T
E
L
KO
M
NIK
A
, V
ol
.
17
,
No.
3,
J
un
e
2
0
19
,
pp
.
1
33
8
~
13
4
3
IS
S
N: 1
69
3
-
6
93
0
,
accr
ed
ited
F
irst
Gr
ad
e b
y K
em
en
r
istekdikti,
Decr
ee
No: 2
1/E/
K
P
T
/20
18
DOI:
10.12928/TE
LK
OM
N
IK
A
.v
1
7
i
3
.
10318
13
38
Rec
ei
v
ed
Nov
e
mb
er
6
, 2
01
8
; Rev
i
s
e
d
F
e
bruar
y
12
,
20
1
9
;
A
c
c
ep
ted
M
arc
h
10
,
20
1
9
Fo
rm
al
exp
ansi
on
met
ho
d
f
or s
ol
v
in
g
a
n elec
trical cir
cuit
mo
del
T
jend
r
o
*
1
, S
u
d
i M
u
n
g
ka
si
2
1
Dep
a
rt
m
e
n
t
o
f
El
e
c
tr
i
c
a
l
En
g
i
n
e
e
ri
n
g
,
F
a
c
u
l
ty
o
f
S
c
i
e
n
c
e
a
n
d
T
e
c
h
n
o
l
o
g
y
,
Sa
n
a
ta
D
h
a
rm
a
Un
i
v
e
rs
i
ty
,
M
ri
c
a
n
,
T
ro
m
o
l
Po
s
2
9
,
Y
o
g
y
a
k
a
rta
5
5
0
0
2
,
I
n
d
o
n
e
s
i
a
2
Dep
a
rt
m
e
n
t
o
f
M
a
th
e
m
a
ti
c
s
,
Fa
c
u
l
ty
o
f
Sc
i
e
n
c
e
a
n
d
T
e
c
h
n
o
l
o
g
y
,
Sa
n
a
ta
D
h
a
rm
a
Un
i
v
e
r
s
i
ty
,
M
ri
c
a
n
,
T
ro
m
o
l
Po
s
2
9
,
Y
o
g
y
a
k
a
rta
5
5
0
0
2
,
I
n
d
o
n
e
s
i
a
*C
o
rre
s
p
o
n
d
i
n
g
a
u
th
o
r
,
e
-
m
a
i
l
:
t
j
e
n
d
ro
@us
d
.a
c
.i
d
1
,
s
u
d
i
@
u
s
d
.a
c
.
i
d
2
Ab
strac
t
W
e
i
n
v
e
s
t
i
g
a
t
e
th
e
v
a
l
i
d
i
t
y
o
f
th
e
fo
rm
a
l
e
x
p
a
n
s
i
o
n
m
e
th
o
d
fo
r
s
o
l
v
i
n
g
a
s
e
c
o
n
d
o
rd
e
r
o
rd
i
n
a
r
y
d
i
ff
e
r
e
n
ti
a
l
e
q
u
a
t
i
o
n
r
a
i
s
e
d
fr
o
m
a
n
e
l
e
c
tri
c
a
l
c
i
r
c
u
i
t
p
ro
b
l
e
m
.
Th
e
f
o
rm
a
l
e
x
p
a
n
s
i
o
n
m
e
th
o
d
a
p
p
r
o
x
i
m
a
t
e
s
th
e
e
x
a
c
t
s
o
l
u
ti
o
n
u
s
i
n
g
a
s
e
ri
e
s
o
f
s
o
l
u
ti
o
n
s
.
An
a
p
p
ro
x
i
m
a
te
fo
rm
a
l
e
x
p
a
n
s
i
o
n
s
o
l
u
t
i
o
n
i
s
a
tru
n
c
a
te
d
v
e
rs
i
o
n
o
f
t
h
i
s
s
e
ri
e
s
.
I
n
th
i
s
p
a
p
e
r,
we
c
o
n
f
i
rm
u
s
i
n
g
s
i
m
u
l
a
ti
o
n
s
th
a
t
th
e
a
p
p
ro
x
i
m
a
te
fo
rm
a
l
e
x
p
a
n
s
i
o
n
s
o
l
u
ti
o
n
i
s
v
a
l
i
d
fo
r
a
s
p
e
c
i
f
i
c
i
n
te
rv
a
l
o
f
d
o
m
a
i
n
o
f
th
e
fre
e
v
a
ri
a
b
l
e
.
Th
e
a
c
c
u
r
a
c
y
o
f
th
e
fo
rm
a
l
e
x
p
a
n
s
i
o
n
a
p
p
ro
x
i
m
a
ti
o
n
i
s
g
u
a
ra
n
te
e
d
o
n
t
h
e
t
i
m
e
-
s
c
a
l
e
1
.
Key
w
ords
:
d
a
m
p
e
d
o
s
c
i
l
l
a
ti
o
n
,
e
l
e
c
tri
c
a
l
c
i
r
c
u
i
t,
f
o
rm
a
l
e
x
p
a
n
s
i
o
n
,
v
a
n
d
e
r
p
o
l
e
q
u
a
t
i
o
n
,
v
i
b
ra
ti
o
n
m
o
d
e
l
Copy
righ
t
©
2
0
1
9
Uni
v
e
rsi
t
a
s
Ahm
a
d
D
a
hl
a
n.
All
rig
ht
s
r
e
s
e
rve
d
.
1.
Int
r
o
d
u
ctio
n
Ma
th
em
ati
c
s
an
d
i
ts
progr
am
m
i
ng
ha
v
e
p
l
a
y
e
d
i
m
po
r
tan
t
r
o
l
es
i
n
s
ol
v
i
ng
as
wel
l
as
de
s
i
g
ni
n
g
ex
p
erim
en
ts
of
el
ec
tr
i
c
al
en
gi
ne
er
i
ng
prob
l
e
m
s
,
f
or
ex
a
m
pl
e,
s
ee
th
e
wor
k
o
f
S
uti
k
no
et
al
.
[1
-
4]
.
T
o
be
s
pe
c
i
f
i
c
,
i
n
thi
s
pa
p
er
we
c
on
s
i
de
r
el
ec
tr
i
c
al
c
i
r
c
ui
t
prob
l
em
s
.
P
r
ob
l
em
s
i
n
el
ec
tr
i
c
a
l
c
i
r
c
ui
ts
are
of
ten
m
od
el
l
ed
i
nto
di
f
f
erenti
a
l
e
qu
at
i
on
s
.
O
n
e
of
the
m
od
el
s
i
s
c
al
l
ed
th
e
v
an
d
er
P
o
l
eq
ua
t
i
on
.
T
hi
s
eq
ua
t
i
on
i
s
d
ue
to
th
e
Dutc
h
ph
y
s
i
c
i
s
t
B
al
t
ha
s
ar
v
an
d
er
P
o
l
i
n
aroun
d
1
92
0
to
d
es
c
r
i
be
os
c
i
l
l
ati
on
s
i
n
a
tr
i
od
e
-
c
i
r
c
ui
t
[
5
].
In
a
s
pe
c
i
f
i
c
s
i
t
ua
t
i
on
wi
th
s
m
al
l
s
ou
r
c
e
i
n
os
c
i
l
l
a
ti
o
ns
,
the
v
an
de
r
P
ol
eq
u
ati
on
b
ec
om
es
a
v
i
brati
on
m
od
el
wi
th
a
l
i
n
ea
r
f
r
i
c
ti
on
term
.
In
thi
s
pa
pe
r
we
s
ol
v
e
th
e
v
i
brat
i
o
n
m
od
el
wi
th
a
l
i
ne
ar
f
r
i
c
ti
on
term
,
whi
c
h
i
s
a
m
od
i
f
i
c
ati
on
of
th
e
v
an
d
er
P
ol
eq
ua
t
i
on
, u
s
i
ng
t
he
f
orm
al
ex
pa
ns
i
on
m
eth
od
.
P
r
ev
i
o
us
r
es
ea
r
c
h
ha
s
be
e
n
c
on
du
c
te
d
b
y
a
n
um
be
r
of
au
tho
r
s
r
el
ati
ng
to
th
e
v
an
de
r
P
ol
eq
ua
t
i
on
[
5
-
8
]
i
n
ph
y
s
i
c
s
[
9
-
10
],
b
i
o
l
og
y
[
11
],
ec
o
no
m
i
c
s
[
12
],
etc
.
[
13
-
1
5
].
A
m
on
gs
t
the
m
,
V
erhu
l
s
t
[
5
]
pro
v
i
d
ed
a
the
orem
ab
ou
t
the
ord
er
of
a
c
c
urac
y
of
the
f
or
m
al
ex
pa
ns
i
on
s
ol
uti
on
wi
th
r
es
pe
c
t
to
the
pe
r
t
urbati
on
f
ac
tor
i
n
th
e
d
am
pi
ng
t
erm
.
Nev
erth
el
es
s
,
i
t
ha
s
n
ot
be
e
n
c
on
f
i
r
m
ed
c
o
m
pu
tat
i
o
na
l
l
y
when
we
us
e
th
i
s
m
eth
od
to
s
ol
v
e
th
e
v
i
brat
i
o
n
m
od
el
w
i
th
a
l
i
ne
ar
da
m
pi
ng
(
f
r
i
c
ti
on
term
)
,
es
pe
c
i
a
l
l
y
th
e
v
al
i
d
i
t
y
of
the
m
eth
od
r
el
at
i
ng
t
o
the
i
nt
e
r
v
al
of
the
f
r
ee
v
ari
ab
l
e.
T
he
r
ef
ore,
th
i
s
p
ap
er
s
ha
l
l
f
i
l
l
thi
s
ga
p
of
r
es
ea
r
c
h
,
th
at
i
s
,
w
e
s
h
al
l
v
a
l
i
d
ate
of
the
f
or
m
al
ex
pa
ns
i
o
n
m
eth
od
c
om
pu
tat
i
on
al
l
y
.
T
he
r
e
s
t
of
thi
s
p
ap
er
i
s
w
r
i
t
te
n
as
f
ol
l
o
w
s
.
W
e
prov
i
d
e
th
e
m
ath
em
ati
c
al
m
od
el
an
d
m
eth
od
i
n
s
ec
ti
on
2
.
A
f
ter tha
t
w
e
pres
e
nt
ou
r
r
es
e
arc
h
r
es
ul
ts
an
d d
i
s
c
us
s
i
on
i
n
s
e
c
ti
on
3
. T
he
pa
pe
r
i
s
c
o
nc
l
u
de
d
wi
th
s
om
e re
m
ar
k
s
i
n
s
ec
ti
on
4
.
2.
M
ath
emat
ica
l
M
o
d
el
and
M
eth
o
d
T
he
v
an
de
r
P
ol
eq
ua
t
i
on
,
as
th
e c
o
ns
i
de
r
ed
m
ath
em
ati
c
a
l
m
od
el
,
i
s
̈
+
=
(
1
−
2
)
̇
where
i
s
a
po
s
i
t
i
v
e
c
on
s
t
an
t
[5]
.
W
he
n
the
f
ac
tor
(
1
−
2
)
i
s
r
ep
l
ac
ed
b
y
−
,
where
i
s
a
s
m
al
l
po
s
i
ti
v
e c
o
ns
tan
t
, th
e
m
od
el
be
c
om
es
x
̈
+
x
=
−
ε
x
̇
whi
c
h
i
s
v
al
i
d f
or
>
1
or
<
−
1
. T
hi
s
m
od
el
i
s
th
e
v
i
brat
i
on
m
od
el
w
i
th
a
l
i
ne
ar f
r
i
c
ti
on
t
erm
.
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NIK
A
IS
S
N: 1
69
3
-
6
93
0
F
ormal
ex
pa
ns
i
on
me
t
ho
d f
or s
ol
v
i
n
g a
n
…
(
T
j
en
dr
o
)
1339
T
he
c
ore
propert
y
i
n
the
f
o
r
m
al
ex
pa
ns
i
on
m
eth
od
i
s
gi
v
en
i
n
a
the
orem
as
f
ol
l
ow
s
d
ue
to
V
erhu
l
s
t
[5]
.
W
e
c
on
s
i
de
r
th
e i
ni
t
i
a
l
v
a
l
ue
prob
l
em
x
̇
=
f
0
(
t
,
x
)
+
ε
f
1
(
t
,
x
)
+
⋯
+
ε
m
f
m
(
t
,
x
)
+
ε
m
+
1
R
(
t
,
x
,
ε
)
where
(
0
)
=
and
|
−
0
|
≤
ℎ
,
∈
,
0
≤
≤
0
.
Her
e
i
s
a
c
on
s
tan
t
,
ℎ
i
s
a
po
s
i
t
i
v
e
c
on
s
tan
t,
i
s
a
do
m
ai
n i
n t
h
e
di
m
en
s
i
on
,
an
d
0
i
s
a
po
s
i
ti
v
e c
on
s
t
an
t.
W
e a
s
s
u
m
e t
ha
t
i
n
thi
s
do
m
ai
n
al
l
f
un
c
ti
on
s
i
nv
ol
v
ed
i
n
th
e
pro
bl
em
are
i
nf
i
ni
tel
y
m
an
y
d
i
f
f
erenti
ab
l
e.
T
he
n
t
he
f
orm
al
ex
pa
ns
i
on
x
0
(
t
)
+
ε
x
1
(
t
)
+
⋯
+
ε
m
x
m
(
t
)
wi
th
0
(
0
)
=
,
(
)
=
0
,
=
1
,
…
,
ap
prox
i
m
ate
s
th
e e
x
a
c
t s
ol
ut
i
on
(
)
w
i
t
h t
h
e p
r
o
pe
r
t
y
‖
x
(
t
)
−
(
x
0
(
t
)
+
ε
x
1
(
t
)
+
⋯
+
ε
m
x
m
(
t
)
)
‖
=
O
(
ε
m
+
1
)
on th
e t
i
m
e
-
s
c
al
e 1
. T
hi
s
m
ea
ns
th
at
t
he
f
orm
al
ex
pa
n
s
i
on
i
s
of
th
e
(
+
1
)
th
ord
er of
ac
c
urac
y
.
3.
Re
sult
s
a
nd
D
isc
u
s
sio
n
F
or
the
c
on
v
en
i
en
c
e
of
w
r
i
ti
ng
an
d
i
n
order
t
o
b
e
c
on
s
i
s
ten
t
wi
th
ou
r
r
ef
erenc
es
(
s
uc
h
as
V
erh
ul
s
t [
5]),
w
e c
o
ns
i
d
er the
m
od
el
x
̈
+
x
=
−
2ε
x
̇
s
up
po
s
e
t
he
i
n
i
ti
al
c
o
nd
i
ti
o
ns
are
(
0
)
=
and
̇
(
0
)
=
0
.
T
he
ex
ac
t s
ol
u
ti
on
to
t
hi
s
pro
bl
em
i
s
x
(
t
)
=
a
e
ε
t
c
os
(
√
1
−
ε
2
t
)
+
ε
a
√
1
−
ε
2
e
−
ε
t
s
in
(
√
1
−
ε
2
t
)
s
ub
s
ti
tut
i
n
g
x
(
t
)
=
x
0
(
t
)
+
ε
x
1
(
t
)
+
ε
2
…
i
nto
th
e
m
od
el
,
w
e
ob
t
ai
n
x
̈
0
+
x
0
=
0
,
x
̈
n
+
x
n
=
−
2
x
̇
n
−
1
,
n
=
1
,
2
,
…
no
w
we p
u
t
x
0
(
0
)
=
a
,
x
̇
0
(
0
)
=
(
0
)
x
n
(
0
)
=
0
,
x
̇
n
(
0
)
=
0
,
n
=
1
,
2
,
…
.
we
ob
tai
n
x
0
(
t
)
=
a
c
os
t
x
1
(
t
)
=
a
s
in
t
−
at
c
os
t
the
r
ef
ore
, o
ur s
ol
u
ti
o
n b
as
e
d o
n
the
f
orm
al
ex
pa
ns
i
on
i
s
x
(
t
)
=
a
c
os
t
+
a
ε
(
s
in
t
−
t
c
os
t
)
+
ε
2
…
that
i
s
, th
e f
i
r
s
t o
r
de
r
f
or
m
al
s
ol
ut
i
on
i
s
y
1
(
t
)
=
a
c
os
t
the
s
ec
on
d o
r
d
er f
or
m
al
s
ol
uti
o
n i
s
y
2
(
t
)
=
a
c
os
t
+
a
ε
(
s
in
t
−
t
c
os
t
)
Rem
ar
k
:
W
e
c
ho
os
e
to
c
on
s
i
d
er
thi
s
prob
l
em
,
be
c
au
s
e
th
i
s
pro
bl
em
ha
s
an
ex
ac
t
s
ol
uti
o
n.
W
e
i
nte
nt
i
on
al
l
y
us
e
the
ex
ac
t
s
ol
u
ti
o
n
t
o
v
erif
y
t
he
v
a
l
i
d
i
t
y
of
f
or
m
al
ex
pa
ns
i
o
n
s
ol
ut
i
o
ns
.
If
the
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N: 16
93
-
6
93
0
T
E
L
KO
M
NIK
A
V
ol
.
17
,
No
.
3,
J
un
e 2
01
9
:
13
38
-
1
34
3
1340
f
or
m
al
ex
pa
ns
i
on
s
ol
u
ti
o
ns
are
v
a
l
i
d
f
or
s
ol
v
i
n
g
prob
l
em
s
ha
v
i
ng
ex
ac
t
s
o
l
ut
i
on
s
,
the
n
we
s
ha
l
l
be
s
ure
to
us
e
t
he
f
orm
al
ex
pa
ns
i
o
n
m
eth
od
to
s
ol
v
e
probl
em
s
w
i
th
th
e
ex
ac
t
s
o
l
ut
i
on
s
are
no
t
k
no
w
n.
No
te
tha
t
i
n
prac
t
i
c
e,
ex
ac
t
s
ol
ut
i
o
ns
are
g
en
era
l
l
y
no
t
k
no
w
n.
N
o
w
f
or
nu
m
eric
al
ex
pe
r
i
m
en
ts
,
w
e
tak
e
=
1
an
d
v
ar
y
t
he
v
a
l
u
es
of
.
T
o
ge
t
c
l
ea
r
i
l
l
us
tr
at
i
on
s
,
w
e
tak
e
=
0
.
5
,
0
.
05
,
0
.
025
r
es
pe
c
ti
v
e
l
y
.
3.1.
S
imu
latio
n
f
o
r
Cas
e
=
.
F
or
the
f
i
r
s
t
c
as
e,
w
e
tak
e
=
0
.
5
.
F
i
gu
r
e
1
s
ho
w
s
th
e
ex
a
c
t
s
ol
uti
on
,
th
e
f
i
r
s
t
order
f
or
m
al
ex
pa
ns
i
o
n
s
ol
uti
on
,
an
d
t
he
s
ec
o
nd
order
f
or
m
al
ex
pa
ns
i
on
s
o
l
ut
i
o
n
o
n
the
i
nt
erv
al
0
≤
≤
1
.
W
e
ob
s
erv
e
tha
t
th
e
s
ec
on
d
order
s
o
l
ut
i
on
ap
prox
i
m
ate
s
the
ex
ac
t
s
ol
uti
on
b
ett
er
tha
n
th
e
f
i
r
s
t
order
do
es
i
n
t
he
do
m
ai
n
0
≤
≤
1
.
H
o
w
e
v
er,
i
f
w
e
ex
ten
d
the
d
om
ai
n
t
o
b
e
0
≤
≤
10
,
the
s
ec
o
n
d
order
s
o
l
ut
i
o
n
be
ha
v
es
po
orl
y
a
nd
ev
en
w
ors
e
tha
n
th
e
f
i
r
s
t
order
s
ol
ut
i
on
,
as
g
i
v
en
i
n F
i
gu
r
e
2.
F
i
gu
r
e
1.
E
x
ac
t, f
i
r
s
t o
r
de
r
,
an
d s
ec
o
nd
or
de
r
s
ol
ut
i
on
s
f
or
=
0
.
5
i
n d
om
ai
n
0
≤
≤
1
F
i
gu
r
e
2.
E
x
ac
t, f
i
r
s
t o
r
de
r
,
an
d s
ec
o
nd
or
de
r
s
ol
ut
i
on
s
f
or
=
0
.
5
i
n d
om
ai
n
0
≤
≤
10
3.2.
S
imu
latio
n
f
o
r
Cas
e
=
.
F
or
the
s
ec
on
d
c
as
e,
w
e
t
ak
e
=
0
.
05
.
F
i
g
ure
3
s
h
o
w
s
the
s
ol
ut
i
on
s
o
n
th
e
i
n
terv
al
0
≤
≤
10
.
S
i
m
i
l
ar
t
o
the
pre
v
i
ou
s
c
as
e,
we
ob
s
er
v
e
t
ha
t
the
s
ec
o
nd
ord
er
s
ol
u
ti
on
ap
prox
i
m
ate
s
the
ex
ac
t
s
o
l
uti
o
n
be
t
ter
th
an
th
e
f
i
r
s
t
order
do
es
i
n
th
e
do
m
ai
n
0
≤
≤
1
an
d
the
ex
ten
d
ed
do
m
ai
n
0
≤
≤
10
.
H
o
w
e
v
er,
i
f
w
e
ex
te
nd
the
d
o
m
ai
n
f
urther
to
be
0
≤
≤
50
,
the
s
ec
on
d o
r
d
er s
ol
u
ti
on
b
eh
a
v
es
w
ors
e
tha
n t
h
e f
i
r
s
t o
r
de
r
s
ol
uti
on
,
as
i
l
l
us
tr
at
ed
i
n F
i
gu
r
e
4.
F
i
gu
r
e
3.
E
x
ac
t, f
i
r
s
t o
r
de
r
,
an
d s
ec
o
nd
or
de
r
s
ol
ut
i
on
s
f
or
=
0
.
05
i
n d
om
ai
n
0
≤
≤
10
F
i
gu
r
e
4.
E
x
ac
t, f
i
r
s
t o
r
de
r
,
an
d s
ec
o
nd
or
de
r
s
ol
ut
i
on
s
f
or
=
0
.
05
i
n d
om
ai
n
0
≤
≤
50
3.3.
S
imu
latio
n
f
o
r
Cas
e
=
.
A
s
the
th
i
r
d
c
as
e,
w
e
f
i
x
=
0
.
025
.
W
e
pl
ot
the
s
ol
uti
on
s
on
th
e
i
nt
erv
a
l
0
≤
≤
10
as
s
ho
w
n
i
n
F
i
gu
r
e
5.
O
nc
e
ag
a
i
n,
w
e
ob
s
er
v
e
th
at
the
s
ec
on
d
order
s
ol
uti
on
ap
prox
i
m
ate
s
the
ex
ac
t
s
ol
ut
i
on
be
tt
er
tha
n
t
he
f
i
r
s
t
order
do
es
i
n
th
e
d
om
ai
n
0
≤
≤
1
an
d
th
e
ex
ten
de
d
d
om
ai
n
0
≤
≤
10
.
Ho
w
e
v
er,
on
c
e
a
ga
i
n,
i
f
we
ex
ten
d
th
e
do
m
ai
n
f
urther
to
be
0
≤
≤
100
,
the
s
ec
o
nd
order
s
ol
uti
on
b
eh
av
es
w
or
s
e t
ha
n
th
e
f
i
r
s
t o
r
de
r
s
ol
uti
on
, a
s
i
l
l
us
tr
at
ed
i
n Fi
gu
r
e
6.
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NIK
A
IS
S
N: 1
69
3
-
6
93
0
F
ormal
ex
pa
ns
i
on
me
t
ho
d f
or s
ol
v
i
n
g a
n
…
(
T
j
en
dr
o
)
1341
F
i
gu
r
e
5.
E
x
ac
t, f
i
r
s
t o
r
de
r
,
an
d s
ec
o
nd
or
de
r
s
ol
ut
i
on
s
f
or
=
0
.
025
i
n d
om
ai
n
0
≤
≤
10
F
i
gu
r
e
6.
E
x
ac
t,
f
i
r
s
t o
r
de
r
,
an
d s
ec
o
nd
order
s
ol
uti
on
s
f
or
=
0
.
025
i
n
do
m
ai
n
0
≤
≤
100
3.4.
S
imu
latio
n
f
o
r
t
h
e
V
al
idit
y
of
O
r
d
er
of
A
c
cur
ac
y
A
s
we
ha
v
e
m
en
ti
o
ne
d
i
n
the
m
ath
em
ati
c
al
m
eth
od
s
ec
ti
on
,
t
he
f
orm
al
ex
pa
ns
i
on
i
s
gu
ara
nte
e
d
to
be
v
a
l
i
d
on
l
y
on
the
ti
m
e
-
s
c
al
e
1.
F
or
an
y
ex
ten
s
i
on
of
the
d
om
a
i
n
l
arger
t
ha
n
0
≤
≤
1
,
the
ac
c
urac
y
i
s
no
t
gu
ara
nte
e
d.
O
b
v
i
ou
s
l
y
f
r
om
the
prev
i
ou
s
s
ub
s
ec
ti
on
s
(
S
ub
s
ec
ti
on
s
3.
1
-
3.3
)
,
we
ob
ta
i
n
th
at
f
or
an
ex
te
nd
ed
do
m
ai
n,
th
e
err
ors
of
the
f
or
m
al
ex
pa
ns
i
on
s
ol
uti
on
s
are
i
n
de
ed
v
er
y
l
ar
ge.
I
n
the
pre
s
en
t
s
ub
s
ec
ti
on
w
e
i
n
v
es
ti
ga
te
t
he
v
al
i
d
i
t
y
of
the
order
of
ac
c
urac
y
of
the
f
or
m
al
ex
pa
ns
i
o
n.
W
e
l
i
m
i
t
ou
r
do
m
ai
n
on
l
y
o
n
the
i
nte
r
v
a
l
of
the
ti
m
e
-
s
c
al
e
1.
W
e
tak
e
a
di
s
c
r
ete
v
ers
i
o
n
of
the
t
i
m
e
do
m
ai
n
to
be
=
0
,
0
.
1
,
0
.
2
,
0
.
3
,
0
.
4
,
0
.
5
,
0
.
6
,
0
.
7
,
0
.
8
,
0
.
9
,
1
.
T
hi
s
m
ea
ns
tha
t
w
e
h
av
e
di
s
c
r
eti
s
ed
t
he
ti
m
e
do
m
ai
n
i
nto
11
po
i
nts
.
E
r
r
or of
an
a
pp
r
ox
i
m
ate
s
ol
ut
i
o
n i
s
qu
a
n
ti
f
i
ed
as
=
1
∑
|
(
)
−
(
)
|
=
1
where
i
s
the
nu
m
be
r
of
di
s
c
r
ete
ti
m
e
po
i
nts
(
i
n
t
h
i
s
c
as
e
=
1
,
2
,
3
,
…
,
wi
th
=
11
)
,
(
)
i
s
th
e
ex
ac
t
s
ol
ut
i
on
,
a
nd
(
)
i
s
the
ap
prox
i
m
ate
s
ol
uti
on
.
F
urtherm
ore,
th
e
order
of
ac
c
urac
y
i
s
c
al
c
ul
a
ted
as
:
=
(
+
1
)
(
+
1
)
the
order
of
ac
c
urac
y
i
s
c
al
c
ul
ate
d
ba
s
ed
o
n
the
th
a
n
d
th
e
(
+
1
)
th
s
i
m
ul
ati
o
ns
,
r
es
p
e
c
ti
v
e
l
y
,
us
i
ng
d
i
f
f
erent
v
a
l
ue
s
of
.
O
ur
r
es
ul
ts
of
err
ors
an
d
order
s
of
ac
c
urac
y
are
s
um
m
a
r
i
s
ed
i
n
T
ab
l
es
1
an
d
2.
T
ab
l
e
1
c
on
ta
i
ns
t
he
err
ors
of
the
f
i
r
s
t
order
f
orm
al
s
ol
uti
on
w
i
th
r
es
p
ec
t
t
o
v
ar
y
i
n
g
on
the
ti
m
e
-
s
c
al
e
1.
A
s
te
nd
s
to
z
ero,
the
o
r
de
r
of
ac
c
urac
y
ap
pro
ac
h
es
1.
T
hi
s
i
s
c
on
s
i
s
ten
t
w
i
th
t
he
the
or
eti
c
a
l
b
ac
k
ground
th
at
th
e
s
ol
uti
on
i
s
of
th
e
f
i
r
s
t
order
.
T
ab
l
e
2
s
u
m
m
aris
es
the
err
ors
of
the
s
ec
on
d
order
f
orm
al
s
ol
ut
i
on
wi
t
h
r
es
pe
c
t
t
o
v
ar
y
i
ng
on
th
e
ti
m
e
-
s
c
al
e
1.
W
e
f
i
nd
tha
t
as
te
nd
s
to
z
ero,
the
order
of
ac
c
urac
y
ap
proac
he
s
2.
T
hi
s
i
s
c
on
s
i
s
ten
t
wi
th
th
e
the
or
y
th
at
as
i
t
i
s
the
s
ec
on
d
order
f
or
m
al
ex
pa
ns
i
o
n
s
ol
ut
i
o
n,
the
o
r
d
e
r
of
ac
c
urac
y
i
s
2
i
n t
h
e t
i
m
e
-
s
c
al
e 1
.
T
ab
l
e 1
.
E
r
r
ors
of
th
e
F
i
r
s
t
O
r
de
r
For
m
al
S
ol
uti
on
wi
t
h
Res
p
ec
t
to
V
a
r
y
i
n
g
on
th
e
T
i
m
e
-
S
c
al
e
1
E
r
r
o
r
Or
d
e
r
o
f
a
c
c
u
r
a
c
y
0
.
5
0
.
0
3
5
1
-
0
.
2
5
0
.
0
1
9
3
0
.
8
6
0
.
1
2
5
0
.
0
1
0
1
0
.
9
3
0
.
0
6
2
5
0
.
0
0
5
2
0
.
9
6
0
.
0
3
1
2
5
0
.
0
0
0
.
9
8
T
ab
l
e 2
.
E
r
r
ors
of
th
e
S
ec
o
nd
O
r
de
r
For
m
al
S
ol
uti
on
wi
t
h
Res
p
ec
t
to
V
a
r
y
i
n
g
on
th
e
T
i
m
e
-
S
c
al
e
1
E
r
r
o
r
Or
d
e
r
o
f
a
c
c
u
r
a
c
y
0
.
5
0
.
0
0
7
5
2
7
-
0
.
2
5
0
.
0
0
2
0
3
7
1
.
8
9
0
.
1
2
5
0
.
0
0
0
5
3
1
1
.
9
4
0
.
0
6
2
5
0
.
0
0
0
1
3
6
1
.
9
7
0
.
0
3
1
2
5
0
.
0
0
0
0
3
4
1
.
9
8
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N: 16
93
-
6
93
0
T
E
L
KO
M
NIK
A
V
ol
.
17
,
No
.
3,
J
un
e 2
01
9
:
13
38
-
1
34
3
1342
A
s
f
i
na
l
r
em
ar
k
s
,
k
no
w
i
ng
the
ac
c
urac
y
of
the
f
or
m
al
ex
pa
ns
i
o
n
m
eth
od
,
we
c
ou
l
d
ex
ten
d
the
a
pp
l
i
c
at
i
on
of
t
hi
s
m
eth
od
t
o
s
o
l
v
e
oth
er
m
ath
e
m
ati
c
al
e
ng
i
ne
erin
g
probl
em
s
,
s
uc
h
as
tho
s
e
s
tud
i
ed
b
y
r
es
ea
r
c
he
r
s
i
n
[16
-
26
].
P
os
s
i
bl
e
oth
er
prob
l
em
s
to
be
s
ol
v
ed
us
i
ng
t
he
f
or
m
al
ex
pa
ns
i
o
n m
eth
od
c
ou
l
d b
e
th
os
e
i
n
[2
7
-
37].
4.
Co
n
clus
ion
W
e
ha
v
e
pro
v
i
de
d
ou
r
r
es
ea
r
c
h
r
es
u
l
ts
o
n
th
e
f
orm
al
ex
pa
ns
i
on
m
eth
od
f
or
s
ol
v
i
ng
an
el
ec
tr
i
c
a
l
c
i
r
c
ui
t
m
od
el
.
T
he
ac
c
urac
y
of
the
f
or
m
al
ex
p
an
s
i
o
n
i
s
gu
arant
ee
d
on
t
h
e
ti
m
e
-
s
c
al
e
1.
W
e
ha
v
e
al
s
o
c
on
f
i
r
m
ed
the
order
of
ac
c
urac
y
f
or
the
f
i
r
s
t
an
d
s
ec
on
d
order
f
orm
al
ex
pa
ns
i
on
s
ol
ut
i
on
us
i
n
g
nu
m
eric
al
ex
pe
r
i
m
en
ts
.
W
e
ob
tai
n
t
ha
t
f
or
the
f
i
r
s
t
order
f
orm
al
ex
pa
ns
i
on
s
ol
ut
i
on
,
as
the
p
ertur
ba
t
i
o
n
f
ac
tor
i
s
ha
l
v
ed
,
th
e
err
or
i
s
al
s
o
ha
l
v
e
d
on
the
t
i
m
e
-
s
c
al
e
1.
F
or
the
s
ec
on
d
order
f
orm
al
ex
pa
ns
i
on
s
o
l
ut
i
on
,
as
the
p
e
r
turbati
on
f
ac
tor
i
s
ha
l
v
ed
,
the
err
or
i
s
qu
arter
e
d
on
the
t
i
m
e
-
s
c
al
e
1.
W
i
th
the
s
e
r
es
ul
ts
,
the
f
or
m
al
ex
pa
ns
i
on
m
eth
od
c
ou
l
d
b
e
us
ed
to
s
ol
v
e
oth
er
pro
bl
em
s
i
n
el
ec
tr
i
c
al
c
i
r
c
ui
ts
f
or
the
t
i
m
e
-
s
c
al
e
1.
W
he
n
the
ti
m
e
-
s
c
al
e
i
s
no
t
eq
ua
l
to
1,
w
e
m
a
y
ne
ed
t
o
do
r
e
-
s
c
al
i
ng
s
o
tha
t
the
ti
m
e
do
m
ai
n
i
s
on
the
ti
m
e
-
s
c
al
e
1.
T
hi
s
c
ou
l
d
be
a f
utu
r
e res
ea
r
c
h
di
r
ec
ti
on
.
A
c
kno
w
ledg
ement
s
T
h
i
s
w
o
r
k
w
a
s
f
i
n
a
n
c
i
a
l
l
y
s
u
p
p
o
r
t
e
d
b
y
a
r
e
s
e
a
r
c
h
g
r
a
n
t
f
r
o
m
D
i
r
e
k
t
o
r
a
t
R
i
s
e
t
d
a
n
P
e
n
g
a
b
d
i
a
n
M
a
s
y
a
r
a
k
a
t
o
f
t
h
e
M
i
n
i
s
t
r
y
o
f
R
e
s
e
a
r
c
h
,
T
e
c
h
n
o
l
o
g
y
,
a
n
d
H
i
g
h
e
r
E
d
u
c
a
t
i
o
n
o
f
t
h
e
R
e
p
u
b
l
i
c
o
f
I
n
d
o
n
e
s
i
a
.
W
e
a
r
e
v
e
r
y
g
r
a
t
e
f
u
l
f
o
r
t
h
e
f
i
n
a
n
c
i
a
l
s
u
p
p
o
r
t
i
n
t
h
e
f
o
r
m
o
f
H
i
b
a
h
P
e
n
e
l
i
t
i
a
n
T
e
r
a
p
a
n
U
n
g
g
u
l
a
n
P
e
r
g
u
r
u
a
n
T
i
n
g
g
i
y
e
a
r
2
0
1
8
w
i
t
h
t
h
e
c
o
n
t
r
a
c
t
n
u
m
b
e
r
1
0
9
/
S
P
2
H
/
L
T
/
D
R
P
M
/
2
0
1
8
.
Ref
er
en
ce
s
[1
]
Su
ti
k
n
o
T
,
Id
ri
s
NR
N,
W
i
d
o
d
o
NS,
J
i
d
i
n
A.
FPGA
Ba
s
e
d
a
P
W
M
T
e
c
h
n
i
q
u
e
fo
r
Pe
r
m
a
n
e
n
t
M
a
g
n
e
t
AC
M
o
to
r Dri
v
e
s
.
I
n
te
rn
a
ti
o
n
a
l
J
o
u
r
n
a
l
o
f
Re
c
o
n
f
i
g
u
ra
b
l
e
a
n
d
Em
b
e
d
d
e
d
Sy
s
te
m
s
.
2
0
1
2
;
1
(
2
):
4
3
-
48.
[2
]
Su
ti
k
n
o
T
,
Id
ri
s
NR
N,
J
i
d
i
n
A,
J
o
p
r
i
M
H.
FPGA
Ba
s
e
d
O
p
ti
m
i
z
e
d
Di
s
c
o
n
ti
n
u
o
u
s
SVP
W
M
Al
g
o
ri
th
m
fo
r
T
h
re
e
P
h
a
s
e
VSI
i
n
AC
Dri
v
e
s
.
I
n
te
rn
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
Po
wer
El
e
c
tro
n
i
c
s
a
n
d
Dri
v
e
S
y
s
t
e
m
.
2
0
1
3
;
3
(2
):
2
2
8
-
240.
[3
]
Su
ti
k
n
o
T
,
Id
ri
s
NR
N,
J
i
d
i
n
AZ.
O
v
e
rv
i
e
w
o
n
Stra
te
g
i
e
s
a
n
d
Ap
p
ro
a
c
h
e
s
fo
r
FPGA
Pro
g
ra
m
m
i
n
g
.
TEL
KO
M
NIKA
Te
l
e
c
o
m
m
u
n
i
c
a
ti
o
n
Co
m
p
u
ti
n
g
El
e
c
tr
o
n
i
c
s
a
n
d
Co
n
tr
o
l
.
2
0
1
4
;
1
2
(
2
):
2
7
3
-
2
8
2
.
[4
]
Su
ti
k
n
o
T
,
J
i
d
i
n
AZ
,
J
i
d
i
n
A,
I
d
ri
s
NRN
.
Stra
te
g
i
e
s
f
o
r F
PG
A I
m
p
l
e
m
e
n
ta
ti
o
n
o
f
Non
-
Re
s
to
ri
n
g
Sq
u
a
r
e
Roo
t
Al
g
o
ri
th
m
.
In
te
rn
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
E
l
e
c
tri
c
a
l
a
n
d
Co
m
p
u
te
r En
g
i
n
e
e
ri
n
g
.
2
0
1
4
;
4
(4
):
5
4
8
-
5
5
6
.
[5
]
Ve
rh
u
l
s
t
F.
Non
l
i
n
e
a
r
Dif
fe
r
e
n
ti
a
l
Eq
u
a
ti
o
n
s
a
n
d
Dy
n
a
m
i
c
a
l
Sy
s
t
e
m
s
.
Se
c
o
n
d
,
Re
v
i
s
e
d
a
n
d
E
x
p
a
n
d
e
d
E
d
i
ti
o
n
.
B
e
rl
i
n
:
Sp
ri
n
g
e
r.
1
9
9
6
.
[6
]
Za
n
e
tt
e
DH
.
Eff
e
c
ts
o
f
No
i
s
e
o
n
th
e
In
te
r
n
a
l
Re
s
o
n
a
n
c
e
o
f
a
Non
l
i
n
e
a
r
O
s
c
i
l
l
a
to
r.
Sc
i
e
n
t
i
f
i
c
Rep
o
rt
s
.
2
0
1
8
;
8
:
5
9
7
6
.
[7
]
Hel
l
e
v
i
k
K,
G
u
d
m
e
s
ta
d
O
T
.
L
i
m
i
t
Cy
c
l
e
O
s
c
i
l
l
a
t
i
o
n
s
a
t
Res
o
n
a
n
c
e
s
.
IOP
Con
fe
re
n
c
e
Se
ri
e
s
:
M
a
te
ri
a
l
s
Sc
i
e
n
c
e
a
n
d
En
g
i
n
e
e
ri
n
g
.
2
0
1
7
;
2
76
:
0
1
2
0
2
0
.
[8
]
Ki
s
s
G
,
L
e
s
s
a
r
d
J
P.
Rap
i
d
l
y
a
n
d
S
l
o
w
l
y
O
s
c
i
l
l
a
ti
n
g
Pe
ri
o
d
i
c
S
o
l
u
t
i
o
n
s
o
f
a
Del
a
y
e
d
v
a
n
d
e
r
P
o
l
O
s
c
i
l
l
a
to
r
.
J
o
u
rn
a
l
o
f
Dy
n
a
m
i
c
s
a
n
d
Di
ff
e
r
e
n
t
i
a
l
E
q
u
a
t
i
o
n
s
.
2
0
1
7
;
2
9
(4
):
1
2
3
3
-
1
2
5
7
.
[9
]
Hus
s
i
n
W
N
W
,
Haru
n
FN,
M
o
h
d
M
H,
Rah
m
a
n
M
AA.
An
a
l
y
ti
c
a
l
M
o
d
e
l
l
i
n
g
Pre
d
i
c
ti
o
n
b
y
U
s
i
n
g
W
a
k
e
O
s
c
i
l
l
a
to
r
M
o
d
e
l
fo
r
Vo
rte
x
-
i
n
d
u
c
e
d
Vi
b
ra
ti
o
n
s
.
J
o
u
rn
a
l
o
f
M
e
c
h
a
n
i
c
a
l
En
g
i
n
e
e
ri
n
g
a
n
d
Sc
i
e
n
c
e
s
.
2
0
1
7
;
1
1
(4
)
:
3
1
1
6
-
3
1
2
8
.
[1
0
]
Herre
ra
L
,
M
o
n
ta
n
o
O
,
O
rl
o
v
Y
.
Hop
f
Bi
fu
rc
a
ti
o
n
o
f
Hy
b
ri
d
v
a
n
d
e
r
Po
l
O
s
c
i
l
l
a
t
o
rs
.
Non
l
i
n
e
a
r
An
a
l
y
s
i
s
:
Hy
b
ri
d
S
y
s
te
m
s
.
2
0
1
7
;
2
6
:
2
2
5
-
238.
[1
1
]
Che
re
v
k
o
AA
,
Bo
r
d
EE,
Kh
e
AK,
Pa
n
a
r
i
n
VA
,
O
rl
o
v
K
J
.
T
h
e
An
a
l
y
s
i
s
o
f
So
l
u
ti
o
n
s
Be
h
a
v
i
o
u
r
o
f
v
a
n
d
e
r
Po
l
Du
ff
i
n
g
Eq
u
a
ti
o
n
De
s
c
ri
b
i
n
g
L
o
c
a
l
Br
a
i
n
He
m
o
d
y
n
a
m
i
c
s
.
J
o
u
rn
a
l
o
f
Ph
y
s
i
c
s
:
Con
fe
re
n
c
e
Se
ri
e
s
.
2
0
1
7
;
8
9
4
(1
)
:
0
1
2
0
1
2
.
[1
2
]
He
L
,
Y
i
L
,
T
a
n
g
P.
Num
e
ri
c
a
l
Sc
h
e
m
e
a
n
d
Dy
n
a
m
i
c
An
a
l
y
s
i
s
f
o
r
Va
ri
a
b
l
e
-
o
rd
e
r
Fra
c
ti
o
n
a
l
v
a
n
d
e
r
Po
l
M
o
d
e
l
o
f
No
n
l
i
n
e
a
r E
c
o
n
o
m
i
c
Cy
c
l
e
.
A
d
v
a
n
c
e
s
i
n
Di
ff
e
re
n
c
e
Eq
u
a
t
i
o
n
s
.
2
0
1
6
;
2
0
1
6
(1
):
1
9
5
.
[1
3
]
Rac
h
u
n
k
o
v
a
I,
T
o
m
e
c
e
k
J
.
A
n
ti
p
e
r
i
o
d
i
c
So
l
u
ti
o
n
s
to
v
a
n
d
e
r
Po
l
Eq
u
a
ti
o
n
s
w
i
th
Sta
t
e
-
d
e
p
e
n
d
e
n
t
Im
p
u
l
s
e
s
.
El
e
c
tr
o
n
i
c
J
o
u
rn
a
l
o
f
Di
ff
e
re
n
ti
a
l
Eq
u
a
ti
o
n
s
.
2
0
1
7
;
2
0
1
7
:
2
4
7
.
[1
4
]
Si
e
w
e
RT
,
T
a
l
l
a
AF,
W
o
a
f
o
P
.
Res
p
o
n
s
e
o
f
a
Res
o
n
a
n
t
T
u
n
n
e
l
l
i
n
g
Dio
d
e
O
p
to
e
l
e
c
tro
n
i
c
O
s
c
i
l
l
a
t
o
r
Cou
p
l
e
d
t
o
a
No
n
-
l
i
n
e
a
r El
e
c
tr
i
c
a
l
C
i
r
c
u
i
t.
IET
Opt
o
e
l
e
c
tro
n
i
c
s
.
2
0
1
6
;
1
0
(6
):
2
0
5
-
210.
[1
5
]
Hov
e
i
j
n
I.
Sta
b
i
l
i
ty
Po
c
k
e
ts
o
f
a
P
e
ri
o
d
i
c
a
l
l
y
Fo
r
c
e
d
O
s
c
i
l
l
a
to
r
i
n
a
M
o
d
e
l
fo
r
Se
a
s
o
n
a
l
i
ty
.
In
d
a
g
a
ti
o
n
e
s
M
a
th
e
m
a
ti
c
a
e
.
2
0
1
6
;
2
7
(5
):
1
2
0
4
-
1
2
1
8
.
[1
6
]
M
e
z
g
h
a
n
i
F,
Ba
rc
h
i
e
s
i
D,
C
h
e
ro
u
a
t
A,
G
ro
s
g
e
s
T
,
Bo
ro
u
c
h
a
k
i
H.
Co
m
p
a
ri
s
o
n
o
f
3
D
Ad
a
p
ti
v
e
Rem
e
s
h
i
n
g
Stra
t
e
g
i
e
s
fo
r
Fi
n
i
t
e
El
e
m
e
n
t
Si
m
u
l
a
ti
o
n
s
o
f
El
e
c
tr
o
m
a
g
n
e
t
i
c
H
e
a
ti
n
g
o
f
G
o
l
d
Nan
o
p
a
rt
i
c
l
e
s
.
Ad
v
a
n
c
e
s
i
n
M
a
th
e
m
a
ti
c
a
l
Ph
y
s
i
c
s
.
2
0
1
5
;
2
0
1
5
.
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NIK
A
IS
S
N: 1
69
3
-
6
93
0
F
ormal
ex
pa
ns
i
on
me
t
ho
d f
or s
ol
v
i
n
g a
n
…
(
T
j
en
dr
o
)
1343
[1
7
]
Dy
m
n
i
k
o
v
a
I,
G
a
l
a
k
ti
o
n
o
v
E,
T
ro
p
p
E.
Ex
i
s
te
n
c
e
o
f
El
e
c
t
ri
c
a
l
l
y
Ch
a
r
g
e
d
Stru
c
tu
re
s
w
i
th
Reg
u
l
a
r
Cen
te
r
i
n
No
n
l
i
n
e
a
r
E
l
e
c
tro
d
y
n
a
m
i
c
s
M
i
n
i
m
a
l
l
y
Cou
p
l
e
d
to
G
ra
v
i
ty
.
Ad
v
a
n
c
e
s
i
n
M
a
th
e
m
a
ti
c
a
l
Ph
y
s
i
c
s
.
2
0
1
5
;
2
0
1
5
.
[1
8
]
Va
fe
a
s
P.
Di
p
o
l
a
r
Ex
c
i
ta
ti
o
n
o
f
a
Pe
r
fe
c
t
l
y
El
e
c
tri
c
a
l
l
y
Con
d
u
c
ti
n
g
Sp
h
e
ro
i
d
i
n
a
L
o
s
s
l
e
s
s
M
e
d
i
u
m
a
t
th
e
L
o
w
-
Fre
q
u
e
n
c
y
Re
g
i
m
e
.
A
d
v
a
n
c
e
s
i
n
M
a
th
e
m
a
ti
c
a
l
Ph
y
s
i
c
s
.
2
0
1
8
;
2
0
1
8
.
[1
9
]
M
o
rá
n
-
L
ó
p
e
z
A,
Córc
o
l
e
s
J
,
Rui
z
-
Cruz
J
A,
M
o
n
te
j
o
-
G
a
ra
i
J
R,
Reb
o
l
l
a
r
J
M
.
El
e
c
tr
o
m
a
g
n
e
t
i
c
Sc
a
tt
e
r
i
n
g
a
t
t
h
e
W
a
v
e
g
u
i
d
e
Ste
p
b
e
tw
e
e
n
E
q
u
i
l
a
te
r
a
l
T
ri
a
n
g
u
l
a
r
W
a
v
e
g
u
i
d
e
s
.
A
d
v
a
n
c
e
s
i
n
M
a
th
e
m
a
ti
c
a
l
Ph
y
s
i
c
s
.
2
0
1
6
;
2
0
1
6
.
[2
0
]
M
u
n
g
k
a
s
i
S.
Ad
a
p
ti
v
e
Fi
n
i
te
Vo
l
u
m
e
M
e
th
o
d
fo
r
th
e
Sh
a
l
l
o
w
W
a
te
r
Eq
u
a
t
i
o
n
s
o
n
T
ri
a
n
g
u
l
a
r
G
ri
d
s
.
Ad
v
a
n
c
e
s
i
n
M
a
th
e
m
a
ti
c
a
l
Ph
y
s
i
c
s
.
2
0
1
6
;
2
0
1
6
.
[2
1
]
G
ó
m
e
z
-
A
g
u
i
l
a
r
J
F
,
E
s
c
a
l
a
n
t
e
-
M
a
r
t
í
n
e
z
J
E
,
C
a
l
d
e
r
ó
n
-
R
a
m
ó
n
C
,
M
o
r
a
l
e
s
-
M
e
n
d
o
z
a
L
J
,
B
e
n
a
v
i
d
e
z
-
C
r
u
z
M
,
G
o
n
z
a
l
e
z
-
L
e
e
M
.
E
q
u
i
v
a
l
e
n
t
C
i
r
c
u
i
t
s
A
p
p
l
i
e
d
i
n
E
l
e
c
t
r
o
c
h
e
m
i
c
a
l
I
m
p
e
d
a
n
c
e
S
p
e
c
t
r
o
s
c
o
p
y
a
n
d
F
r
a
c
t
i
o
n
a
l
D
e
r
i
v
a
t
i
v
e
s
w
i
t
h
a
n
d
w
i
t
h
o
u
t
S
i
n
g
u
l
a
r
K
e
r
n
e
l
.
A
d
v
a
n
c
e
s
i
n
M
a
t
h
e
m
a
t
i
c
a
l
P
h
y
s
i
c
s
.
2
0
1
6
; 2
016
;
1
-
1
5
.
[2
2
]
G
ó
m
e
z
-
Ag
u
i
l
a
r
J
F
,
Ros
a
l
e
s
-
G
a
rc
ía
J
,
Es
c
o
b
a
r
-
J
i
m
é
n
e
z
RF,
L
ó
p
e
z
-
L
ó
p
e
z
M
G
,
Al
v
a
ra
d
o
-
M
a
rtí
n
e
z
V
M
,
O
l
i
v
a
re
s
-
Pe
re
g
r
i
n
o
VH.
O
n
th
e
Po
s
s
i
b
i
l
i
ty
o
f
th
e
J
e
r
k
Deri
v
a
ti
v
e
i
n
El
e
c
tri
c
a
l
Cir
c
u
i
ts
.
Ad
v
a
n
c
e
s
i
n
M
a
th
e
m
a
ti
c
a
l
Ph
y
s
i
c
s
.
2
0
1
6
;
2
0
1
6
; 1
-
8
.
[2
3
]
Dy
m
n
i
k
o
v
a
I
,
G
a
l
a
k
t
i
o
n
o
v
E.
Ba
s
i
c
G
e
n
e
r
i
c
Pro
p
e
r
ti
e
s
o
f
R
e
g
u
l
a
r
Rot
a
ti
n
g
Bl
a
c
k
Hol
e
s
a
n
d
So
l
i
to
n
s
.
Ad
v
a
n
c
e
s
i
n
M
a
th
e
m
a
ti
c
a
l
Ph
y
s
i
c
s
.
2
0
1
7
;
2
0
1
7
;
1
-
10
.
[2
4
]
Su
n
D,
Ba
o
W
,
L
i
X
.
An
a
l
y
ti
c
Cal
c
u
l
a
ti
o
n
o
f
T
ra
n
s
m
i
s
s
i
o
n
Fi
e
l
d
i
n
Ho
m
o
g
e
n
e
o
u
s
l
y
L
a
y
e
re
d
M
e
d
i
u
m
s
E
x
c
i
te
d
b
y
EM
P.
Ad
v
a
n
c
e
s
i
n
M
a
th
e
m
a
ti
c
a
l
Ph
y
s
i
c
s
.
2
0
1
7
;
2
017
; 1
-
8
.
[2
5
]
G
a
o
S,
Che
n
S,
J
i
Z,
T
i
a
n
W
,
Che
n
J
.
DC
G
l
o
w
Dis
c
h
a
rg
e
i
n
A
x
i
a
l
M
a
g
n
e
ti
c
Fi
e
l
d
a
t
L
o
w
Pre
s
s
u
re
s
.
Ad
v
a
n
c
e
s
i
n
M
a
th
e
m
a
ti
c
a
l
Ph
y
s
i
c
s
.
2
0
1
7
;
2
0
1
7
;
1
-
8
.
[2
6
]
T
a
o
B.
M
o
d
e
l
Eq
u
a
t
i
o
n
s
f
o
r
T
h
re
e
-
Dim
e
n
s
i
o
n
a
l
Non
l
i
n
e
a
r
W
a
t
e
r
W
a
v
e
s
u
n
d
e
r
T
a
n
g
e
n
ti
a
l
El
e
c
tr
i
c
Fi
e
l
d
.
Ad
v
a
n
c
e
s
i
n
M
a
th
e
m
a
ti
c
a
l
Ph
y
s
i
c
s
.
2
0
1
7
;
2
0
1
7
;
1
-
8
.
[2
7
]
S
u
p
r
i
y
a
d
i
B
,
M
u
n
g
k
a
s
i
S
.
F
i
n
i
t
e
V
o
l
u
m
e
N
u
m
e
r
i
c
a
l
S
o
l
v
e
r
s
f
o
r
N
o
n
-
L
i
n
e
a
r
E
l
a
s
t
i
c
i
t
y
i
n
H
e
t
e
r
o
g
e
n
e
o
u
s
M
e
d
i
a
.
I
n
t
e
r
n
a
t
i
o
n
a
l
J
o
u
r
n
a
l
f
o
r
M
u
l
t
i
s
c
a
l
e
C
o
m
p
u
t
a
t
i
o
n
a
l
E
n
g
i
n
e
e
r
i
n
g
.
2
0
1
6
;
1
4
(
5
)
:
4
7
9
-
488.
[2
8
]
Su
z
u
k
i
Y
,
T
a
k
a
h
a
s
h
i
M
.
M
u
l
ti
s
c
a
l
e
Se
a
m
l
e
s
s
-
Dom
a
i
n
M
e
t
h
o
d
Ba
s
e
d
o
n
De
p
e
n
d
e
n
t
V
a
ri
a
b
l
e
a
n
d
Dep
e
n
d
e
n
t
-
Va
r
i
a
b
l
e
G
ra
d
i
e
n
t
s
.
In
te
r
n
a
ti
o
n
a
l
J
o
u
rn
a
l
fo
r
M
u
l
ti
s
c
a
l
e
Com
p
u
t
a
ti
o
n
a
l
En
g
i
n
e
e
r
i
n
g
.
2
0
1
6
;
1
4
(6
):
6
0
7
-
6
3
0
.
[2
9
]
Kro
w
c
z
y
n
s
k
i
M
,
Cec
o
t
W
.
A
Fa
s
t
T
h
re
e
-
L
e
v
e
l
Up
s
c
a
l
i
n
g
fo
r
S
h
o
rt
Fi
b
e
r
-
Rei
n
fo
rc
e
d
Com
p
o
s
i
te
s
.
In
te
rn
a
ti
o
n
a
l
J
o
u
r
n
a
l
fo
r M
u
l
ti
s
c
a
l
e
Co
m
p
u
ta
t
i
o
n
a
l
En
g
i
n
e
e
ri
n
g
.
2
0
1
7
;
1
5
(1
):
1
9
-
3
4
.
[3
0
]
Pa
n
d
a
N,
Bu
tl
e
r
T
,
Es
te
p
D,
G
ra
h
a
m
L
,
D
a
w
s
o
n
C.
A
St
o
c
h
a
s
ti
c
In
v
e
r
s
e
Pro
b
l
e
m
fo
r
M
u
l
ti
s
c
a
l
e
M
o
d
e
l
s
.
In
te
r
n
a
t
i
o
n
a
l
J
o
u
rn
a
l
f
o
r M
u
l
ti
s
c
a
l
e
Co
m
p
u
ta
ti
o
n
a
l
E
n
g
i
n
e
e
ri
n
g
.
2
0
1
7
;
1
5
(3
)
:
2
6
5
-
2
8
3
.
[3
1
]
Roj
e
k
J
,
Nos
e
w
i
c
z
S,
Chm
i
e
l
e
w
s
k
i
M
.
M
i
c
ro
-
M
a
c
ro
Rel
a
t
i
o
n
s
h
i
p
s
fro
m
Dis
c
re
te
El
e
m
e
n
t
S
i
m
u
l
a
ti
o
n
s
o
f
Si
n
t
e
ri
n
g
.
In
te
rn
a
ti
o
n
a
l
J
o
u
r
n
a
l
f
o
r M
u
l
ti
s
c
a
l
e
C
o
m
p
u
ta
t
i
o
n
a
l
En
g
i
n
e
e
ri
n
g
.
20
1
7
;
1
5
(4
):
3
2
3
-
3
4
2
.
[3
2
]
Dan
i
e
l
Y
S,
Az
i
z
ZA,
Is
m
a
i
l
Z,
Sa
l
a
h
F.
En
tr
o
p
y
An
a
l
y
s
i
s
o
f
Uns
te
a
d
y
M
a
g
n
e
to
h
y
d
ro
d
y
n
a
m
i
c
Nan
o
fl
u
i
d
o
v
e
r
Stre
t
c
h
i
n
g
Sh
e
e
t
w
i
th
El
e
c
tri
c
Fi
e
l
d
.
In
te
r
n
a
ti
o
n
a
l
J
o
u
rn
a
l
fo
r
M
u
l
ti
s
c
a
l
e
Co
m
p
u
ta
ti
o
n
a
l
En
g
i
n
e
e
ri
n
g
.
2
0
1
7
;
1
5
(6
)
:
5
4
5
-
565.
[3
3
]
S
u
n
W
,
F
i
s
h
J
,
D
h
i
a
H
B
.
A
V
a
r
i
a
n
t
o
f
t
h
e
S
-
V
e
r
s
i
o
n
o
f
t
h
e
F
i
n
i
t
e
E
l
e
m
e
n
t
M
e
t
h
o
d
f
o
r
C
o
n
c
u
r
r
e
n
t
M
u
l
t
i
s
c
a
l
e
C
o
u
p
l
i
n
g
.
I
n
t
e
r
n
a
t
i
o
n
a
l
J
o
u
r
n
a
l
f
o
r
M
u
l
t
i
s
c
a
l
e
C
o
m
p
u
t
a
t
i
o
n
a
l
E
n
g
i
n
e
e
r
i
n
g
.
2
0
1
8
;
1
6
(
2
)
:
1
8
7
-
207.
[3
4
]
M
u
n
g
k
a
s
i
S
,
M
a
g
d
a
l
e
n
a
I
,
Pu
d
j
a
p
ra
s
e
ty
a
SR, W
i
ry
a
n
to
L
H,
Rob
e
rts
SG
.
A
St
a
g
g
e
re
d
M
e
th
o
d
f
o
r
t
h
e
Sh
a
l
l
o
w
W
a
te
r
Eq
u
a
t
i
o
n
s
In
v
o
l
v
i
n
g
Va
ry
i
n
g
Cha
n
n
e
l
W
i
d
th
a
n
d
T
o
p
o
g
ra
p
h
y
.
In
te
r
n
a
t
i
o
n
a
l
J
o
u
rn
a
l
fo
r
M
u
l
ti
s
c
a
l
e
Co
m
p
u
ta
ti
o
n
a
l
En
g
i
n
e
e
ri
n
g
.
2
0
1
8
;
1
6
(3
)
:
2
3
1
-
2
4
4
.
[3
5
]
M
o
y
e
d
a
A,
Fi
s
h
J
.
M
u
l
ti
s
c
a
l
e
An
a
l
y
s
i
s
o
f
Pre
s
t
re
s
s
e
d
Con
c
re
te
Stru
c
t
u
re
s
.
In
te
rn
a
ti
o
n
a
l
J
o
u
rn
a
l
f
o
r
M
u
l
ti
s
c
a
l
e
Co
m
p
u
ta
ti
o
n
a
l
En
g
i
n
e
e
ri
n
g
.
2
0
1
8
;
1
6
(3
)
:
2
8
5
-
3
0
1
.
[3
6
]
P
u
s
z
k
a
r
z
A
K
,
Kr
u
c
i
n
s
k
a
I
.
S
i
m
u
l
a
t
i
o
n
s
o
f
A
i
r
P
e
r
m
e
a
b
i
l
i
t
y
o
f
M
u
l
t
i
l
a
y
e
r
T
e
x
t
i
l
e
s
b
y
t
h
e
C
o
m
p
u
t
a
t
i
o
n
a
l
F
l
u
i
d
D
y
n
a
m
i
c
s
.
I
n
t
e
r
n
a
t
i
o
n
a
l
J
o
u
r
n
a
l
f
o
r
M
u
l
t
i
s
c
a
l
e
C
o
m
p
u
t
a
t
i
o
n
a
l
E
n
g
i
n
e
e
r
i
n
g
.
2
0
1
8
;
1
6
(
6
)
:
5
0
9
-
526.
[3
7
]
L
i
D
,
F
i
s
h
J
,
Y
u
a
n
Z
F
.
T
w
o
-
S
c
a
l
e
a
n
d
T
h
r
e
e
-
S
c
a
l
e
C
o
m
p
u
t
a
t
i
o
n
a
l
C
o
n
t
i
n
u
a
M
o
d
e
l
s
o
f
C
o
m
p
o
s
i
t
e
C
u
r
v
e
d
B
e
a
m
s
.
I
n
t
e
r
n
a
t
i
o
n
a
l
J
o
u
r
n
a
l
f
o
r
M
u
l
t
i
s
c
a
l
e
C
o
m
p
u
t
a
t
i
o
n
a
l
E
n
g
i
n
e
e
r
i
n
g
.
2
0
1
8
;
1
6
(
6
)
:
5
27
-
554.
Evaluation Warning : The document was created with Spire.PDF for Python.