Wave–diffusion dualism of the neutral-fractional processes

---

See discussions, stats, and author profiles for this publication at: 
https://www.researchgate.net/publication/273854691
Wave–diffusion dualism of the neutral-fractional processes
Article
in
Journal of Computational Physics · June 2014
DOI: 10.1016/j.jcp.2014.06.005
CITATIONS
42
READS
320
1 author:
Yuri Luchko
Beuth Hochschule für Technik Berlin
211
PUBLICATIONS
10,748
CITATIONS
SEE PROFILE
All content following this page was uploaded by 
Yuri Luchko
on 28 November 2019.
The user has requested enhancement of the downloaded file.

---

Journal of Computational Physics 293 (2015) 40–52
Contents lists available at
ScienceDirect
Journal
of
Computational
Physics
www.elsevier.com/locate/jcp
Wave–diffusion
dualism
of
the
neutral-fractional
processes
Yuri Luchko
Department
of
Mathematics
II,
Beuth
Technical
University
of
Applied
Sciences
Berlin,
Luxemburger
Str.
10,
13353
Berlin,
Germany
a
r
t
i
c
l
e
i
n
f
o
a
b
s
t
r
a
c
t
Article
history:
Received
24
April
2014
Received
in
revised
form
29
May
2014
Accepted
4
June
2014
Available
online
11
June
2014
Keywords:
Caputo
fractional
derivative
Riesz
fractional
derivative
Neutral-fractional
equation
Mellin
transform
Damped
waves
Propagation
velocities
Entropy
Entropy
production
rate
Wave–diffusion
dualism
In
this
paper,
a
neutral-fractional
equation
is
introduced
and
analyzed.
In
contrast
to
the
general
time- and
space-fractional
diffusion
equation,
the
neutral-fractional
equation
contains
fractional
derivatives
of
the
same
order
α
,
1
≤
α
≤
2 both
in
space
and
in
time.
As
it
has
been
shown
earlier,
solutions
of
the
neutral-fractional
equation
can
be
interpreted
as
damped
waves
with
the
constant
propagation
velocities
that
means
that
this
equation
inherits
some
characteristics
of
the
wave
equation.
Otherwise,
the
first
fundamental
solution
of
the
one-dimensional
neutral-fractional
equation
is
known
to
be
a
spatial
probability
density
function
evolving
in
time
and
is
thus
related
to
the
diffusion
processes.
In
this
paper,
we
investigate
the
entropy
and
the
entropy
production
rate
of
the
neutral-fractional
equation
and
show
that
both
of
them
are
strongly
connected
to
those
of
the
diffusion
processes.
Thus
a
wave–diffusion
dualism
of
the
processes
described
by
the
neutral-fractional
equation
is
established.
©
2014
Elsevier
Inc.
All rights reserved.
1.
Introduction
Within
the
last
few
decades
a
lot
of
results
related
both
to
the
theory
of
the
fractional
differential
equations
and
their
applications
in
physics,
chemistry,
engineering,
medicine,
biology,
etc.
have
been
obtained
(see
e.g.
 
[3,9–17,21,24,25,28,36]
to
mention
only
few
of
many
recent
publications).
In
particular,
several
models
of
the
anomalous
transport
processes
in
the
form
of
the
time- and/or
space-fractional
diffusion–wave
equations
have
been
considered
by
a
number
of
researchers.
Anomalous
transport
processes
include
the
anomalous
diffusion
(sub- and
supper-diffusion),
the
anomalous
wave
propagation,
and
the
diffusion–wave
processes
that
were
described
until
now
as
some
intermediate
processes
between
the
diffusion
and
the
wave
propagation.
In
this
paper,
another
interpretation
of
the
processes
described
by
some
partial
differential
equations
of
fractional
order
is
suggested.
Namely,
we
show
that
they
are
not
a
mixture
of
a
diffusion
process
and
a
wave
propagation,
but
rather
a
new
phenomena
that
behaves
as
a
diffusion
with
respect
to
some
physical
characteristics
whereas
it
looks
like
a
wave
regarding
other
characteristics.
Thus
one
can
speak
about
a
wave–diffusion
dualism
of
the
processes
described
by
these
equations
that
we
call
the
neutral-fractional
equations
and
that
were
previously
referred
to
as
the
neutral-fractional
diffusion
equations
or
the
fractional
wave
equations
depending
on
what
characteristics
one
was
interested
in.
The
neutral-fractional
equations
we
consider
in
this
paper
contain
fractional
derivatives
of
the
same
order
α
,
1
≤
α
≤
2
both
in
space
and
in
time.
The
fractional
derivative
in
time
is
interpreted
in
the
Caputo
sense
whereas
the
space-fractional
derivative
is
taken
in
form
of
an
inverse
operator
to
the
fractional
Riesz
potential
(Riesz
fractional
derivative).
As
has
been
shown
in
 
[14]
,
both
a
maximum
location
and
the
“gravity”- and
“mass”-centers
of
the
first
fundamental
solution
to
the
neutral-fractional
equation
propagate
with
the
constant
velocities
as
the
solutions
to
the
wave
equation
E-mail
address:
[e-mail]
.
http://dx.doi.org/10.1016/j.jcp.2014.06.005
0021-9991/
©
2014
Elsevier
Inc.
All rights reserved.

---

Y. Luchko / Journal of Computational Physics 293 (2015) 40–52
41
(
α
=
2),
but
in
contrast
to
the
wave
equation
these
velocities
are
different
from
each
other
for
a
fixed
value
of
α
,
1
<
α
<
2.
Let
us
mention
that
the
propagation
velocity
v
of
a
maximum
location
of
the
first
fundamental
solution
to
the
time-fractional
diffusion–wave
equation
of
the
order
α
is
determined
by
the
formula
(see
e.g.
 
[20,22]
)
v
(
t
,
α
)
=
C
α
t
α
2
−
1
.
For
1
<
α
<
2,
the
propagation
velocity
v
depends
on
time
t
and
is
a
decreasing
function
that
varies
from
+∞
at
time
t
=
0
+
to
zero
as
t
→ +∞
that
makes
it
difficult
to
interpret
solutions
to
the
time-fractional
diffusion–wave
equation
as
some
waves.
It
is
well
known
that
the
anomalous
transport
processes
can
be
modeled
in
terms
of
the
continuous
time
random
walk
processes
and
described
by
the
time- and/or
space-fractional
differential
equations
that
are
derived
from
the
stochastic
models
for
a
special
choice
of
the
jump
probability
density
functions
with
the
infinite
first
or/and
second
moments
(see
e.g.
 
[11,17,28]
).
The
neutral-fractional
equation
can
be
obtained
from
the
continuous
time
random
walk
model,
too.
In
 
[16]
,
the
case
of
the
waiting
time
probability
density
function
and
the
jump
length
probability
density
function
with
the
same
power
law
asymptotic
behavior
has
been
considered.
Under
some
standard
assumptions,
the
neutral-fractional
equation
can
be
asymptotically
derived
from
the
continuous
time
random
walk
model
mentioned
above
(see
 
[16]
for
details).
Thus
it
is
not
a
surprise
that
the
fundamental
solution
to
the
neutral-fractional
equation
can
be
interpreted
as
a
probability
density
function.
In
this
paper,
we
investigate
the
entropy
and
the
entropy
production
rate
of
the
neutral-fractional
equation
and
show
that
both
of
them
are
strongly
connected
to
those
of
the
diffusion
processes.
The
concept
of
entropy
was
first
introduced
in
the
macroscopic
thermodynamics
and
then
extended
for
description
of
some
phenomena
in
statistical
mechanics,
information
theory,
ergodic
theory
of
dynamical
systems,
etc.
Historically,
many
definitions
of
entropy
were
proposed
and
applied
in
different
knowledge
areas.
In
this
paper,
we
employ
the
statistical
concept
of
entropy
that
goes
back
to
Shannon
and
was
introduced
by
him
in
the
theory
of
communication
and
transmission
of
information
(see
 
[34]
).
The
entropy
of
the
processes
governed
by
the
time- and
space-fractional
diffusion
equations
was
considered
in
 
[10,12]
and
 
[30,31]
,
respectively.
From
the
mathematical
viewpoint,
the
neutral-fractional
equation
we
deal
with
in
this
paper
was
considered
for
the
first
time
in
 
[5]
,
where
an
explicit
formula
for
the
fundamental
solution
of
the
one-dimensional
neutral-fractional
equation
was
derived.
In
 
[26]
,
a
space–time
fractional
diffusion–wave
equation
with
the
Riesz–Feller
derivative
of
order
α
∈
(
0
,
2
]
and
skewness
θ
and
with
the
Caputo
fractional
derivative
of
order
β
∈
(
0
,
2
]
was
investigated
in
detail.
A