Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
14,
No.
5,
Month
2024,
pp.
5738
∼
5743
ISSN:
2088-8708,
DOI:
10.11591/ijece.v14i5.pp5738-5743
❒
5738
An
efcient
appr
oximate
method
f
or
solving
Bratu’
s
boundary
v
alue
pr
oblem
Kamel
Al-Khaled
1
,
Mahmood
Shar
eef
Ajeel
2
,
Issam
Ab
u-Irwaq
1
,
Hala
K.
Al-Khalid
3
1
Department
of
Mathematics
and
Statistics,
Jordan
Uni
v
ersity
of
Science
and
T
echnology
,
Irbid,
Jordan
2
Department
of
Material
Engineering
-
Colle
ge
of
Engineering,
Shatrah
Uni
v
ersity
,
Thi-Qar
,
Iraq
3
Department
of
Mathematics,
W
estern
Michig
an
Uni
v
ersity
,
Kalamazoo,
United
States
of
America
Article
Inf
o
Article
history:
Recei
v
ed
Mar
19,
2024
Re
vised
Jul
10,
2024
Accepted
Jul
17,
2024
K
eyw
ords:
Adomian
decomposition
method
Bratu’
s
equation
Green’
s
function
Numerical
solutions
Second
order
boundary
v
alue
problems
ABSTRA
CT
W
e
compute
the
numerical
solution
of
Bratu’
s
boundary
v
alue
problem
(BVP).
T
o
achie
v
e
this,
we
apply
a
ne
w
and
usef
ul
approach
to
solv
e
Bratu’
s
bound-
ary
v
alue
problem
by
using
Green’
s
function
and
a
ne
w
inte
gral
operator
,
along
with
a
modied
v
ersion
of
the
Adomian
decomposition
method.
This
process
produces
solutions
that
call
for
the
boundary
conditions
to
be
applied
e
xplicitly
.
Statistical
results
demonstrating
the
rob
ustness
and
ef
cienc
y
of
the
proposed
scheme
are
included.
An
e
xact
and
approximate
solution
comparison
is
made
with
kno
wn
results.
The
quantitati
v
e
outcomes
sho
wcase
our
no
v
el
approach’
s
high
numerical
precision
and
consistenc
y
acr
oss
a
range
of
parameter
congu-
rations.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Kamel
Al-Khaled
Department
of
Mathematics
and
Statistics,
Jordan
Uni
v
ersity
of
Science
and
T
echnology
Irbid,
P
.O.Box
3030,
Jordan
Email:
kamel@just.edu.jo
1.
INTR
ODUCTION
Ordinary
dif
ferential
equations
that
are
nonlinear
are
the
result
of
mathematica
l
modeling
of
v
arious
ph
ysical
systems.
T
o
analyze
mathematical
modeling
that
yields
solut
ions
that
match
ph
ysical
rea
lity
(that
is,
the
real
w
orld
of
ph
ysics)
an
ef
cient
method
is
needed.
As
a
result,
we
need
to
be
able
to
solv
e
strongly
nonlinear
ordinary
dif
ferential
equations.
The
Chandrasekhar
model
of
uni
v
erse
e
xpansion,
fuel
ignition
in
thermal
comb
ustion
theory
,
and
Bratu’
s
problem,
which
w
as
used
to
model
a
comb
ustion
problem
in
a
numer
-
ical
slab,
are
e
xamples
of
specic
models.
It
encourages
a
thermal
reaction
in
a
stif
f
material
that
depe
n
ds
on
the
equilibrium
between
heat
produced
chemically
and
heat
transferred
through
conduction
[1]–[4].
W
e
e
xamine
the
elliptic
nonlinear
partial
dif
ferential
equation
with
homogeneous
Dirichlet
boundary
conditions
at
the
boundary
kno
wn
as
the
classical
Bratu
problem
[1].
The
Bratu’
s
model
has
been
computa-
tionally
handled
by
a
number
of
methods,
including
the
Lapl
ace
transform
decomposition
numerical
algorithm
[5],
[6],
dif
ferential
transform
method
[7],
weighted
residual
method
[8],
shooting
method
[9],
nite
dif
ference
method,
nite
element
approximation,
and
Adomian
decomposition
method
(ADM)
[10],
[11].
DTM
[12]
pro
vide
an
ef
fecti
v
e
numerical
technique
for
approximating
geometric
boundary
v
alue
problems.
The
hardest
component
of
the
Bratu’
s
equation
is
the
nonlinear
term,
e
y
.
Therefore,
by
applying
the
strate
gy
co
v
ered
in
[12],
Sinc
geometric
can
readily
solv
e
Braut’
s
equation.
Th
e
v
ariation
iteration
method
is
used
by
Anakira
et
al.
[13]
to
e
xam
ine
and
modify
an
approximate
analytical
solution
for
solving
the
fuzzy
Bratu’
s
equation.
A
numerical
approach
based
on
Chebyshe
v
w
a
v
elet
approximations
for
the
one-dimensional
Bratu’
s
problem
is
presented
in
[14].
In
[15],
Bratu’
s
boundary
v
alue
problem
is
solv
ed
using
a
ne
w
inte
gral
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
❒
5739
operator
by
utilizing
the
adomian
dec
o
m
position
method
and
the
restarted
adomian
decomposition
method
with
ne
w
techniques.
A
h
ybrid
Adomian
decomposition
approach
and
an
additional
inte
gral
transform
called
the
Kashuri
Fundo
transform
were
used
to
solv
e
Bratu’
s
equation
[16].
The
primary
goal
of
the
numerical
solution
of
the
Bratu’
s
boundary
v
alue
problem
on
a
Banach
space
setting
is
to
incorporate
a
Green’
s
function
into
a
no
v
el
tw
o-step
iteration
scheme,
as
demonstrated
in
[17].
A
numerical
solution
for
the
tw
o-dimensional
Bratu’
s
problem
is
obtained
in
[18]
through
the
de
v
elopment
of
a
ne
w
ite
rati
v
e
nite
dif
ference
algorithm.
Con
v
ersely
,
the
y
calculate
tw
o-branched
numerical
solutions
for
the
nonlinear
Bratu’
s
problem
in
[19].
Bratu’
s
problem
has
a
closed-form
solution
that
w
as
created
in
[20].
Using
articial
neural
netw
ork
architecture
and
a
soft
computing
technique
that
mak
es
use
of
the
symbiotic
or
g
anism
search
algorithm,
an
ef
cient
meta-heuristic,
better
solutions
for
Bratu’
s
dif
ferential
equation
are
obtained
in
[21].
This
paper
sho
ws
that
nonlinear
second-order
boundary
v
alue
problems
can
be
ef
fecti
v
ely
solv
ed
numerically
through
manipulation
of
the
Adomian
decomposition
method.
d
2
y
dx
2
=
λ
(
x
)
e
µy
,
y
(0)
=
y
(1)
=
0
(1)
in
which
λ
(
x
)
>
0
may
be
an
e
xponential,
rational,
or
polynomial
function,
and
µ
>
0
is
a
constant.
Owing
to
the
wide
range
of
possibilities,
we
shall
tak
e
λ
(
x
)
to
represent
an
y
analytic
function
of
x
with
a
po
wer
series
e
xpansion.
Dif
fusion
theory
and
celestial
machines
routinely
use
(1),
for
instance
in
mechanical
problems
where
dissipation
is
absent.
2.
BRA
TU’S
BOUND
AR
Y
V
ALUE
PR
OBLEM
This
section
presents
a
numerical
algorithm
to
solv
e
a
general
boundary
v
alue
problem
using
the
Adomian
decomposition
technique.
d
2
y
dx
2
=
λ
(
x
)
F
(
y
)
,
y
(0)
=
α
,
y
(1)
=
β
.
(2)
where
F
(
y
)
,
the
nonlinear
function,
can
be
de
v
eloped
in
a
series
of
the
form
F
(
y
)
=
P
∞
n
=0
a
n
y
n
since
it
is
assumed
to
be
analytic.
Equation
(1)
presents
the
same
problem
for
us
to
discuss
as
a
special
case,
if
F
(
y
)
=
e
µy
.
The
basic
concept
is
to
use
Green’
s
function
to
determine
the
inte
gral
representation
of
(2),
and
then
modify
Adomian
decomposition
to
produce
the
nonlinear
inte
gral
equation
of
the
follo
wing
form.
y
(
x
)
=
Z
1
0
λ
(
x
)
G
(
x,
ξ
)
F
(
y
(
ξ
))
dξ
+
(1
−
x
)
α
+
xβ
.
(3)
The
well
kno
wn
Green’
s
function
for
(2)
is
gi
v
en
by
,
G
(
x,
ξ
)
=
ξ
(1
−
x
)
,
0
≤
ξ
≤
x
x
(1
−
ξ
)
,
x
≤
ξ
≤
1
.
ADM
[22],
[23]
assumes
a
series
solution
for
(2)
of
the
form,
y
(
x
)
=
∞
X
n
=0
y
i
(
x
)
(4)
The
abo
v
e
innite
series
con
v
er
ges
if
P
∞
n
=0
(1
+
ϵ
)
n
|
y
n
|
<
∞
,
for
some
small
number
ϵ
.Ph
ysical
issues
typically
satisfy
this
requirement.
Assume
that
the
nonlinear
function
F
(
y
)
has
the
T
aylor
e
xpansion
around
y
0
as
follo
ws
to
demonstrate
Adomian’
s
scheme.
F
(
y
)
=
F
(
y
0
)
+
F
′
(
y
0
)(
y
−
y
0
)
+
1
2!
F
′′
(
y
0
)(
y
−
y
0
)
2
+
...
(5)
Substituting
the
dif
ference
y
−
y
0
=
y
1
+
y
2
+
...
from
(4)
into
(5),
to
obtain,
F
(
y
)
=
F
(
y
0
)
+
F
′
(
y
0
)(
y
1
+
y
2
+
...
)
+
1
2!
F
′′
(
y
0
)(
y
1
+
y
2
+
...
)
2
+
...
An
ef
cient
appr
oximate
method
for
solving
Br
atu’
s
boundary
value
pr
oblem
(Kamel
Al-Khaled)
Evaluation Warning : The document was created with Spire.PDF for Python.
5740
❒
ISSN:
2088-8708
After
simplifying
the
abo
v
e
equation,
we
get,
F
(
y
)
=
F
(
y
0
)
+
F
′
(
y
0
)(
y
1
+
y
2
+
...
)
+
1
2!
F
′′
(
y
0
)(
y
2
1
+
2
y
1
y
2
+
2
y
1
y
3
+
y
2
2
+
2
y
2
y
3
+
y
2
3
+
...
)
(6)
No
w
,
reordering
and
rearranging
of
the
terms
in
the
abo
v
e
equation,
we
arri
v
e
at
Adomian’
s
polynomials
as
[24]:
A
0
=
F
(
y
0
)
,
A
1
=
y
1
F
′
(
y
0
)
,
A
2
=
y
2
F
′
(
y
0
)
+
1
2!
y
2
1
F
′′
(
y
0
)
and
so
on
for
more
terms.
Substituting
these
A
n
’
s
together
with
the
series
in
(3),
we
arri
v
e
at,
y
0
(
x
)
+
y
1
(
x
)
+
y
2
(
x
)
+
...
=
Z
1
0
λ
(
x
)
G
(
x,
ξ
)(
A
0
+
A
1
+
A
2
+
...
)
dξ
+
(1
−
x
)
α
+
xβ
.
If
the
series
is
con
v
er
gent,
then
we
determine
each
term
of
the
series
solution
in
(3)
recursi
v
ely
as,
y
0
(
x
)
=
(1
−
x
)
α
+
xβ
y
1
(
x
)
=
Z
1
0
λ
(
x
)
G
(
x,
ξ
)
A
0
(
y
0
)
dξ
.
.
.
y
n
(
x
)
=
Z
1
0
λ
(
x
)
G
(
x,
ξ
)
A
n
−
1
(
y
0
,
y
1
,
...,
y
n
−
1
)
dξ
The
abo
v
e
algorithm
determines
the
y
i
s
and
hence
determines
the
approximate
solution
y
a
(
x
)
=
y
0
(
x
)
+
y
1
(
x
)
+
...
+
y
n
=
li
m
M
→∞
P
M
n
=0
y
n
(
x
)
,
where
M
is
the
number
of
terms
that
we
found.
It
is
possible
to
nd
an
e
xact
solution
in
closed
form
in
some
circumstances.
Moreo
v
er
,
the
solutions
of
decomposition
series
usually
con
v
er
ge
quickly
.
The
con
v
er
gence
of
the
decomposition
series
has
been
studied
by
a
number
of
writers
[25],
[26].
The
discussion
that
w
as
pre
viously
presented
can
be
clearly
summarized
by
looking
at
the
numerical
results
belo
w
.
3.
RESUL
TS
AND
DISCUSSION
Here,
we
report
our
numerical
ndings
for
Bratu’
s
problem.
T
o
demonstrate
the
rob
ustness
of
the
schemes,
we
will
in
v
estig
at
e
v
arious
parameter
re
gimes
for
the
eigen
v
alue
λ
.
The
calculations
sho
w
ho
w
simple
it
is
to
assemble
the
method,
run
it,
and
sel
ect
parameters.
The
numerical
results
demonstrate
the
con
v
er
gence
and
accurac
y
of
our
approach.
Examine
Bratu’
s
boundary
v
alue
problem.
y
′′
(
x
)
=
−
λe
y
(
x
)
,
y
(0)
=
y
(1)
=
0
.
(7)
Bratu’
s
equation
appeared
rst
in
the
theory
of
comb
ustion,
after
that,
the
equation
appeared
in
man
y
mathe-
matical
models
concerned
with
ph
ysical
applications.
Specically
,
it
conta
ins
the
solid
fuel
ignition
model
for
the
thermal
reaction
process
in
a
comb
ustible,
nondeformable
material
of
constant
density
during
the
ignition
period
[2],
and
the
Chandrashekhar
model
for
the
e
xpansion
of
the
uni
v
erse
[3].
As
stated
in
[2],
(7)
has
an
e
xact
solution
pro
vided
by
(8),
y
(
x
)
=
−
2
log
h
cosh(0
.
5(
x
−
0
.
5)
θ
)
cosh(0
.
25
θ
)
i
(8)
pro
vided
that
θ
is
a
solution
of
θ
=
√
2
λ
cosh(
θ
/
4)
.
T
o
nd
the
critical
v
alue
of
λ
,
dif
ferentiating
both
sides
we
obtain
1
=
1
4
√
2
λ
sinh(
θ
/
4)
,
in
which
the
critical
v
al
u
e
of
λ
is
λ
c
=
8
sinh
2
(
θ
/
4)
.
It
follo
ws
that
when
λ
>
λ
c
,
λ
=
λ
c
and
0
<
λ
<
λ
c
,
where
λ
c
=
3
.
5138
is
the
critical
v
al
ue,
Bratu’
s
problem
has
zero
solution,
a
unique
solution,
and
tw
o
bifurcated
solutions,
respecti
v
ely
.
F
or
the
ADM
solution,
(7)
can
be
written
in
an
inte
gral
equation
form
as,
y
(
x
)
=
λ
R
1
0
G
(
x,
ξ
)
e
y
(
ξ
)
dξ
Int
J
Elec
&
Comp
Eng,
V
ol.
14,
No.
5,
Month
2024:
5738-5743
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
❒
5741
where
G
(
x,
ξ
)
is
the
Green’
s
function.
The
nonlinear
function
e
y
may
be
e
xpanded
as
an
innite
sum
of
Adomian
polynomials,
as
e
y
=
P
∞
i
=0
A
i
=
A
0
+
A
1
+
A
2
+
...
,
where
A
0
=
1
,
A
1
=
y
1
e
y
0
,
A
2
=
y
2
+
1
2!
y
2
1
,
A
3
=
y
3
+
y
1
y
2
+
1
3!
y
3
1
,
A
4
=
y
4
+
y
1
y
3
+
1
2!
y
2
2
+
1
2
y
2
1
y
2
+
1
4!
y
4
1
.
According
to
the
analysis
in
the
pre
vious
section,
we
start
with
y
0
(
x
)
=
0
,
the
ADM
solution
of
Bratu’
s
equation
can
be
written
as:
y
k
+1
(
x
)
=
λ
(1
−
x
)
R
x
0
ξ
A
k
(
ξ
)
dξ
+
λx
R
1
x
(1
−
ξ
)
A
k
(
ξ
)
dξ
,
k
=
0
,
1
,
2
...
Using
Mathematica,
the
rst
v
e
iterations
in
y
are
calculated,
their
sum
is
denoted
by
y
a
(
x
)
,
we
list
the
rst
tw
o
terms:
y
1
(
x
)
=
λx
2
2
(1
−
x
)
+
λx
(
1
2
−
x
+
x
2
2
)
y
2
(
x
)
=
λ
(1
−
x
)(
λx
3
6
−
λx
4
8
)
+
λx
(
λ
24
−
λx
2
4
+
λx
3
3
−
λx
4
8
)
.
The
approximation
using
v
e
t
erms
in
equation
is
carried
out
for
Bratu’
s
model
with
λ
=
1
at
x
=
0
.
1
,
0
.
2
,
...,
0
.
9
.
T
able
1
e
xhibits
the
results
of
the
approximate
solution,
and
the
e
xact
solution
as
gi
v
en
in
the
closed
form
(8).
The
mathematical
formulat
ion
of
electrospinning
process
is
link
ed
also
to
the
f
amous
Bratu’
s
equation
as
gi
v
en
by
(9),
d
2
y
dx
2
=
λ
(
x
)
e
y
,
y
(0)
=
y
(1)
=
0
(9)
where
y
=
−
6
ln
u
and
u
represents
the
jet’
s
axial
v
elocity
.
After
solving
the
mass,
linear
momentum,
and
electric
char
ge
balance
equations,
W
an
et
al.
[27]
calculated
the
v
alue,
λ
=
18
E
2
ρ
2
r
4
(
I
−
r
2
K
E
)
.
The
parameters
I
,
K
,
E
,
ρ,
and
r
represent
the
jet’
s
current,
conducti
vity
,
v
oltage,
density
,
and
radius,
respec-
ti
v
ely
[28].
Three
dif
ferent
sel
ections
of
the
parameters
E
,
K
,
ρ,
I
,
r
were
chosen
such
that
the
y
made
the
v
alue
of
λ
is
equal
to
1
,
sometimes
less
than
1
,
and
sometimes
greater
than
1
,
and
Figures
1
and
2
summarise
our
results.
Numerical
e
xperiments
indicate
that
dif
ferent
eigen
v
alues
beha
v
e
similarly
[29].
It
is
interesting
to
point
out
that
the
solution
is
bounded
for
all
v
alues
of
its
domain.
T
able
1.
ADM
approximation
for
y
′′
=
λe
y
for
λ
=
1
x
Exact
solution
ADM
Error
0
.
1
0
.
049846
0
.
049817
2.92373E-05
0
.
2
0
.
089189
0
.
089133
5.64677E-05
0
.
3
0
.
117609
0
.
117530
7.90006E-05
0
.
4
0
.
134790
0
.
134696
9.39895E-05
0
.
5
0
.
140539
0
.
140440
9.92575E-05
0
.
6
0
.
134790
0
.
134696
9.39895E-05
0
.
7
0
.
117609
0
.
117530
7.90006E-05
0
.
8
0
.
089189
0
.
089133
5.64677E-05
0
.
9
0
.
049846
0
.
049817
2.92373E-05
0.2
0.4
0.6
0.8
1.0
x
0.02
0.04
0.06
0.08
0.10
0.12
0.14
y
0.2
0.4
0.6
0.8
1.0
x
0.2
0.4
0.6
0.8
1.0
y
Figure
1.
The
ADM
solution
when
λ
=
1
.
0
(left)
and
(right)
when
λ
=
3
.
51
for
y
′′
=
λe
y
An
ef
cient
appr
oximate
method
for
solving
Br
atu’
s
boundary
value
pr
oblem
(Kamel
Al-Khaled)
Evaluation Warning : The document was created with Spire.PDF for Python.
5742
❒
ISSN:
2088-8708
Figure
2.
The
ADM
solution
when
λ
=
1
.
25
(left)
and
(right)
when
λ
=
0
.
75
for
y
′′
=
λe
y
4.
CONCLUSION
In
order
to
obtain
approximate
analytical
solutions
for
Bratu-type
equations,
this
paper
succ
essfully
applies
the
ADM.
W
ithout
linearization
or
restricti
v
e
presumptions,
the
method
w
as
used
directly
.
ADM
is
con
v
enient
and
ef
fecti
v
e,
as
demonstrated
by
comparisons
to
the
precise
solution.
F
or
a
wide
range
of
linear
and
nonlinear
dif
ferential
equations,
ADM
is
useful
in
pro
viding
both
analytical
and
numerical
solutions.
F
or
ph
ysical
problems,
ADM
pro
vides
f
ast-con
v
er
gent,
real
istic
series
solutions.
The
outcomes
presented
in
this
paper
sho
w
that,
with
v
ery
little
computational
ef
fort,
our
method
can
be
used
to
obtain
precise
numerical
solutions
of
nonlinear
boundary
v
alue
probl
ems.
In
a
ree
xamination,
the
Bratu’
s
problem
can
be
solv
ed
by
substituting
the
fractional
deri
v
ati
v
e
of
order
α
for
the
second
deri
v
ati
v
e,
where
1
<
α
≤
2
.
In
the
e
v
ent
that
the
deri
v
ati
v
e
in
(1)
is
mar
ginally
less
than
2,
this
will
allo
w
the
eigen
v
alue
beha
vior
to
be
determined.
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Evaluation Warning : The document was created with Spire.PDF for Python.
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ISSN:
2088-8708
❒
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BIOGRAPHIES
OF
A
UTHORS
Kamel
Al-Khaled
recei
v
ed
a
Ph.D
de
gree
in
mathematics
from
Uni
v
ersity
of
Nebraska-
Lincoln,
Lincoln-Nebraska,
USA,
August
1996.
He
joined
the
De
partment
of
Mathematics
and
Statistics
at
Jordan
Uni
v
ersity
of
Science
and
T
echnology
in
1996.
Since
2009,
Dr
.
Al-Khaled
has
held
the
position
of
a
full
professor
in
the
Department
of
Mathematics
and
Statistics
at
the
same
uni
v
ersity
.
His
research
interests
include
applied
mat
hematical
modeling,
approximation
theory
,
op-
eration
research,
numerical
analysis,
fractional
cal
culus,
and
the
dif
ferential
equations
of
applied
mathematics.
He
w
as
recognized
among
the
‘w
orld’
s
top
2%
scientists’
in
Stanford
Uni
v
ersity’
s
2021,
2022,
2023
ranking.
He
can
be
contacted
at
email:
kamel@just.edu.jo.
Mahmood
Shar
ee
f
Ajeel
is
assistant
professor
at
Department
of
Material
Engineer
ing
-
Colle
ge
of
Engineering,
Shatrah
Uni
v
ersity
,
Thi-Qar
64001
-
Iraq.
His
research
interests
include
numerical
analysis,
partial
dif
ferential
equations
and
computational
methods.
He
can
be
contacted
at
email:
mahmoodshareef@shu.edu.iq.
Issam
Ab
u-Irwaq
is
instructor
at
Department
of
Mathematics
and
Statistics,
F
aculty
of
Science
and
Arts,
Jordan
Uni
v
ersity
of
Science
and
T
echnology
,
Irbid
22110,
Jordan.
He
recei
v
ed
M.Sc.
de
gree
in
mathematics
from
Carleton
Uni
v
ersit
y
,
Otta
w
a,
Canada
in
2001
and
and
B.Sc.
de
gree
with
the
rst
rank
in
applied
mathe
matics
from
Jordan
Uni
v
ersity
of
Science
and
T
echnology
,
Irbid
22110,
J
ordan
in
1998.
His
research
interests
include
fractional
calculus
and
partial
dif
ferential
equations.
He
can
be
contacted
at
email:
imab
uirw
aq@just.edu.jo.
Hala
K.
Al-Khalid
recei
v
ed
B.Sc.
de
gree
with
highest
honor
a
v
erage
in
mathematics
from
Jordan
Uni
v
ersity
of
Science
and
T
echnology
,
Irbid
22110,
Jordan
in
January
2020
and
M.Sc.
in
mathematics
at
W
estern
Michig
an
Uni
v
ersity
,
Kalamazoo,
MI
49008-5248,
USA
in
2023.
Miss
Al-Khalid
has
a
fello
wship
from
W
e
stern
Michig
an
Uni
v
ersity
to
pursue
her
graduate
studies.
She
is
currently
a
Ph.D.
student
in
the
Department
of
Mathematics
at
W
estern
Michig
an
Uni
v
ersity
,
Kalama-
zoo,
MI
49008-5248,
USA.
Her
research
interes
ts
include
analysis
and
partial
dif
ferential
equations.
She
can
be
contacted
at
email:
halakamelmustaf
a.alkhalid@wmich.edu.
An
ef
cient
appr
oximate
method
for
solving
Br
atu’
s
boundary
value
pr
oblem
(Kamel
Al-Khaled)
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