Indonesian
J
our
nal
of
Electrical
Engineering
and
Computer
Science
V
ol.
11,
No.
3,
September
2018,
pp.1228
1235
ISSN:
2502-4752,
DOI:10.11591/ijeecs.v11.i3.pp1228-1235
1228
A
Study
of
Some
Iterati
v
e
Methods
f
or
Solving
Fuzzy
V
olterra-Fr
edholm
Integral
Equations
Ahmed
A.
Hamoud
1
,
Ali
Dhur
gham
Azeez
2
,
and
Kirtiwant
P
.
Ghadle
3
1,3
Department
of
Mathematics,,
Dr
.
Babasaheb
Ambedkar
Marathw
ada
Uni
v
ersity
,,
Aurang
abad-431004
(M.S.)
India.
1
Department
of
Mathematics,
T
aiz
Uni
v
ersity
,
T
aiz,
Y
emen.
2
Master
of
Mathematics,
Thi
Qar
Directorates
of
Education,
Iraq
Article
Inf
o
Article
history:
Recei
v
ed
May
9,
2018
Re
vised
July
3,
2018
Accepted
July
16,
2018
K
eyw
ord:
Adomian
Decomposition
Method
V
ariational
Iteration
Method
Homotop
y
Analysis
Method
Fuzzy
V
olterra-Fredholm
Inte
gral
Equation.
ABSTRA
CT
This
paper
mainly
focuses
on
the
recent
adv
ances
in
the
some
approximated
methods
for
solving
fuzzy
V
olterra-Fredholm
inte
gral
equations,
namely
,
Adomian
decomposition
method,
v
ariati
onal
iteration
method
and
homotop
y
analysis
method.
W
e
con
v
erted
fuzzy
V
olterra-Fredholm
inte
gral
equation
to
a
system
of
V
olterra-Fredholm
inte
gral
equations
in
crisp
case.
The
approximated
methods
using
to
find
the
approximate
solut
ions
of
this
system
and
hence
obtain
an
approximation
for
the
fuzzy
solution
of
the
fuzzy
V
olterra-Fredholm
inte
gral
equation.
T
o
assess
the
accurac
y
of
each
method,
algorithms
with
Mathematica
6
according
is
used.
Also,
numerical
e
xample
i
s
included
to
demonstrate
the
v
alidity
and
applicability
of
the
proposed
techniques.
Copyright
c
2018
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Ahmed
A.
Hamoud
Department
of
Mathematics,
Dr
.
Babasaheb
Ambedkar
Marathw
ada
Uni
v
ersity
,
Aurang
abad-431004
(M.S.)
India.
Email:
drahmed985@yahoo.com
1.
INTR
ODUCTION
Recently
,
the
topics
of
fuzzy
inte
gral
equations
which
a
ttracted
increasing
interest,
in
particular
in
relation
to
fuzzy
control,
ha
v
e
been
rapidly
de
v
eloped.
The
concept
of
fuzzy
numbers
and
arithmetic
operations
fi
rstly
in-
troduced
by
Zadeh
[1],
and
then
by
Dubois
and
Prade
[2]
.
Also,
the
y
ha
v
e
introduced
the
concept
of
inte
gration
of
fuzzy
functions.
The
fuzzy
mapping
function
w
as
introduced
by
Cheng
and
Zadeh
[1].
Moreo
v
er
,
[3]
presented
an
elementary
fuzzy
calculus
based
on
the
e
xte
n
s
ion
principle.
Later
,
Goetschel
and
V
oxman
[4]
preferred
a
Riemann
inte
gral
type
approach.
Kale
v
a
[5]
chose
to
define
the
inte
gral
of
fuzzy
function,
using
the
Lebesgue-type
concept
for
inte
gration.
One
of
the
first
applications
of
the
fuzzy
inte
gral
equation
w
as
gi
v
en
by
Ma
and
W
u
who
in
v
estig
ated
the
fuzzy
Fredholm
inte
gral
equation
of
the
second
kind.
Recently
,
some
mathematicians
ha
v
e
studied
fuzzy
inte
gral
and
inte
gro-dif
ferential
equation
by
numerical
techniques
[6,
7,
8,
9,
10,
11,
12,
13,
14,
15].
As
we
kno
w
the
fuzzy
inte
gral
and
dif
ferential
equations
are
one
of
the
important
parts
of
the
fuzzy
analysis
theory
that
play
a
main
role
in
the
numerical
analysis.
In
this
w
ork,
we
will
suggests
recent
adv
ances
in
the
some
approximated
methods
for
solving
fuzzy
V
olterra-
Fredholm
inte
gral
equations
of
the
s
econd
kind,
namely
,
Adomian
decomposition
method,
v
ariational
iteration
method
and
homotop
y
analysis
method.
2.
FUZZY
V
OL
TERRA-FREDHOLM
INTEGRAL
EQ
U
A
TION
The
fuzzy
V
olterra-Fredholm
inte
gral
equation
of
the
second
kind
is
as
follo
ws:
~
u
(
x
)
=
~
f
(
x
)
+
1
Z
x
a
K
1
(
x;
t
)
G
1
(
t;
~
u
(
t
))
dt
+
2
Z
b
a
K
2
(
x;
t
)
G
2
(
t;
~
u
(
t
))
dt;
(1)
J
ournal
Homepage:
http://iaesjournal.com/online/inde
x.php/IJECE
J
o
u
rn
al
h
o
m
epag
e:
h
t
t
p
:
/
/
i
aesc
o
re.
c
o
m
/
jo
u
rn
al
s
/
i
n
d
e
x.
p
h
p
/
i
je
ec
s
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
1229
where
1
;
2
0
,
~
f
(
x
)
is
a
fuzzy
function
of
x
;
a
x
b
,
and
K
i
(
x;
t
)
;
G
i
(
t;
~
u
(
t
))
;
i
=
1
;
2
;
are
analytic
functions
on
[
a;
b
]
:
F
or
solving
in
parametric
form
of
Eq.(1),
consider
(
f
(
x;
r
)
;
f
(
x;
r
))
and
(
u
(
x;
r
)
;
u
(
x;
r
))
;
0
r
1
and
t
2
[
a;
b
]
are
parametric
form
of
~
f
(
x
)
and
~
u
(
x
)
,
respecti
v
ely
.
then,
parametric
form
of
Eq.(1)
is
as
follo
ws:
u
(
x;
r
)
=
f
(
x;
r
)
+
1
Z
x
a
K
1
(
x;
t
)
G
1
(
t;
u
(
t;
r
))
dt
+
2
Z
b
a
K
2
(
x;
t
)
G
2
(
t;
u
(
t;
r
))
dt;
(2)
u
(
x;
r
)
=
f
(
x;
r
)
+
1
Z
x
a
K
1
(
x;
t
)
G
1
(
t;
u
(
t;
r
))
dt
+
2
Z
b
a
K
2
(
x;
t
)
G
2
(
t;
u
(
t;
r
))
dt;
(3)
Let
for
a
t
b
,
we
ha
v
e
H
1
(
t;
u
;
u
)
=
m
in
f
G
1
(
t;
)
j
u
(
t;
r
)
u
(
t;
r
)
g
;
H
2
(
t;
u
;
u
)
=
m
in
f
G
2
(
t;
)
j
u
(
t;
r
)
u
(
t;
r
)
g
;
F
1
(
t;
u
;
u
)
=
m
ax
f
G
1
(
t;
)
j
u
(
t;
r
)
u
(
t;
r
)
g
;
F
2
(
t;
u
;
u
)
=
max
f
G
2
(
t;
)
j
u
(
t;
r
)
u
(
t;
r
)
g
:
Then,
K
1
(
x;
t
)
G
1
(
t;
u
(
t;
r
))
=
K
1
(
x;
t
)
H
1
(
t;
u
;
u
)
;
K
1
(
x;
t
)
0
;
K
1
(
x;
t
)
F
1
(
t;
u
;
u
)
;
K
1
(
x;
t
)
<
0
:
K
2
(
x;
t
)
G
2
(
t;
u
(
t;
r
))
=
K
2
((
x;
t
))
H
2
(
t;
u
;
u
)
;
K
2
(
x;
t
)
0
;
K
2
((
x;
t
))
F
2
(
t;
u
;
u
)
;
K
2
(
x;
t
)
<
0
:
K
1
(
x;
t
)
G
1
(
t;
u
(
t;
r
))
=
K
1
(
x;
t
)
F
1
(
t;
u
;
u
)
;
K
1
(
x;
t
)
0
;
K
1
(
x;
t
)
H
1
(
t;
u
;
u
)
;
K
1
(
x;
t
)
<
0
:
K
2
(
x;
t
)
G
2
(
t;
u
(
t;
r
))
=
K
2
(
x;
t
)
F
2
(
t;
u
;
u
)
;
K
2
(
x;
t
)
0
;
K
2
(
x;
t
)
H
2
(
t;
u
;
u
)
;
K
2
(
x;
t
)
<
0
:
F
or
each
0
r
1
and
a
x
b
.
W
e
can
see
that
Eq.(1)
con
v
er
t
to
a
system
of
V
olterra-Fredholm
inte
gral
equations
in
crisp
case
for
each
0
r
1
and
a
t
b
.
No
w
,
we
e
xplain
Adomian
decomposition
method,
v
ariational
it
eration
method
and
homotop
y
analysis
method
for
approximating
solution
of
this
system
of
inte
gral
equations
in
crisp
case.
Then,
we
find
approximate
solutions
for
~
u
(
x
)
;
a
x
b:
3.
DESCRIPTION
OF
THE
METHODS
Here
we
will
highlight
briefly
on
some
reliable
methods
for
solving
this
type
of
equations,
where
details
can
be
found
in
[16,
17,
21,
22,
23].
3.1.
Adomian
Decomposition
Method
(ADM)
The
Adomian
decomposition
method
has
been
applied
to
a
wild
class
of
functional
equations
[16,
19,
20,
21]
by
scientists
and
engineers
since
the
be
ginning
of
the
1980s.
Adomian
gi
v
es
the
solution
as
a
infinite
series
usually
con
v
er
ging
to
a
solution
consider
the
follo
wing
fuzzy
Fredholm-V
olterra
inte
gral
equation
of
the
form
u
(
x;
r
)
=
f
(
x;
r
)
+
1
Z
x
a
K
1
(
x;
t
)
G
1
(
t;
u
(
t;
r
))
dt
+
2
Z
b
a
K
2
(
x;
t
)
G
2
(
t;
u
(
t;
r
))
dt;
u
(
x;
r
)
=
f
(
x;
r
)
+
1
Z
x
a
K
1
(
x;
t
)
G
1
(
t;
u
(
t;
r
))
dt
+
2
Z
b
a
K
2
(
x;
t
)
G
2
(
t;
u
(
t;
r
))
dt;
(4)
The
ADM
assume
an
infinite
series
solution
for
the
unkno
wns
functions
[
u
;
u
]
,
gi
v
en
by
u
(
x
)
=
1
X
i
=0
u
i
(
x
)
;
u
(
x
)
=
1
X
i
=0
u
i
(
x
)
:
(5)
A
Study
of
Some
Iter
ative
Methods
for
Solving
Fuzzy
V
olterr
a-F
r
edholm...
(Ahmed
A.
Hamoud)
Evaluation Warning : The document was created with Spire.PDF for Python.
1230
ISSN:
2502-4752
The
nonlinear
operators
G
1
(
t;
u
(
t
))
;
G
1
(
t;
u
(
t
))
;
G
2
(
t;
u
(
t
))
;
G
2
(
t;
u
(
t
))
into
an
infinite
series
of
polynomi-
als
gi
v
en
by
G
1
(
t;
u
(
t
))
=
1
X
i
=0
A
n
;
G
1
(
t;
u
(
t
))
=
1
X
i
=0
A
n
;
G
2
(
t;
u
(
t
))
=
1
X
i
=0
B
n
;
G
2
(
t;
u
(
t
))
=
1
X
i
=0
B
n
:
(6)
where
the
~
A
n
=
[
A
n
;
A
n
]
;
~
B
n
=
[
B
n
;
B
n
]
;
n
0
;
are
the
so-called
Adomian
polynomial.
Substituting
Eqs.(5)
and
Eqs.(6)
into
Eq.(4),
we
get
u
0
=
f
(
x;
r
)
;
u
1
=
1
Z
x
a
K
1
(
x;
t
)
A
0
dt
+
2
Z
b
a
K
2
(
x;
t
)
B
0
dt;
u
n
+1
=
1
Z
x
a
K
1
(
x;
t
)
A
n
dt
+
2
Z
b
a
K
2
(
x;
t
)
B
n
dt:
and
u
0
=
f
(
x;
r
)
;
u
1
=
1
Z
x
a
K
1
(
x;
t
)
A
0
dt
+
2
Z
b
a
K
2
(
x;
t
)
B
0
dt;
u
n
+1
=
1
Z
x
a
K
1
(
x;
t
)
A
n
dt
+
2
Z
b
a
K
2
(
x;
t
)
B
n
dt:
W
e
approximate
~
u
(
x;
r
)
=
[
u
(
x;
r
)
;
u
(
x;
r
)]
by
'
n
=
n
1
X
i
=0
u
i
(
x;
r
)
;
'
n
=
n
1
X
i
=0
u
i
(
x;
r
)
;
where,
lim
n
!1
'
n
=
u
(
x;
r
)
;
lim
n
!1
'
n
=
u
(
x;
r
)
:
3.2.
V
ariational
Iteration
Method
(VIM)
The
v
ariational
iteration
method
(VIM)
is
proposed
by
(He
1997)
[18,
23]
as
a
modification
of
a
general
Lagrange
multiplier
method.
This
method
has
been
sho
wn
to
solv
e
ef
fecti
v
ely
,easily
and
accurately
a
lar
ge
class
of
nonlinear
problems
with
approximations
con
v
er
ging
rapidly
to
a
accurate
solutions.
T
o
illustrate
its
basic
idea
of
the
technique,
we
consider
follo
wing
general
nonlinear
system:
L
[
u
(
x
)]
+
N
[
u
(
x
)]
=
g
(
x
)
;
(7)
Where
L
is
linear
operator
,
N
is
a
nonlinear
operator
,
and
g
(
x
)
is
gi
v
en
continuous
function.
The
basic
character
of
method
is
to
a
correction
functional
for
system
Eq.(7)
which
u
n
+1
(
x
)
=
u
n
(
x
)
+
Z
x
0
(
)
f
Lu
n
(
)
+
N
~
u
n
(
)
g
(
)
g
d
;
(8)
Where
(
)
is
a
general
Lagrangian
multiplier
(Kale
v
a
1987)
which
can
be
identified
optimally
via
v
ariational
theory
,
the
subscript
n
denotes
the
n
th
-order
approximation
and
~
u
n
is
consider
a
restricted
v
ariation,
i.e.
~
u
n
=
0
where
Indonesian
J
Elec
Eng
&
Comp
Sci
V
ol.
11,
No.
3,
September
2018:
1228
-1235
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
1231
L
=
d
dt
.
F
or
the
inte
gral
equation
(1),
let
w
(
x
)
be
a
function
such
that
w
0
(
x
)
=
~
u
(
x
)
,
noting
that
~
u
(
x
)
is
continuous.
Then
we
ha
v
e
w
0
(
x
)
=
~
f
(
x
)
+
1
Z
x
a
K
1
(
x;
t
)
G
1
(
t;
w
0
(
t
))
dt
+
2
Z
b
a
K
2
(
x;
t
)
G
2
(
t;
w
0
(
t
))
dt:
(9)
Consider
1
Z
x
a
K
1
(
x;
t
)
G
1
(
t;
w
0
(
t
))
dt
+
2
Z
b
a
K
2
(
x;
t
)
G
2
(
t;
w
0
(
t
))
dt;
(10)
as
a
restricted
v
ariation;
we
ha
v
e
the
iteration
sequence
w
n
+1
=
w
n
+
Z
x
0
"
w
0
n
(
s
)
1
Z
s
a
K
1
(
s;
t
)
G
1
(
t;
w
0
(
t
))
dt
2
Z
b
a
K
2
(
s;
t
)
G
2
(
t;
w
0
(
t
))
dt
~
f
(
s
)
#
ds:
T
aking
the
v
ariation
with
respect
to
the
independent
v
ariable
w
n
and
noticing
that
w
n
(0)
=
0
,
we
get
w
n
+1
=
w
n
+
(
s
)
w
n
j
s
=
x
Z
x
0
0
(
s
)
w
n
ds
=
0
(11)
Then
we
apply
the
follo
wing
stationary
conditions:
1
+
(
s
)
j
s
=
x
=
0
;
0
(
s
)
j
s
=
x
=
0
;
The
general
Lagrange
multiplier
,
therefore,
can
be
readily
identified:
=
1
and,
as
a
result,
we
obtain
the
follo
wing
iteration
formula:
w
n
+1
=
w
n
Z
x
0
"
w
0
n
(
s
)
~
f
(
s
)
1
Z
s
a
K
1
(
s;
t
)
G
1
(
t;
w
0
(
t
))
dt
2
Z
b
a
K
2
(
s;
t
)
G
2
(
t;
w
0
(
t
))
dt
#
ds:
Therefore,
we
can
write
the
follo
wing
iteration
formulas
u
n
+1
(
x;
r
)
=
u
n
(
x;
r
)
Z
x
0
"
u
n
(
s;
r
)
f
(
s;
r
)
1
Z
s
a
K
1
(
s;
t
)
G
1
(
t;
u
(
t;
r
))
dt
2
Z
b
a
K
2
(
s;
t
)
G
2
(
t;
u
(
t;
r
))
dt
#
ds:
u
n
+1
(
x;
r
)
=
u
n
(
x;
r
)
Z
x
0
"
u
n
(
s;
r
)
f
(
s;
r
)
1
Z
s
a
K
1
(
s;
t
)
G
1
(
t;
u
(
t;
r
))
dt
2
Z
b
a
K
2
(
s;
t
)
G
2
(
t;
u
(
t;
r
))
dt
#
ds:
3.3.
Homotopy
Analysis
Method
(HAM)
Consider
,
N
[
~
u
]
=
0
;
where
N
is
a
nonlinear
operator
,
~
u
=
[
u
(
x;
r
)
;
u
(
x;
r
)]
are
unkno
wn
functions
and
x
is
an
independent
v
ariable
[22].
Let
u
0
(
x;
r
)
;
u
0
(
x;
r
)
denote
an
initial
guess
of
the
e
xact
solution
u
(
x;
r
)
;
u
(
x;
r
)
,
h
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
[
u
]
=
0
;
as
follo
ws:
(1
q
)
L
[
(
x
;
q
;
r
)
u
0
(
x;
r
)]
q
hH
1
(
x
)
N
[
(
x
;
q
;
r
)]
=
^
H
[
(
x
;
q
;
r
);
u
0
(
x;
r
)
;
H
1
(
x
)
;
h;
q
]
:
(12)
It
should
be
emphasized
that
we
ha
v
e
great
freedom
to
choose
the
initial
guess
u
0
(
x;
r
)
;
the
auxiliary
linear
operator
L
,
the
non-zero
auxiliary
parameter
h
,
and
the
auxiliary
function
H
1
(
x
)
:
Enforcing
the
homotop
y
Eq.(12)
to
be
zero,
i.e.,
^
H
1
[
(
x
;
q
;
r
);
u
0
(
x;
r
)
;
H
1
(
x
)
;
h;
q
]
=
0
;
(13)
A
Study
of
Some
Iter
ative
Methods
for
Solving
Fuzzy
V
olterr
a-F
r
edholm...
(Ahmed
A.
Hamoud)
Evaluation Warning : The document was created with Spire.PDF for Python.
1232
ISSN:
2502-4752
we
ha
v
e
the
so-called
zero-order
deformation
equation
(1
q
)
L
[
(
x
;
q
;
r
)
u
0
(
x;
r
)]
=
q
hH
1
(
x
)
N
[
(
x
;
q
;
r
)]
:
(14)
when
q
=
0
,
the
zero-order
deformation
Eq.(14)
becomes
(
x
;
0
;
r
)
=
u
0
(
x;
r
)
:
(15)
and
when
q
=
1
;
since
h
6
=
0
and
H
1
(
x
)
6
=
0
,
the
zero-order
deformation
Eq.(14)
is
equi
v
alent
to
(
x
;
1
;
r
)
=
u
(
x;
r
)
:
(16)
Thus,
according
to
Eqs.(15)
and
(16),
as
the
embedding
parameter
q
increases
from
0
to
1
,
(
x
;
q
;
r
)
v
aries
continuously
from
the
initial
approximation
u
0
(
x;
r
)
to
the
e
xact
solution
u
(
x;
r
)
:
Such
a
kind
of
continuous
v
ariation
is
called
deformation
in
homotop
y
.
Due
to
T
aylor’
s
theorem,
(
x
;
q
;
r
)
can
be
e
xpanded
in
a
po
wer
series
of
q
as
follo
ws
(
x
;
q
;
r
)
=
u
0
(
x;
r
)
+
1
X
m
=1
u
m
(
x;
r
)
q
m
;
(17)
where,
u
m
(
x;
r
)
=
1
m
!
@
m
(
x
;
q
;
r
)
@
q
m
j
q
=0
;
(18)
Let
the
initial
guess
u
0
(
x;
r
)
,
the
auxiliary
linear
parameter
L
,
the
nonzero
auxiliary
parameter
h
and
the
auxiliary
function
H
1
(
x
)
be
properly
chosen
so
that
the
po
wer
series
Eq.(17)
of
(
x
;
q
;
r
)
con
v
er
ges
at
q
=
1
,
then,
we
ha
v
e
under
these
assumptions
the
solution
series
u
(
x;
r
)
=
(
x
;
1
;
r
)
=
u
0
(
x;
r
)
+
1
X
m
=1
u
m
(
x;
r
)
:
(19)
From
Eq.(17)
,
we
can
write
Eq.(14)
as
follo
ws:
(1
q
)
L
[
(
x
;
q
;
r
)
u
0
(
x;
r
)]
=
(1
q
)
L
[
1
X
m
=1
u
m
(
x;
r
)
q
m
]
(20)
=
q
hH
1
(
x
)
N
[
(
x
;
q
;
r
)]
then,
L
[
1
X
m
=1
u
m
(
x;
r
)
q
m
]
q
L
[
1
X
m
=1
u
m
(
x;
r
)
q
m
]
=
q
hH
1
(
x
)
N
[
(
x
;
q
;
r
)]
:
(21)
By
dif
ferentiating
Eq.(20)
m
times
with
respect
to
q
,
we
obtain
f
L
[
1
X
m
=1
u
m
(
x;
r
)
q
m
]
q
L
[
1
X
m
=1
u
m
(
x;
r
)
q
m
]
g
(
m
)
=
hH
1
(
x
)
m
@
m
1
N
[
(
x
;
q
;
r
)]
@
q
m
1
j
q
=0
:
Therefore,
L
[
u
m
(
x;
r
)
m
u
m
1
(
x;
r
)]
=
hH
1
(
x
)
<
m
(
u
m
1
(
x;
r
))
;
(22)
where,
<
m
(
u
m
1
(
x
))
=
1
(
m
1)!
@
m
1
N
[
'
(
x
;
q
)]
@
q
m
1
j
q
=0
;
(23)
and
m
=
(
0
m
1
;
1
m
>
1
:
Note
that
the
high-order
deformation
Eq.(22)
is
go
v
erning
the
linear
operator
L
,
and
the
term
<
m
(
u
m
1
(
x;
r
))
can
be
e
xpressed
simply
by
Eq.(23)
for
an
y
nonlinear
operator
N
:
Indonesian
J
Elec
Eng
&
Comp
Sci
V
ol.
11,
No.
3,
September
2018:
1228
-1235
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
1233
T
o
obtain
the
approximation
solution
of
Eq.(2),
according
to
HAM
R
m
(
u
m
1
(
x;
r
))
=
u
m
1
(
x;
r
)
f
(
x;
r
)
1
Z
x
a
K
1
(
x;
t
)
G
1
(
t;
u
(
t;
r
))
dt
2
Z
b
a
K
2
(
x;
t
)
G
2
(
t;
u
(
t;
r
))
dt
(1
m
)
f
(
x;
r
)
;
m
1
(24)
Substituting
Eq.(24)
into
Eq.
(22)
L
[
u
m
(
x;
r
)
m
u
m
1
(
x;
r
)]
=
hH
1
(
x
)[
u
m
1
(
x;
r
)
1
Z
x
a
K
1
(
x;
t
)
G
1
(
t;
u
(
t;
r
))
dt
2
Z
b
a
K
2
(
x;
t
)
G
2
(
t;
u
(
t;
r
))
dt
(1
m
)
f
(
x;
r
)]
:
(25)
W
e
tak
e
an
initial
guess
u
0
(
x;
r
)
=
f
(
x;
r
)
,
an
auxiliary
linear
operator
Lu
=
u
;
a
nonzero
auxiliary
parameter
h
=
1
;
and
auxiliary
function
H
1
(
x
)
=
1
.
This
is
substituted
into
Eq.(25)
to
gi
v
e
the
recurrence
relation
u
0
(
x;
r
)
=
f
(
x;
r
)
u
n
+1
(
x;
r
)
=
1
Z
x
a
K
1
(
x;
t
)
G
1
(
t;
u
n
(
t;
r
))
dt
+
2
Z
b
a
K
2
(
x;
t
)
G
2
(
t;
u
n
(
t;
r
))
dt;
n
0
:
(26)
Similarly
,
we
can
construct
a
homotop
y
when
consider
,
N
[
u
]
=
0
;
to
gi
v
e
the
recurrence
relation
u
0
(
x;
r
)
=
f
(
x;
r
)
u
n
+1
(
x;
r
)
=
1
Z
x
a
K
1
(
x;
t
)
G
1
(
t;
u
n
(
t;
r
))
dt
+
2
Z
b
a
K
2
(
x;
t
)
G
2
(
t;
u
n
(
t;
r
))
dt;
n
0
:
(27)
From
Eqs.(26),
and
Eqs.(27)
we
approximate
~
u
(
x;
r
)
=
[
u
(
x;
r
)
;
u
(
x;
r
)]
by
u
(
x;
r
)
=
lim
n
!1
u
n
;
u
(
x;
r
)
=
lim
n
!1
u
n
:
4.
NUMERICAL
EXAMPLE
In
this
section,
we
solv
e
t
h
e
fuzzy
V
olterra-Fredholm
inte
gral
equation
of
the
second
kind
by
the
ADM,
VIM
and
HAM.
Example
4.1
Consider
the
fuzzy
V
olterra-Fredholm
inte
gral
equation
of
the
second
kind
as
follo
ws:
~
u
(
x
)
=
~
f
(
x
)
+
Z
x
0
sin
(
x
)
sin
(
t
2
)
~
u
3
(
t
)
dt
+
Z
0
:
6
0
sin
(
x
2
)
sin
(
t
)(1
+
~
u
2
(
t
))
dt;
(28)
whear
,
f
(
x;
r
)
=
s
in
(
x
2
)(
13
15
(
r
2
+
r
)
+
2
15
(4
r
3
r
)
;
f
(
x;
r
)
=
s
in
(
x
2
)(
2
15
(
r
2
+
r
)
+
13
15
(4
r
3
r
)
;
and,
r
=
0
:
3
;
=
10
2
;
0
x;
t
0
:
6
:
A
Study
of
Some
Iter
ative
Methods
for
Solving
Fuzzy
V
olterr
a-F
r
edholm...
(Ahmed
A.
Hamoud)
Evaluation Warning : The document was created with Spire.PDF for Python.
1234
ISSN:
2502-4752
x
AD
M
(
n
=11)
V
I
M
(
n
=4)
H
AM
(
n
=4)
0.1
0.2203548375
0.220466127
0.2204663982
0.2
0.3062332542
0.306329751
0.3063488741
0.3
0.4035946723
0.403659665
0.4037996457
0.4
0.5233741235
0.523379658
0.5234862764
0.5
0.5964831157
0.614656263
0.6259432736
0.6
0.6523678927
0.652356871
0.6524855123
T
able
1.
The
Obtained
Solutions
for
Example
4.1
The
abo
v
e
table
sho
w
comparison
between
the
approximate
soluti
ons
by
using
ADM,
VIM
and
HAM
for
results
of
the
e
xample
4.1
.
5.
CONCLUSION
W
e
discussed
the
dif
ferent
methods
for
solving
fuzzy
V
olterra-Fredholm
inte
gral
equations,
namely
,
Adomian
decomposition
method,
v
ariational
iteration
method
and
homotop
y
analysis
method.
T
o
assess
the
accurac
y
of
each
method,
the
test
e
xample
with
kno
wn
e
xact
solution
is
used.
The
results
sho
w
that
these
methods
are
v
ery
ef
ficient,
con
v
enient
and
can
be
adapted
to
fit
a
lar
ger
class
of
problems.
The
comparison
re
v
eals
that
although
t
h
e
numerical
results
of
these
methods
are
similar
approximately
,
HAM
is
the
easiest,
the
most
ef
ficient
and
con
v
enient.
REFERENCES
[1]
S.
Chang
and
L.
Zadeh,
”On
Fuzzy
mapping
and
control,
”
IEEE
T
rans.
Systems,
Man
c
ybernet
,
(1972),
2,
pp.
30–34.
[2]
D.
Dubois
and
H.
Prade,
”Operations
on
fuzzy
numbers,
”
Int.
J.
systems
Science
,
(1978),
9,
pp.
613–626.
[3]
D.
Dubois
and
H.
Prade,
”T
o
w
a
rds
fuzzy
dif
ferential
calculus.
I:
inte
gration
of
fuzzy
mappings,
”
Fuzzy
Sets
and
Systems
,
(1982),
8,
pp.
1–17.
[4]
R.
Goetschel
and
W
.
V
oxman,
”Elementary
calculus,
”
Fuzzy
Sets
and
Systems
,
(1986),
18,
pp.
31–43.
[5]
O.
Kale
v
a,
”Fuzzy
dif
ferential
equation,
”
Fuzzy
Sets
and
Systems
,
(1987),
24,
pp.
301–317.
[6]
E.
Babolian
and
A.
Da
v
ari,
”Numerical
impl
ementation
of
Adomian
decomposition
method,
”
Appl.
Math.
Com-
put.
,
(2004),
pp.
301-305.
[7]
A.
Hamoud
and
K.
Ghadle,
”The
approximate
solutions
of
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A
Study
of
Some
Iter
ative
Methods
for
Solving
Fuzzy
V
olterr
a-F
r
edholm...
(Ahmed
A.
Hamoud)
Evaluation Warning : The document was created with Spire.PDF for Python.