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.
View publication stats
Document Outline - Wave-diffusion dualism of the neutral-fractional processes
- 1 Introduction
- 2 Neutral-fractional equation
- 3 Fundamental solution as a probability density function
- 4 Fundamental solution as a damped wave
- 5 Discussions and open problems
- Acknowledgements
- References
Dostları ilə paylaş: |