Wave–diffusion dualism of the neutral-fractional processes

0
for
t
>
0.
For
x
=
0
,
the
asymptotics
 
(12)
of
the
Mittag-Leffler
function
and
the
known
asymptotics
of
the
Bessel
function
lead
to
the
restriction
n
<
2
α
+
1 for
the
conditional
and
to
the
restriction
n
<
2
α
−
1 for
the
absolute
convergence
of
the
integral
at
the
right-hand
side
of
the
representation
 
(15)
.
These
restrictions
mean
that
the
integral
at
the
right-hand
side
of
 
(15)
is
conditionally
convergent
at
least
for
n
=
1
,
2
,
3 and
absolutely
convergent
at
least
for
n
=
1 for
all
α
,
1
<
α
<
2.
Using
the
technique
of
the
Mellin
integral
transform
(see
 
[13]
or
 
[19]
),
the
representation
G
α
,
n
(
x
,
t
)
=
1
απ
n
/
2
|
x
|
n
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
t
|
x
|
−
s
ds
(16)
of
the
fundamental
solution
G
α
,
n
in
terms
of
the
Mellin–Barnes
integral
(Fox
H-function)
can
be
derived.
In
the
one-
dimensional
case
this
representation
can
be
rewritten
in
the
form
G
α
,
1
(
x
,
t
)
=
1
α
|
x
|
1
2
π
i
γ
+
i
∞
γ
−
i
∞
sin
(
π
s
/
2
)
sin
(
π
s
/
α
)
t
|
x
|
−
s
ds
,
0
<
γ
<
α
(17)
by
employing
the
duplication
and
reflection
formulas
for
the
Euler
Gamma
function.
Finally,
we
mention
that
the
representation
 
(16)
along
with
the
known
formula
Γ (
1
+
z
)
=
z
Γ (
z
)
for
the
Gamma
function
and
some
elementary
properties
of
the
Mellin
integral
transform
(see
 
[19]
or
 
[27]
)
lead
to
the
relation
G
α
,
3
(
r
,
t
)
=
1
2
π
r
2
G
α
,
1
(
r
,
t
)
+
t
∂
∂
t
G
α
,
1
(
r
,
t
)
,
r
= |
x
| =
0
(18)
between
the
three- and
the
one-dimensional
fundamental
solutions
G
α
,
3
and
G
α
,
1
.
2.3.
Particular
cases
of
the
fundamental
solution
In
the
one-dimensional
case
(
n
=
1),
the
formula
G
α
,
1
(
x
,
t
)
=
1
π
∞
0
E
α
−
τ
α
t
α
cos
τ
|
x
|
d
τ
,
x
∈
R
,
t
>
0
(19)
easily
follows
from
the
representation
 
(15)
because
of
the
formula
J
−
1
/
2
(
z
)
=
2
π
z
cos
(
z
).
The
formula
(19)
was
used
in
 
[5]
and
then
in
more
detailed
form
in
 
[14]
to
get
a
nice
explicit
form
of
the
fundamental
solution
in
terms
of
the
elementary
functions:
G
α
,
1
(
x
,
t
)
=
1
π
|
x
|
α
−
1
t
α
sin
(
π α
/
2
)
t
2
α
+
2
|
x
|
α
t
α
cos
(
π α
/
2
)
+ |
x
|
2
α
,
t
>
0
,
x
∈
R
,
1
≤
α
<
2
.
(20)
For
α
=
1 (modified
advection
equation
 
(6)
),
the
fundamental
solution
G
α
,
1
is
the
well
known
Cauchy
kernel
G
1
,
1
(
x
,
t
)
=
1
π
t
t
2
+
x
2
that
is
a
spatial
probability
density
function
evolving
in
time.
For
α
=
2 (wave
equation),
the
Green
function
G
α
,
1
is
known
to
be
given
by
the
formula
G
2
,
1
(
x
,
t
)
=
1
2
δ(
x
−
t
)
+
δ(
x
+
t
)
.
Some
plots
of
G
α
,
1
with
α
=
1
.
5 are
presented
in
Fig. 1
.

---

Y. Luchko / Journal of Computational Physics 293 (2015) 40–52
45
Fig. 1.
Plots of
G
α,
1
(
x
,
t
)
with
α
=
1
.
5 and for different values of
t
.
Fig. 2.
Plots of
G
α,
2
(
r
,
t
),
r
= |
x
|
>
0 with
α
=
1
.
5 at the time instant
t
=
1.
In
the
two-dimensional
case
(
n
=
2),
the
representation
 
(15)
has
the
form
G
α
,
2
(
x
,
t
)
=
1
2
π
∞
0
τ
E
α
−
τ
α
t
α
J
0
τ
|
x
|
d
τ
,
x
=
0
,
t
>
0
,
(21)
where
the
Bessel
function
with
the
index
0 can
be
represented
as
e.g.
J
0
(
z
)
=
∞
0
cos
z
sin
(φ)
d
φ
or
in
form
of
the
series
J
0
(
z
)
=
∞
k
=
0
(
−
1
)
k
k
!
k
!
z
2
2
k
.
In
Fig. 2
,
a
plot
of
the
fundamental
solution
G
α
,
2
is
presented.
To
produce
the
plot,
the
integral
representation
 
(21)
and
the
MATLAB-programs
 
[39]
for
numerical
evaluation
of
the
Mittag-Leffler
function
E
α
that
implement
the
algorithms
suggested
in
 
[6]
were
employed.
As
one
can
see
on
the
plot,
G
α
,
2
is
negative
for
some
values
of
the
variables
x
and
t
and
therefore
cannot
be
interpreted
as
a
probability
density
function.
Another
important
point
is
that
G
α
,
2
is
not
unimodal
and
has
many
(probably
infinitely
many)
local
minimum
and
maximum
points.
In
the
three-dimensional
case,
the
representation
 
(15)
can
be
rewritten
in
the
form
G
α
,
3
(
x
,
t
)
=
1
2
π
2
|
x
|
∞
0
E
α
−
τ
α
t
α
τ
sin
τ
|
x
|
d
τ
,
x
=
0
,
t
>
0
(22)
because
of
the
formula
J
1
/
2
(
z
)
=
2
π
z
sin
(
z
).
The
representation
 
(20)
of
the
fundamental
solution
G
α
,
1
along
with
the
relation
 
(18)
between
the
fundamental
solutions
G
α
,
1
and
G
α
,
3
leads
to
a
closed
form
formula
for
the
fundamental
solution
of
the
three-dimensional
neutral-fractional
equation
(
x
=
0
,
t
>
0):

---

46
Y. Luchko / Journal of Computational Physics 293 (2015) 40–52
Fig. 3.
Plots of
G
α,
3
(
r
,
t
)
with
α
=
1
.
5 at the time instants
t
=
0
.
2
,
0
.
3
,
0
.
4.
Fig. 4.
Plots of the fundamental solution
G
α,
3
(
r
,
t
),
r
= |
x
|
>
0 with
α
=
1
.
5 at the points
r
=
0
.
3
,
0
.
5
,
0
.
7.
G
α
,
3
(
x
,
t
)
=
sin
(
π α
/
2
)
2
π
2
(
−
(
α
−
1
)
t
2
α
+
2
|
x
|
α
t
α
cos
(
π α
/
2
)
+
(
1
+
α
)
|
x
|
2
α
)
|
x
|
3
−
α
t
−
α
(
t
2
α
+
2
|
x
|
α
t
α
cos
(
π α
/
2
)
+ |
x
|
2
α
)
2
.
(23)
In
Figs. 3 and
 
4
,
several
plots
of
the
fundamental
solution
G
α
,
3
with
α
=
1
.
5 and
for
some
values
of
t
and
|
x
|
=
r
,
respec-
tively,
are
presented.
As
we
can
see
in the
plots
of
 
Fig. 3
,
for
a
fixed
value
of
t
,
the
fundamental
solution
G
α
,
3
(
r
,
t
)
has
only
one
maximum
point
that
depends
on
the
time
instant
t
(and
of
course
on
the
value
of
the
parameter
α
).
The
plots
of
 
Fig. 4
demonstrate
that
the
profiles
of
the
fundamental
solution
G
α
,
3
look
like
some
damped
waves.
This
interpretation
will
be
discussed
in
Section
4
in
more
details.