Indonesian
J
our
nal
of
Electrical
Engineering
and
Computer
Science
V
ol.
19,
No.
1,
July
2020,
pp.
371
379
ISSN:
2502-4752,
DOI:
10.11591/ijeecs.v19i1.pp371-379
r
371
New
r
eliable
modifications
of
the
homotopy
methods
Khawlah
H.
Hussain
Department
of
Mechanical
T
echnology
,
Basra
T
echnical
Institute
Southern
T
echnical
Uni
v
ersity
,
Iraq
Article
Inf
o
Article
history:
Recei
v
ed
Dec
14,
2019
Re
vised
Feb
3,
2020
Accepted
Feb
20,
2020
K
eyw
ords:
Caputo
deri
v
ati
v
e
Homotop
y
analysis
method
Homotop
y
perturbation
method
Inte
gro-dif
ferential
equation
ABSTRA
CT
In
this
article,
ne
w
modifications
of
the
homotop
y
methods
are
presented
and
applied
to
non-homogeneous
fractional
V
olterra
inte
gro-dif
ferential
equations
with
boundary
conditions.
A
comparati
v
e
study
between
the
ne
w
modified
hom
otop
y
perturbation
method
(MHPM)
and
the
ne
w
modified
homotop
y
analysis
method
(MHAM).
Se
v
eral
illustrati
v
e
e
xamples
are
gi
v
en
to
demonstrate
the
ef
fecti
v
eness
and
reliability
of
the
methods.
Copyright
©
2020
Insitute
of
Advanced
Engineeering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Kha
wlah
H.
Hussain,
Department
of
Mechanical
T
echnology
,
Basra
T
echnical
Institute
Southern
T
echnical
Uni
v
ersity
,
AL-Basrah,
Iraq.
Email:
kha
wlah.hussain@stu.edu.iq
1.
INTR
ODUCTION
In
this
paper
,
we
shall
be
concerned
with
the
non-homogeneous
fractional
V
olterra
inte
gro-dif
ferential
equations
of
the
second
kind
of
the
form:
Z
x
0
(
x;
t
)(
y
(
t
))
dt;
(1)
c
D
y
(
x
)
=
f
(
x
)
y
(
x
)
+
(
x
)
+
y
(
k
)
(0)
=
d
k
;
(2)
y
(
k
)
(1)
=
c
k
;
k
=
1
;
;
n;
n
1
<
n;
0
x
1
;
n
2
N
;
(3)
where
c
D
denotes
a
dif
ferential
operator
with
fractional
order
,
and
the
(
x
)
;
f
(
x
)
and
(
x;
t
)
are
holo-
morphic
functions,
(
y
(
t
))
is
a
polynomial
of
y
(
t
)
with
constant
coef
ficients.
The
homotop
y
analysis
method
(HAM)
proposed
by
Liao
in
1992
and
the
homotop
y
perturbation
method
(HPM)
proposed
by
He
in
1998
are
compared
through
an
e
v
olution
equation
used
as
the
second
e
xam-
ple
in
a
recent
paper
by
Ganji
et
al.
It
is
found
that
the
HPM
is
a
special
case
of
the
HAM
when
~
=
1
.
The
well-kno
wn
and
po
werful
HAM
is
based
on
both
Homotop
y
in
topology
and
the
McLaurin
series.
In
one
of
his
pioneering
articles,
he
claimed
that
the
method
does
not
require
either
small
or
lar
ge
parameters
comparing
with
the
perturbation
techniques.
The
general
concept
of
this
method
has
been
considered
by
man
y
researchers
in
their
published
w
orks
[1–5].
The
fractional
inte
gro-dif
ferential
equations
ha
v
e
attracted
much
more
interest
of
mathematicians
a
n
d
ph
ysicists
which
pro
vides
an
ef
ficienc
y
for
the
descripti
on
of
man
y
practical
dynamical
arising
in
engineer
-
ing
and
scientific
disciplines
such
as,
ph
ysics,
biology
,
electrochemistry
,
chemistry
,
economy
,
electromagnetic,
J
ournal
homepage:
http://ijeecs.iaescor
e
.com
with the boundary conditions
Evaluation Warning : The document was created with Spire.PDF for Python.
372
r
ISSN:
2502-4752
control
theory
and
viscoelasticit
y
[1,
2,
6–15].
In
recent
years,
man
y
authors
focus
on
the
de
v
elopment
of
numerical
and
analyti
cal
techniques
for
fractional
i
nte
gro-dif
ferential
equations.
F
or
instance,
we
can
recall
the
follo
wing
w
orks.
Al-Samadi
and
Gumah
[16]
applied
the
HAM
for
fractional
SEIR
epidemic
model,
Zurig
at
et
al.
[17]
applied
HAM
for
system
of
fractional
inte
gro-dif
ferential
equations.
Y
ang
and
Hou
[18]
applied
the
Laplace
decomposition
method
to
solv
e
the
fractional
inte
gro-dif
ferential
equations,
Mittal
and
Nig
am
[19]
applied
the
Adomian
decomposition
method
to
approximate
solutions
for
fractional
inte
gro-
dif
ferential
equations.
Ma
and
Huang
[20]
applied
h
ybrid
collocation
method
to
study
inte
gro-dif
ferential
equations
of
fractional
order
.
Moreo
v
er
,
properties
of
the
fractional
inte
gro-dif
ferential
equations
ha
v
e
been
studied
by
se
v
eral
authors
[7,
21–27].
The
main
objecti
v
e
of
the
present
paper
is
to
study
the
beha
vior
of
the
solution
that
can
be
formally
determined
by
analytical
approximated
methods
as
the
MHAM
and
MHPM.
2.
PRELIMIN
ARIES
In
this
section,
we
gi
v
e
f
fractional
calculus
theory
which
are
further
used
in
this
paper
[2,
7,
25,
28,
29].
Definition
2..1
A
r
eal
function
f
(
x
)
;
x
>
0
;
is
said
to
be
in
the
space
C
";
"
2
R
;
if
ther
e
e
xists
a
r
eal
number
p
>
"
suc
h
that
f
(
x
)
=
x
p
f
1
(
x
)
,
wher
e
f
1
(
x
)
2
C
[0
;
1)
:
Clearly
C
"
C
!
if
!
":
Definition
2..2
A
function
f
(
x
)
;
x
>
0
,
is
said
to
be
in
the
space
C
n
"
;
n
2
N
[
f
0
g
,
if
f
(
n
)
2
C
"
:
Definition
2..3
[2]
The
Riemann-Liouville
fr
actional
inte
gr
al
of
or
der
>
0
of
a
function
f
2
C
"
;
"
1
is
defined
as
J
f
(
x
)
=
1
(
)
Z
x
0
(
x
t
)
1
f
(
t
)
dt;
x
>
0
;
2
R
+
;
J
0
f
(
x
)
=
f
(
x
)
;
(4)
wher
e
R
+
is
the
set
of
positive
r
eal
number
s.
Definition
2..4
[21]
The
fr
actional
derivative
of
f
(
x
)
2
C
n
1
;
n
2
N
[
f
0
g
in
the
Caputo
sense
is
defined
by
c
D
f
(
x
)
=
J
n
D
n
f
(
x
)
=
8
>
<
>
:
1
(
n
)
R
x
0
(
x
t
)
n
1
d
n
f
(
t
)
dt
n
dt;
n
1
<
<
n;
d
n
f
(
x
)
dx
n
;
=
n;
(5)
wher
e
the
par
ameter
is
the
or
der
of
the
derivative
,
in
g
ener
al
it
is
r
eal
or
e
ven
comple
x.
But
in
this
c
hapter
,
we
will
consider
as
positive
r
eal.
Hence
,
we
have
the
following
pr
operties:
1.
J
J
v
f
=
J
+
v
f
;
;
v
>
0
:
2.
J
x
=
(
+1)
(
+
+1)
x
+
;
>
0
;
>
1
;
x
>
0
:
3.
J
D
f
(
x
)
=
f
(
x
)
P
m
1
k
=0
f
(
k
)
(0
+
)
x
k
k
!
;
m
1
<
m:
Definition
2..5
[8]
The
Riemann-Liouville
fr
actional
derivative
of
or
der
>
0
is
normally
defined
as
D
f
(
x
)
=
D
m
J
m
f
(
x
)
;
m
1
<
m:
(6)
Theor
em
2..1
[2]
(Banac
h
cont
raction
principle).
Let
(
X
;
d
)
be
a
complete
metric
space
,
then
eac
h
contr
ac-
tion
mapping
T
:
X
!
X
has
a
unique
fixed
point
x
of
T
in
X
i.e
.
T
x
=
x:
Theor
em
2..2
[2]
(Sc
hauder’
s
fix
ed
point
theorem).
Let
X
be
a
Banac
h
space
and
let
A
a
con
ve
x,
closed
subset
of
X
.
If
T
:
A
!
A
be
the
map
suc
h
that
the
set
f
T
y
:
y
2
A
g
is
r
elatively
compact
i
n
X
(or
T
is
continuous
and
completely
continouous).
Then
T
has
at
least
one
fixed
point
y
2
A
:
T
y
=
y
:
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
19,
No.
1,
July
2020
:
371
–
379
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
373
3.
DESCRIPTION
OF
THE
METHOD
Some
po
werful
methods
ha
v
e
been
focusing
on
the
de
v
elopment
of
more
adv
anced
and
ef
ficient
methods
for
inte
gro-dif
ferential
equat
ions
such
as
the
HAM
and
HPM
[2,
27,
30,
31].
W
e
will
describe
these
methods
in
this
section:
3.1.
Homotopy
analysis
method
(HAM)
The
basic
concept
behind
the
HAM
is
illustrated
by
using
the
follo
wing
nonlinear
equation:
N
[
y
]
=
0
;
where
N
is
a
nonlinear
operator
,
y
(
x
)
is
unkno
wn
function
and
x
is
an
independent
v
ariable.
Let
y
0
(
x
)
denote
an
initial
guess
of
the
e
xact
solution
y
(
x
)
;
~
6
=
0
an
auxiliary
parameter
,
H
1
(
x
)
6
=
0
an
auxiliary
function,
and
L
an
auxiliary
linear
operator
with
the
property
L
[
s
(
x
)]
=
0
when
s
(
x
)
=
0
.
Then
using
q
2
[0
;
1]
as
an
embedding
parameter
,
we
can
construct
a
homotop
y
when
consider
,
N
[
y
]
=
0
;
as
follo
ws
[2]:
(1
q
)
L
[
(
x
;
q
)
y
0
(
x
)]
q
~
H
1
(
x
)
N
[
(
x
;
q
)]
=
^
H
[
(
x
;
q
);
y
0
(
x
)
;
H
1
(
x
)
;
~
;
q
]
:
(7)
It
should
be
emphasized
that
we
ha
v
e
great
freedom
to
choose
the
initial
guess
y
0
(
x
)
;
the
auxiliary
linear
operator
L
,
the
non-zero
auxiliary
parameter
~
,
and
the
auxiliary
function
H
1
(
x
)
:
Enforcing
the
homotop
y
(7)
to
be
zero,
i.e.,
^
H
(8)
(9)
1
[
(
x
;
q
);
y
0
(
x
)
;
H
1
(
x
)
;
~
;
q
]
=
0
;
(
x
;
0)
=
y
0
(
x
)
;
(10)
and
when
q
=
1
;
since
~
6
=
0
and
H
1
(
x
)
6
=
0
,
the
zero-order
deformation
(9)
is
equi
v
alent
to
(
x
;
1)
=
y
(
x
)
:
(11)
Thus,
according
to
Eqs.(10)
and
(11),
as
the
embedding
parameter
q
increases
from
0
to
1
,
(
x
;
q
)
v
aries
continuously
from
the
initial
approximation
y
0
(
x
)
to
the
e
xact
solution
y
(
x
)
:
Such
a
kind
of
continuous
v
ariation
is
called
deformation
in
homotop
y
[31].
Due
to
T
aylor’
s
theorem,
(
x
;
q
)
can
be
e
xpanded
in
a
po
wer
series
of
q
as
follo
ws:
(
x
;
q
)
=
y
0
(
x
)
+
1
X
m
=1
y
m
(
x
)
q
m
;
(12)
where,
y
m
(
x
)
=
1
m
!
@
m
(
x
;
q
)
@
q
m
j
q
=0
:
(13)
Let
the
initial
guess
y
0
(
x
)
,
the
auxiliary
linear
parameter
L
,
the
nonzero
auxiliary
parameter
~
and
the
auxiliary
function
H
1
(
x
)
be
properly
chosen
so
that
the
po
wer
series
(12)
of
(
x
;
q
)
con
v
er
ges
at
q
=
1
,
then,
we
ha
v
e
under
these
assumptions
the
solution
series,
y
(
x
)
=
(
x
;
1)
=
y
0
(
x
)
+
1
X
m
=1
y
m
(
x
)
:
(14)
Ne
w
r
eliable
modifications
of
...
(Khawlah
H.
Hussain)
we have the so-called zero-order deformation equation
(1 − q)L[φ(x; q) − y0(x)] = qŸH1(x)N[φ(x; q)],
when q = 0, the zero-order deformation (9) becomes
Evaluation Warning : The document was created with Spire.PDF for Python.
374
r
ISSN:
2502-4752
From
(12),
we
can
write
(9)
as
follo
ws:
(1
q
)
L
[
(
x
;
q
)
y
0
(
x
)]
=
(1
q
)
L
[
1
X
m
=1
y
m
(
x
)
q
m
]
(15)
=
q
~
H
1
(
x
)
N
[
(
x
;
q
)]
;
then,
L
[
1
X
m
=1
y
m
(
x
)
q
m
]
q
L
[
1
X
m
=1
y
m
(
x
)
q
m
]
=
q
~
H
1
(
x
)
N
[
(
x
;
q
)]
:
(16)
By
dif
ferentiating
(16)
m
times
with
respect
to
q
,
we
obtain,
f
L
[
1
X
m
=1
y
m
(
x
)
q
m
]
q
L
[
1
X
m
=1
y
m
(
x
)
q
m
]
g
(
m
)
=
q
~
H
1
(
x
)
N
[
(
x
;
q
)]
(
m
)
=
m
!
L
[
y
m
(
x
)
y
m
1
(
x
)]
=
~
H
1
(
x
)
m
@
m
1
N
[
(
x
;
q
)]
@
q
m
1
j
q
=0
:
Therefore,
L
[
y
m
(
x
)
m
y
m
1
(
x
)]
=
~
H
1
(
x
)
<
m
(
!
y
m
1
(
x
))
;
(17)
where,
<
m
(
!
y
m
1
(
x
))
=
1
(
m
1)!
@
m
1
N
[
'
(
x
;
q
)]
@
q
m
1
j
q
=0
;
(18)
and
m
=
(
0
m
1
;
1
m
>
1
:
4.
HOMO
T
OPY
PER
TURB
A
TION
METHOD
(HPM)
The
homotop
y
perturbation
method
first
proposed
by
He
[1].
T
o
illustrate
the
basic
idea
of
this
method,
we
consider
the
follo
wing
nonlinear
dif
ferential
equation
A
(
y
)
f
(
r
)
=
0
;
r
2
;
(19)
under
the
boundary
conditions
B
y
;
@
y
@
n
=
0
;
r
2
;
(20)
where
A
is
a
general
differential
operator,
B
is
a
boundary
operator,
f
(
r
)
is
a
known
analytic
function,
is
the
boundary
of
the
domain
.
In
general,
the
operator
A
can
be
divided
into
two
parts
L
and
N
,
where
L
is
linear,
while
N
is
nonlinear.
(19)
therefore
can
be
rewritten
as
follows
L
(
y
)
+
N
(
y
)
f
(
r
)
=
0
:
(21)
By
the
homotop
y
technique
(Liao
1992,
1997).
W
e
construct
a
homotop
y
v
(
r
;
p
)
:
[0
;
1]
!
R
which
satisfies
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
19,
No.
1,
July
2020
:
371
–
379
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
375
H
(
v
;
p
)
=
(1
p
)[
L
(
v
)
L
(
y
0
)]
+
p
[
A
(
v
)
f
(
r
)]
=
0
;
p
2
[0
;
1]
:
(22)
or
H
(
v
;
p
)
=
L
(
v
)
L
(
y
0
)
+
pL
(
y
0
)]
+
p
[
N
(
v
)
f
(
r
)]
=
0
;
(23)
where
p
2
[0
;
1]
is
an
embedding
parame
ter
,
y
0
is
an
initial
approximat
ion
of
(19)
which
satisfies
the
boundary
conditions.
From
(22),
(23)
we
ha
v
e
(24)
H
(
v
;
0)
=
L
(
v
)
L
(
y
0
)
=
0
;
H
(
v
;
1)
=
A
(
v
)
f
(
r
)
=
0
:
(25)
The
changing
in
the
process
of
p
from
zero
to
unity
is
just
that
of
v
(
r
;
p
)
from
y
0
(
r
)
to
y
(
r
)
.
In
topology
this
is
called
deformation
and
L
(
v
)
L
(
y
0
)
,
and
A
(
v
)
f
(
r
)
are
called
homotopic.
No
w
,
assume
that
the
solution
of
Eqs.
(22),
(23)
can
be
e
xpressed
as
(26)
The
approximate
solution
of
(19)
can
be
obtained
by
Setting
p
=
1
.
u
=
lim
p
!
1
v
=
v
0
+
v
1
+
v
2
+
(27)
5.
THE
MAIN
RESUL
TS
In
this
section,
we
shall
gi
v
e
an
uniqueness
result
of
(1)
,
with
the
condition
(2)
and
pro
v
e
it.
Before
starting
and
pro
ving
the
main
results,
we
introduce
the
follo
wing
h
ypotheses:
(A1)
There
e
xists
a
constant
L
>
0
such
that,
for
an
y
y
1
;
y
2
2
C
(
J
;
R
)
j
(
y
1
)
(
y
2
)
j
L
j
y
1
y
2
j
(A2)
There
e
xists
a
function
2
C
(
D
;
R
+
)
;
the
set
of
all
positi
v
e
function
continuous
on
D
=
f
(
x;
t
)
2
R
R
:
0
t
x
1
g
such
that
=
sup
x;t
2
[0
;
1]
R
x
0
j
(
x;
t
)
j
dt
<
1
:
(A3)
The
tw
o
functions
f
;
:
J
!
R
are
continuous.
Lemma
5..1
If
y
0
(
x
)
2
C
(
J
;
R
)
;
then
y
(
x
)
2
C
(
J
;
R
+
)
is
a
solution
of
the
pr
oblem
(1)
(2)
if
f
y
satisfying
y
(
x
)
=
y
0
+
1
(
)
Z
x
0
(
x
s
)
1
f
(
s
)
y
(
s
)
ds
+
1
(
)
Z
x
0
(
x
s
)
1
(
s
)
ds
+
1
(
)
Z
x
0
(
x
s
)
1
Z
s
0
(
s;
)(
y
(
))
d
ds;
wher
e
y
0
=
P
n
1
k
=0
d
k
x
k
k
!
:
Our
result
is
based
on
the
Banach
contraction
principle.
Theor
em
5..2
Assume
that
(A1),
(A2)
and
(A3)
hold.
If
k
f
k
1
+
L
<
1
:
(28)
Then
there
exists
a
unique
solution
y
(
x
)
2
C
(
J
)
to
(1)
(2)
:
Ne
w
r
eliable
modifications
of
...
(Khawlah
H.
Hussain)
v = v0 + pv1 + p2v2 + · · ·
Γ(α + 1)
Evaluation Warning : The document was created with Spire.PDF for Python.
376
r
ISSN:
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Pr
oof
5..3
By
Lemma
5..1.
we
know
that
a
function
y
is
a
solution
to
(1)
(2)
if
f
y
satisfies
y
(
x
)
=
y
0
+
1
(
)
Z
x
0
(
x
s
)
1
f
(
s
)
y
(
s
)
ds
+
1
(
)
Z
x
0
(
x
s
)
1
(
s
)
ds
+
1
(
)
Z
x
0
(
x
s
)
1
Z
s
0
(
s;
)(
y
(
))
d
ds:
Let
the
oper
ator
T
:
C
(
J
;
R
)
!
C
(
J
;
R
)
be
defined
by
(
T
y
)(
x
)
=
y
0
+
1
(
)
Z
x
0
(
x
s
)
1
f
(
s
)
y
(
s
)
ds
+
1
(
)
Z
x
0
(
x
s
)
1
(
s
)
ds
+
1
(
)
Z
x
0
(
x
s
)
1
Z
s
0
(
s;
)(
y
(
))
d
ds;
we
can
see
that,
If
y
2
C
(
J
;
R
)
is
a
fixed
point
of
T
,
then
y
is
a
solution
of
(1)
(2)
.
Now
we
pr
o
ve
T
has
a
fixed
point
y
in
C
(
J
;
R
)
.
F
or
that,
let
y
1
;
y
2
2
C
(
J
;
R
)
and
for
any
x
2
[0
;
1]
suc
h
that
y
1
(
x
)
=
y
0
+
1
(
)
Z
x
0
(
x
s
)
1
f
(
s
)
y
1
(
s
)
ds
+
1
(
)
Z
x
0
(
x
s
)
1
(
s
)
ds
+
1
(
)
Z
x
0
(
x
s
)
1
Z
s
0
(
s;
)(
y
1
(
))
d
ds;
and,
y
2
(
x
)
=
y
0
+
1
(
)
Z
x
0
(
x
s
)
1
f
(
s
)
y
2
(
s
)
ds
+
1
(
)
Z
x
0
(
x
s
)
1
(
s
)
ds
+
1
(
)
Z
x
0
(
x
s
)
1
Z
s
0
(
s;
)(
y
2
(
))
d
ds:
Consequently
,
we
g
et
j
(
T
y
1
)(
x
)
(
T
y
2
)(
x
)
j
1
(
)
Z
x
0
(
x
s
)
1
j
f
(
s
)
j
j
y
1
(
s
)
y
2
(
s
)
j
ds
+
1
(
)
Z
x
0
(
x
s
)
1
R
s
0
j
(
s;
)
j
j
(
y
1
(
))
(
y
2
(
))
j
d
ds
k
f
k
1
(
+
1)
j
y
1
(
x
)
y
2
(
x
)
j
+
L
(
+
1)
j
y
1
(
x
)
y
2
(
x
)
j
(
+
1)
j
y
1
(
x
)
y
2
(
x
)
j
=
k
f
k
1
+
L
(
+
1)
j
y
1
(
x
)
y
2
(
x
)
j
:
F
r
om
the
inequality
(28)
we
have
k
T
y
1
T
y
2
k
1
k
f
k
1
+
L
(
+
1)
k
y
1
y
2
k
1
:
This
means
that
T
is
contr
action
map.
By
the
Banac
h
contr
action
principle
,
we
can
conclude
that
T
has
a
unique
fixed
p
oint
y
i
n
C
(
J
;
R
)
:
6.
NUMERICAL
EXAMPLE
In
this
section,
we
proposed
a
numerical
solution
for
nonlinear
fractional
V
olterra
integro-differential
equations
by
using
the
MHAM
and
MHPM
,
as
shown
in
T
ables
1-3
and
F
igure
1.
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
19,
No.
1,
July
2020
:
371
–
379
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
377
Example
1.
Consider
the
follo
wing
fractional
V
olterra
inte
gro-dif
ferential
equation
with
the
boundary
conditions:
c
D
y
(
x
)
=
y
(
x
)
+
x
(1
+
e
x
)
+
3
e
x
Z
x
0
y
(
t
)
dt;
3
<
4
;
x
2
[0
;
1]
;
(29)
with
the
boundary
conditions:
y
(0)
=
1
;
y
(1)
=
1
+
e;
y
00
(0)
=
2
;
y
00
(1)
=
3
e:
(30)
Therefore
the
e
xact
solution
is
y
(
x
)
=
1
+
xe
x
;
for
=
4
T
able
1.
Numerical
results
of
the
e
xample
1
x
Exact
MHAM
MHPM
0.0
1.000000000
1.000000000
1.000000000
0.1
1.110517092
1.107047479
1.102647336
0.2
1.244280552
1.237721115
1.229286556
0.3
1.404957642
1.395985741
1.384226779
0.4
1.596729879
1.586238703
1.572164823
0.5
1.824360635
1.813384918
1.798234919
0.6
2.093271280
2.082918367
2.068063429
0.7
2.409626895
2.401010221
2.387829229
0.8
2.780432743
2.774603867
2.784330528
0.9
3.213642800
3.211517152
3.205058944
1.0
3.718281828
3.718552210
3.718281829
T
able
2.
V
alues
of
A
and
B
for
dif
ferent
v
alues
of
by
MHAM
=
4
=
3
:
5
A
0.96461853025138
1.06172444793295
B
3.43572350417823
1.60468681403168
T
able
3.
V
alues
of
A
and
B
for
dif
ferent
v
alues
of
by
MHPM
=
4
=
3
:
5
A
0.9200006577
0.9855569381
B
3.806600712
2.306799253
Figure
1.
Numerical
Results
of
the
Example
1
Ne
w
r
eliable
modifications
of
...
(Khawlah
H.
Hussain)
Evaluation Warning : The document was created with Spire.PDF for Python.
378
r
ISSN:
2502-4752
7.
CONCLUSION
In
this
paper
,
ne
w
modifications
of
the
homotop
y
perturbation
method
(HPM)
and
the
homotop
y
analysis
method
(HAM)
are
presented
and
applied
to
non-homogeneous
fractional
V
olterra
inte
gro-dif
ferential
equations
with
boundary
conditions.
A
comparati
v
e
study
between
the
ne
w
modified
of
homotop
y
perturbation
method
(MHPM)
and
the
ne
w
modified
of
homotop
y
analysis
method
(MHAM).
F
or
this
purpose,
we
sho
wed
that
the
MHAM
is
more
rapid
con
v
er
gence
than
the
MHPM.
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