Wave–diffusion dualism of the neutral-fractional processes

particular,
we
see
that
whereas
the
entropy
production
rate
of
a
wave
propagation
is
equal
to
zero,
it
is
strictly
positive
for
all
t
>
0 for
any
neutral-fractional
process
with
1
≤
α
<
2.
The
only
symptom
that
indicates
a
transition
to
a
wave
propagation
is
that
the
constant
B
α
in
the
formula
 
(35)
for
the
entropy
tends
to
−∞
as
α
→
2 as
follows
from
the
formula
 
(34)
.
Therefore
for
a
fixed
t
>
0 and
α
→
2,
the
entropy
S
(
α
,
t
)
can
be
interpreted
as
−∞
and
thus
as
a
constant
in
the
generalized
sense.
In the
plot
of
 
Fig. 5
,
we
can
see
that
B
α
first
changes
very
slowly
(say,
B
α
≈
2
.
5310 for
α
=
1 and
B
α
≈ −
6
.
2286 for
α
=
1
.
9999)
and
starts
to
really
tend
to
−∞
only
in
a
very
small
vicinity
of
the
point
α
=
2.
4.
Fundamental
solution
as
a
damped
wave
The
Mellin–Barnes
representation
 
(16)
shows
that
the
fundamental
solution
G
α
,
n
(
x
,
t
)
can
be
represented
via
an
auxiliary
function
of
a
single
argument:
G
α
,
n
(
x
,
t
)
=
G
α
,
n
|
x
|
,
t
=
G
α
,
n
(
r
,
t
)
=
r
−
n
L
α
,
n
r
t
,
r
>
0
(39)
with
L
α
,
n
(
r
)
=
1
απ
n
/
2
1
2
π
i
L
Γ
s
α
Γ
1
−
s
α
Γ
n
2
−
s
2
Γ (
1
−
s
)
2
s
Γ
s
2
(
r
)
s
ds
.
(40)
It
easily
follows
from
the
representation
 
(39)
that
for
a
fixed
t
>
0 the
fundamental
solution
G
α
,
n
attains
its
local
maximum
or
minimum
at
the
point
r
∗
(
t
,
α
,
n
)
=
r
∗
t
=
c
(
α
,
n
)
t
,
(41)
where
r
∗
=
c
(
α
,
n
)
is
a
minimum
or
maximum
point
of
the
single-variable
function
r
−
n
L
α
,
n
(
r
)
.
The
formula
 
(41)
means
that
the
maximum
and
minimum
locations
(if
any)
of
the
fundamental
solution
to
the
neutral-diffusion
equation
(1)
all
propagate
with
the
constant
velocities
and
thus
G
α
,
n
can
be
interpreted
as
a
damped
wave.
In
the
next
subsections,
some
more
characteristics
of
G
α
,
n
that
are
usually
related
to
a
wave
propagation
are
discussed.
Let
us
mention
that
the
propa-
gation
velocity
of
a
maximum
location
of
the
first
fundamental
solution
to
the
time-fractional
diffusion–wave
equation
of
the
order
α
,
1
<
α
<
2 depends
on
time
t
that
makes
it
difficult
to
interpret
solutions
to
the
time-fractional
diffusion–wave
equation
as
some
waves
(see
e.g.
 
[20,22]
).
4.1.
Some
physical
characteristics
of
the
fundamental
solution
In
this
section,
we
calculate
some
physical
characteristics
of
the
damped
waves
that
can
be
described
by
the
fundamental
solutions
to
the
neutral-fractional
equation
with
the
focus
on
the
one-dimensional
case.

---

50
Y. Luchko / Journal of Computational Physics 293 (2015) 40–52
The
location
of
the
“gravity”-center
x
g
α
(
t
)
of
G
α
,
1
is
defined
by
the
formula
x
g
α
(
t
)
=
∞
0
rG
α
,
1
(
r
,
t
)
dr
∞
0
G
α
,
1
(
r
,
t
)
dr
.
(42)
For
1
<
α
<
2,
the
formulas
 
(34)
and
 
(37)
lead
to
the
following
result:
x
g
α
(
t
)
=
2
t
α
sin
(
π
/
α
)
.
(43)
If
α
=
1,
the
mean
value
of
G
α
,
1
does
not
exists
and
thus
the
“gravity”-center
x
g
1
(
t
)
of
G
α
,
1
is
located
at
+∞
for
any
t
>
0.
The
“mass”-center
x
m
α
(
t
)
of
G
α
,
1
that
is
defined
by
the
formula
(see
e.g.
 
[7]
)
x
m
α
(
t
)
=
∞
0
r
(
G
α
,
1
(
r
,
t
))
2
dr
∞
0
(
G
α
,
1
(
r
,
t
))
2
dr
(44)
can
be
represented
in
the
form
x
m
α
(
t
)
=
v
m
(
α
)
t
,
v
m
(
α
)
=
∞
0
τ
−
1
L
2
α
(
τ
)
d
τ
∞
0
L
2
α
(
τ
)
d
τ
(45)
where
the
function
L
α
is
defined
as
in
 
(34)
.
Another
important
characteristic
of
the
wave
propagation
processes
is
location
of
their
energy
that
in
our
case
is
defined
as
the
time
corresponding
to
the
centroid
of
G
α
,
1
in
the
time
domain
and
is
given
by
the
formula
(see
 
[2]
)
t
c
α
(
x
)
=
∞
0
t
(
G
α
,
1
(
x
,
t
))
2
dt
∞
0
(
G
α
,
1
(
x
,
t
))
2
dt
.
(46)
For
1
<
α
<
2,
both
integrals
at
the
right-hand
side
of
(46)
converge
and
the
finite
location
of
the
energy
of
G
α
,
1
is
represented
in
the
form
t
c
α
(
x
)
=
x
v
c
(
α
)
,
v
c
(
α
)
=
∞
0
L
2
α
(
τ
)
d
τ
∞
0
τ
L
2
α
(
τ
)
d
τ
,
(47)
where
the
function
L
α
is
defined
as
in
 
(34)
.
4.2.
The
velocities
of
the
damped
waves
It
is
well
known
(see
e.g.
[2,7,35,37,38]
)
that
several
different
definitions
of
the
wave
propagation
velocities
can
be
introduced.
In
this
section,
we
calculate
some
of
them
for
the
damped
waves
that
are
described
by
the
one-dimensional
neutral-fractional
equation
 
(1)
.
It
turns
out
that
all
of
these
velocities
are
constant
in
time
and
depend
only
on
the
order
α
of
the
neutral-fractional
equation.
We
start
with
the
propagation
velocity
v
p
(
α
)
of
the
maximum
location
of
G
α
,
1
that
can
be
interpreted
as
the
phase
velocity
and
calculate
it
using
the
formula
 
(26)
v
p
(
α
)
=
dx

α
(
t
)
dt
=
−
cos
(
π α
/
2
)
+
α
2
−
sin
2
(
π α
/
2
)
α
+
1
1
α
.
(48)
For
α
=
1 (modified
convection
equation
 
(6)
),
the
phase
velocity
of
G
α
,
1
is
equal
to
zero
(the
maximum
point
stays
at
the
point
x
=
0)
whereas
for
α
=
2 (wave
equation)
the
maximum
point
propagates
with
the
constant
velocity
1.
The
propagation
velocity
v
g
(
α
)
of
the
“gravity”-center
of
G
α
,
1
is
given
by
the
formula
v
g
(
α
)
:=
dx
g
α
(
t
)
dt
=
2
α
sin
(
π
/
α
)
(49)
that
easily
follows
from
the
formula
 
(43)
.
v
g
(
α
)
is
thus
time-independent
and
depends
only
on
the
order
α
of
the
neutral-
fractional
equation.
Evidently,
v
g
(
2
)
=
1 and
v
g
(
α
)
→ +∞
as
α
→
1
+
0.
The
velocity
v
m
(
α
)
of
the
“mass”-center
of
G
α
,
1
or
its
pulse
velocity
can
be
obtained
from
the
formula
 
(45)
and
is
given
by
the
formula
v
m
(
α
)
=
dx
m
α
(
t
)
dt
=
∞
0
τ
−
1
L
2
α
(
τ
)
d
τ
∞
0
L
2
α
(
τ
)
d
τ
,
(50)
where
the
function
L
α
is
defined
as
in
 
(34)
.

---

Y. Luchko / Journal of Computational Physics 293 (2015) 40–52
51
Fig. 6.
Plots of the “gravity”-center velocity
v
g
, the pulse velocity
v
m
, the phase velocity
v
p
, and the centrovelocity
v
c
of
G
α,
1
for 1
≤
α
≤
2.
The
second
centrovelocity
v
2
(
α
)
is
defined
as
the
mean
pulse
velocity
computed
from
the
time
0 to
the
time
t
(see
e.g.
 
[2]
).
The
formulas
 
(45)
and
 
(50)
show
that
for
the
damped
wave
that
is
described
by
the
fundamental
solution
of
the
one-dimensional
neutral-fractional
equation
the
second
centrovelocity
is
equal
to
its
pulse
velocity
v
m
(
α
)
:
v
2
(
α
)
=
x
m
α
(
t
)
t
=
v
m
(
α
)
=
∞
0
τ
−
1
L
2
α
(
τ
)
d
τ
∞
0
L
2
α
(
τ
)
d
τ
.
(51)
The
Smith
centrovelocity
v
c
(
α
)
of
a
wave
propagation
process
describes
the
motion
of
the
first
moment
of
its
energy
distribution
(see
e.g.
 
[35]
).
We
calculate
it
in
explicit
form
using
the
formula
 
(47)
:
v
c
(
α
)
=
dt
c
α
(
x
)
dx
−
1
=
∞
0
L
2
α
(
τ
)
d
τ
∞
0
τ
L
2
α
(
τ
)
d
τ
,
(52)
where
the
function
L
α
is
defined
is
defined
as
in
 
(34)
.
For
α
→
1,
the
Smith
centrovelocity
tends
to
0.
The
first
centrovelocity
v
1
(
α
)
is
defined
as
the
mean
centrovelocity
from
0 to
x
(see
e.g.
 
[2]
).
It
follows
from
the
formulas
(47)
and
 
(52)
that
the
first
centrovelocity
of
G
α
,
1
is
equal
to
the
Smith
centrovelocity
v
c
(
α
)
:
v
1
(
α
)
=
x
t
c
α
(
x
)
=
v
c
(
α
)
=
∞
0
L
2
α
(
τ
)
d
τ
∞
0
τ
L
2
α
(
τ
)
d
τ
.
(53)
Because
all
velocities
introduced
above
are
constant
in
time
and
depend
only
on
the
order
α
of
the
one-dimensional
neutral-fractional
equation,
its
solutions
can
be
interpreted
as
a
kind
of
the
damped
waves.
Some
velocity
plots
of
G
α
,
1
are
presented
in
Fig. 6
.
In
the
two-dimensional
case,
the
fundamental
solution
G
α
,
2
has
many
(probably
infinitely
many)
local
minima
and
maxima
as
can
be
seen
in the
plot
of
 
Fig. 2
.
Finally,
the
fundamental
solution
G
α
,
3
has
only
one
maximum
point
as
can
be
seen
in the
plots
of
 
Fig. 3
and
is
proved
by
direct
calculations
based
on
the
closed
form
formula
 
(23)
.
According
to
the
formula
 
(41)
all
of
them
propagate
with
the
finite
velocities
that
depend
only
on
the
equation
order
α
.