Indonesian
J
our
nal
of
Electrical
Engineering
and
Computer
Science
V
ol.
11,
No.
3,
September
2018,
pp.857
867
ISSN:
2502-4752,
DOI:10.11591/ijeecs.v11.i3.pp857-867
857
Homotopy
Analysis
Method
f
or
the
First
Order
Fuzzy
V
olterra-Fr
edholm
Integr
o-Differ
ential
Equations
Ahmed
A.
Hamoud
1
and
Kirtiwant
P
.
Ghadle
2
1,2
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
Article
Inf
o
Article
history:
Recei
v
ed,
April
28,
2018
Re
vised,
Jul
22,
2018
Accepted,
Aug
4,
2018
K
eyw
ord:
Homotop
y
analysis
method
Fuzzy
V
olterra-Fredholm
inte
gro-dif
ferential
equation
Existence
and
uniqueness
results.
ABSTRA
CT
A
fuzzy
V
olterra-Fredholm
inte
gro-dif
ferential
equation
(FVFIDE)
in
a
parametric
case
is
con
v
erted
to
its
related
crisp
case.
W
e
use
homotop
y
analysis
method
to
find
the
approxi-
mate
solution
of
this
system
and
hence
obtain
an
approximation
for
the
fuzzy
solution
of
the
FVFIDE.
This
paper
discusses
e
xistence
and
uniqueness
results
and
con
v
er
gence
of
the
proposed
method.
Copyright
c
2018
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Ahmed
A.
Hamoud
Department
of
Mathematics,
T
aiz
Uni
v
ersity
,
T
aiz,
Y
emen.
Email:
drahmed985@yahoo.com
1.
INTR
ODUCTION
In
recent
years,
the
topics
of
fuzzy
inte
gral
equat
ions
which
attracted
increasing
interest,
in
particular
in
relation
to
fuzzy
control,
ha
v
e
been
rapidly
de
v
eloped.
The
concept
of
fuzzy
numbers
and
arithmetic
operat
ions
firstly
introduced
by
Zadeh
[1],
and
t
hen
by
Dubois
and
Prade.
Also,
in
[2]
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
,
Dubois
and
Prade
[3]
presented
an
elementary
fuzzy
calculus
based
on
the
e
xtension
principle.
The
fuzzy
inte
gro-dif
ferential
equations
are
a
natural
w
ay
to
model
uncertainty
of
dynamical
systems.
Kale
v
a
[4]
chose
to
define
the
inte
gral
of
the
fuzzy
function,
using
the
Lebesgue-type
concept
for
inte
gration.
Recently
,
Hence
v
arious
other
methods
for
solving
them
such
as
using
homotop
y
perturbation
method
[5],
e
xpansion
method
[6],
Laplace
transformation
method
[7],
homotop
y
analysis
method
[8],
dif
ferential
transform
method
[9],
fix
ed
point
theorems
[10],
v
ariational
iteration
method
[11].
Also,
some
mathematicians
ha
v
e
studied
fuzzy
inte
gral
and
inte
gro-dif
ferential
equation
by
numerical
techniques
[12]-
[21]
[23,
26].
As
we
kno
w
the
fuzzy
inte
gral
and
inte
gro-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
e
xamine
HAM
to
approximate
the
solution
of
the
fuzzy
V
olterra-Fredholm
inte
gro-
dif
ferential
equat
ion
of
the
second
kind.
The
structure
of
this
paper
is
or
g
anized
as
follo
ws:
In
Section
2,
we
state
some
kno
wn
notations
and
definitions
and
also
some
theorems
which
are
used
throughout
thi
s
paper
.
In
Section
3,
the
fuzzy
V
olterra-Fredholm
inte
gro-dif
ferential
equation
of
the
second
kind
is
briefly
presented.
In
Section
4,
we
con
v
ert
a
fuzzy
V
olterra-Fredholm
inte
gro-dif
ferential
equation
of
the
second
kind
to
the
system
of
V
olterra-Fredholm
inte
gro-dif
ferential
equation
of
the
second
kind
in
a
crisp
case
and
approximate
with
HAM.
In
Section
5,
the
e
xistence
and
uniqueness
results
and
con
v
er
gence
of
the
proposed
method
is
pro
v
ed.
In
Section
6,
the
analytical
e
xample
is
presented
illustrate
the
accurac
y
of
this
method.
Finally
,
we
will
gi
v
e
a
report
on
our
paper
and
a
brief
conclusion
in
Section
7.
J
ournal
Homepage:
http://iaescor
e
.com/journals/inde
x.php/IJEECS
Evaluation Warning : The document was created with Spire.PDF for Python.
858
ISSN:
2502-4752
2.
PRELIMIN
ARIES
The
concept
of
fuzzy
numbers
is
generalized
of
classical
real
numbers
and
we
can
say
that
a
fuzzy
number
is
a
fuzzy
subset
of
the
real
li
ne
which
has
some
additional
properties.
The
concept
of
fuzzy
number
is
vital
for
fuzzy
analysis,
fuzzy
inte
gral
equations
and
fuzzy
dif
ferential
equations,
and
a
v
ery
helpful
tool
in
dif
ferent
applications
of
fuzzy
sets.
Basic
definition
of
fuzzy
numbers
is
gi
v
en
in
[1,
2,
3,
27].
Definition
2..1
[2]
Let
us
denote
by
R
F
the
class
of
fuzzy
subsets
of
the
r
eal
axis
u
:
R
!
I
=
[0
;
1]
;
satisfying
the
following
pr
operties:
u
is
upper
semi-continuous
function,
u
is
fuzzy
con
ve
x,i.e
,
u
(
x
+
(1
)
y
)
min
f
u
(
x
)
;
u
(
y
)
g
for
all
x;
y
2
R
;
2
[0
;
1]
,
u
is
normal,
i.e
,
9
x
0
2
R
for
whic
h
u
(
x
0
)
=
1
,
sup
u
=
f
x
2
R
j
u
(
x
)
>
0
g
is
the
support
of
the
u
,
and
its
closur
e
cl
(sup
u
)
is
compact.
Let
E
be
the
set
of
all
fuzzy
numbers
on
R
F
.
The
(
cut
)
-le
v
el
set
of
a
fuzzy
number
u
2
E
;
0
1
,
denoted
by
[
u
]
,
is
defined
as
[
u
]
=
f
x
2
R
:
u
(
x
)
g
;
0
<
1
;
cl
(sup
u
)
;
=
0
:
where
cl
(sup
u
=
x
2
R
j
u
(
x
)
>
0)
denotes
the
closure
of
the
support
of
u
.
It
is
clear
that
the
-le
v
el
set
of
a
fuzzy
number
is
a
closed
and
bounded
interv
al
[
u
(
)
;
u
(
)]
,
where
u
(
)
denotes
the
left-hand
end
point
of
[
u
]
,
and
u
(
)
denotes
the
right-hand
end
point
of
[
u
]
.
Since
each
u
2
R
can
be
re
g
arded
as
a
fuzzy
number
~
u
defined
by:
~
u
(
t
)
=
1
;
t
=
u
0
;
t
6
=
u:
An
equi
v
alent
parametric
definition
is
also
gi
v
en
in
[1]
as:
Definition
2..2
[2]
A
fuzzy
number
~
u
in
par
ametric
form
is
a
pair
(
u
;
u
)
of
functions
u
(
)
,
u
(
)
;
0
1
,
whic
h
satisfy
the
following
r
equir
ements:
u
(
)
is
a
bounded
non-decr
easing
left
continuous
function
in
(0
;
1]
,
and
right
continuous
at
0
,
u
(
)
is
a
bounded
non-incr
easing
left
continuous
function
in
(0
;
1]
,
and
right
continuous
at
0
,
u
(
)
u
(
)
;
0
1
.
A
cri
sp
number
is
simply
r
epr
esented
by
u
(
)
=
u
(
)
=
;
0
1
.
W
e
r
ecall
that
for
a
<
b
<
c
whic
h
a;
b;
c
2
R
;
the
triangular
fuzzy
number
u
=
(
a;
b;
c
)
determined
by
a;
b;
c
ar
e
given
suc
h
that
u
(
)
=
a
+
(
b
a
)
and
u
(
)
=
c
(
c
b
)
ar
e
the
end
points
of
the
-le
vel
sets,
for
all
2
[0
;
1]
.
The
Hausdorf
f
distance
between
fuzzy
numbers
gi
v
en
by
D
:
R
F
R
F
!
R
+
[
f
0
g
:
D
(
u;
)
=
sup
2
[0
;
1]
max
fj
u
(
)
(
)
j
;
j
u
(
)
(
)
jg
wehre
u
=
(
u
(
)
;
u
(
))
,
=
(
(
)
;
(
))
R
is
utilized
in
[1].
Then,
it
is
easy
to
see
that
d
is
a
metric
in
E
and
has
the
follo
wing
properties:
D
(
u
+
;
+
)
=
D
(
u;
)
;
8
u;
;
2
E
,
D
(
k
u;
k
)
=
j
k
j
D
(
u;
)
;
8
k
2
R
;
u;
2
E
;
D
(
!
+
;
+
e
)
D
(
!
;
)
+
d
(
;
e
)
;
8
!
;
;
;
e
2
E
,
(
D
;
E
)
is
a
complete
metric
space.
Indonesian
J
Elec
Eng
&
Comp
Sci
V
ol.
11,
No.
3,
September
2018:
857
-867
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
859
Definition
2..3
The
function
f
:
[
a;
b
]
!
R
F
is
called
a
Lipsc
hitz
function
if
ther
e
e
xists
a
r
eal
constant
L
0
suc
h
that,
for
all
x;
t
2
[
a;
b
]
D
(
f
(
x
)
;
f
(
t
))
L
j
x
t
j
:
W
e
r
efer
to
L
as
the
Lipsc
hitz
constant
of
the
function
f
:
Remark
2..1
Let
u
(
)
=
(
u
(
)
;
u
(
))
;
be
a
fuzzy
number
,
we
tak
e
u
c
(
)
=
u
(
)
+
u
(
)
2
;
u
d
(
)
=
u
(
)
u
(
)
2
:
It
is
clear
that
u
d
(
)
0
and
u
(
)
=
u
c
(
)
u
d
(
)
and
u
(
)
=
u
c
(
)
+
u
d
(
)
;
also
a
fuzzy
number
u
2
E
is
said
symmetric
if
u
c
(
)
is
independent
of
for
all
0
1
:
Definition
2..4
Let
f
:
R
!
E
be
a
fuzzy
valued
function.
If
for
arbitr
ary
fixed
t
0
2
R
and
8
>
0
;
9
>
0
suc
h
that
j
t
t
0
j
<
=
)
j
f
(
t
)
f
(
t
0
)
j
<
,
f
is
said
to
be
continuous.
Theor
em
2..2
Let
f
(
x
)
be
a
fuzzy-valued
function
on
[
a;
1
)
and
it
is
r
epr
esented
by
(
f
(
x;
)
;
f
(
x;
))
.
F
or
any
fixed
t
2
[0
;
1]
assume
f
(
x;
)
and
f
(
x;
)
ar
e
Riemann-inte
gr
able
on
[
a;
b
]
for
e
very
b
a
,
and
assume
ther
e
ar
e
two
positive
M
(
)
and
M
(
)
suc
h
that
R
b
a
f
(
x;
)
dx
M
(
)
and
R
b
a
f
(
x;
)
dx
M
(
)
for
e
very
b
a
.
Then
f
(
x
)
is
impr
oper
fuzzy
Riemann-inte
gr
able
on
[
a;
1
)
and
the
impr
oper
fuzzy
Riemann-inte
gr
al
is
a
fuzzy
number
.
Furthermor
e
,
we
have:
Z
1
a
f
(
x
)
dx
=
Z
1
a
f
(
x;
)
dx;
Z
1
a
f
(
x;
)
dx
Pr
oposition
2..3
[25].
If
eac
h
of
f
(
x
)
and
g
(
x
)
is
fuzzy-valued
function
and
fuzzy
Riemman
inte
gr
able
on
=
[
a;
1
)
then
f
(
x
)
+
g
(
x
)
is
fuzzy
Riemman
inte
gr
able
on
.
Mor
eo
ver
,
we
have:
Z
(
f
(
x
)
+
g
(
x
))
dx
=
Z
f
(
x
)
dx
+
Z
g
(
x
)
dx
Definition
2..5
[25]
The
inte
gr
al
of
a
fuzzy
function
was
define
by
using
the
Riemann
inte
gr
al
concept.
Let
f
:
[
a;
b
]
!
E
,
for
eac
h
partition
P
=
t
0
;
t
1
;
:::;
t
n
of
[
a;
b
]
and
for
arbitr
ary
i
2
[
t
i
1
;
t
i
]
;
1
i
n
,
suppose
R
p
=
n
X
i
=1
f
(
i
)(
t
i
t
i
1
)
:=
max
j
t
i
t
i
1
j
;
1
i
n:
The
definite
inte
gr
al
of
f
(
t
)
o
ver
[
a;
b
]
is
Z
b
a
f
(
t
)
dt
=
lim
!
0
R
p
:
Pr
o
vided
that
this
limit
e
xists
in
the
metric
d
.
If
the
fuzzy
function
f
(
t
)
is
continuous
in
the
metric
d
,
its
definit
e
inte
gr
al
e
xists,
and
also
Z
b
a
f
(
t;
r
)
dt
=
Z
b
a
f
(
t;
r
)
dt;
Z
b
a
f
(
t;
r
)
dt
=
Z
b
a
f
(
t;
r
)
dt:
More
details
about
the
properties
of
the
fuzzy
inte
gral
are
gi
v
en
in
[2,
26,
25].
Theor
em
2..4
[28]
(Banac
h
contr
action
principle).
Let
(
X
;
d
)
be
a
complete
metric
space
,
then
eac
h
contr
act
ion
mapping
T
:
X
!
X
has
a
unique
fixed
point
x
of
T
in
X
i.e
.
T
x
=
x:
Theor
em
2..5
[24]
(Sc
hauder’
s
fixed
point
theor
em).
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
u
:
u
2
A
g
is
r
elatively
compact
in
X
(or
T
is
continuous
and
completely
continouous).
Then
T
has
at
least
one
fixed
point
u
2
A
:
T
u
=
u
:
Homotopy
Analysis
Method
for
the
F
ir
st
Or
der
Fuzzy
V
olterr
a-F
r
edholm...
(Ahmed
A.
Hamoud)
Evaluation Warning : The document was created with Spire.PDF for Python.
860
ISSN:
2502-4752
3.
FUZZY
V
OL
TERRA-FREDHOLM
INTEGR
O-DIFFERENTIAL
EQ
U
A
TION
In
this
section
we
consider
the
fuzzy
V
olterra-Fredholm
inte
gro-dif
ferential
equation
~
u
0
(
x
)
=
~
f
(
x
)
+
Z
x
a
k
1
(
x;
t
)
F
1
(
~
u
(
t
))
dt
+
Z
a
k
2
(
x;
t
)
F
2
(
~
u
(
t
))
dt;
(1)
with
initial
condition
~
u
0
(0)
=
~
u
(0)
;
(2)
where
;
2
R
,
f
(
x
)
;
k
1
;
k
2
and
F
1
(
~
u
(
t
))
are
analytical
functions
k
1
;
k
2
:
C
([0
;
]
2
)
!
R
+
,
that
ha
v
e
suitable
deri
v
ati
v
es
on
an
interv
al
0
t
x
and
~
u
(
x
)
is
unkno
wn
function.
The
solution
is
e
xpressed
in
the
form:
~
u
(
x
)
=
1
X
i
=0
~
u
i
(
x
)
:
(3)
Let
~
u
(
x;
t
)
=
(
u
(
x;
t
)
;
u
(
x;
t
))
;
~
f
(
x;
t
)
=
(
f
(
x;
t
)
;
f
(
x;
t
))
:
and
~
u
0
(
x;
t
)
=
(
u
0
(
x;
t
)
;
u
0
(
x;
t
))
;
~
f
0
(
x;
t
)
=
(
f
0
(
x;
t
)
;
f
0
(
x;
t
))
:
Therefore,
the
related
fuzzy
inte
gro-dif
ferential
equation
(1)
can
be
written
as
follo
ws
u
0
(
x;
t
)
=
f
(
x;
t
)
+
Z
x
a
k
1
(
x;
s
)
F
1
(
u
(
s;
t
))
ds
+
Z
a
k
2
(
x;
s
)
F
2
(
u
(
s;
t
))
ds
(4)
u
0
(
x;
t
)
=
f
(
x;
t
)
+
Z
x
a
k
1
(
x;
s
)
F
1
(
u
(
s;
t
))
ds
+
Z
a
k
2
(
x;
s
)
F
2
(
u
(
s;
t
))
ds
(5)
Similar
to
Remark
2.1,
let
u
c
(
x;
t
)
=
u
(
x;
t
)
+
u
(
x;
t
)
2
;
u
d
(
x;
t
)
=
u
(
x;
t
)
u
(
x;
t
)
2
:
(6)
and
f
c
(
x;
t
)
=
f
(
x;
t
)
+
f
(
x;
t
)
2
;
f
d
(
x;
t
)
=
f
(
x;
t
)
f
(
x;
t
)
2
:
(7)
then
(4)
and
(5)
can
be
written
as
u
0
c
(
x;
t
)
=
f
c
(
x;
t
)
+
Z
x
a
k
1
(
x;
s
)
F
1
(
u
c
(
s;
t
))
ds
+
Z
a
k
2
(
x;
s
)
F
2
(
u
c
(
s;
t
))
ds
(8)
u
0
d
(
x;
t
)
=
f
d
(
x;
t
)
+
Z
x
a
k
1
(
x;
s
)
F
1
(
u
d
(
s;
t
))
ds
+
Z
a
k
2
(
x;
s
)
F
2
(
u
d
(
s;
t
))
ds
(9)
and
u
c
(0
;
t
)
=
u
(0
;
t
)
+
u
(0
;
t
)
2
;
u
d
(0
;
t
)
=
u
(0
;
t
)
u
(0
;
t
)
2
:
(10)
4.
HOMO
T
OPY
AN
AL
YSIS
METHOD
(HAM)
In
this
section,
we
shall
describe
the
solution
approaches
bas
ed
on
HAM
for
fuzzy
V
olterra-Fredholm
inte
gro-
dif
ferential
equations.
F
or
this,
we
consider
the
first
equation
of
(8)
namely
we
apply
HAM
for
finding
u
c
(
x;
t
)
;
and
second
equation
approach
is
similar
to
the
first
one
[22].
Consider
u
0
c
(
x;
t
)
=
f
c
(
x;
t
)
+
Z
x
a
k
1
(
x;
s
)
F
1
(
u
c
(
s;
t
))
ds
+
Z
a
k
2
(
x;
s
)
F
2
(
u
c
(
s;
t
))
ds;
(11)
Indonesian
J
Elec
Eng
&
Comp
Sci
V
ol.
11,
No.
3,
September
2018:
857
-867
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
861
with
initial
condition
u
c
(0
;
t
)
:
W
e
first
construct
the
zero-order
deformation
equation
(1
q
)[
'
(
x;
t
;
q
)
u
c
0
(
x;
t
)]
=
q
~
N
[
'
(
x;
t
;
q
)]
;
(12)
subject
to
the
initial
condition
'
(0
;
t
;
q
)
=
u
c
0
(
x;
t
)
=
u
c
(0
;
t
)
;
(13)
where
q
2
[0
;
1]
is
the
embedding
parameter
and
~
6
=
0
is
an
auxiliary
parameter
and
[
'
(
x;
t
;
q
)]
=
@
[
'
(
x;
t
;
q
)]
@
x
;
(14)
with
the
property
[
C
]
=
0
;
(15)
where
C
is
inte
gral
constant.
Also
from
(11),
we
can
define
N
[
'
(
x;
t
;
q
)]
=
@
[
'
(
x;
t
;
q
)]
@
x
f
c
(
x;
t
)
Z
x
a
k
1
(
x;
s
)
F
1
(
'
(
s;
t
;
q
))
ds
Z
a
k
2
(
x;
s
)
F
2
(
'
(
s;
t
;
q
))
ds;
(16)
When
parameter
of
q
increases
from
0
to
1
,
then
homotop
y
solution
'
(
x;
t
;
q
)
v
aries
from
u
c
0
(
x;
t
)
to
solution
u
c
(
t;
r
)
of
the
original
equation
(11).
Using
the
parameter
q
;
'
(
x;
t
;
q
)
can
be
e
xpanded
in
T
aylor
series
as
follo
ws
'
(
x;
t
;
q
)
=
u
c
0
(
x;
t
)
+
1
X
m
=0
u
c
m
(
x;
t
)
q
m
;
where
u
c
m
(
x;
t
)
=
1
m
!
@
m
[
'
(
x;
t
;
q
)]
@
m
q
j
q
=0
:
Assuming
that
auxiliary
parameter
~
is
properly
selected
so
that
the
abo
v
e
series
is
con
v
er
gent
when
q
=
1
;
then
the
solution
u
c
(
x;
t
)
can
be
gi
v
en
by
u
c
(
x;
t
)
=
u
c
0
(
x;
t
)
+
1
X
m
=0
u
c
m
(
x;
t
)
:
Dif
ferentiating
(12)
and
initial
condition
(13)
m-times
with
respect
to
q
,
then
setting
q
=
0
;
and
finally
di
viding
them
by
m
!
,
we
g
ain
the
m
th
-order
deformation
equation
[
u
c
m
(
x;
t
)
m
u
c
m
1
(
x;
t
)]
=
~
<
m
!
(
u
c
m
1
)
;
(17)
subject
to
the
follo
wing
initial
conditions,
u
c
m
(0
;
t
)
=
0
;
(18)
where
<
m
!
(
u
c
m
1
)
=
1
(
m
1)!
@
m
1
N
[
'
(
x;
t
;
q
)]
@
m
1
q
j
q
=0
(19)
=
@
u
c
m
1
(
x;
t
)
@
x
(1
m
)
f
c
(
x;
t
)
Z
x
a
k
1
(
x;
s
)
F
1
(
u
c
m
1
(
s;
t
))
ds
Z
a
k
2
(
x;
s
)
F
2
(
u
c
m
1
(
s;
t
))
ds;
and
m
=
(
0
m
1
;
1
m
>
1
:
Note
that
the
high-order
deformation
Eq.(17)
is
go
v
erning
the
linear
operator
L
,
and
the
term
<
m
(
!
u
c
m
1
)
can
be
e
xpressed
simply
by
Eq.(19)
for
an
y
nonlinear
operator
N
:
Similarly
,
we
can
apply
HAM
for
finding
u
d
(
x;
t
)
:
Homotopy
Analysis
Method
for
the
F
ir
st
Or
der
Fuzzy
V
olterr
a-F
r
edholm...
(Ahmed
A.
Hamoud)
Evaluation Warning : The document was created with Spire.PDF for Python.
862
ISSN:
2502-4752
5.
MAIN
RESUL
TS
A
function
~
u
is
a
solution
of
t
he
initial
v
alue
problem
(1)–(2)
if
and
only
if
it
is
continuous
and
satisfies
the
inte
gral
equation
~
u
(
x
)
=
~
u
0
+
Z
x
a
~
f
(
t
)
dt
+
Z
x
a
Z
t
a
k
1
(
t;
s
)
F
1
(
~
u
(
s
))
dsdt
+
Z
x
a
Z
a
k
2
(
t;
s
)
F
2
(
~
u
(
s
))
dsdt;
By
changing
the
order
of
the
inte
gration,
we
ha
v
e
~
u
(
x
)
=
~
u
0
+
Z
x
a
~
f
(
t
)
dt
+
Z
x
a
Z
x
s
k
1
(
t;
s
)
F
1
(
~
u
(
s
))
dtds
+
Z
x
a
Z
a
k
2
(
t;
s
)
F
2
(
~
u
(
s
))
dsdt;
~
u
(
x
)
=
~
u
0
+
Z
x
a
~
f
(
t
)
dt
+
Z
x
a
Z
x
s
k
1
(
t;
s
)
F
1
(
~
u
(
s
))
dtds
+
Z
a
Z
x
a
k
2
(
t;
s
)
F
2
(
~
u
(
s
))
dtds;
Since
the
function
k
1
and
k
1
are
with
no
sign
changes
by
assumption,
we
ha
v
e
Z
x
s
k
1
(
t;
s
)
F
1
(
~
u
(
s
))
dt
=
Z
x
s
k
1
(
t;
s
)
dt
F
1
(
~
u
(
s
))
Z
x
a
k
2
(
t;
s
)
F
2
(
~
u
(
s
))
dt
=
Z
x
a
k
2
(
t;
s
)
dt
F
2
(
~
u
(
s
))
thus
~
u
(
x
)
=
~
g
(
x
)
+
Z
x
a
K
1
(
x;
s
)
F
1
(
~
u
(
s
))
ds
+
Z
a
K
2
(
x;
s
)
F
2
(
~
u
(
s
))
ds;
(20)
where
~
g
(
x
)
=
~
u
0
+
Z
x
a
~
f
(
t
)
dt;
K
1
(
x;
s
)
=
Z
x
s
k
1
(
t;
s
)
dt;
K
2
(
x;
s
)
=
Z
x
a
k
2
(
t;
s
)
dt:
Lemma
5..1
Let
k
1
(
t;
s
)
be
continuous
in
(
t;
s
)
and
Lipsc
hitz
with
r
espect
to
s:
Then
K
1
(
x;
s
)
is
Lipsc
hitz
with
r
espect
to
s:
Proof.
Let
0
s
1
s
2
x:
Then
j
K
1
(
x;
s
1
)
K
1
(
x;
s
2
)
j
=
j
Z
x
s
1
k
1
(
t;
s
1
)
dt
Z
x
s
2
k
1
(
t;
s
2
)
dt
j
=
j
Z
s
2
s
1
k
1
(
t;
s
1
)
dt
+
Z
x
s
2
k
1
(
t;
s
1
)
dt
Z
x
s
2
k
1
(
t;
s
2
)
dt
j
Z
s
2
s
1
j
k
1
(
t;
s
1
)
j
dt
+
Z
x
s
2
j
k
1
(
t;
s
1
)
k
1
(
t;
s
2
)
j
dt
M
j
s
1
s
2
j
+
L
j
s
1
s
2
j
(
x
s
2
)
(
M
+
L
(
a
))
j
s
1
s
2
j
;
where
M
=
max
(
x;s
)
2
G
j
k
1
(
x;
s
)
j
,
G
:=
f
(
x;
t
)
j
x
2
J
;
t
2
[
a;
x
]
g
J
J
;
and
L
is
the
Lipschitz
constant
of
k
1
and
thus
K
1
satisfies
in
Lipschitz
condition.
Similarly
,
we
can
proof
the
procedure
of
K
2
(
x;
s
)
is
Lipschitz
with
respect
to
s:
Before
starting
and
pro
ving
the
main
results,
we
introduce
the
follo
wing
h
ypotheses:
(A1)
There
e
xist
tw
o
constants
M
1
;
M
2
>
0
such
that,
for
an
y
u
1
;
u
2
2
C
(
J
;
R
)
D
"
(
u
c
1
;
u
c
2
)
:=
sup
x
2
J
e
"M
1
x
D
(
u
c
1
(
s
)
;
u
c
2
(
s
))
;
"
1
;
as
a
metric
on
X
.
(A2)
There
e
xist
tw
o
functions
K
1
;
K
2
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
M
1
=
max
x;s
2
G
j
K
1
(
x;
t
)
j
<
1
,
M
2
=
max
x;s
2
G
j
K
2
(
x;
t
)
j
<
1
;
Indonesian
J
Elec
Eng
&
Comp
Sci
V
ol.
11,
No.
3,
September
2018:
857
-867
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
863
(A3)
The
function
g
:
J
!
R
is
continuous.
Theor
em
5..1
Assume
that
(A1),
(A2)
and
(A3)
hold.
If
e
"M
1
x
e
"M
1
a
"
+
e
"M
2
e
"M
2
a
"
<
1
:
Then
the
initial
value
pr
oblem
(1)
-
(2)
has
a
unique
solution.
Proof.
Let
the
op
e
rator
A
:
X
!
X
:
T
o
do
this,
it
is
e
vident
that
(
Au
c
)(
x
)
2
R
F
for
all
x
2
J
and
thus
Au
c
:
J
!
R
F
be
defined
by
(
Au
c
)(
x
)
=
g
c
(
x
)
+
Z
x
a
K
1
(
x;
s
)
F
1
(
u
c
(
s
))
ds
+
Z
a
K
2
(
x;
s
)
F
2
(
u
c
(
s
))
ds;
D
(
Au
c
1
(
x
)
;
Au
c
2
(
x
))
=
D
(
g
c
(
x
)
+
Z
x
a
K
1
(
x;
s
)
F
1
(
u
c
1
(
s
))
ds
+
Z
a
K
2
(
x;
s
)
F
2
(
u
c
1
(
s
))
ds;
g
c
(
x
)
+
Z
x
a
K
1
(
x;
s
)
F
1
(
u
c
2
(
s
))
ds
+
Z
a
K
2
(
x;
s
)
F
2
(
u
c
2
(
s
))
ds
)
;
=
D
(
Z
x
a
K
1
(
x;
s
)
F
1
(
u
c
1
(
s
))
ds
+
Z
a
K
2
(
x;
s
)
F
2
(
u
c
1
(
s
))
ds;
Z
x
a
K
1
(
x;
s
)
F
1
(
u
c
2
(
s
))
ds
+
Z
a
K
2
(
x;
s
)
F
2
(
u
c
2
(
s
))
ds
)
Z
x
a
D
(
K
1
(
x;
s
)
F
1
(
u
c
1
(
s
))
;
K
1
(
x;
s
)
F
1
(
u
c
2
(
s
)))
ds
+
Z
a
D
(
K
2
(
x;
s
)
F
2
(
u
c
1
(
s
))
;
K
2
(
x;
s
)
F
2
(
u
c
2
(
s
)))
ds
Z
x
a
M
1
D
(
u
c
1
(
s
)
;
u
c
2
(
s
))
ds
+
Z
a
M
2
D
(
u
c
1
(
s
)
;
u
c
2
(
s
))
ds
=
Z
x
a
M
1
e
"M
1
s
e
"M
1
s
D
(
u
c
1
(
s
)
;
u
c
2
(
s
))
ds
+
Z
a
M
2
e
"M
2
s
e
"M
2
s
D
(
u
c
1
(
s
)
;
u
c
2
(
s
))
ds
Z
x
a
M
1
e
"M
1
s
D
"
(
u
c
1
;
u
c
2
)
ds
+
Z
a
M
2
e
"M
2
s
D
"
(
u
c
1
;
u
c
2
)
ds
e
"M
1
x
e
"M
1
a
"
D
"
(
u
c
1
;
u
c
2
)
+
e
M
2
e
"M
2
a
"
D
"
(
u
c
1
;
u
c
2
)
=
e
"M
1
x
e
"M
1
a
"
+
e
"M
2
e
"M
2
a
"
D
"
(
u
c
1
;
u
c
2
)
:
Since
e
"M
1
x
e
"M
1
a
"
+
e
"M
2
e
"M
2
a
"
<
1
;
the
operator
A
is
a
cont
raction
mapping.
By
the
Banach
fix
ed
point
theorem
we
conclude
that
the
initial
v
alue
problem
(1)-(2)
has
a
unique
solution.
Similarly
,
we
can
proof
the
procedure
of
u
d
(
x;
t
)
.
No
w
,
we
will
discuss
the
con
v
er
gence
of
HAM
for
Eq.(1)
in
the
fuzzy
case
which
di
vides
into
tw
o
crisp
inte
gro-dif
ferential
equations
as
Eqs.(8)-(9).
Theor
em
5..2
Let
the
series
P
1
m
=0
u
c
m
(
x;
t
)
con
ver
g
e
to
u
c
(
x;
t
)
,
wher
e
u
c
m
(
x;
t
)
is
pr
oduced
by
the
m
-or
der
def
o
r
-
mation
(17)
,
and
besides
P
1
m
=0
u
c
m
(
x;
t
)
con
ver
g
es,
then
u
c
(
x;
t
)
is
the
e
xact
solution
of
V
olterr
a-F
r
edholm
inte
gr
o-
dif
fer
ential
equation
(8)
when
using
HAM.
Proof.
W
e
assume
P
1
m
=0
u
c
m
(
x;
t
)
con
v
er
ge
uniformly
to
u
c
(
x;
t
)
then
lim
m
!1
u
c
m
(
x;
t
)
=
0
;
8
0
x
;
0
t
1
:
Homotopy
Analysis
Method
for
the
F
ir
st
Or
der
Fuzzy
V
olterr
a-F
r
edholm...
(Ahmed
A.
Hamoud)
Evaluation Warning : The document was created with Spire.PDF for Python.
864
ISSN:
2502-4752
W
e
can
write,
n
X
m
=1
[
u
c
m
(
x;
t
)
m
u
c
m
1
(
x;
t
)]
=
u
c
1
(
x;
t
)
+
(
u
c
2
(
x;
t
)
u
c
1
(
x;
t
))
+(
u
c
3
(
x;
t
)
u
c
2
(
x;
t
))
+
:
:
:
+(
u
c
n
(
x;
t
)
u
c
n
1
(
x;
t
))
=
u
c
n
(
x;
t
)
:
(21)
Hence,
from
Eq.(21)
lim
n
!1
u
c
n
(
x
)
=
0
:
(22)
So,
using
Eq.(22),
we
ha
v
e
1
X
m
=1
[
u
c
m
(
x;
t
)
m
u
c
m
1
(
x;
t
)]
=
1
X
m
=1
[
u
c
m
(
x;
t
)
m
u
c
m
1
(
x;
t
)]
=
0
:
Therefore
from
Eq.(22),
we
can
obtain
that
1
X
m
=1
[
u
c
m
(
x;
t
)
m
u
c
m
1
(
x;
t
)]
=
~
1
X
m
=1
<
m
1
(
!
u
c
m
1
(
x;
t
))
=
0
:
Since
~
6
=
0
and
we
ha
v
e
1
X
m
=1
<
m
1
(
!
u
c
m
1
(
x;
t
))
=
0
:
(23)
By
substituting
<
m
1
(
!
u
c
m
1
)
into
the
relation
(23)
and
simplifying
it,
we
ha
v
e
<
m
1
(
!
u
c
m
1
(
x;
t
))
=
1
X
m
=1
[
@
u
c
m
1
(
x;
t
)
@
x
Z
x
a
k
1
(
x;
s
)
F
1
(
u
c
m
1
(
s;
t
))
ds
Z
a
k
2
(
x;
s
)
F
2
(
u
c
m
1
(
s;
t
))
ds
(1
m
)
f
c
(
x;
t
)]
;
=
(
1
X
m
=1
@
u
c
m
1
(
x;
t
)
@
x
Z
x
a
k
1
(
x;
s
)[
1
X
m
=1
F
1
(
u
c
m
1
(
s;
t
))]
ds
Z
a
k
2
(
x;
s
)[
1
X
m
=1
F
2
(
u
c
m
1
(
s;
t
))]
ds
1
X
m
=1
(1
m
)
f
c
(
x;
t
)
;
=
@
u
c
(
x;
t
)
@
x
Z
x
a
k
1
(
x;
s
)
F
1
(
u
c
(
s;
t
))
ds
Z
a
k
2
(
x;
t
)
F
2
(
u
c
(
s;
t
))
ds
f
c
(
x;
t
)
:
(24)
From
Eq.(23)
and
Eq.(24),
we
ha
v
e
@
u
c
(
x;
t
)
@
x
=
f
c
(
x;
t
)
+
Z
x
a
k
1
(
x;
s
)
F
1
(
u
c
(
s;
t
))
ds
+
Z
a
k
2
(
x;
s
)
F
2
(
u
c
(
s;
t
))
ds;
therefore,
u
c
(
x;
t
)
must
be
the
e
xact
solution
of
Eq.(8).
Similarly
,
we
can
proof
the
procedure
of
u
d
(
x;
t
)
.
Then,
u
(
x;
t
)
must
be
the
e
xact
solution
of
Eq.(1),
and
the
proof
is
complete.
6.
ILLUSTRA
TIVE
EXAMPLE
In
this
section,
we
present
the
analytical
technique
based
on
HAM
to
solv
e
fuzzy
V
olterra-Fredholm
inte
gro-
dif
ferential
equation.
Indonesian
J
Elec
Eng
&
Comp
Sci
V
ol.
11,
No.
3,
September
2018:
857
-867
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
865
Example
1.
Let
us
consider
fuzzy
V
olterra-Fredholm
inte
gro-dif
ferential
equation:
@
~
u
(
x;
t
)
@
x
+
~
u
(
x;
t
)
=
~
f
(
x;
t
)
+
Z
x
0
xs
2
~
u
(
s;
t
)
ds
+
Z
1
0
xs
~
u
(
s;
t
)
ds;
(25)
where
~
u
(
x;
t
)
=
(
u
(
x;
t
)
;
u
(
x;
t
))
;
~
f
(
x;
t
)
=
(
f
(
x;
t
)
;
f
(
x;
t
))
f
(
x;
t
)
=
t
2
x
3
+
1
;
f
(
x;
t
)
=
2
tx
3
4
x
t
+
8
;
=
0
;
=
1
:
with
initial
conditions
u
(0
;
t
)
=
0
;
u
(0
;
t
)
=
8
:
Exact
solution
of
this
fuzzy
V
olterra-Fredholm
inte
gro-dif
ferential
equation
is
gi
v
en
by
~
u
(
x;
t
)
=
(
xt;
8
xt
)
:
From
Eq.(7),
we
ha
v
e
f
c
(
x;
t
)
=
f
(
x;
t
)
+
f
(
x;
t
)
2
=
4
2
x;
f
d
(
x;
t
)
=
f
(
x;
t
)
f
(
x;
t
)
2
=
4
t
2
x
2
tx
3
:
The
e
xact
solution
of
related
crisp
equations
are
as
follo
ws
u
c
(
x;
t
)
=
u
(
x;
t
)
+
u
(
x;
t
)
2
=
4
;
u
d
(
x;
t
)
=
u
(
x;
t
)
u
(
x;
t
)
2
=
4
tx:
From
Eqs.(16)
and(25)
can
be
written
N
[
'
(
x;
t
;
q
)]
=
@
[
'
(
x;
t
;
q
)]
@
x
+
'
(
x;
t
;
q
)
4
+
2
x
Z
1
0
k
2
(
x;
s
)
F
2
(
'
(
s;
t
;
q
))
ds;
No
w
,
using
m
th
-order
deformation
equation
and
initial
conditions,
we
recursi
v
ely
obtain
u
c
0
(
x;
t
)
=
u
c
(0
;
t
)
=
u
(0
;
t
)
+
u
(0
;
t
)
2
=
4
u
c
1
(
x;
t
)
=
0
;
u
c
2
(
x;
t
)
=
0
;
:
:
:
Thus
the
approximate
HAM
solution
Y
c
m
(
x;
t
)
=
m
X
n
=0
u
c
n
(
x;
t
)
=
4
The
approximate
solution
same
as
e
xact
solution.
Similarly
,
to
approximate
u
d
(
t;
r
)
;
N
[
'
(
x;
t
;
q
)]
=
@
[
'
(
x;
t
;
q
)]
@
x
+
'
(
x;
t
;
q
)
4
+
t
+
2
x
+
2
tx
3
Z
1
0
k
2
(
x;
s
)
F
2
(
'
(
s;
t
;
q
))
ds;
No
w
,
using
m
th
-order
deformation
equation
and
initial
conditions,
we
recursi
v
ely
obtain
u
d
0
(
x;
t
)
=
u
d
(0
;
t
)
=
u
(0
;
t
)
u
(0
;
t
)
2
=
4
;
u
d
1
(
x;
t
)
=
~
3
xt
(3
+
x
)
;
u
d
2
(
x;
t
)
=
~
72
xt
(
~
(72
+
45
x
+
8
x
2
)
+
72
+
24
x
)
;
:
:
:
Homotopy
Analysis
Method
for
the
F
ir
st
Or
der
Fuzzy
V
olterr
a-F
r
edholm...
(Ahmed
A.
Hamoud)
Evaluation Warning : The document was created with Spire.PDF for Python.
866
ISSN:
2502-4752
Thus
the
approximate
HAM
solution
when
~
=
1
Y
d
m
(
x;
t
)
=
m
X
n
=0
u
d
n
(
x;
t
)
=
u
d
0
(
x;
t
)
+
u
d
1
(
x;
t
)
+
+
u
d
m
(
x;
t
)
4
tx:
Note
that,
we
can
control
the
con
v
er
gence
re
gion
of
HAM
series
solution
by
the
auxiliary
parameter
~
.
7.
CONCLUSION
Homotop
y
analysis
method
has
been
performed
to
find
approximate
analytical
s
olutions
for
fuzzy
V
olterra-
Fredholm
inte
gro-dif
ferential
equations.
The
reliability
of
the
method
and
reduction
in
the
size
of
the
computational
w
ork
gi
v
e
this
method
a
wider
applicability
.
The
method
is
v
ery
po
werful
and
ef
ficient
in
finding
analytical
as
well
as
numerical
solutions
for
wide
classes
of
linear
and
nonlinear
fuzzy
inte
gro-dif
ferent
ial
equations.
Obtained
results
sho
w
that
we
can
control
the
con
v
er
gence
re
gion
of
HAM
series
solution
by
the
auxiliary
parameter
~
.
The
illustrati
v
e
e
xample
and
con
v
er
gence
theorem
sho
w
the
ef
ficienc
y
and
accurac
y
of
the
HAM.
REFERENCES
[1]
S.
Chang
and
L.
Zadeh,
”On
fuzzy
mapping
and
control,
”
IEEE
T
r
ans.
Systems,
Man
cybernet,
(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
ards
fuzzy
dif
ferential
calculus.
Inte
grati
on
of
fuzzy
mappings,
”
Fuzzy
Sets
and
Systems,
(1982),
8,
pp.
1–17.
[4]
O.
Kale
v
a,
”Fuzzy
dif
ferential
equation,
”
Fuzzy
Sets
and
Systems,
(1987),
24,
pp.
301–317.
[5]
S.
Narayanamoorth
y
and
S.
Sathiyapriya,
”A
pertinent
approach
to
s
olv
e
nonlinear
fuzzy
inte
gro-dif
ferential
equa-
tions,
”
Spring
er
Plus.
(2016),
5,
pp.
1–17.
[6]
T
.
Allahviranloo,
S.
Abbasbandy
and
S.
Hashemzehi,
”Approximating
the
solution
of
the
linear
and
nonlinear
fuzzy
V
olterra
inte
gro-dif
ferential
equations
using
e
xpansion
method,
”
Abstr
.
Appl.
Anal.
(2014),
pp.
1–8.
[7]
M.
Das
and
D.
T
alukdar
,
”Method
for
solving
fuzzy
inte
gro-dif
ferential
equations
by
us
ing
fuzzy
Laplace
trans-
formation,
”
Int.
J
.
Sci.
T
ec
h.
(2014),
5,
pp.
291–295.
[8]
E.
Hussain
and
A.
Ali,
”Homotop
y
analysis
method
for
solving
fuzzy
int
e
gro-di
f
ferential
equations,
”
Modern
Appl.
Sci.
(2013),
7,
pp.
15–25.
[9]
S.
Behiry
and
S.
Mohamed,
”Solving
high-order
nonlinear
V
olterra–Fredholm
inte
gro-dif
ferential
equations
by
dif
ferential
transform
method,
”
Nat.
Sci.
(2012),
8,
pp.
581–587.
[10]
H.
Rahimi,
M.
Khezerloo
and
S.
Khezerloo,
”Approximating
the
fuzzy
solution
of
the
non-linear
fuzzy
V
olterra
inte
gro-dif
ferential
equation
using
fix
ed
point
theorems,
”
Int
J
Indus
Math.
(2011),
3(3),
pp.
227–236.
[11]
M.
Hashemi
and
S.
Abbas
bandy
,
”The
solution
of
fuzzy
V
olterra-Fredholm
inte
gro-dif
ferential
equations
using
v
ariational
iteration
method,
”
Int.
J
.
Math.
Comput.
(2011),
11,
pp.
29–38.
[12]
A.
Hamoud
and
K.
Ghadle,
”modified
Adomian
decomposition
method
for
s
olving
fuzzy
V
olterra-Fredholm
inte
gral
equations,
”
J
ournal
of
the
Indian
Math.
Soc.
(2018),
85(1-2),
pp.
01–17.
[13]
A.
Hamoud
and
K.
Ghadle,
”Existence
and
uniqueness
of
solutions
for
fractional
mix
ed
V
olterra-Fredholm
inte
gro-dif
ferential
equations,
”
Indian
J.
Math.
(2018),
60(3),
pp.
375–395.
[14]
A.
Hamoud,
K.
Ghadle,
M.
Bani
Issa
and
Ginisw
amy
,
”Existence
and
uniqueness
t
heorems
for
fractional
V
olterra-Fredholm
inte
gro-dif
ferential
equations,
”
Int.
J.
Appl.
Math.
(2018),
31(3),
pp.
333–348.
[15]
A.
Hamoud,
K.
Ghadle
and
S.
Atshan,
”The
approximate
solutions
of
frac
tional
inte
gro-dif
ferential
equations
by
using
modified
Adomian
decomposition
method,
”
Khayyam
J.
Math.
(2019),
5(1),
pp.
21–39.
[16]
A.
Hamoud
and
K.
Ghadle,
”Existence
and
uniqueness
of
the
solution
for
V
olterra-Fredholm
inte
gro-dif
ferential
equations,
”
Journal
of
Siberian
Federal
Uni
v
ersity
.
Mathematics
&
Ph
ysics
,
(2018),
11(6),
pp.
692–701.
[17]
A.
Hamoud
and
K.
Ghadle,
”Some
ne
w
e
xistence,
uniqueness
and
con
v
er
gence
results
for
fractional
V
olterra-
Fredholm
inte
gro-dif
ferential
equations,
”
J.
Appl.
Comput.
Mech.
(2019),
5(1),
pp.
58–69.
[18]
A.
Hamoud,
M.
Bani
Issa
and
K.
Ghadle,
”Existence
and
uniqueness
results
for
nonlinear
V
olterra-Fredholm
inte
gro-dif
ferential
equations,
”
Nonlinear
Functional
Analysis
and
Applications
,
(2018),
23(4),
pp.
797–805.
[19]
A.
Hamoud
and
K.
Ghadle,
”Usage
of
the
homotop
y
analysis
method
for
solving
fractional
V
olterra-Fredholm
inte
gro-dif
ferential
equation
of
the
second
kind,
”
T
amkang
Journal
of
Mathematics
,
(2018),
49(4),
pp.
301–315.
[20]
A.
Hamoud
and
K.
Ghadle,
”Recent
adv
ances
on
reliable
methods
for
solving
V
olterra-Fredholm
inte
gral
and
inte
gro-dif
ferential
equations,
”
Asian
J.
Math.
Comput.
Res.
(2018),
24,
pp.
128–157.
[21]
Y
.
Chalco-Cano
and
H.
Roman-Flores,
”On
ne
w
solutions
of
fuzzy
dif
ferential
equations,
”
Chaos,
Solitons
and
F
r
actals
,
(2008),
38,
pp.
112–119.
Indonesian
J
Elec
Eng
&
Comp
Sci
V
ol.
11,
No.
3,
September
2018:
857
-867
Evaluation Warning : The document was created with Spire.PDF for Python.