Wave–diffusion dualism of the neutral-fractional processes

5.
Discussions
and
open
problems
In
this
paper,
some
analytical
and
physical
properties
of
the
first
fundamental
solution
G
α
,
n
to
the
neutral-fractional
equation
were
discussed.
In
the
one-dimensional
case,
the
fundamental
solution
can
be
interpreted
both
as
a
diffusion
process
and
as
a
wave
propagation
that
emphasizes
its
wave–diffusion
dualism.
In
particular,
we
showed
that
G
α
,
1
behaves
as
a
damped
wave
with
the
constant
propagation
velocities
of
its
maximum
location,
“mass”-center,
and
the
“gravity”-center.
Otherwise,
G
α
,
1
is
a
probability
density
function
whose
entropy
and
the
entropy
production
rate
are
very
similar
to
those
of
the
conventional
diffusion.
Of
course,
it
would
be
interesting
to
investigate
other
kinds
of
entropy
as
the
Tsallis
and
the
Rényi
entropies
and
the
corresponding
entropy
production
rates
and
to
compare
the
results
with
the
known
findings
for
both
the
conventional
diffusion
equation
and
for
the
time- and
the
space-fractional
diffusion
equations.
As
to
the
three- and
especially
the
two-dimensional
cases
of
the
neutral-fractional
equation,
some
questions
are
still
open.
In
these
cases,
the
fundamental
solution
is
not
always
nonnegative
and
thus
cannot
be
interpreted
as
a
probabil-
ity
density
function
(by
the
way,
the
first
fundamental
solution
of
the
time-fractional
diffusion–wave
equation
shares
this
property
and
is
also
a
pdf
only
in
the
one-dimensional
case,
but
not
in
the
two- or
three-dimensional
ones).
As
we
have
established,
the
phase
velocity
of
the
fundamental
solution
G
α
,
n
is
a
constant
that
depends
only
on
the
equation
order
α
.
Further
investigations
of
the
physical
properties
of
the
solutions
to
the
two- and
three-dimensional
neutral-fractional
equa-
tions
should
include
a
treatment
of
other
velocities
of
the
damped
waves
that
are
described
by
these
equations
as
the
velocity
of
the
“mass”-center,
the
“gravity”-center,
the
pulse
velocity,
the
group
velocity,
the
centrovelocities,
etc.

---

52
Y. Luchko / Journal of Computational Physics 293 (2015) 40–52
Another
potentially
interesting
research
topic
would
be
to
employ
the
neutral-fractional
equation
we
dealt
with
in
this
paper
in
the
theory
of
the
fractional
Schrödinger
equation
(see
e.g.
 
[1]
or
 
[15]
for
more
details).
In
the
literature,
different
kinds
of
the
space-,
time-,
and
space–time-fractional
Schrödinger
equations
were
already
introduced
and
analyzed
and
it
would
be
interesting
to
investigate
if
a
neutral-fractional
Schrödinger
equation
could
contribute
to
explanation
of
some
new
quantum
effects.
Finally,
we
mention
a
very
recent
research
field
of
fractional
calculus
that
deals
with
the
inverse
problems
for
the
fractional
differential
equations
(see
e.g.
 
[23]
and
references
there).
The
inverse
problems
for
the
multi-dimensional
neutral-
fractional
equation
and
their
applications
would
be
worth
to
be
considered,
too.
Acknowledgements
The
author
is
thankful
to
the
anonymous
referees
for
some
constructive
remarks
and
suggestions
that
helped
to
improve
the
quality
of
the
paper.
References
[1]
B.
 
Al-Saqabi,
 
L.
 
Boyadjiev,
 
Yu.
 
Luchko,
 
Comments
 
on
 
employing
 
the
 
Riesz–Feller
 
derivative
 
in
 
the
 
Schrödinger
 
equation,
 
Eur.
 
Phys.
 
J.
 
Spec.
 
Top.
 
222
 
(2013)
 
1779–1794.
[2]
J.M.
 
Carcione,
 
D.
 
Gei,
 
S.
 
Treitel,
 
The
 
velocity
 
of
 
energy
 
through
 
a
 
dissipative
 
medium,
 
Geophysics
 
75
 
(2010)
 
T37–T47.
[3]
A.
 
Freed,
 
K.
 
Diethelm,
 
Yu.
 
Luchko,
 
Fractional-order
 
viscoelasticity
 
(FOV):
 
constitutive
 
development
 
using
 
the
 
fractional
 
calculus,
 
NASA’s
 
Glenn
 
Research
Center,
 
Ohio,
 
2002.
[4]
R.
 
Gorenflo,
 
F.
 
Mainardi,
 
Random
 
walk
 
models
 
for
 
space-fractional
 
diffusion
 
processes,
 
Fract.
 
Calc.
 
Appl.
 
Anal.
 
1
 
(1998)
 
167–191.
[5]
R.
 
Gorenflo,
 
A.
 
Iskenderov,
 
Yu.
 
Luchko,
 
Mapping
 
between
 
solutions
 
of
 
fractional
 
diffusion–wave
 
equations,
 
Fract.
 
Calc.
 
Appl.
 
Anal.
 
3
 
(2000)
 
75–86.
[6]
R.
 
Gorenflo,
 
J.
 
Loutchko,
 
Yu.
 
Luchko,
 
Computation
 
of
 
the
 
Mittag-Leffler
 
function
 
and
 
its
 
derivatives,
 
Fract.
 
Calc.
 
Appl.
 
Anal.
 
5
 
(2002)
 
491–518.
[7]
I.
 
Gurwich,
 
On
 
the
 
pulse
 
velocity
 
in
 
absorbing
 
and
 
nonlinear
 
media
 
and
 
parallels
 
with
 
the
 
quantum
 
mechanics,
 
Prog.
 
Electromagn.
 
Res.
 
33
 
(2001)
69–96.
[8]
A.
 
Hanyga,
 
Multi-dimensional
 
solutions
 
of
 
space–time-fractional
 
diffusion
 
equations,
 
Proc.
 
R.
 
Soc.
 
Lond.
 
Ser.
 
A,
 
Math.
 
Phys.
 
Sci.
 
458
 
(2002)
 
429–450.
[9]
R.
 
Hilfer
 
(Ed.),
 
Applications
 
of
 
Fractional
 
Calculus
 
in
 
Physics,
 
World
 
Scientific,
 
Singapore,
 
2000.
[10]
K.H.
 
Hoffmann,
 
C.
 
Essex,
 
C.
 
Schulzky,
 
Fractional
 
diffusion
 
and
 
entropy
 
production,
 
J.
 
Non-Equilib.
 
Thermodyn.
 
23
 
(1998)
 
166–175.
[11]
R.
 
Klages,
 
G.
 
Radons,
 
I.M.
 
Sokolov
 
(Eds.),
 
Anomalous
 
Transport:
 
Foundations
 
and
 
Applications,
 
Wiley-VCH,
 
Weinheim,
 
2008.
[12]
X.
 
Li,
 
C.
 
Essex,
 
M.
 
Davison,
 
K.H.
 
Hoffmann,
 
C.
 
Schulzky,
 
Fractional
 
diffusion,
 
irreversibility
 
and
 
entropy,
 
J.
 
Non-Equilib.
 
Thermodyn.
 
28
 
(2003)
 
279–291.
[13]
Yu.
 
Luchko,
 
Multi-dimensional
 
fractional
 
wave
 
equation
 
and
 
some
 
properties
 
of
 
its
 
fundamental
 
solution,
 
e-print,
 
arXiv:1311.5920
 
[math-ph],
 
2013.
[14]
Yu.
 
Luchko,
 
Fractional
 
wave
 
equation
 
and
 
damped
 
waves,
 
J.
 
Math.
 
Phys.
 
54
 
(2013)
 
031505.
[15]
Yu.
 
Luchko,
 
Fractional
 
Schrödinger
 
equation
 
for
 
a
 
particle
 
moving
 
in
 
a
 
potential
 
well,
 
J.
 
Math.
 
Phys.
 
54
 
(2013)
 
012111.
[16]
Yu.
 
Luchko,
 
Models
 
of
 
the
 
neutral–fractional
 
anomalous
 
diffusion
 
and
 
their
 
analysis,
 
in:
 
AIP
 
Conf.
 
Proc.,
 
vol. 1493,
 
2012,
 
pp. 626–632.
[17]
Yu.
 
Luchko,
 
Anomalous
 
diffusion
 
models
 
and
 
their
 
analysis,
 
Forum
 
Berl.
 
Math.
 
Ges.
 
19
 
(2011)
 
53–85.
[18]
Yu.
 
Luchko,
 
Operational
 
method
 
in
 
fractional
 
calculus,
 
Fract.
 
Calc.
 
Appl.
 
Anal.
 
2
 
(1999)
 
463–489.
[19]
Yu.
 
Luchko,
 
V.
 
Kiryakova,
 
The
 
Mellin
 
integral
 
transform
 
in
 
fractional
 
calculus,
 
Fract.
 
Calc.
 
Appl.
 
Anal.
 
16
 
(2013)
 
405–430.
[20]
Yu.
 
Luchko,
 
F.
 
Mainardi,
 
Some
 
properties
 
of
 
the
 
fundamental
 
solution
 
to
 
the
 
signalling
 
problem
 
for
 
the
 
fractional
 
diffusion–wave
 
equation,
 
Cent.
 
Eur.
 
J.
 
Phys.
 
11
 
(2013)
 
666–675.
[21]
Yu.
 
Luchko,
 
A.
 
Punzi,
 
Modeling
 
anomalous
 
heat
 
transport
 
in
 
geothermal
 
reservoirs
 
via
 
fractional
 
diffusion
 
equations,
 
Int.
 
J.
 
Geomath.
 
1
 
(2011)
 
257–276.
[22]
Yu.
 
Luchko,
 
F.
 
Mainardi,
 
Yu.
 
Povstenko,
 
Propagation
 
speed
 
of
 
the
 
maximum
 
of
 
the
 
fundamental
 
solution
 
to
 
the
 
fractional
 
diffusion–wave
 
equation,
 
Comput.
 
Math.
 
Appl.
 
66
 
(2013)
 
774–784.
[23]
Yu.
 
Luchko,
 
W.
 
Rundell,
 
M.
 
Yamamoto,
 
L.
 
Zuo,
 
Uniqueness
 
and
 
reconstruction
 
of
 
an
 
unknown
 
semilinear
 
term
 
in
 
a
 
time-fractional
 
reaction–diffusion
equation,
 
Inverse
 
Probl.
 
29
 
(2013)
 
065019.
[24]
F.
 
Mainardi,
 
Fractional
 
relaxation–oscillation
 
and
 
fractional
 
diffusion–wave
 
phenomena,
 
Chaos
 
Solitons
 
Fractals
 
7
 
(1996)
 
1461–1477.
[25]
F.
 
Mainardi,
 
Fractional
 
Calculus
 
and
 
Waves
 
in
 
Linear
 
Viscoelasticity,
 
Imperial
 
College
 
Press,
 
London,
 
2010.
[26] F.
Mainardi,
Yu.
Luchko,
G.
Pagnini,
The
fundamental
solution
of
the
space–time
fractional
diffusion
equation,
Fract.
Calc.
Appl.
Anal.
4
(2001)
153–192,
http://arxiv.org/abs/cond-mat/0702419
.
[27]
O.I.
 
Marichev,
 
Handbook
 
of
 
Integral
 
Transforms
 
of
 
Higher
 
Transcendental
 
Functions,
 
Theory
 
and
 
Algorithmic
 
Tables,
 
Ellis
 
Horwood,
 
Chichester,
 
1983.
[28]
R.
 
Metzler,
 
J.
 
Klafter,
 
The
 
restaurant
 
at
 
the
 
end
 
of
 
the
 
random
 
walk:
 
recent
 
developments
 
in
 
the
 
description
 
of
 
anomalous
 
transport
 
by
 
fractional
 
dynamics,
 
J.
 
Phys.
 
A,
 
Math.
 
Gen.
 
37
 
(2004)
 
R161–R208.
[29]
R.
 
Metzler,
 
T.F.
 
Nonnenmacher,
 
Space- and
 
time-fractional
 
diffusion
 
and
 
wave
 
equations,
 
fractional
 
Fokker–Planck
 
equations,
 
and
 
physical
 
motivation,
Chem.
 
Phys.
 
284
 
(2002)
 
67–90.
[30]
J.
 
Prehl,
 
C.
 
Essex,
 
K.H.
 
Hoffmann,
 
The
 
superdiffusion
 
entropy
 
production
 
paradox
 
in
 
the
 
space-fractional
 
case
 
for
 
extended
 
entropies,
 
Physica
 
A
 
389
(2010)
 
214–224.
[31]
J.
 
Prehl,
 
C.
 
Essex,
 
K.H.
 
Hoffmann,
 
Tsallis
 
relative
 
entropy
 
and
 
anomalous
 
diffusion,
 
Entropy
 
14
 
(2012)
 
701–716.
[32]
A.
 
Saichev,
 
G.
 
Zaslavsky,
 
Fractional
 
kinetic
 
equations:
 
solutions
 
and
 
applications,
 
Chaos
 
7
 
(1997)
 
753–764.
[33]
S.G.
 
Samko,
 
A.A.
 
Kilbas,
 
O.I.
 
Marichev,
 
Fractional
 
Integrals
 
and
 
Derivatives:
 
Theory
 
and
 
Applications,
 
Gordon
 
and
 
Breach,
 
New
 
York,
 
1993.
[34]
C.
 
Shannon,
 
A
 
mathematical
 
theory
 
of
 
communication,
 
Bell
 
Syst.
 
Tech.
 
J.
 
27
 
(1948),
 
379–423,
 
623–656.
[35]
R.L.
 
Smith,
 
The
 
velocities
 
of
 
light,
 
Am.
 
J.
 
Phys.
 
38
 
(1970)
 
978–984.
[36]
V.V.
 
Uchaikin,
 
Background
 
and
 
Theory,
 
vol.
 
I,
 
Applications,
 
vol.
 
II,
 
Fractional
 
Derivatives
 
for
 
Physicists
 
and
 
Engineers,
 
Springer,
 
Heidelberg,
 
2012.
[37]
E.
 
van
 
Groesen,
 
F.
 
Mainardi,
 
Balance
 
laws
 
and
 
centrovelocity
 
in
 
dissipative
 
systems,
 
J.
 
Math.
 
Phys.
 
30
 
(1990)
 
2136–2140.
[38]
E.
 
van
 
Groesen,
 
F.
 
Mainardi,
 
Energy
 
propagation
 
in
 
dissipative
 
systems,
 
Part
 
I:
 
centrovelocity
 
for
 
linear
 
systems,
 
Wave
 
Motion
 
11
 
(1989)
 
201–209.
[39] Matlab
File
Exchange,
Matlab-Code
that
calculates
the
Mittag-Leffler
function
with
desired
accuracy,
Available
for
download
at
 
http://www.mathworks.
com/matlabcentral/fileexchange/8738-mittag-leffler-function
,
2005.
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Document Outline

• Wave-diffusion dualism of the neutral-fractional processes
  • 1 Introduction
  • 2 Neutral-fractional equation
    • 2.1 Problem formulation
    • 2.2 Fundamental solution to the neutral-fractional equation
    • 2.3 Particular cases of the fundamental solution
  • 3 Fundamental solution as a probability density function
    • 3.1 Entropy and the entropy production rate
  • 4 Fundamental solution as a damped wave
    • 4.1 Some physical characteristics of the fundamental solution
    • 4.2 The velocities of the damped waves
  • 5 Discussions and open problems
  • Acknowledgements
  • References