RESEARCH PAPER
FUNDAMENTAL SOLUTION OF A DISTRIBUTED
ORDER TIME-FRACTIONAL DIFFUSION-WAVE
EQUATION AS PROBABILITY DENSITY
Rudolf Gorenflo 1 , Yuri Luchko 2 , Mirjana Stojanović
3
Dedicated to Professor Francesco Mainardi
on the occasion of his 70th anniversary
Abstract
In this paper, the Cauchy problem for the spatially one-dimensional
distributed order diffusion-wave equation
2
∂2
p(β) Dtβ u(x, t) dβ =
u(x, t)
∂x2
0
is considered. Here, the time-fractional derivative Dtβ is understood in
the Caputo sense and p(β) is a non-negative weight function with support
somewhere in the interval [0, 2]. By employing the technique of the Fourier
and Laplace transforms, a representation of the fundamental solution of the
Cauchy problem in the transform domain is obtained. The main focus is
on the interpretation of the fundamental solution as a probability density
function of the space variable x evolving in time t. In particular, the fundamental solution of the time-fractional distributed order wave equation
(p(β) ≡ 0, 0 ≤ β < 1) is shown to be non-negative and normalized. In
the proof, properties of the completely monotone functions, the Bernstein
functions, and the Stieltjes functions are used.
MSC 2010 : Primary 26A33; Secondary 33E12, 35S10, 45K05
c 2013 Diogenes Co., Sofia
pp. 297–316 , DOI: 10.2478/s13540-013-0019-6
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298
Key Words and Phrases: time-fractional diffusion-wave equation of distributed order, Laplace transform, Fourier transform, probability density,
completely monotone functions, Bernstein functions, complete Bernstein
functions, Stieltjes functions
1. Introduction
The growing interest in fractional differential equations without or with
distributed orders can be explained among other things by their applications as models in anomalous diffusion and relaxation phenomena that are
typical in biophysics, plasma physics, econophysics, etc. (see e.g. [9] and
[35] and references therein). Besides, these equations have applications in
visco-elasticity ([23], [27]) as well as in statistical physics for description of
fractional diffusion processes that are limits of random walks with ”power
law” waiting time. In [7], [10], and [35], integro-differential equations with
distributed order temporal derivatives were treated and employed for kinetic description of anomalous diffusion and relaxation phenomena. In
particular, the retarding sub-diffusion was considered there. In [3] and [8],
fractional diffusion in inhomogeneous media was investigated.
The distributed order fractional derivative in the form
2
p(β)
f )(t) =
p(β)(Dβ f )(t) dβ,
(D
0
D β being the Caputo fractional derivative of order β and p(β) a nonnegative weight function or non-negative generalized function, was first
introduced by Caputo in [6] to generalize the stress-strain relation of inelastic media.
For the theory of the general linear evolution equations with temporal
fractional derivatives of distributed order we refer the reader e.g. to [17]
and [36], where some existence and uniqueness results for the initial-value
problems for these equations were given. Let us also cite the early paper
[30].
In [19], boundary value problems for the generalized time-fractional
diffusion equation of distributed order over an open bounded domain G ×
[0, T ], G ∈ IRn were considered. To show the uniqueness of the solution of the problem, an appropriate maximum principle for the generalized
time-fractional diffusion equation of distributed order was formulated and
proved there. In [1] and [2], time-fractional distributed order diffusion-wave
equations have been considered in spaces of generalized functions.
Starting with the pioneering paper [34] for many years the researchers
concentrated their interest on the use of the special weight function p(β) =
δ(β − α), 0 < α ≤ 2, for which this equation reduces to the single order
time-fractional diffusion-wave equation (see e.g. [11], [13], [20], [22], [24],
[25], [32], to mention only a few of many relevant publications):
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FUNDAMENTAL SOLUTION OF A DISTRIBUTED . . .
299
∂2
u(x, t).
∂x2
We refer to this equation as the fractional diffusion equation in the case
0 < α ≤ 1, and as the fractional wave equation in the case 1 < α ≤ 2.
The fundamental solution G(x, t) of the Cauchy problem for this equation can be interpreted as a probability density (if 0 < α ≤ 1 as the density
of the sojourn probability of a diffusing particle to be in point x at time
instant t), it is non-negative (indeed positive for t > 0) and normalized:
∞
G(x, t)dx = 1.
Dtα u(x, t) =
−∞
This has already been shown by Schneider and Wyss in [34], and this result
was later generalized in [7] to the one-dimensional time-fractional diffusion
equation of distributed order with an arbitrary non-negative weight function
satisfying the condition p(β) ≡ 0, 1 < β ≤ 2.
Schneider and Wyss also considered the special case p(β) = δ(β − α),
1 < α ≤ 2, and showed that again the first fundamental solution G1 (x, t)
can be viewed as a probability density in space x, evolving in time t, but
remarkably the generalization of this result to more than one spatial dimension is not possible.
In this paper, our main aim is to extend this result of Schneider and
Wyss to the one-dimensional time-fractional wave equation of distributed
order with an arbitrary non-negative weight function satisfying the condition p(β) ≡ 0, 0 ≤ β ≤ 1. To achieve our goal, we employ the basic theory
of completely monotone functions, Stieltjes functions, Bernstein functions,
and complete Bernstein functions presented in e.g. [33], for brevity in the
sequel referring to these classes of functions simply as special type functions. Using this technique, we give a new proof for the time-fractional
diffusion equation of distributed order (p(β) ≡ 0, 1 < β ≤ 2), too.
As to the general case of the weight function p(β), i.e. the case when this
function does not identically vanish both for 0 ≤ β ≤ 1 and for 1 < β ≤ 2,
only few results regarding interpretation of the first fundamental solution
G1 (x, t) as a probability density in space x, evolving in time t are known.
To the best knowledge of the authors, this problem was considered just for
a very special choice of the weight function, namely for p(β) = δ(β − 2α) +
2λδ(β − α), 0 < α ≤ 1. In this case, the distributed order equation is called
the time-fractional telegraph equation that is a fractional generalization of
the telegraph equation (α = 1):
∂2
u(x, t).
Dt2α u(x, t) + 2λDtα u(x, t) =
∂x2
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R. Gorenflo, Yu. Luchko, and M. Stojanović
In [31], it was shown that the first fundamental solution G1 to the timefractional telegraph equation can be interpreted as a probability density in
space x, evolving in time t. In case α is a rational number, explicit expressions for the probability distribution of a fractional telegraph process with
a random time and for the characteristic function of a fractional telegraph
process stopped at stable-distributed times were obtained in [4].
The rest of our paper is organized as follows: Section 2 is devoted to
the problem formulation. In Section 3, we sketch how to find the FourierLaplace transform of the first fundamental solution G1 (x, t) of the Cauchy
problem for the one-dimensional time-fractional diffusion-wave equation
of distributed order. In Section 4, we first discuss the fundamental solution G1 (x, t) in the general situation and then consider the time-fractional
diffusion equation of distributed order (p(β) ≡ 0, 1 < β ≤ 2). To show
that G1 (x, t) can be interpreted as a pdf, we first reproduce the arguments
from [7] via the subordination integral using the basic theory of completely
monotone functions. Then we present a new proof of this fact by employing
some basic properties of our special type functions. Finally, we apply the
same technique to the time-fractional wave equation of distributed order
(p(β) ≡ 0, 0 ≤ β ≤ 1) and show that its first fundamental solution can be
interpreted as a pdf. For the reader’s convenience, we list definitions and
basic properties of our special type functions (excerpted from the excellent
book [33]) in the Appendix. Let us remark that in [18] by use of these
properties Kochubei has devised and investigated a powerful generalization
of the Caputo fractional derivative comprising distributed order fractional
derivatives relevant in distributed order fractional diffusion.
2. Problem formulation
Let us now turn our attention to the Cauchy problem for the onedimensional distributed order time-fractional equation
2
∂2
p(β)Dtβ u(x, t)dβ =
u(x, t), t > 0, x ∈ IR
(2.1)
∂x2
0
with the Caputo fractional derivative
(2.2)
(D β f )(t) := (J n−β f (n) )(t), n − 1 < β ≤ n, n ∈ IN,
α
J being the Riemann-Liouville fractional integral
⎧
t
⎨ 1
(t − τ )α−1 f (τ ) dτ, α > 0,
(J α f )(t) := Γ(α) 0
⎩
f (t), α = 0,
and with a non-negative weight function p(β) that satisfies the conditions
2
p(β) dβ < +∞.
(2.3)
0 ≤ p(β), p(β) ≡ 0, β ∈ [0, 2], 0 <
0
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We note that the weight function may be a generalized function in the
sense of Gel’fand and Shilov ([14]) that represents a non-negative measure
h
in (2.1). To avoid unpleasant delicacies, we assume that 0 p(β)dβ and
2
2−h p(β)dβ tend to zero as h → 0. For theory and applications of the
Caputo derivative and the Riemann-Liouville integral we refer the reader
e.g. to [11], [15], or [32].
In what follows, we mainly deal with two important particular cases of
the equation (2.1): (i) p(β) ≡ 0 for 1 < β ≤ 2,
(2.4)
(ii) p(β) ≡ 0 for 0 ≤ β ≤ 1.
In case (i), the equation (2.1) is referred to as the distributed order timefractional diffusion equation, in case (ii) as the time-fractional wave equation. When the support of the weight function is not specified we speak
generally of the distributed order fractional diffusion-wave equation. This
naming comprises both cases (i) and (ii) as well as the third case (iii) we
now give by the conditions
p(β) dβ and 0 <
(iii) 0 <
(0,1]
p(β) dβ.
(2.5)
(1,2]
It seems that the first fundamental solution to the equation (2.1) is a pdf
in this case, too, but we were unable to prove this.
In general, the equation (2.1) possesses an infinite number of solutions.
In the real world situations that are modeled with the equation (2.1), certain conditions that describe an initial state of the corresponding process
and possibly the observations of its visible parts ensure the deterministic
character of the process. In this paper, the initial-value or Cauchy problem
for (2.1) is considered. According to the definition (2.2) of the Caputo fractional derivative, the required initial conditions are different for the case
(i):
u(x, 0) = f (x), x ∈ IR
(2.6)
and for the cases (ii) and (iii):
(2.7)
u(x, 0) = f (x), ut (x, 0) = g(x), x ∈ IR.
We are mainly interested in investigation of the fundamental solution to the
Cauchy problem for the equation (2.1). Let us denote by G1 the solution of
the Cauchy problem (2.6) with f (x) = δ(x) or the solution of the Cauchy
problem (2.7) with f (x) = δ(x), g(x) ≡ 0 and by G2 the solution of the
Cauchy problem (2.7) with f (x) ≡ 0, g(x) = δ(x). Then the solution
u(x, t) of the Cauchy problem (2.6) or (2.7) for the equation (2.1) can be
written in the form
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R. Gorenflo, Yu. Luchko, and M. Stojanović
+∞
u(x, t) =
−∞
or
+∞
u(x, t) =
−∞
G1 (x − ξ, t)f (ξ) dξ
G1 (x − ξ, t)f (ξ) dξ +
+∞
−∞
G2 (x − ξ, t)g(ξ) dξ,
respectively. For completeness, in Section 3 we shortly consider the second
fundamental solution G2 , too.
3. Fundamental solution in the Fourier-Laplace domain
Like in the case of the time-fractional diffusion or diffusion-wave equations (see e.g. [24]), the technique of the Fourier and the Laplace transforms
is usually employed to get a representation of the fundamental solution to
the Cauchy problem (2.6) or (2.7) for the equation (2.1) in the FourierLaplace domain.
For a generic function v(x), x ∈ IR, its Fourier transform is defined as
∞
eiκx v(x)dx, κ ∈ IR
(3.1)
F{v(x); κ} = v̂(κ) :=
−∞
and for a generic function w(t), t ∈ IR+ , its Laplace transform is given by
∞
e−st w(t)dt, s > s0 .
L{w(t); s} = w̃(s) :=
0
Of course, both the Fourier and the Laplace transforms are defined for
functions for which the corresponding integrals converge in one or another
sense; in the following we always suppose that the convergence conditions
are satisfied. In particular, for the δ-function it is well-known that
δ̂(κ) ≡ 1, δ̃(s) ≡ 1.
(3.2)
We now determine the Fourier-Laplace transform of the fundamental solution G1 of the equation (2.1) with the initial condition G1 (x, 0) = δ(x)
∂
G1 (x, 0) = 0 in the general case. To be
in case (i) and G1 (x, 0) = δ(x),
∂t
able to write down this transform, let us first define the auxiliary functions
β
p(β)s dβ, B2 (s) :=
p(β)sβ dβ
B1 (s) :=
(0,1]
and
(1,2]
B(s) := B1 (s) + B2 (s) =
2
p(β)sβ dβ
(3.3)
0
with, for convenience, B2 (s) ≡ 0 in case (i), so that B(s) ≡ B1 (s) in case
(i).
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Applying now in succession first the Laplace transform and then the
Fourier transform to (2.1), we obtain in case (i) the equation
ˆ (κ, s) − B1 (s) = −κ2 G̃
ˆ (κ, s)
B1 (s)G̃
1
1
s
ˆ of the first fundamental solution G
for the Fourier-Laplace transform G̃
1
1
and then the representation
ˆ (κ, s) =
G̃
1
B1 (s)/s
B(s)/s
=
.
2
B1 (s) + κ
B(s) + κ2
In the general case we first get
ˆ (κ, s) −
B(s)G̃
1
B(s)
ˆ (κ, s)
= −κ2 G̃
1
s
and then again
B(s)/s
,
B(s) + κ2
whereas for the second fundamental solution, the function G2 , we have
ˆ (κ, s) =
G̃
1
ˆ (κ, s) −
B(s)G̃
2
B(s)
ˆ (κ, s),
= −κ2 G̃
2
s2
from which follows
2
ˆ (κ, s) = B(s)/s .
G̃
2
B(s) + κ2
To get the formulas presented above, the well-known formula (see e.g. [15]
or [32])
n−1
u(k) (0+ )sβ−1−k , n−1 < β ≤ n, n ∈ IN (3.4)
L D β u(t); s = sβ ũ(s)−
k=0
for the Laplace transform of the Caputo fractional derivative (we used it
with n = 1 and n = 2) along with the standard formulas for the Fourier
transform of the second derivative and of the Dirac δ-function (see (3.2))
were employed.
Being essentially interested in the first fundamental solution G1 we note
that both in the case (i) and in the general case we have
ˆ (κ, s) =
G̃
1
B(s)/s
.
B(s) + κ2
(3.5)
As to the second fundamental solution G2 , from the formulas above we first
get the relation
ˆ (κ, s)
ˆ (κ, s) = 1 G̃
G̃
2
1
s
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R. Gorenflo, Yu. Luchko, and M. Stojanović
and then the representation
G2 (x, t) =
t
0
G1 (x, τ ) dτ.
In contrast to the case of the time-fractional diffusion or diffusion-wave
equations, it is not possible to invert the Fourier-Laplace transform given by
(3.5) for an arbitrary weight function p(β) in explicit form. What we can do
ˆ (say,
is to try to reflect some properties of the Fourier-Laplace transform G̃
1
its asymptotical behavior) to the original function G1 or to apply the inverse
Fourier transform to (3.5) and study properties of the Laplace transform
G̃1 . In the rest of this paper we demonstrate both these techniques. But
first let us consider some important particular cases of the weight function
p(β).
Example 3.1. In the special case p(β) = δ(β − α), 0 < α ≤ 2, the
equation (2.1) is reduced to the time-fractional diffusion equation (0 < α <
1) or to the diffusion equation (α = 1), or to the diffusion-wave equation
(1 < α < 2), or to the wave equation (α = 2) that all are well studied. In
this case the equation (3.5) reads
ˆ (κ, s) =
G̃
1
sα−1
.
sα + κ2
ˆ can be inverted to the space-time domain
For 0 < α < 2, the function G̃
1
and presented in terms of the Mainardi function that is a particular case
of the Wright function (see e.g. [25]). It is well-known that G1 can be
interpreted as a probability density with respect to the spatial variable x
evolving in time t for 0 < α ≤ 2 (see e.g. [34]).
Example 3.2. For the weight function in the form p(β) = m
k=1 bk δ(β−
)
with
0
<
α
<
·
·
·
<
α
<
2,
the
function
B
is
given
by
B(s) =
α
1
m
k
m
αk and the equation (3.5) reads
b
s
k=1 k
m
bk sαk −1
ˆ
.
G̃1 (κ, s) = mk=1 α
k + κ2
k=1 bk s
The equation (2.1) is called in this case the multi-term time-fractional diffusion (αm ≤ 1) or diffusion-wave (1 < αm ) equation, respectively. For
the theory of the initial-boundary-value problems for the multi-term timefractional diffusion equation we refer the reader to e.g. [21] (see also references mentioned in this article).
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Example 3.3. Let us now consider the equation (2.1) with a continuous weight function p(β). We start with the case (i) and p(β) ≡ 1, 0 ≤
β ≤ 1. Then we get
(s − 1)/s
s−1 ˆ
, G̃1 (κ, s) =
.
B(s) =
ln(s)
(s − 1) + ln(s)κ2
In this case, the fundamental solution G1 can be interpreted as a probability
density, too (see e.g. [7]). For an explicit formula for the variance of the
fundamental solution G1 as well as its asymptotical behavior see [10].
In the case (iii) and with p(β) ≡ 1, 0 ≤ β ≤ 2, we get
B(s) =
(s2 − 1)/s
s2 − 1 ˆ
, G̃1 (κ, s) = 2
.
ln(s)
(s − 1) + ln(s)κ2
It should be noted that this case is still not well studied in the literature.
In particular, to the best knowledge of the authors, there is no proof known
that the fundamental solution G1 can be interpreted as a probability density.
Of paramount interest in fractional diffusion processes is the spread
of the probability (of diffusing substance) in space, evolving in time formally taken as the variance (second moment) of the fundamental solution. For analysis of this question one can directly use (3.5) or its partial inversions. This has been done explicitly for simple and two-fold
orders, in particular
∞ 2 asymptotic properties for t of the second moment
2
< x (t) >= −∞ x G1 (x, t) dx near zero and near infinity have been found
by aid of Tauber-Karamata theory (see e.g. [5] or [12]). This technique
uses the fact that the second moment of the fundamental solution G1 can
be evaluated as the inverse Laplace transform of the function
−
∂2 1 (κ = 0, s) = 2
G
2
∂κ
sB(s)
(see e.g. [10] or [28]).
Let us remark that the Fourier-Laplace transform technique can also
be applied for a variant of the equation (2.1), where the distributed order
space-fractional derivative appears on the right hand side of the equation:
2
∂ α u(t, x)
∂u(x, t)
=
K(α)
dα, u(x, 0) = δ(x),
∂t
∂|x|α
0
where K(α) is a non-negative weight function and the Riesz fractional
derivative is defined by its Fourier transform:
α
∂ u
; κ = −|κ|α û(κ).
F
∂|x|α
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This and other forms of the time- and/or space-fractional equations of
distributed order were considered in e.g. [26], [28], and [35].
4. Fundamental solution as a probability density function
In this section, we consider some important properties of the fundamental solution G1 of the time-fractional equation (2.1) of distributed order. In
the general situation, i.e., without any restrictions on the weight function
p, we show that G1 is a normalized function and deduce its Laplace transform. This Laplace transform along with some properties of the special type
functions is used to give a new proof of the known fact that the fundamental solution G1 of the time-fractional diffusion equation (2.1) of distributed
order (case (i) in (2.4)) admits an interpretation as a probability density
function in x evolving with time t (see [7]). The same technique is employed
to prove the main result of our paper, namely that the fundamental solution G1 of the time-fractional wave equation (2.1) of distributed order (case
(ii) in (2.4)) can be interpreted as a pdf, too. For the reader’s convenience,
definitions and basic properties of our special type functions are shortly presented in the Appendix. Unfortunately, we were not able to show that G1
is a pdf in case (iii), the case required for the general situation; to do this,
probably another technique is needed. In particular, we mention here the
articles [4] and [31], where this problem was solved for the so called timefractional telegraph equation that corresponds to a very special choice of
the weight function, namely for p(β) = δ(β − 2α) + 2λδ(β − α), 0 < α ≤ 1.
The approach suggested in [4] and [31] can be probably extended to other
special cases of the multi-term time-fractional diffusion-wave equation, in
particular to the equation that corresponds to the weight function p in the
form p(β) = δ(β − α) + δ(β − 2α) + δ(β − 3α) + ... + δ(β − nα).
To show that G1 is normalized, we use the fact that the normalization
∞
G(x, t) dx = 1, t ≥ 0
−∞
means in the Fourier transform domain that
Ĝ1 (κ = 0, t) ≡ 1, t ≥ 0.
A shortcut via (3.5) indeed yields
ˆ (κ = 0, s) = 1/s,
G̃
1
and then
Ĝ1 (κ = 0, t) ≡ 1, t ≥ 0
what we wanted to prove.
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Now let us determine the Laplace transform of the fundamental solution
G1 . By the Fourier inversion of (3.5) and using the well-known formula
2a
, a > 0,
F{exp(−a|x|); κ} = 2
a + κ2
we get
B(s)
exp (−|x| B(s))
(4.1)
G̃1 (x, s) =
2s
with the function B defined by (3.3). Let us note that for our proofs we
need just a restriction of the function B(s) of the complex argument s to the
real semi-axis s = λ > 0. Evidently, for real positive values of s the function
B is positive, too, so that formula (4.1) well holds for s = λ > 0. Formula
(3.5) and of course formula (4.1) show that the fundamental solution G1 is
an even function in x, so that we can restrict ourselves to the values x ≥ 0
while dealing with this function.
The formula (4.1) will be substantially used in our proofs in the rest of
this section.
4.1. Fundamental solution of the time-fractional diffusion
equation of distributed order
To start with, we first outline the known proof that the fundamental
solution G1 can be interpreted as a probability density function in case (i)
(see [10]).
We remind that in this case we call equation (2.1) ”time-fractional
diffusion equation of distributed order”. It reads
1
∂2
p(β)Dtβ u(x, t)dβ =
u(x, t), t > 0, x ∈ IR,
∂x2
0
and our task is to prove (a) and (b):
∞
G1 (x, t)dx = 1 for t ≥ 0, (b) G1 (x, t) ≥ 0 for t > 0, x ≥ 0.
(a)
−∞
The statement (a) has been already proved above for the general case.
To prove (b), we use the subordination trick and some basic properties of
the completely monotone functions (see Appendix).
A non-negative function f defined on the positive semi-axes is completely monotone if and only if it can be represented in the form
∞
e−λt φ(t) dt, λ > 0
f (λ) =
0
with a non-negative (generalized) function φ. In particular, we note that
the restriction of the Laplace transform f˜ of a non-negative function f given
by
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R. Gorenflo, Yu. Luchko, and M. Stojanović
∞
f˜(s) =
0
e−st f (t) dt, s > 0
to the real half-line s = λ > 0 is completely monotone. Vice versa, we can
deduce that the inverse Laplace transform of a function f˜ whose restriction
to the real half-line s = λ > 0 is completely monotone is a non-negative
function. We use this fact in our proofs.
Taking into account the representation (3.3) with s > 0 we observe
that
∞
1
=
exp (−r(B(s) + κ2 )dr
B(s) + κ2
0
and re-write (3.5) in form of a subordination integral
∞
B(s)
ˆ
exp (−rB(s)).
exp (−rκ2 )R̃(r, s)dr with R̃(r, s) =
G̃1 (κ, s) =
s
0
(4.2)
This implies
∞
x2
1
√ exp (− )R(r, t)dr.
(4.3)
G1 (x, t) =
2 πr
4r
0
To complete the proof, we convince ourselves of the fact that R(r, t) is
non-negative for all r ≥ 0, t > 0 by exhibiting it (with respect to t) as
the inverse Laplace transform (for all r > 0) of a function of the variable s
whose restriction to the real semi-axis s = λ > 0 is completely monotone.
1
B(λ)
=
p(β)λβ−1 dβ and B (λ)
Indeed, we first note that the functions
λ
0
are completely monotone. To prove this, we remark that all functions
λβ−1 , 0 < β < 1 (as can be seen by forming all their derivatives) as well
as their linear combinations with non-negative weights (p(β) ≥ 0), their
pointwise limits and therefore the integral of p(β)λβ−1 over the interval
(0, 1] are completely monotone (see Appendix). Because B(λ) ≥ 0 and
B (λ) is a completely monotone function, the function exp (−rB(λ)) is
completely monotone, too. Finally, the function R̃(r, λ) in its dependence
on λ is completely monotone for all r ≥ 0 as product of two completely
monotone functions. Hence, R(r, t) is non-negative as an inverse Laplace
transform of a function that is completely monotone on the real positive
half-axis and (4.3) finally reveals that (b) (non-negativity of G1 ) is true.
Unfortunately, this proof does not work in case (ii) because λβ−1 is no
longer completely monotone if 1 < β ≤ 2. Instead, we give now another
proof of the fact that G1 is non-negative in the case (i) that works with
some modifications for the case (ii), too. In this new proof, we essentially
use some properties of our special type functions that the reader can consult
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309
in Appendix. For a well developed theory of these functions and for their
applications we refer e.g. to the book [33].
We start now with the representation (4.1) of the Laplace
transform
and
consider
the
functions
B(s)/s and
of
the
fundamental
solution
G
1
B(s) on the real half-axis s = λ > 0.
First we prove that the function B(λ) is a complete Bernstein function.
Indeed, all functions λβ , 0 < β < 1 as well as their linear combinations
with non-negative weights (p(β) ≥ 0), their pointwise limits and therefore
the integral of p(β)λβ over the interval√(0, 1], i.e. the function B(λ), are
complete Bernstein functions. Because λ is a complete Bernstein function
and a composition of two complete Bernstein
functions is a complete Bernstein function, too, we have proved that B(λ) is a complete Bernstein
function.
Then the function f1 (λ) := exp(−|x| B(λ)) is a completely monotone function for x > 0 as a composition
of completely monotone function
exp(−|x|λ) and the Bernstein function B(λ) (of course, a complete Bernstein function
belongs to the class of Bernstein functions).
Because
B(λ) ≥ 0 and is a complete Bernstein function, the function
B(λ)/λ is a Stieltjes function (please note that in the next
f2 (λ) :=
subsection we use the vice versa implication that holds true, too) and thus
a completely monotone function.
The Laplace transform of the fundamental solution G1 given by (4.1)
is a completely monotone function if restricted to the positive semi-axis
as product of two completely monotone functions (f1 and f2 ) that again
shows that G1 is non-negative for x ≥ 0.
4.2. Fundamental solution of the time-fractional wave equation
of distributed order
In this subsection, we present a proof of a result which to our best
knowledge is new, namely, that the fundamental solution G1 to the timefractional wave equation of distributed order
∂2
p(β)Dtβ u(x, t)dβ =
u(x, t), t > 0, x ∈ IR
∂x2
(1,2]
can be interpreted as a pdf. At the beginning of Section 4 we have already
∞
proved the normalization property −∞ G1 (x, t)dx = 1. It remains to show
that G1 (x, t) ≥ 0 for t > 0, x ∈ IR.
Like in the previous subsection, we show that both factors
B(s)
f1 (s) = exp (−|x| B(s)) and f2 (s) =
s
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of the Laplace transform (4.1) of the fundamental solution G1 are completely monotone if restricted to the positive real semi-axis s = λ > 0,
hence their product. As we know this implies G1 (x, t) ≥ 0.
This time,
with the function f2 and consider first its square
2 we start
2
β−2
dβ. Because each λβ−2 , 1 < β < 2 is a Stieltjes
f2 (s), i.e. 1 p(β)λ
function, so also the function f22 as a pointwise limit of positive linear
combinations of Stieltjes functions. Because λ1/2 is a complete Bernstein
function and a composition of a complete Bernstein function and a Stieltjes
function is again a Stieltjes function we deduce that the function f2 is a
Stieltjes function.
On the one hand, this means that f2 is a completely monotone
function. On the other hand, we conclude that the function λf2 (λ) = B(λ)
is a complete Bernstein function (we have used the vice versa statement
in the proof
in the previous subsection). Then the function f1 (λ) =
exp(−|x| B(λ)) is a completely monotone function for x > 0 as a composition of the completely
monotone function exp(−|x|λ) and the complete
Bernstein function B(λ).
This means that the Laplace transform (4.1) of the fundamental solution G1 is completely monotone on the positive real semi-axis and the proof
is completed.
5. Conclusions and open questions
In this paper, we have dealt with the problem of proving that the first
fundamental solution G1 of the one-dimensional distributed order diffusionwave equation (2.1) can be interpreted as a spatial pdf evolving in time
and in particular is non-negative. The non-negativity of solutions to any
equations that pretend to be mathematical models of certain real processes
is extremely important in many situations. In our case, the distributed
order diffusion-wave equation is supposed to describe anomalous diffusion of
a substance and therefore its solutions that are interpreted as concentration
of the substance in a point x at the time instance t have to be non-negative.
A contribution to the subject we made in this paper was to prove that
G1 is a spatial pdf evolving in time (and therefore non-negative) in the
case of the distributed order time-fractional wave equation. A particular
case of this problem, namely the case of the time-fractional wave equation
(p(β) = δ(β − α), 1 < α ≤ 2), has been solved by Schneider and Wyss in
[34]. They also considered the case of the time-fractional diffusion equation
(p(β) = δ(β − α), 0 < α ≤ 1) and proved that its fundamental solution can
be interpreted as a spatial pdf evolving in time. Whereas this last result
was later generalized in [7] to the one-dimensional time-fractional diffusion
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FUNDAMENTAL SOLUTION OF A DISTRIBUTED . . .
311
equation of distributed order, the generalization of their result for the timefractional wave equation (p(β) = δ(β − α), 1 < α ≤ 2) to the case of the
distributed order time-fractional wave equation (p(β) ≡ 0, 0 ≤ β ≤ 1)
required a new proof technique that we introduced in this paper.
We have shown that our technique can be applied to the case of the
time-fractional diffusion equation of distributed order (p(β) ≡ 0, 1 < β ≤
2), too. Unfortunately, our method does not work in the general case,
i.e., for the weight function p(β) that does not identically vanish both for
0 ≤ β ≤ 1 and for 1 < β ≤ 2. To the best knowledge of the authors, the
general case is still an open problem that probably requires a completely
different approach and would be worth to be investigated. We mention here
that a particular case of this problem (the fractional telegraph equation
with the weight function p(β) = δ(β − 2α) + 2λδ(β − α), 0 < α ≤ 1) has
been already solved in [4] and [31]. In our present investigation the sub-case
0 < α ≤ 1/2 is already contained.
The technique used in these papers can be probably applied for some
other special cases of the weight function like e.g. p(β) = δ(β − α) + δ(β −
2α) + δ(β − 3α) + ... + δ(β − nα), but hardly for the general case.
Another problem worth to be considered is to find conditions for nonnegativity of the fundamental solutions to the space- and time-space-fractional distributed order diffusion-wave equations. Mainardi, Luchko, and
Pagnini have solved this problem in [25] for the ”single”-order equation with
the Riesz-Feller space-fractional derivative of order α ∈ (0, 2] and skewness
θ, |θ| ≤ min{α, 2−α} and the Caputo time-derivative of order β, β ∈ (0, 2].
Using a subordination integral (the authors called it ”composition”), the
first fundamental solution was shown to be a spatial pdf evolving in time
in the cases {0 < α ≤ 2} ∩ {0 < β ≤ 1} and 1 < β ≤ α ≤ 2. It seems
that the first fundamental solution to the space-time-fractional distributed
order diffusion-wave equation can be interpreted as a spatial pdf evolving
in time under some suitable conditions but a proof is still missing.
Finally we mention that the question whether the spatial pdf evolving in
time that is determined by the first fundamental solution of the distributed
order diffusion-wave equation (2.1) in the cases (ii) or (iii) does represent
a stochastic process is still open. We can speak of a stochastic process if
there is a mechanism known that produces it, e.g. as a limit of a random
walk process.
Until now, up to our knowledge, such random walks have not yet been
constructed. We leave the problem of constructing such ones or proving
their impossibility as a challenge to the reader.
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6. Appendix
For the reader’s convenience, definitions and basic properties of the
special type functions that we have used in our paper are listed in this appendix. For the proofs, more properties, and applications of these functions
we refer to [12], [29] and [33].
Definition 6.1. A non-negative function ϕ : (0, ∞) → IR is called a
completely monotone function if it is of class C ∞ and (−1)n ϕ(n) (λ) ≥ 0 for
all n ∈ IN and λ > 0.
The basic properties of completely monotone functions that we needed
in this paper are the following ones:
• A function ϕ : (0, ∞) → IR is completely monotone if and only if
it can be represented as the Laplace transform of a non-negative
measure (non-negative function or generalized function).
• If ϕ and ψ are completely monotone then their product ϕψ is
completely monotone, too. Linear combinations with non-negative
weights and integrals with non-negative weights of completely monotone functions are completely monotone.
α
The functions e−aλ , 0 ≤ a, α ≤ 1 and Eα,β (−λ), 0 < α ≤ 1, α ≤ β
are well-known examples of completely monotone functions. Here Eα,β (z)
denotes the generalized Mittag-Leffler function defined by
∞
zk
, α > 0, β > 0.
Eα,β (z) :=
Γ(αk + β)
k=0
Below we present (in our words) definitions and properties of our special
type functions.
Loosely speaking, a function defined on (0, ∞) is a Stieltjes function if
it can be written as a restriction of the Laplace transform of a completely
monotone function to the real positive semi-axis. Of course, a Stieltjes
function belongs to the class of completely monotone functions.
Linear combinations of Stieltjes functions with non-negative weights,
their pointwise limits and therefore integrals of Stieltjes functions with nonnegative weights are again Stieltjes functions. Basic examples of Stieltjes
α−1 for 0 ≤ λ ≤ 1 (among these are the
functions are the functions λ
functions 1 and λ1 ), √1λ arctan √1λ , and λ1 log(1 + λ).
Definition 6.2. A non-negative function ϕ : (0, ∞) → IR is called
a Bernstein function if it is of class C ∞ and (−1)n−1 ϕ(n) (λ) ≥ 0 for all
n ∈ IN and λ > 0.
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Loosely speaking, a non-negative function is a Bernstein function if its
derivative is completely monotone. The Bernstein functions possess many
nice properties; we mention here just a few of them that are connected with
the completely monotone functions:
• If ϕ and ψ are Bernstein functions, then their linear combinations
with non-negative weights as well as their composition are Bernstein
functions. Moreover, the pointwise limit of a convergent sequence
of Bernstein functions is a Bernstein function.
• If ϕ is completely monotone and ψ is a Bernstein function, then
ϕ(ψ) is completely monotone. In particular, the function e−ψ is
completely monotone if ψ is a Bernstein function.
• If ϕ is a Bernstein function, then ϕ(λ)/λ is completely monotone.
The complete Bernstein functions constitute a very important subclass
of Bernstein functions. There are several ways how to introduce them. In
particular, a non-negative function ϕ on (0, ∞) is a complete Bernstein
function if and only if the function ϕ(λ)/λ is a Stieltjes function.
Other important properties we have used are the following ones:
• A not identically vanishing function ϕ is a complete Bernstein function if and only if 1/ϕ is a not identically vanishing Stieltjes function.
• The set of complete Bernstein functions is a convex cone, i.e., if ϕ
and ψ are complete Bernstein functions then their linear combinations with non-negative weights are complete Bernstein functions.
• The set of complete Bernstein functions is closed under compositions and under pointwise limits.
• If ϕ is a complete Bernstein function and ψ is a Stieltjes function
then ϕ(ψ) is again a Stieltjes function.
The book [33] contains a large list of complete Bernstein functions. For
our proofs we needed the fact that the functions ϕ(λ) = λα for 0 < α < 1
are complete Bernstein functions.
Acknowledgements
We are very grateful to Prof. R.L. Schilling for pointing out to us the
great usefulness of the theory of Bernstein and Stieltjes functions and for
providing us the proof that the square root of a positive linear combination
of Stieltjes functions is a Stieltjes function too. We appreciate the helpful
discussions with Prof. F. Mainardi on the subject of this work.
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1
Department of Mathematics and Informatics
Free University of Berlin
Arnimallee 3, D-14195 Berlin, GERMANY
e-mail: gorenfl[email protected]
Received: September 24, 2012
2
Department of Mathematics
Beuth Technical University of Applied Sciences Berlin
Luxemburger Str. 10, D-13353 Berlin, GERMANY
e-mail: [email protected]
3
Department of Mathematics and Informatics
Faculty of Science, University of Novi Sad
Trg D.Obradovića 4, 21 000 Novi Sad, SERBIA
e-mail: [email protected]
Please cite to this paper as published in:
Fract. Calc. Appl. Anal., Vol. 16, No 2 (2013), pp. 297–316;
DOI:10.2478/s13540-013-0019-6
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