeneralofseriesofthesameform(seebelow)canbeprovedbyusingthesametechnique
as in Appendix C of [40]. Indeed the function (34) is in fact a repeated series:
(cid:88)∞ (cid:0)−Kk2(cid:1)ntµn (cid:88)∞ (ωtρ)rΓ(r+γ(n+1−ν))
uˆ(k,t)=gˆ(k)t−ν(µ−1)−1+µ . (35)
Γ(γ(n+1−ν)) r!Γ(ρr+µ(n+1)−ν(µ−1))
n=0 r=0
7
Sincethethree-parametergeneralizedMittag–Lefflerisanentirefunction,inordertoprovetheabsolute
convergence of (34) it is sufficient to show that for each r ∈N∪{0}
(cid:88)∞ (−Kk2tµ)nΓ(r+γ(n+1−ν))
(36)
Γ(γ(n+1−ν))Γ(ρr+µ(n+1)−ν(µ−1))
n=0
converges absolutely. We need to study the ratio
(cid:12)(cid:12) (−Kk2tµ)n+1Γ(r+γ(n+2−ν)) Γ(γ(n+1−ν))Γ(ρr+µ(n+1)−ν(µ−1))(cid:12)(cid:12)
(cid:12) (cid:12) (37)
(cid:12)Γ(γ(n+2−ν))Γ(ρr+µ(n+2)−ν(µ−1)) (−Kk2tµ)nΓ(r+γ(n+1−ν)) (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)
=(cid:12)(cid:12)(−Kk2tµ)(cid:12)(cid:12)(cid:12)(cid:12)Γ(γ(n+1−ν)+γ+r)(cid:12)(cid:12)(cid:12)(cid:12) Γ(γ(n+1−ν)) (cid:12)(cid:12)(cid:12)(cid:12) Γ(ρr+µ(n+1)−ν(µ−1)) (cid:12)(cid:12)
(cid:12) Γ(γ(n+1−ν)+γ) (cid:12)(cid:12)Γ(γ(n+1−ν)+r)(cid:12)(cid:12)Γ(ρr+µ(n+1)−ν(µ−1)+µ)(cid:12)
≈(cid:12)(cid:12)(−Kk2tµ)(cid:12)(cid:12)[ρr+µ(n+1)−ν(µ−1)]−µ.
Last step of the above formula is valid for large values of k and can be determined by means of the
well-known asymptotics
(cid:20) (cid:21)
Γ(z+a) (a−b)(a+b−1)
=za−b 1+ +O(z−2) , (38)
Γ(z+b) 2z
for |z| → ∞, |arg(z)| ≤ π −ε, |arg(z +a)| ≤ π −ε, 0 < ε < π. Formula (37) is zero for n → ∞
implying absolute convergence of (36) and therefore of (34).
To conclude the proof of the theorem, by applying the inverse Fourier transform to (34) we obtain the
claimed result.
We now discuss the case with the regularized Hilfer–Prabhakar derivative CDγ,µ .
ρ,ω,0+
Theorem 5.2. The solution to the Cauchy problem
CDγ,µ u(x,t)=K ∂2 u(x,t), t>0, x∈R,
 ρ,ω,0+ ∂x2
u(x,0+)=g(x), (39)
lim u(x,t)=0,
x→±∞
with µ∈(0,1), ω ∈R, K,ρ>0, γ ≥0, is given by
u(x,t)=(cid:90) +∞dke−ikxgˆ(k) 1 (cid:88)∞ (−Ktµ)nEγn (ωtρ)k2n. (40)
2π ρ,µn+1
−∞ n=0
Proof. Taking the Fourier–Laplace transform of (39), by formula (26), we have that
sµ(1−ωs−ρ)γuˆ˜(k,s)−sµ−1(1−ωs−ρ)γgˆ(k)=−Kk2uˆ˜(k,s), (41)
so that
sµ−1(1−ωs−ρ)γgˆ(k) (cid:18) Kk2 (cid:19)−1
uˆ˜(k,s)= =s−1gˆ(k) 1+ (42)
sµ(1−ωs−ρ)γ +Kk2 sµ(1−ωs−ρ)γ
= (cid:88)∞ (cid:0)−Kk2(cid:1)ns−µn−1(1−ωs−ρ)−γngˆ(k), (cid:12)(cid:12)(cid:12) Kk2 (cid:12)(cid:12)(cid:12)<1.
(cid:12)sµ(1−ωs−ρ)γ(cid:12)
n=0
Inverting first the Laplace transform (see the proof of Theorem 5.1 for more information on the
inversion) it yields
∞
uˆ(k,t)= (cid:88)(−Ktµ)nEγn (ωtρ)k2ngˆ(k). (43)
ρ,µn+1
n=0
By applying the inverse Fourier transform we obtain the claimed result.
8
5.2. Fractional free electron laser equation
The free electron laser integro-differential equation
(cid:40)dy =−iπg(cid:82)x(x−t)eiη(x−t)y(t)dt, g,η ∈R, x∈(0,1],
dx 0 (44)
y(0)=1,
describes the unsaturated behavior of the free electron laser (FEL) (see for example [6]). In recent
years many attempts to solve the generalized fractional integro-differential FEL equation have been
proposed (see for example [19]). Here we consider the following fractional generalization of the FEL
equation, involving Hilfer–Prabhakar derivatives.

Dγ,µ,ν y(x)=λE(cid:36) y(x)+f(x), x∈(0,∞), f(x)∈L1[0,∞),
ρ,ω,0+ ρ,µ,ω,0+
(cid:104) (cid:105) (45)
E−γ(1−ν) y(x) =κ, κ≥0,
 ρ,(1−ν)(1−µ),ω,0+
x=0+
where µ ∈ (0,1), ν ∈ [0,1], ω,λ ∈ C, ρ > 0, γ,(cid:36) ≥ 0. This generalizes the problem studied in [19],
correspondingtoν =γ =0. Heref(x)isagivenfunction. TheoriginalFELequationisthenretrieved
for γ =0, ν =0, µ→1, f ≡0, λ=−iπg, ω =iη, ρ=(cid:36) =κ=1.
We have the following
Theorem 5.3. The solution to the Cauchy problem (45) is given by
∞ ∞
y(x)=κ(cid:88)λkxν(1−µ)+µ+2µk−1Eγ+k((cid:36)+γ)−γν (ωxρ)+(cid:88)λkEγ+k((cid:36)+γ) f(x). (46)
ρ,ν(1−µ)+µ+2kµ ρ,µ(2k+1),ω,0+
k=0 k=0
Proof. By taking the Laplace transform of (45) (see (22)) we get
sµ(1−ωs−ρ)γL[y](s)−κs−ν(1−µ)(1−ωs−ρ)γν =λL[xµ−1E(cid:36) (ωxρ)](s)·L[y](s)+L[f](s), (47)
ρ,µ
so that
κs−ν(1−µ)−µ(1−ωs−ρ)γν−γ s−µ(1−ωs−ρ)−γL[f](s)
L[y](s)= + (48)
1−λs−2µ(1−ωs−ρ)−(cid:36)−γ 1−λs−2µ(1−ωs−ρ)−(cid:36)−γ
∞
(cid:88)
=κ λks−ν(1−µ)−µ−2µk(1−ωs−ρ)γν−γ−k((cid:36)+γ)
k=0
∞
(cid:88)
+ λks−µ(2k+1)(1−ωs−ρ)−γ−k((cid:36)+γ)L[f](s).
k=0
Laststepisvalidfor|λs−2µ(1−ωs−ρ)−(cid:36)−γ|. InvertingtheLaplacetransform(seetheproofofTheorem
5.1 for more information) and using the convolution theorem, we obtain the claimed result.
Example5.1. LetusconsidertheCauchyproblem (45)withκ=0,f(x)=xm−1. Bydirectcalculation
we have that
Eγ+k((cid:36)+γ) xm−1 =Γ(m)xµ(2k+1)+m−1Eγ+k((cid:36)+γ) (ωxρ), (49)
ρ,µ(2k+1),ω,0+ ρ,µ(2k+1)+m
and the explicit solution of the Cauchy problem is given by
∞
y(x)=Γ(m)xµ+m−1(cid:88)(λx2µ)kEγ+k((cid:36)+γ) (ωxρ). (50)
ρ,µ(2k+1)+m
k=0
9
Example5.2. LetusconsidertheCauchyproblem (45)withκ=0,f(x)=xm−1Eσ (ωxρ). Recalling
ρ,m
that
Eγ+k((cid:36)+γ) xm−1Eσ (ωxρ)=xµ(2k+1)+m−1Eγ+k((cid:36)+γ)+σ(ωxρ), (51)
ρ,µ(2k+1),ω,0+ ρ,m ρ,µ(2k+1)+m
we obtain
∞
y(x)=xµ+m−1(cid:88)(λx2µ)kEγ+k((cid:36)+γ)+σ(ωxρ). (52)
ρ,µ(2k+1)+m
k=0
5.3. Fractional Poisson processes involving Hilfer–Prabhakar derivatives
InthissectionwepresentageneralizationofthehomogeneousPoissonprocessforwhichthegoverning
difference-differentialequationscontaintheregularizedHilfer–Prabhakardifferentialoperatoractingin
time. Theconsideredframeworkgeneralizesalsothetime-fractionalPoissonprocesswhichintherecent
yearshasbecomesubjectofintenseresearch. Itiswellknownthatthestateprobabilitiesoftheclassical
Poisson process and its time-fractional generalization can be found by solving an infinite system of
difference-differential equations. We solve an analogous infinite system and find the corresponding
state probabilities that we give in form of an infinite series and in integral form. As the zero state
probability of a renewal process coincides with the residual time probability, we can characterize our
process also by its waiting distribution (the common way of characterizing a renewal process). We
willseeinthefollowingthatthestateprobabilitiesofthegeneralizedPoissonprocessareexpressedby
functions which in fact generalize the classical Mittag–Leffler function. The Mittag–Leffler function
appeared as residual waiting time between events in renewal processes already in the Sixties of the
past century, namely processes with properly scaled thinning out the sequence of events in a power
law renewal process (see [10] and [27]). Such a process in essence is a fractional Poisson process. It
must however be said that Gnedenko and Kovalenko did their analysis only in the Laplace domain,
not recognizing their result as the Laplace transform of a Mittag–Leffler type function. Balakrishnan
in1985[2]alsofoundthisLaplacetransformashighlyrelevantforanalysisoftime-fractionaldiffusion
processes, but did not identify it as arising from a Mittag–Leffler type function. In the Nineties of the
past century the Mittag–Leffler function arrived at its deserved honour, more and more researchers
became aware of it and used it. Let us only sketch a few highlights. Hilfer and Anton [15] were the
first authors who explicitly introduced the Mittag–Leffler waiting-time density
d
f (t)=− E (−tµ)=tµ−1E (−tµ) (53)
µ dt µ µ,µ
(writing it in form of a Mittag–Leffler function with two indices) into the theory of continuous time
random walk. They showed that it is needed if one wants to get as evolution equation for the so-
journ density the fractional variant of the Kolmogorov–Feller equation. In modern terminology they
subordinated a random walk to the fractional Poisson process. By completely different argumentation
the authors of [26] also discussed the relevance of f (t) in theory of continuous time random walk.
µ
However, all these authors did not treat the fractional Poisson process as a subject of study in its own
rightbutsimplyasusefulforgeneralanalysisofcertainstochasticprocesses. Thedetailedanalyticand
probabilistic investigation was started (as far as we know) in 2000 by Repin and Saichev [37]. More
and more researchers then, often independently of each other, investigated this renewal process. Let
ushereonlyrecallthefewrelevantpapers[12,13,42,4,3,5,22,31,34,30]andseealsothereferences
cited therein.
Let us thus start with the governing equations for the state probabilities. In view of Section 4 we
define the following Cauchy problem involving the regularized operator CDγ,µ .
ρ,ω,0+
10