Wave–diffusion dualism of the neutral-fractional processes

particular
case
of
this
equation
that
for
θ
=
0 corresponds
to
our
one-dimensional
neutral-fractional
wave
equation
was
mentioned
in
[26]
,
too.
In
[29]
,
a
fundamental
solution
to
the
one-dimensional
neutral-fractional
equation
was
deduced
and
analyzed
in
terms
of
the
Fox
H-function.
In
 
[14]
,
the
one-dimensional
neutral-fractional
equation
was
investigated
from
the
viewpoint
of
an
interpretation
of
its
solutions
as
the
damped
waves.
Its
fundamental
solution
was
derived
in
terms
of
elementary
functions
for
all
values
of
α
,
1
≤
α
<
2.
For
the
fundamental
solution,
both
its
maximum
location
and
its
maximum
value
were
determined
in
closed
form
as
well
as
the
propagation
velocities
of
the
maximum
location
and
the
“gravity”- and
“mass”-centers
of
the
fundamental
solution.
In
 
[13]
,
a
multi-dimensional
neutral-fractional
equation
with
a
special
focus
given
to
the
three-dimensional
case
was
analyzed.
The
fundamental
solution
to
this
equation
is
a
spherically
symmetric
function
that
possesses
some
nice
integral
representations
and
can
be
even
written
down
in
explicit
form
in
terms
of
elementary
functions
in
the
one- and
three-dimensional
cases.
In
contrast
to
the
one-dimensional
case,
the
fundamental
solution
cannot
be
interpreted
as
a
probability
density
function
in
the
two- and
three-dimensional
cases
and
thus
these
equations
cannot
be
employed
for
modeling
of
any
diffusion
processes.
Instead,
their
fundamental
solutions
can
be
interpreted
as
some
damped
waves
with
the
constant
phase
velocities
that
depend
only
on
the
order
α
of
the
neutral-fractional
equation.
In
this
paper,
we
mainly
deal
with
the
one-dimensional
neutral-fractional
equation
with
a
special
focus
given
to
an in-
terpretation
of
its
solutions
as
some
diffusion
processes.
For
the
sake
of
completeness,
the
multi-dimensional
case
and
some
results
regarding
an
interpretation
of
the
solutions
as
the
damped
waves
are
also
mentioned.
The
rest
of
the
paper
is
organized
as
follows.
In
Section
2
,
the
basic
definitions,
problem
formulation,
and
some
analytical
results
for
the
initial-value
problems
for
the
multi-dimensional
neutral-fractional
equation
are
presented.
In
particular,
the
explicit
formulas
for
the
fundamental
solutions
for
the
one- and
three-dimensional
equations
are
derived
in
terms
of
the
elementary
functions
for
all
values
of
α
,
1
≤
α
<
2.
In
the
two-dimensional
case,
such
simple
explicit
formulas
seem
to
be
not
available.
Section
3
is
devoted
to
a
probabilistic
interpretation
of
the
solutions
to
the
one-dimensional
neutral-fractional
equation.
In
particular,
the
Shannon
entropy
and
the
entropy
production
rate
are
calculated.
As
in
the
case
of
the
diffusion
equation,
the
entropy
production
rate
decreases
with
the
time
as
t
−
1
.
Surprisingly,
the
law
for
the
entropy
production
rate
does
not
depend
on
the
equation
order
α
and
it
is
twice
as
much
as
the
entropy
production
rate
of
the
diffusion
equation.
In
Section
4
,
an
interpretation
of
the
solutions
to
the
neutral-fractional
equation
as
damped
waves
is
presented.
In
particular,
we
show
that
the
phase
velocity
of
the
fundamental
solution
of
the
neutral-fractional
equation
is
a
constant
that
depends
only
on
the
equation
order
α
.
To
illustrate
analytical
findings,
some
results
of
numerical
calculations,
plots,
their
physical
interpretation
and
discussion
are
presented.
In
Section
5
,
some
conclusions
and
open
problems
for
further
research
are
formulated.

---

42
Y. Luchko / Journal of Computational Physics 293 (2015) 40–52
2.
Neutral-fractional
equation
2.1.
Problem
formulation
In
this
paper,
we
consider
the
neutral-fractional
equation
in
the
form
D
α
t
u
(
x
,
t
)
= −
(
−
)
α
2
u
(
x
,
t
),
x
∈
R
n
,
t
∈
R
+
,
1
≤
α
≤
2
,
(1)
where
u
=
u
(
x
,
t
),
x
∈
R
n
,
t
∈
R
+
is
a
real
field
variable,
−
(
−
)
α
2
is
the
Riesz
space-fractional
derivative
and
D
α
t
is
the
Caputo
time-fractional
derivative.
For
a
readers
convenience,
the
definitions
of
the
fractional
derivatives
are
given
be-
low:
D
α
t
u
(
t
)
=
I
n
−
α
∂
n
u
∂
t
n
(
t
),
n
−
1
<
α
≤
n
,
n
∈
N
,
(2)
I
α
,
α
≥
0 being
the
Riemann–Liouville
fractional
integral
I
α
u
(
t
)
=
1
Γ (
α
)
t
0
(
t
−
τ
)
α
−
1
u
(
x
,
τ
)
d
τ
,
α
>
0
,
I
0
u
(
t
)
=
u
(
x
,
t
)
and
Γ
the
Euler
Gamma
function.
It
follows
from
the
definition
of
the
Caputo
fractional
derivative
that
it
coincides
with
the
standard
derivative
of
order
n
for
α
=
n
,
n
∈
N
.
Of
course,
α
=
n
is
in
fact
a
singular
case
because
for
α
=
0 the
Riemann–Liouville
fractional
integral
is
defined
by
a
different
formula
compared
to
that
for
α
>
0.
Thus
one
cannot
expect
a
smooth
transition
of
the
properties
of
the
neutral-fractional
equation
 
(1)
to
the
properties
of
the
wave
equation
as
α
→
2
that
will
be
demonstrated
in
the
following
sections.
The
Riesz
space-fractional
derivative
of
order
α
,
0
<
α
≤
2 is
defined
for
a
sufficiently
well-behaved
function
f
:
R
n
→
R
as
a
pseudo-differential
operator
with
the
symbol
−|
κ
|
α
[32,33]
:
F
−
(
−
)
α
2
f
(
κ
)
= −|
κ
|
α
(
F
f
)(
κ
),
(3)
(
F
f
)(
κ
)
being
the
Fourier
transform
of
a
function
f
at
the
point
κ
∈
R
n
that
is
defined
by
(
F
f
)(
κ
)
= ˆ
f
(
κ
)
=
R
n
e
i
κ
·
x
f
(
x
)
dx
.
(4)
For
0
<
α
<
2,
α
=
1,
the
Riesz
space-fractional
derivative
can
be
also
represented
as
a
hypersingular
integral
−
(
−
)
α
2
f
(
x
)
= −
1
d
n
,
l
(
α
)
R
n
(
l
h
f
)(
x
)
|
h
|
n
+
α
dh
with
the
finite
differences
operator
(
l
h
f
)(
x
)
=
l
k
=
0
(
−
1
)
k
l
k
f
(
x
−
kh
)
and
a
suitable
normalization
constant
d
n
,
l
(
α
)
(see
[33]
for
more
details).
In
particular,
in
the
one-dimensional
case
the
representation
−
(
−
)
α
2
f
(
x
)
=
−
1
2
Γ (
−
α
)
cos
(
απ
)
∞
0
f
(
x
+
ξ )
−
2
f
(ξ )
+
f
(
x
−
ξ )
ξ
α
+
1
d
ξ
(5)
holds
true
for
0
<
α
<
2,
α
=
1.
If
α
=
1,
the
representation
 
(5)
is
not
valid
and
has
to
be
replaced
by
the
following
one
(see
e.g.
 
[4]
):
−
(
−
)
1
2
f
(
x
)
= −
1
π
d
dx
+∞
−∞
f
(ξ )
x
−
ξ
d
ξ,
where
the
integral
is
understood
in
the
sense
of
the
Cauchy
principal
value.
Thus
in
the
one-dimensional
case
the
neutral-
fractional
equation
 
(1)
with
α
=
1 has
the
form
∂
u
∂
t
(
x
,
t
)
= −
1
π
d
dx
+∞
−∞
u
(ξ,
t
)
x
−
ξ
d
ξ
(6)
that
is
of
course
not
the
conventional
advection
equation.
For
α
=
2,
Eq.
(1)
is
the
standard
wave
equation
that
is
well
studied
in
the
literature,
so
that
in
the
following
sections
we
focus
on
the
case
1
≤
α
<
2.

---

Y. Luchko / Journal of Computational Physics 293 (2015) 40–52
43
For
Eq.
(1)
,
we
consider
the
initial-value
problem
u
(
x
,
0
)
=
ϕ
(
x
),
∂
u
∂
t
(
x
,
0
)
=
0
,
x
∈
R
n
.
(7)
Let
us
mention
that
even
in
the
case
α
<
2 (but
α
>
1),
two
initial
conditions
are
required
to
guarantee
the
uniqueness
of
a
solution
to
the
initial-value
problem
 
(7)
for
the
neutral-fractional
equation
 
(1)
.
To
ensure
a
kind
of
transition
from
the
case
α
=
1 to
the
case
α
>
1,
we
chose
the
second
initial
condition
to
be
identically
zero.
In
this
paper,
we
are
mostly
interested
in
behavior
and
properties
of
the
first
fundamental
solution
(Green
function)
G
α
,
n
of
Eq.
(1)
,
i.e.
its
solution
with
the
initial
condition
ϕ
(
x
)
=
n
i
=
1
δ(
x
i
),
x
=
(
x
1
,
. . . ,
x
n
)
T
∈
R
n
,
δ
being
the
Dirac
delta
function.
2.2.
Fundamental
solution
to
the
neutral-fractional
equation
Because
the
Riesz
fractional
derivative
is
defined
in
terms
of
the
Fourier
transform,
we
apply
the
multi-dimensional
Fourier
transform
 
(4)
with
respect
to
the
spatial
variable
x
∈
R
n
to
Eq.
(1)
with
1
<
α
<
2 and
to
the
initial
conditions
 
(7)
with
ϕ
(
x
)
=
n
i
=
1
δ(
x
i
)
and
obtain
the
initial-value
problem
⎧
⎨
⎩
ˆ
G
α
,
n
(
κ
,
0
)
=
1
,
∂
ˆ
G
α
,
n
∂
t
(
κ
,
0
)
=
0
(8)
for
the
fractional
differential
equation
D
α
t
ˆ
G
α
,
n
(
κ
,
t
)
(
t
)
+ |
κ
|
α
ˆ
G
α
,
n
(
κ
,
t
)
=
0
(9)
for
the
Fourier
transform
ˆ
G
α
,
n
of
the
fundamental
solution.
The
unique
solution
of
the
problem
(8)
–
(9)
is
given
by
the
formula
(see
e.g.
 
[18]
)
ˆ
G
α
,
n
(
κ
,
t
)
=
E
α
−|
κ
|
α
t
α
,
(10)
where
the
Mittag-Leffler
function
is
defined
by
a
convergent
series
E
α
(
z
)
=
∞
k
=
0
z
k
Γ (
1
+
α
k
)
,
α
>
0
.
(11)
Due
to
the
well
known
asymptotic
formula
E
α
(
−
x
)
= −
m
k
=
1
(
−
x
)
−
k
Γ (
1
−
α
k
)
+
O
|
x
|
−
1
−
m
,
m
∈
N
,
x
→ +∞
,
0
<
α
<
2
,
(12)
the
Fourier
transform
ˆ
G
α
,
n
of
the
fundamental
solution
belongs
to
the
functional
space
L
1
(
R
n
)
with
respect
to
κ
for
1
<
α
<
2.
Therefore
we
can
apply
the
inverse
Fourier
transform
to
the
representation
 
(10)
and
thus
obtain
the
formula
G
α
,
n
(
x
,
t
)
=
1
(
2
π
)
n
R
n
e
−
i
κ
·
x
E
α
−|
κ
|
α
t
α
d
κ
,
x
∈
R
n
,
t
>
0
(13)
for
the
fundamental
solution
G
α
,
n
.
Because
E
α
(
−|
κ
|
α
t
α
)
is
a
radial
function
(spherically
symmetric
function)
in
κ
,
the
formula
(see
e.g.
 
[33]
)
1
(
2
π
)
n
R
n
e
−
i
κ
·
x
φ
|
κ
|
d
κ
=
|
x
|
1
−
n
2
(
2
π
)
n
2
∞
0
φ (
τ
)
τ
n
2
J
n
2
−
1
τ
|
x
|
d
τ
(14)
for
the
inverse
Fourier
transform
of
the
radial
functions
can
be
applied
under
the
condition
that
the
integral
at
the
right
hand
side
of
 
(14)
is
conditionally
or
absolutely
convergent.
In
this
formula,
J
ν
denotes
the
Bessel
function
with
the
index
ν
.
Thus
the
fundamental
solution
G
α
,
n
can
be
represented
in
the
form
G
α
,
n
(
x
,
t
)
=
|
x
|
1
−
n
2
(
2
π
)
n
2
∞
0
E
α
−
τ
α
t
α
τ
n
2
J
n
2
−
1
τ
|
x
|
d
τ
(15)
under
the
conditions
which
are
formulated
below
and
guarantee
that
the
integral
at
the
right
hand
side
of
this
formula
converges
conditionally
or
absolutely.

---

44
Y. Luchko / Journal of Computational Physics 293 (2015) 40–52
In
the
case
x
=
(
0
,
0
,
. . . ,
0
)
=
0
that
corresponds
to
the
case
|
x
|
=
0,
simple
direct
calculations
show
that
G
α
,
n
(
0
,
t
)
=
1
α
t
n
1
2
n
−
1
π
n
2
Γ
n
2
Γ
n
α
Γ
1
−
n
α
Γ (
1
−
n
)
under
the
convergence
condition
0
<
n
<
α
.
For
1
<
α
<
2,
this
condition
means
that
the
fundamental
solution
G
α
,
n
(
0
,
t
)
is
finite
only
in
the
one-dimensional
case
n
=
1.
Because
Γ (
z
)
has
a
pole
at
the
point
z
=
0,
in
the
one-dimensional
case
we
get
G
α
,
1
(
0
,
t
)
=
0
.
For
n
∈
N
,
n
>
1,
the
fundamental
solution
of
the
neutral-fractional
equation
 
(1)
is
infinite
at
the
point
x
=
(
0
,
0
,
. . . ,
0
)
=