Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
10,
No.
1,
February
2020,
pp.
512
520
ISSN:
2088-8708,
DOI:
10.11591/ijece.v10i1.pp512-520
r
512
Sinc
collocation
link
ed
with
finite
differ
ences
f
or
K
orteweg-de
Vries
Fractional
Equation
Kamel
Al-Khaled
Department
of
Mathematics
and
Statistics,
Jordan
Uni
v
ersity
of
Science
and
T
echnology
,
Jordan
Article
Inf
o
Article
history:
Recei
v
ed
Apr
9,
2019
Re
vised
Jul
28,
2019
Accepted
Aug
30,
2019
K
eyw
ords:
Fractional
deri
v
ati
v
e
K
orte
we
g-de
Vries
Equation
Numerical
solutions
Sinc-Collocation
ABSTRA
CT
A
no
v
el
numerical
method
is
proposed
for
K
orte
we
g-de
Vries
Fractional
Equation.
The
fractional
deri
v
ati
v
es
are
described
based
on
the
Caputo
sense.
W
e
construct
the
solution
using
dif
ferent
approach,
that
is
based
on
using
collocation
techniques.
The
method
combining
a
finite
dif
ference
approach
in
the
time-fract
ional
direction,
and
the
Sinc-Collocation
in
the
space
direction,
where
the
deri
v
ati
v
es
are
replaced
by
the
necessary
matrices,
and
a
system
of
algebraic
equations
is
obtained
to
approximate
solution
of
the
problem.
The
numerical
results
are
sho
wn
to
demonstrate
the
ef
ficienc
y
of
the
ne
wly
proposed
method.
Easy
and
economical
implementation
is
the
strength
of
this
method.
Copyright
c
2020
Insitute
of
Advanced
Engineeering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Kamel
Al-Khaled,
Jordan
Uni
v
ersity
of
Science
and
T
echnology
,
Irbid,
P
.O.Box
3030,
Jordan.
Email:
kamel@just.edu.jo
1.
INTR
ODUCTION
Nonlinear
partial
dif
ferential
equations
appear
in
man
y
branches
of
ph
ysics,
engineering
and
applied
mathematics.
In
recent
years,
there
has
been
a
gro
wing
interest
in
the
field
of
fractional
calculus.
Oldham
and
Spanier
[1],
Miller
and
Ross
[2],
and
Podlubn
y
[3]
pro
vide
the
history
and
a
comprehensi
v
e
treatment
of
this
subject.
Fractional
calculus
is
the
field
of
mathematical
analysis,
which
deals
with
the
in
v
estig
ation
and
appli-
cations
of
inte
grals
and
deri
v
ati
v
es
of
arbitrary
order
,
which
can
be
real
or
comple
x.
The
subject
of
fractional
calculus
has
g
ained
im
p
or
tance
during
the
past
three
decades
due
mainly
to
its
demonstrat
ed
applications
in
dif-
ferent
areas
of
ph
ysics
and
engineering.
Se
v
eral
fields
of
applications
of
fractional
dif
ferentiation
and
fractional
inte
gration
are
already
well
established,
some
others
just
started.
Man
y
applications
of
fractional
calculus
can
be
found
in
turb
ulence
and
fluid
dynamics,
stochastic
dynamical
systems,
plasma
ph
ysics
and
controlled
ther
-
monuclear
fusion,
nonlinear
control
theory
,
image
processing,
nonlinear
biological
systems.
It
is
important
to
solv
e
time
fractional
partial
dif
ferential
equations.
It
w
as
found
that
fractional
time
deri
v
ati
v
es
arise
generally
as
infinitesimal
generators
of
the
time
e
v
olution
when
taking
along
time
scaling
limit.
Hence,
the
importance
of
i
n
v
est
ig
ating
fractional
equations
arises
from
the
necessity
to
sharpen
the
concepts
of
equilibrium,
stability
states,
and
time
e
v
olution
in
the
long
time
limit.
There
has
been
some
attempt
to
solv
e
linear
problems
with
multiple
fractional
deri
v
ati
v
es.
In
[4],
an
approximate
solution
based
on
the
decomposition
method
is
gi
v
en
for
the
generalized
fractional
dif
fusion-w
a
v
e
equation.
In
[5],
the
authors
used
the
Sinc-Le
gendre
collocation
method
to
a
numerical
solution
for
a
class
of
fractional
con
v
ection-dif
fusion
equation.
The
theory
of
nonlinear
dispersi
v
e
w
a
v
e
motion
has
recently
under
gone
much
study
,
especially
by
Whitham
[6].
It
can
be
sho
wn
that
the
t
heory
of
w
ater
w
a
v
es
for
the
case
of
shallo
w
w
ater
and
w
a
v
es
of
small
amplitude
can
be
approximately
described
by
the
K
orte
we
g-de
Vries
equation
J
ournal
homepage:
http://ijece
.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
513
@
u
(
x;
t
)
@
t
+
(
c
+
u
(
x;
t
))
@
u
(
x;
t
)
@
x
+
@
3
u
(
x;
t
)
@
x
3
=
0
;
(
x;
t
)
2
I
R
(0
;
T
)
(1)
where
c
and
are
gi
v
en
constants,
and
u
gi
v
es
the
height
of
a
w
a
v
e
abo
v
e
some
equilibrium
l
e
v
el
.
Since
the
amplitude
of
these
w
a
v
es
is
assumed
to
be
small,
it
can
serv
e
as
a
perturbation
parameter
.
These
probl
ems
ha
v
e
been
studied
by
man
y
authors
[7-9].
Ho
we
v
er
,
the
y
used
a
formal
perturbation
technique.
Sometimes
called
multiscale
e
xpansion,
or
,
using
e
v
ens
functions
techniques,
as
in
[10].
One
aspect
that
has
been
in
v
estig
ated
is
the
linearized
form
of
Equation
(1):
@
u
(
x;
t
)
@
t
+
c
@
u
(
x;
t
)
@
x
+
@
3
u
(
x;
t
)
@
x
3
=
0
;
(
x;
t
)
2
I
R
(0
;
T
)
(2)
which
has
tra
v
eling
w
a
v
e
solutions
u
(
x;
t
)
=
a
cos(
k
x
!
t
)
;
where
a
is
constant
and
!
=
!
(
k
)
=
ck
k
3
.
The
e
xistence
of
tra
v
eling
w
a
v
e
solutions
to
(2)
already
has
been
studied
in
[11].
If
we
drop
the
third
deri
v
ati
v
e
term
in
(1),
we
ha
v
e
@
u
(
x;
t
)
@
t
+
(
c
+
u
(
x;
t
))
@
u
(
x;
t
)
@
x
+
@
3
u
(
x;
t
)
@
x
3
=
0
;
(
x;
t
)
2
I
R
(0
;
T
)
which
is
a
quasi-linear
first-order
w
a
v
e
equation
whose
w
a
v
e
speed
depends
on
the
amplitude
and
has
the
implicit
solutions
u
(
x;
t
)
=
a
cos[
k
x
k
(
c
+
u
)
t
]
.
If
c
=
0;
=
1
in
Equation
k(1)
we
get
another
form
of
K
orte
we
g-de
Vries
equation
@
u
(
x;
t
)
@
t
+
u
(
x;
t
)
@
u
(
x;
t
)
@
x
+
@
3
u
(
x;
t
)
@
x
3
=
0
;
(3)
This
nonlinear
equation
admits
tra
v
eling
w
a
v
e
solutions
of
dif
ferent
types.
One
particular
type
of
tra
v
eling
w
a
v
e
that
arises
from
the
K
orte
we
g-de
Vries
equation
is
the
soliton,
or
solitary
w
a
v
e.
The
same
e
qu
a
tion
(3)
has
also
come
up
in
the
theory
of
plasma
and
se
v
eral
other
branches
of
ph
ysics.
In
recent
years,
there
has
been
a
gro
wing
interest
in
the
field
of
fractional
calculus.
Oldham
and
Spanier
[1],
Miller
and
Ross
[2],
and
Podlubn
y
[3]
pro
vide
the
history
and
a
comprehensi
v
e
treatment
of
this
subject.
Fractional
calculus
is
the
field
of
mat
hematical
analysis,
which
deals
with
the
in
v
estig
ation
and
applications
of
inte
grals
and
deri
v
ati
v
es
of
arbitrary
order
,
which
can
be
real
or
comple
x.
Man
y
applica-
tions
of
fractional
calculus
can
be
found
in
turb
ulence
and
fluid
dynamics,
stochastic
dynamical
systems,
plasma
ph
ysics
and
controlled
thermonuclear
fusion,
nonlinear
control
theory
,
image
processing,
nonlinear
biological
systems,
for
more
see
[12]
and
the
references
therein.
Indeed,
it
pro
vides
se
v
eral
potentially
useful
tools
for
solving
dif
ferential
equations
.
It
is
important
to
solv
e
time
fractional
partial
dif
ferential
equations.
It
w
as
found
that
fractional
time
deri
v
ati
v
es
arise
generally
as
infinitesimal
generators
of
the
time
e
v
olution
when
taking
along
time
scaling
limit.
Hence,
the
importance
of
in
v
estig
ating
fractional
equations
arises
from
the
necessity
to
sharpen
the
concepts
of
equilibrium,
stability
states,
and
time
e
v
olution
in
the
long
time
limit.
In
general,
there
e
xists
no
method
that
yields
an
e
xact
solution
for
nonlinear
fractional
partial
dif
ferential
equations.
There
has
been
some
attempt
to
solv
e
linear
problems
with
multiple
fractional
deri
v
ati
v
es.
In
the
present
paper
,
we
consider
the
fractional
K
orte
we
g-de
Vries
Equation
@
u
(
x;
t
)
@
t
+
(
c
+
u
(
x;
t
))
@
u
(
x;
t
)
@
x
+
@
3
u
(
x;
t
)
@
x
3
=
0
;
(
x;
t
)
2
I
R
(0
;
T
)
(4)
with
the
initial
condition
u
(
x;
0)
=
f
(
x
)
;
x
2
I
R
(5)
F
ollo
wing
[13],
we
construct
the
solution
using
dif
ferent
approac
h
,
that
is
based
on
using
collocation
techniques.
The
method
combining
a
finite
dif
ference
approach
in
the
time-fractional
direction,
and
the
Sinc-Collocation
in
the
space
direction,
where
the
deri
v
ati
v
es
are
replaced
by
the
necessary
matrices,
and
a
system
of
algebraic
equations
is
obtained
to
approximate
solution
of
the
problem.
Man
y
researchers
ha
v
e
used
v
arious
numerical
methods
to
solv
e
K
orte
we
g-de
Vries
Equation.
Al-Khaled
[14],
uses
Sinc-Galerkin
method
to
find
a
numerical
solution
of
the
K
orte
we
g-de
Vries
Equation.
The
method
results
in
an
iterati
v
e
scheme
of
an
error
of
order
O
(exp(
c=h
))
for
some
positi
v
e
constants
c;
h
.
Sinc
collocation
link
ed
with
finite
dif
fer
ences
for
...
(Kamel
Al-Khaled)
Evaluation Warning : The document was created with Spire.PDF for Python.
514
r
ISSN:
2088-8708
In
[15],
the
KdV
equation
is
transformed
into
an
equi
v
alent
inte
gral
equation,
and
a
Sinc-collocation
procedure
is
de
v
eloped
for
the
inte
gral
equation.
In
this
paper
,
follo
wing
the
same
idea
as
in
[13],
we
will
use
Sinc-methodology
t
o
study
the
solution
of
equation
(4).
W
e
will
present
a
simple
numerical
method
that
uses
finite
dif
ferences
to
replace
the
first-order
time
deri
v
ati
v
e
with
a
fractional
deri
v
ati
v
e
of
order
,
with
0
<
1
.
The
ph
ysical
interpretation
of
the
fractional
deri
v
ati
v
e
is
that
it
represents
a
de
gree
of
memory
in
the
dif
fusing
material.
The
Sinc-collocation
method
will
be
used
in
the
space
direction.
The
main
idea
is
to
replace
dif
ferential
and
inte
gral
equations
by
their
Sinc
approximations.
The
ease
of
implementation
coupled
with
the
e
xponential
con
v
er
gence
rate
ha
v
e
demonstrated
by
viability
of
this
method.
T
o
enable
us
to
follo
w
the
solution
of
the
Fractional
Bur
gers’
equation
[17,
20,
23],
man
y
definitions
and
studies
of
fractional
calculus
ha
v
e
been
proposed
in
the
last
tw
o
centuries.
These
definitions
include,
Riemman-Liouville,
W
e
yl,
Reize,
Campos,
Caputa,
and
Nishimoto
fractional
operator
.
The
Riemann-Liouville
definition
of
fractional
deri
v
ati
v
e
operator
J
a
which
is
defined
in
[1].
The
Riemann-Liouville
deri
v
ati
v
e
has
certain
disadv
antages
when
trying
to
model
real-w
orld
phenomena
with
fractional
dif
ferential
equations.
Therefore,
we
shall
introduce
a
modified
fractional
dif
ferentiation
operator
D
proposed
by
Caputo’
s
(see,
[2]).
Sinc
function
that
will
be
used
in
this
project,
are
discussed
in
Stenger
[15]
and
by
Lund
[24].
2.
CONSTR
UCTION
OF
THE
SCHEME
In
this
section,
finite
dif
ference
method
scheme
and
Sinc-Collocation
method
is
used
for
solving
the
Fractional
Bur
gers’
equation
(4).
In
the
analysis
of
the
numerical
method
that
follo
ws,
we
will
assume
that
problem
(4)-(5)
has
a
unique
and
suf
ficiently
smooth
solution.
2.1.
Fractional
time-deri
v
ati
v
e
This
sub-section
is
de
v
oted
to
a
description
of
the
operational
properties
of
the
purpose
of
acquaint-
ing
with
suf
ficient
fractional
calculus
theory
.
Man
y
definitions
and
studies
of
fractional
calculus
ha
v
e
been
proposed
in
the
last
tw
o
centuries.
These
definitions
include,
Riemman-Liouville,
W
e
yl,
Reize,
Campos,
Caputa,
and
Nishimoto
fractional
operator
.
As
mentioned
in
[19],
the
Riemann-Liouville
deri
v
ati
v
e
has
certain
disadv
antages
when
trying
to
model
real-w
orld
phenomena
with
fractional
dif
ferential
equations.
Therefore,
we
shall
introduce
no
w
a
modified
fractional
dif
ferentiation
operator
D
proposed
by
Caputo
in
his
w
ork
on
the
theory
of
viscoelasticity
[22].
The
Caputo
fractional
deri
v
ati
v
e
is
considered
in
the
Caputo
sense.
F
or
more
details
on
the
geometric
and
ph
ysical
interpretation
for
fractional
deri
v
ati
v
es
of
both
Riemann-Liouville
and
Caputo
types
see
[22].
Definition
1
F
or
m
to
be
the
smallest
inte
g
er
that
e
xceeds
,
the
Caputo
fr
actional
derivatives
of
or
der
>
0
is
defined
as
D
u
(
x;
t
)
=
@
u
(
x;
t
)
@
t
=
8
>
<
>
:
1
(
m
)
R
t
0
(
t
)
m
1
@
m
u
(
x;
)
@
m
d
;
m
1
<
<
m
@
m
u
(
x;t
)
@
t
m
;
=
m
2
N
F
or
mathematical
properties
of
fractional
deri
v
ati
v
es
and
inte
grals
one
can
consult
the
abo
v
e
mentioned
references.
2.2.
Discr
etization
the
time-fractional
deri
v
ati
v
e
Consider
the
one-dimensional
time-fractional
Bur
gers’
equation
(4).
F
ollo
wing
[16],
we
introduce
a
finite
dif
ference
approximation
to
discretize
the
time-fractional
deri
v
ati
v
e.
Let
t
k
=
k
t;
k
=
0
;
1
;
:::K
,
where
t
=
T
=K
is
the
time
step.
No
w
by
using
the
definition
of
Caputo
time-fractional
deri
v
ati
v
e,
and
since
0
<
1
,
in
Definition
1,
we
tak
e
m
=
1
,
then
for
k
=
0
;
1
;
:::;
K
,
we
ha
v
e
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
1,
February
2020
:
512
–
520
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
515
@
u
(
x;
t
k
+1
)
@
t
=
1
(1
)
Z
t
k
+1
0
@
u
(
x;
s
)
@
s
ds
(
t
k
+1
s
)
=
1
(1
)
k
X
j
=0
Z
t
j
+1
t
j
@
u
(
x;
s
)
@
s
ds
(
t
k
+1
s
)
=
1
(1
)
k
X
j
=0
u
(
x;
t
j
+1
)
u
(
x;
t
j
)
t
Z
t
j
+1
t
j
ds
(
t
k
+1
s
)
+
r
k
+1
t
where
r
k
+1
t
is
the
truncation
error
,
that
tak
es
the
form
r
k
+1
t
c
u
1
(1
)
k
X
j
=0
Z
t
j
+1
t
j
(
t
j
+1
t
j
2
s
)
(
t
k
+1
s
)
ds
+
O
(
t
2
)
It
has
been
pro
v
ed
in
[16]
that
r
k
+1
t
c
u
t
2
,
where
c
u
is
a
constant
depending
only
on
u
.
Set
=
t
j
+1
s
,
and
since
t
k
=
k
t;
k
=
0
;
1
;
:::;
K
,
we
ha
v
e:
As
s
=
t
k
,
then
=
t
k
+1
t
k
=
t
(
k
j
)
=
t
k
j
,
and
as
s
=
t
j
+1
,
then
=
t
k
j
.
Therefore,
@
u
(
x;
t
k
+1
)
@
t
1
(1
)
k
X
j
=0
u
(
x;
t
j
+1
)
u
(
x;
t
j
)
t
Z
t
k
j
+1
t
k
j
d
=
1
(1
)
k
X
j
=0
u
(
x;
t
k
j
+1
)
u
(
x;
t
k
j
)
t
Z
t
j
+1
t
j
d
=
1
(2
)
k
X
j
=0
u
(
x;
t
k
j
+1
)
u
(
x;
t
k
j
)
t
h
(
t
j
+1
)
1
(
t
j
)
1
i
=
1
(2
)
k
X
j
=0
u
(
x;
t
k
j
+1
)
u
(
x;
t
k
j
)
t
h
(
t
)
1
[(
j
+
1)
1
(
j
)
1
]
i
T
o
simplify
the
abo
v
e
result,
we
introduce
the
notations
b
j
=
(
j
+
1)
1
j
1
;
j
=
0
;
1
;
2
;
:::;
K
,
and
we
define
the
discrete
fractional
dif
ferential
operator
@
u
(
x;
t
k
+1
)
@
t
1
(
t
)
(2
)
k
X
j
=0
b
j
h
u
(
x;
t
k
j
+1
)
u
(
x;
t
k
j
)
i
(6)
In
equation
(4),
replace
t
by
t
k
+1
,
and
plug
in
into
equation
(6),
we
obtain
the
approximation
1
(2
)(
t
)
k
X
j
=0
b
j
h
u
(
x;
t
k
j
+1
)
u
(
x;
t
k
j
)
i
=
@
3
u
(
x;
t
k
+1
)
@
x
3
(
c
+
u
(
x;
t
k
+1
))
@
u
(
x;
t
k
+1
)
@
x
;
k
=
0
;
1
;
:::;
K
;
or
,
k
X
j
=0
b
k
j
h
u
(
x;
t
j
+1
)
u
(
x;
t
j
)
i
=
(
t
)
(2
)
@
3
u
(
x;
t
k
+1
)
@
x
3
(
c
+
u
(
x;
t
k
+1
))
@
u
(
x;
t
k
+1
)
@
x
(7)
Let
u
k
(
x
)
be
an
approximation
to
u
(
x;
t
k
)
,
and
call
=
(2
)(
t
)
,
then
for
k
=
0
;
1
;
:::;
K
,
the
abo
v
e
equation
becomes
k
X
j
=0
b
k
j
u
j
+1
(
x
)
=
k
X
j
=0
b
k
j
u
j
(
x
)
+
d
3
dx
3
u
k
+1
(
x
)
(
c
+
u
k
+1
(
x
))
d
dx
u
k
+1
(
x
)
(8)
Sinc
collocation
link
ed
with
finite
dif
fer
ences
for
...
(Kamel
Al-Khaled)
Evaluation Warning : The document was created with Spire.PDF for Python.
516
r
ISSN:
2088-8708
2.3.
Sinc-Collocation
The
goal
of
this
sub-section
is
to
recall
notations
and
definitions
of
the
Sinc
function
that
will
be
used
in
this
paper
.
These
are
discussed
in
[15,
24,
21].
The
sinc
function
is
defined
on
the
whole
real
line,
by
sinc
(
z
)
sin(
z
)
z
;
z
6
=
0
;
1
;
z
=
0
:
(9)
F
or
h
>
0
and
k
=
0
;
1
;
2
;
:::
,
the
translated
sinc
function
with
e
v
enly
spaces
nodes
are
gi
v
en
by
S
(
k
;
h
)(
z
)
8
>
<
>
:
sin[(
h
)(
z
k
h
)]
[(
h
)(
z
k
h
)]
;
z
6
=
k
h;
1
;
z
=
k
h:
(10)
Definition
2
Let
d
>
0
,
and
let
D
d
denote
the
r
e
gion
f
z
=
x
+
iy
:
j
y
j
<
d
g
in
the
comple
x
plane
I
C
,
and
the
conformal
map
of
a
simply
connected
domain
D
in
the
comple
x
domain
onto
D
d
suc
h
that
(
a
)
=
1
and
(
b
)
=
1
,
wher
e
a
and
b
ar
e
boundary
points
of
D
,
i.e
.,
a;
b
2
@
D
.
Let
denote
the
in
ver
se
map
of
,
and
let
the
ar
c
,
with
endpoints
a
and
b
(
a;
b
=
2
)
,
given
by
=
(
1
;
1
)
:
F
or
h
>
0
,
let
the
points
x
k
in
be
given
by
x
k
=
(
k
h
)
;
k
2
Z
Z
;
(
z
)
=
exp
(
(
z
))
:
The
sinc-collocation
procedure
for
equation
(8),
be
gins
by
selecting
composite
s
inc
functions,
appro-
priate
to
the
interv
al
(
a;
b
)
,
as
the
basis
function
for
the
e
xpansion
of
the
approximate
solution
for
u
(
x
)
.
F
or
the
present
paper
the
interv
al
in
the
abo
v
e
definition
is
(
1
;
1
)
.
Therefore,
to
approximate
the
first
and
third
deri
v
ati
v
e
we
tak
e
(
x
)
=
x
.
The
basis
functions
are
deri
v
ed
from
the
composite
translated
sinc
functions
S
i
(
x
)
=
S
(
i;
h
)
(
x
)
=
sin
c
[(
(
x
)
ih
)
=h
]
(11)
S
i
(
x
)
=
S
(
i;
h
)
(
x
)
=
S
(
i;
h
)(
x
)
in
equation
(11)
define
the
basis
element
for
equation
(8)
on
the
interv
al
(
1
;
1
)
.
Here
h
is
the
mesh
size,
the
sinc
g
r
id
points
x
n
2
(
1
;
1
)
will
be
denoted
by
x
n
because
the
y
are
real.
The
in
v
erse
images
of
the
equispaced
grids
are
x
n
=
1
(
nh
)
.
Also
for
positi
v
e
inte
ger
N
,
define
C
N
(
f
;
h
)(
x
)
=
N
X
i
=
N
f
(
ih
)
S
(
i;
h
)
(
x
)
=
N
X
i
=
N
f
(
ih
)
S
(
i;
h
)(
x
)
:
(12)
T
o
approximate
the
deri
v
ati
v
es
of
a
function
f
(
x
)
by
the
sinc
e
xpansion,
the
deri
v
ati
v
es
of
sinc
functions
be
e
v
aluated
at
the
nodes
will
be
needed
[15,
24].
In
particular
,
the
follo
wing
con
v
enient
notation
will
be
useful
in
formulating
the
discrete
system.
(0)
k
j
=
8
<
:
1
;
j
=
k
0
;
j
6
=
k
;
(1)
k
j
=
h
d
dx
[
S
(
j
;
h
)
(
x
)
x
=
x
k
=
8
>
<
>
:
0
;
j
=
k
(
1)
j
k
j
k
;
j
6
=
k
and,
(3)
k
j
=
h
3
d
3
dx
3
[
S
(
j
;
h
)
(
x
)
x
=
x
k
=
8
>
<
>
:
0
;
j
=
k
(
1)
j
k
(
j
k
)
3
[6
2
(
j
k
)
2
]
;
j
6
=
k
No
w
,
we
e
xpand
u
k
(
x
)
;
k
=
0
;
1
;
:::;
K
1
by
Sinc
function
u
k
(
x
)
=
N
X
i
=0
C
k
i
S
i
(
x
)
;
k
=
1
;
:::;
K
(13)
where
C
k
0
;
C
k
1
;
:::;
C
k
N
are
unkno
wn
coef
ficients
to
be
determined.
Substit
ute
equation
(13)
into
equation
(8),
for
k
=
0
;
1
;
:::;
K
1
,
we
ha
v
e
k
X
j
=0
N
X
i
=0
C
j
+1
i
S
i
(
x
)
b
k
j
=
k
X
j
=0
N
X
i
=0
C
j
i
S
i
(
x
)
b
k
j
N
X
i
=0
C
k
+1
i
S
000
i
(
x
)
c
+
N
X
i
=0
C
k
+1
i
S
i
(
x
)
N
X
i
=0
C
k
+1
i
S
0
i
(
x
)
:
(14)
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
1,
February
2020
:
512
–
520
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
517
T
o
find
the
unkno
wn
coef
ficients
C
k
0
;
C
k
1
;
:::;
C
k
N
,
Sinc
collocation
method
with
collocation
points
x
n
is
applied
to
equation
(14),
and
for
k
=
1
;
:::;
K
yield
k
X
j
=0
N
X
i
=0
C
j
+1
i
S
i
(
x
n
)
b
k
j
=
k
X
j
=0
N
X
i
=0
C
j
i
S
i
(
x
n
)
b
k
j
N
X
i
=0
C
k
+1
i
S
000
i
(
x
n
)
c
+
N
X
i
=0
C
k
+1
i
S
i
(
x
n
)
N
X
i
=0
C
k
+1
i
S
0
i
(
x
n
)
:
(15)
where
u
0
(
x
)
can
be
obtained
from
the
initial
condition
as
follo
ws:
u
0
(
x
)
=
u
(
x;
t
0
)
=
f
(
x
)
:
(16)
Equations
(15)-(16)
generate
a
set
of
N
+
1
algebraic
equations
which
can
be
solv
ed
to
find
the
unkno
wn
coef
ficients
C
k
0
;
C
k
1
;
:::;
C
k
N
.
2.4.
Matrix
F
orm
of
the
pr
oposed
method
In
order
to
find
the
matrix
form
of
the
proposed
method,
define
T
oeplitz
matrices
I
(
q
)
j
k
;
q
=
0
;
1
;
3
whose
j
k
th
entry
is
gi
v
en
by
(
q
)
k
j
.
Note
that
the
matrices
I
(1)
j
k
,
I
(3)
j
k
are
sk
e
w
symmetric,
and
the
matrix
I
(0)
j
k
is
an
identity
matrix.
By
separating
the
k
th
term
from
the
first
term
of
left
hand
side
of
equation
(15),
and
for
n
=
0
;
1
;
:::;
N
1
,
we
obtain
N
X
i
=0
C
k
+1
i
S
i
(
x
n
)
b
0
+
N
X
i
=0
C
k
+1
i
S
000
i
(
x
n
)
=
k
X
j
=0
N
X
i
=0
C
j
i
S
i
(
x
n
)
b
k
j
k
1
X
j
=0
N
X
i
=0
C
j
+1
i
S
i
(
x
n
)
b
k
j
+
c
+
N
X
i
=0
C
k
+1
i
S
i
(
x
n
)
N
X
i
=0
C
k
+1
i
S
0
i
(
x
n
)
:
(17)
In
the
abo
v
e
equation,
collect
terms,
and
making
the
same
upper
indices,
we
ha
v
e
N
X
i
=0
C
k
+1
i
h
S
i
(
x
n
)
b
0
S
000
i
(
x
n
)
i
=
k
X
j
=0
N
X
i
=0
C
j
i
S
i
(
x
n
)
h
b
k
j
b
k
+1
j
i
+
c
+
N
X
i
=0
C
k
+1
i
S
i
(
x
n
)
N
X
i
=0
C
k
+1
i
S
0
i
(
x
n
)
:
(18)
The
matrix
form
can
be
obtained
for
k
=
0
;
1
;
:::;
K
1
and
n
=
1
;
2
;
:::;
N
1
as
M
[
C
]
k
+1
=
k
X
`
=0
(
b
k
`
b
k
`
+1
)
A
d
[
C
]
`
:
where,
M
=
[
A
d
B
A
([
C
]
k
+1
)
T
(
c
[
C
]
k
+1
)]
;
[
C
]
k
=
[
C
k
0
;
C
k
1
;
:::;
C
k
N
]
T
,
here
T
is
the
transpose.
A
d
=
[
I
(0)
ij
:
i
=
2
;
:::;
N
1
;
j
=
1
;
:::;
N
;
and
0
else
where
]
N
N
,
A
=
[
I
(1)
ik
:
i
=
2
;
:::;
N
1
;
j
=
1
;
:::;
N
;
and
0
else
where
]
N
N
,
and
B
=
[
I
(3)
ik
:
i
=
2
;
:::;
N
1
;
j
=
1
;
:::;
N
;
and
0
else
where
]
N
N
.
Sinc
collocation
link
ed
with
finite
dif
fer
ences
for
...
(Kamel
Al-Khaled)
Evaluation Warning : The document was created with Spire.PDF for Python.
518
r
ISSN:
2088-8708
The
symbol
in
matrix
M
means
the
componentwise
multiplication.
W
e
can
obtain
the
coef
ficients
C
k
i
;
i
=
0
;
1
;
:::;
N
of
the
approximate
solution
by
solving
the
linear
system
using
an
iterati
v
e
technique.
The
con
v
er
gence
of
the
of
the
series
u
k
to
u
with
increasing
t
h
e
number
of
collocation
points
N
,
we
require
tw
o
necessary
conditions,
first,
the
function
u
must
belong
to
the
P
ale
y
W
iener
space
[15],
and
second,
u
must
defined
on
the
whole
real
line.
F
or
the
analysis
of
the
stability
for
the
Sinc
method
for
solving
the
fractional
Bur
gers’
equation,
we
may
refer
readers
to
resemble
similar
proof
in
[13].
3.
NUMERICAL
RESUL
TS
Here,
we
obtain
some
numerical
results
for
the
solutions
of
the
fractional
KdV
equation
(2).
W
e
use
the
par
ameters,
d
=
=
2
;
N
=
16
to
check
the
performance
for
the
solution
of
the
fract
ional
KdV
equation.
The
computations
associated
with
the
e
xample
were
performed
using
Mathematica.
Example
1
Consider
the
non-linear
fr
actional
KdV
equation
@
u
(
x;
t
)
@
t
+
6
u
(
x;
t
)
@
u
(
x;
t
)
@
x
+
@
3
u
(
x;
t
)
@
x
3
=
0
;
0
x
1
;
0
<
1
;
t
>
0
(19)
It
is
be
noted
that
the
e
xact
solution
for
any
has
the
closed
form
[18]
u
(
x;
t
;
)
=
sec
h
2
1
p
2
h
x
2
t
[1
+
]
i
(20)
In
order
to
illustrate
the
approximate
solution
is
ef
ficienc
y
and
accurate.
Some
numerical
v
alues
for
gi
v
en
e
xplicit
v
alues
of
the
parameters
t
and
for
fix
ed
x
=
0
:
2
are
depicted
in
T
able
1
.
From
the
numerical
v
alues
in
T
able
1
,
it
can
be
seen
that
the
e
xact
solution
(
=
1
)
is
quite
close
to
the
approximate
solution
when
=
0
:
95
.
Also,
in
Figure
1
it
is
observ
ed
that
the
v
alues
of
the
approximate
solution
at
dif
ferent
’
s
has
the
same
beha
vior
as
those
obtained
using
equation
(20)
for
which
=
1
.
This
sho
ws
the
approximate
solution
is
ef
ficienc
y
.
In
the
theory
of
fractional
calculus,
it
is
ob
vious
that
when
the
fractional
deri
v
ati
v
e
(
m
1
<
m
)
tends
to
positi
v
e
inte
ger
m
,
then
the
approximate
solution
continuously
tends
to
the
e
xact
solution
of
the
problem
with
deri
v
ati
v
e
m
=
1
.
A
closer
look
at
the
v
alues
in
T
ables
1
and
2
,
we
observ
e
that
our
approach
do
ha
v
e
this
characteristic.
Figures
2
and
3
sho
ws
the
approximate
solution
for
=
0
:
25
and
=
1
respecti
v
ely
.
Comparison
of
Figures
2
and
3
sho
ws
that
the
solution
continuously
depends
on
the
fractional
deri
v
ati
v
es.
T
able
1.
Numerical
Results
obtained
by
equation
(20)
for
v
arious
v
alues
of
when
x
=
2
t
=
1
=
0
:
95
=
0
:
75
=
0
:
50
=
0
:
25
=
0
:
10
0
:
1
0
:
269740
0
:
279333
0
:
336412
0
:
480254
0
:
759333
0
:
947435
0
:
2
0
:
341487
0
:
358411
0
:
450034
0
:
634307
0
:
874187
0
:
978219
0
:
3
0
:
426435
0
:
449797
0
:
566152
0
:
756832
0
:
935571
0
:
990783
0
:
4
0
:
523486
0
:
551760
0
:
680689
0
:
852402
0
:
970527
0
:
996662
0
:
5
0
:
629290
0
:
666666
0
:
786983
0
:
922553
0
:
989626
0
:
999259
a
=
1
a
=
0.9
a
=
0.1
a
=
0.5
-
6
-
4
-
2
2
4
6
x
0.2
0.4
0.6
0.8
1.0
u
Figure
1.
The
approximate
solution
when
t
=
0
:
2
,
for
dif
ferent
v
alues
of
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
1,
February
2020
:
512
–
520
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
519
-
5
0
5
0.00
0.05
0.10
0.0
0.5
1.0
Figure
2.
The
approximate
solution
when
=
0
:
25
-
5
0
5
0.00
0.05
0.10
0.0
0.5
1.0
Figure
3.
The
approximate
solution
when
=
1
:
0
4.
DISCUSSION
AND
CONCLUSIONS
The
Sinc-Collocation
method
appears
to
be
v
ery
promising
for
solving
the
fractional
Bur
gers’
equation.
An
important
adv
antage
to
be
g
ained
from
the
use
of
this
method
is
the
ability
to
produce
v
ery
accurate
results.
The
e
xample
presented
demonstrate
the
accurac
y
of
the
m
ethod,
which
is
an
impro
v
ement
o
v
er
current
methods
such
as
finite
elements
and
finite
dif
ference
methods.
This
feature
sho
w
the
method
to
be
an
attracti
v
e
for
numerical
solutions
to
the
fractional
Bur
gers’
equation.
W
e
conclude,
with
confi-
dence,
that
the
collocation
using
Sinc
basis
can
be
consi
dered
as
a
beneficial
method
for
solving
a
broad
class
of
fractional
nonli
near
partial
dif
ferential
equations.
The
study
of
these
equations
will
be
the
matter
of
furthers
in
v
estig
ations.
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BIOGRAPHY
OF
A
UTHOR
Kamel
Al-Khaled
Full
Professor
,
Department
of
Mathematics
and
Statisti
cs,
F
aculty
of
Science
and
Arts,
Jordan
Uni
v
ersity
of
Science
and
T
echnology
,
Irbid
22110,
Jordan.
email:kamel@just.edu.jo
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
1,
February
2020
:
512
–
520
Evaluation Warning : The document was created with Spire.PDF for Python.