Wave–diffusion dualism of the neutral-fractional processes

3.
Fundamental
solution
as
a
probability
density
function
As
has
been
already
mentioned,
for
α
=
1 (modified
advection
equation
 
(6)
),
the
fundamental
solution
G
α
,
1
is
the
Cauchy
kernel
G
1
,
1
(
x
,
t
)
=
1
π
t
t
2
+
x
2
(24)
that
is
a
spatial
probability
density
function
evolving
in
time.
For
1
<
α
<
2,
the
Green
function
 
(20)
is
a
spatial
probability
density
function
evolving
in
time,
too,
as
has
been
shown
for
the
first
time
in
 
[14]
.
This
pdf
possesses
all
finite
moments
up
to
the
order
α
.
In
particular,
the
mean
value
of
G
α
,
1
(its
first
moment)
exists
for
all
α
>
1 (we
note
that
the
Cauchy
kernel
does
not
possess
a
mean
value).
The
moments
of
the
order
β,
|
β
|
<
α
of
G
α
,
1
can
be
evaluated
via
the
Mellin
integral
transform
(see
 
[14]
for
details):
∞
0
G
α
,
1
(
r
,
t
)
r
β
dr
=
t
β
α
sin
(
π
β/
2
)
sin
(
π
β/
α
)
.
(25)
It
is
known
(see
e.g.
 
[14]
)
that
the
fundamental
solution
G
α
,
1
(
r
,
t
)
is
connected
with
the
stable
unimodal
distributions
and
is
unimodal,
too,
i.e.,
it
has
only
one
maximum
point
that
we
denote
by
r

α
.
This
maximum
point
can
be
determined
in
explicit
form
using
the
representation
 
(20)
:
r

α
(
t
)
=
v
p
(
α
)
t
,
v
p
(
α
)
=
c
p
(
α
)
1
α
,
c
p
(
α
)
=
−
cos
(
π α
/
2
)
+
α
2
−
sin
2
(
π α
/
2
)
α
+
1
.
(26)

---

Y. Luchko / Journal of Computational Physics 293 (2015) 40–52
47
For
the
maximum
value
G

α
of
G
α
,
1
we
get
the
formula
G

α
(
t
)
=
m
α
t
,
m
α
=
G

α
(
1
)
=
1
π
v
p
(
α
)
c
α
sin
(
π α
/
2
)
1
+
2
c
α
cos
(
π α
/
2
)
+
c
2
α
,
(27)
where
v
p
(
α
)
and
c
α
are
defined
as
in
 
(26)
.
Because
of
the
scaling
symmetry
of
G
α
,
1
(
r
,
t
)
,
the
product
p
α
of
the
maximum
value
G

α
,
1
(
t
)
and
the
maximum
location
r

α
(
t
)
is
time-independent
for
a
fixed
value
of
α
,
1
<
α
<
2:
p
α
=
G

α
(
t
)
·
r

α
(
t
)
=
1
π
c
α
sin
(
π α
/
2
)
1
+
2
c
α
cos
(
π α
/
2
)
+
c
2
α
.
(28)
In
the
two-dimensional
case,
the
plot
presented
in
 
Fig. 2
clearly
shows
that
the
fundamental
solution
G
α
,
2
is
negative
for
some
values
of
the
variables
x
and
t
and
therefore
cannot
be
interpreted
as
a
probability
density
function.
In
the
three-dimensional
case,
it
follows
from
the
formula
(23)
that
G
α
,
3
is
not
a
probability
density
function,
too,
because
it
is
not
everywhere
nonnegative.
Moreover,
because
of
the
relation
G
α
,
3
(
r
,
t
)
= −
1
2
π
r
∂
∂
r
G
α
,
1
(
r
,
t
),
r
= |
x
| =
0
(29)
between
the
one- and
the
three-dimensional
fundamental
solutions
and
the
formula
(26)
,
the
following
inequalities
are
valid
(
r
= |
x
|
>
0):
G
α
,
3
(
r
,
t
) <
0 for
r
<
v
p
(
α
)
t
,
G
α
,
3
(
r
,
t
)
=
0 for
r
=
v
p
(
α
)
t
,
G
α
,
3
(
r
,
t
) >
0 for
r
>
v
p
(
α
)
t
,
where
v
p
(
α
)
is
defined
as
in
 
(26)
.
The
relation
(29)
easily
follows
from
the
representations
 
(19)
and
 
(22)
(see
 
[13]
).
It
is
worth
to
mention
that
the
same
relation
has
been
deduced
in
 
[8]
for
the
time-space-fractional
diffusion
equation
with
the
time
derivative
of
order
α
,
0
<
α
<
1 and
the
space
derivative
of
order
β,
0
< β <
2,
β
=
1 by
using
a
different
method.
3.1.
Entropy
and
the
entropy
production
rate
In
this
section,
we
restrict
our
attention
to
the
one-dimensional
neutral-fractional
equation
because
only
in
this
case
its
fundamental
solution
is
a
pdf
as
we
have
seen
in
the
previous
section.
In
particular,
we
consider
the
entropy
and
the
entropy
production
rate
of
a
one-dimensional
neutral-fractional
process
that
is
governed
by
the
neutral-fractional
equation
(1)
.
To
better
understand
a
meaning
of
the
entropy,
following
Shannon
 
[34]
we
first
define
it
for
a
discrete
random
variable
X
with
values
in
a
finite
set
X
that
are
taken
with
the
probability
p
(
x
),
x
∈
X
:
S
(
p
)
= −
k
x
∈
X
p
(
x
)
ln
p
(
x
)
,
where
k
is
a
constant
that
we
set
equal
to
one
for
the
sake
of
convenience
(in
statistical
mechanics,
the
constant
k
is
the
Boltzmann
constant
k
B
).
It
can
be
easily
seen
that
the
entropy
is
nonnegative
and
attains
its
minimum
S
(
p
)
=
0 for
a random
variable
with
a
determined
outcome
and
its
maximum
S
(
p
)
=
ln
(
|
X
|
)
for
a
uniform
distribution.
Thus
the
entropy
is
a
measure
for
the
unevenness
of
the
probability
distributions
and
can
also
serve
as
a
measure
for
uncertainty
of
the
physical
processes
that
are
connected
with
the
corresponding
random
variables.
In
the
case
of
a
one-dimensional
continuous
random
variable
with
the
probability
density
function
p
(
x
),
x
∈
X
⊆
R
,
we
adopt
the
following
definition
of
the
entropy:
S
(
p
)
= −
k
∞
−∞
p
(
x
)
ln
p
(
x
)
dx
,
(30)
where
the
constant
k
is
again
set
to
be
equal
to
one.
Of
course,
we
are
aware
of
the
fact,
that
the
entropy
definition
 
(30)
is
a
special
case
of
the
more
general
definitions
by
Tsallis
or
Rényi
(see
e.g.
 
[12,31]
),
but
in
this
paper
we
are
mainly
interested
in
a
qualitative
picture
and
restrict
our
investigation
to
the
Shannon
entropy
given
by
 
(30)
.
It
is
well
known
that
the
entropy
of
a
Gaussian
random
variable
defined
by
the
pdf
N
μ
;
σ
2
=
1
√
2
π σ
2
exp
−
(
x
−
μ
)
2
2
σ
2
has
the
form
S
N
μ
;
σ
2
=
1
2
1
+
ln
2
π σ
2
.
(31)

---

48
Y. Luchko / Journal of Computational Physics 293 (2015) 40–52
Thus
the
entropy
increases
with
the
width
σ
2
of
the
probability
density
function
N
(
μ
;
σ
2
)
,
i.e.,
the
broader
the
distribution
(uncertainty
of
the
event),
the
larger
the
entropy,
so
that
the
entropy
can
again
be
interpreted
as
a
measure
of
uncertainty
of
an
event
that
is
governed
by
the
pdf
p
(
x
)
.
In
the
case,
a
probability
density
function
depends
on
the
time
variable,
i.e.
it
describes
a
time-dependent
stochastic
process,
the
entropy
 
(30)
depends
on
the
time,
too:
S
(
p
,
t
)
= −
∞
−∞
p
(
x
,
t
)
ln
p
(
x
,
t
)
dx
.
(32)
For
such
processes,
the
entropy
production
rate
R
defined
by
R
(
p
,
t
)
=
d
dt
S
(
p
,
t
)
is
a
very
important
characteristic
that
can
be
interpreted
as
a
natural
measure
of
the
irreversibility
of
a
process.
Say,
in
the
case
of
a
diffusion
process
that
is
described
by
the
one-dimensional
diffusion
equation
with
the
diffusion
coefficient
taken
to
be
equal
to
one
and
that
is
therefore
governed
by
the
Gaussian
distribution
N
(
0
;
2
t
)
,
the
formula
 
(31)
leads
to
the
following
result:
R
N
(
0
;
2
t
)
=
d
dt
S
N
(
0
;
2
t
)
=
1
2
t
,
(33)
i.e.,
the
entropy
production
rate
is
strictly
positive
for
t
>
0 and
the
diffusion
process
can
be
classified
as
an
irreversible
process.
Otherwise,
a
wave
propagation
described
by
the
wave
equation
(
α
=
2 in
 
(1)
)
is
a
reversible
process
with
the
entropy
production
rate
equal
to
zero
for
t
>
0.
It
is
worth
mentioning
that
the
entropy
production
rates
for
the
time- and
the
space-fractional
diffusion
equations
that
were
calculated
in
 
[10]
and
 
[30]
,
respectively,
depend
on
the
derivative
order
α
and
increase
with
increasing
of
α
from
1
(diffusion)
to
2
(wave
propagation)
that
results
in
the
so
called
entropy
production
paradox
(see
 
[10]
and
 
[30]
for
resolving
of
this
paradox).
In
the
case
of
the
one-dimensional
neutral-fractional
equation,
we
first
get
the
representation
(
t
>
0
,
x
=
0
,
1
≤
α
<
2)
G
α
,
1
(
x
,
t
)
=
1
|
x
|
L
α
t
|
x
|
,
L
α
(
τ
)
=
1
π
τ
α
sin
(
π α
/
2
)
1
+
2
τ
α
cos
(
π α
/
2
)
+
τ
2
α
(34)
for
its
fundamental
solution
(pdf)
in
terms
of
an
auxiliary
function
L
α
depending
of
the
scaling
argument
t
/
|
x
|
that
easily
follows
from
 
(20)
.
Substituting
this
representation
into
the
formula
 
(32)
,
after
some
elementary
transformations
we
obtain
for
the
entropy
S
(
α
,
t
)
of
the
neutral-fractional
process
of
order
α
the
following
result:
S
(
α
,
t
)
= −
∞
−∞
1
|
x
|
L
α
t
|
x
|
ln
1
|
x
|
L
α
t
|
x
|
dx
=
A
α
ln
(
t
)
+
B
α
,
(35)
where
A
α
=
2
∞
0
L
α
(
τ
)
τ
d
τ
,
B
α
= −
2
∞
0
L
α
(
τ
)
τ
ln
L
α
(
τ
)
τ
d
τ
.
(36)
The
constant
A
α
can
be
evaluated
in
explicit
form.
Indeed,
the
integral
that
defines
A
α
can
be
interpreted
as
the
Mellin
transform
of
the
function
L
α
at
the
point
s
=
0.
The
formula
 
(17)
provides
us
with
the
inverse
Mellin
transform
of
L
α
and
thus
its
Mellin
transform
is
given
by
the
formula
L
∗
α
(
s
)
=
∞
0
L
α
(
τ
)
τ
s
−
1
d
τ
=
1
α
sin
(
π
s
/
2
)
sin
(
π
s
/
α
)
.
(37)
For
s
=
0,
we
get
A
α
=
2
∞
0
L
α
(
τ
)
τ
d
τ
=
2
α
lim
s
→
0
sin
(
π
s
/
2
)
sin
(
π
s
/
α
)
=
1
.
This
formula
along
with
the
formula
 
(35)
leads
to
the
following
simple
expression
for
the
entropy
production
rate
R
(
t
)
of
a neutral-fractional
process
described
by
the
neutral-fractional
equation
 
(1)
:
R
(
t
)
=
d
dt
S
(
α
,
t
)
=
1
t
.
(38)

---

Y. Luchko / Journal of Computational Physics 293 (2015) 40–52
49
Fig. 5.
Plot of the constant
B
α
.
This
formula
shows
that
R
(
t
)
does
not
depend
on
the
equation
order
α
.
Another
interesting
property
of
the
entropy
pro-
duction
rate
R
(
t
)
is
that
it
is
exactly
twice
as
much
as
the
entropy
production
rate
of
the
diffusion
equation
for
any
α
,
1
≤
α
<
2.
Because
the
entropy
production
rate
R
(
t
)
does
not
vanish
for
any
t
>
0,
the
neutral-fractional
processes
are
irreversible
that
is
a
typical
characteristic
of
the
diffusion
processes.
Another
important
point
is
to
understand
what
happens
in
the
limiting
case
α
→
2.
As
has
been
mentioned
in
Sec-
tion
2.1
,
the
case
α
=
2 is
a
singular
case
of
the
neutral-fractional
equation
 
(1)
and
one
cannot
expect
a
smooth
transition
of
its
properties
to
the
ones
of
the
wave
equation
as
α
→
2.
In