Pub
.
Ma
.
UAB
N° 22
No
.
1980
Ac es
VII
JMH1
SOME
EXPERIMENTE
WITH
INTERVALSHOOTINGMETHODS
FORTWO
POINT
BOUNDARY
VALÚE
PROBLEMS
IN
ORDINARY
DIFFERENTIAL
EQUATIONS
.
Ma ía
Raquel
G
.P
.
Valenga
A ea
de
Ma emá ica
Uni e sidade
do
Minho
Abs ac
The
pu pose
o his
communica ion
is o
gi e
some
discussion,
suppo ed
by
nume ical
esul s
o
in e alshoo ing echniques
o 2
p B
.V
.P
.
wi h

a
commen a y
on
he
ela i e
ad an ages
and disa an ages
o
hese
me hods
.
1
.
The
Aim
o
In e alAnalysis
The aim
o
in e al
analysis
ís
he
simul aneous
machine
compu a ion
o
gua an eed
bounds
on
he
exac solu ion
o a
p oblem
.
Quan i ies
a e
ep esen ed
by
in e als
con aining
hei
exac alue
and
echniques
a e
de eloped
such ha
h oughou
compu a ion
ound-o ,
un-
ca ionand
p opaga ion
o
ini ial
e o
a e
all accoun ed
o
.
2
.
In e al
A i hme ic
In
he
sequel
we
use
he
no a ion
x
I
o
an
in e al
wi h
lowe
and
uppe
bounds
x_
,
x
such ha
xI=

V
x

,
x
i

,

x
6
x

x_
.x

E

.
The
wid h
o an
in e al
is
deno ed
by
W(X
I
)
=
x
-
x
"
A i hme icope a ions
wi h
in e als
a e
de ined
in such
a
way
ha
he
esul ing
in e al
will always
con ain
he
exac
esul
o
he
ope a ion,
Moo e
(3)
In
a ious
con ex s
we
shall
ha e
o
compu e
in e als
which
a e
unc ions
o
in e als,Moo e
(3)
.
3
.
Shoo ing
Me hods
o
Linea
P oblems
We
conside
i s
he
Linea
p oblem
o
second
o de
gi en
by
Y''=p(x)y'+q(x)Y+ (x),x
E
[a,b]

(3
.1)
oge he
wi hbounda y
condi ions
a
ly(a)
-
a2y'

(a)
=d-
,
bl
Y(b)+b
2
y'

(b)
=/
1
3
(3
.2)
whe e
ai,
bi,
i=1,2,y'(a),
y'(a),y(b),y'(b),o¿
.q3

may deno e
in e als
.
We
assume
ha
p
(x)
,
q(x)
and
(x)
a e
con inuous
on
(a,b)
and
ha
he
homogeneous
p oblemde i ed
om
(3
.1), (3
.2)
has
only
he
i ial
solu ion
y(x)
=0
in
which
case
(3 .1), (3
.2)
has
a
unique
solu ion
.
In heo y
we
can
ind
his
by
combining
he
solu ions
o
wo
ini ial
alue p oblems
o
sa is y
he
pai
o
bounda y
condi ions
.
A
s aigh , o wa d
in e al
ex ension
o
he
classical
shoo ing
echnique
was
de eloped,Valenga
(5)
.
The
associa ed
ini ial
alue
p oblems
a e
sol ed
by
in e al
ini ial
alue me hods,Moo e
(3),
Valenga
(5),
Hansen
(2)
.
An
es ima e
o
he
o de
o
accu acy
ob ained
is
ound
Valenga
(5)
in
e ms
o
he
o de
o
he
in e al
ini ial
alue
me hods
used
.
h=0
.1
max
W'(Y
I
(x))
=
Nhk
Whe e
N
is a
cons an ,
h
he
in eg a ion
s ep
and
K
he
o de
o
he
ini iál
alue
me hóds
.
Example
3
.1
.
Y(2)
=0
.Y(3)=0
.
A
ou h
o de in e al
ini ial
alue
me hod
p oducedwi h
w(y
I
)<
2
.0x10
-5
max
y
(x
)
:z
0
.
55
The
co esponding
esul s
wi h
h=0,2
had
a
wid hwhich
we e
a
mul iple
e y
close
o 14 .6
o
hose
o h=0
.1
.
This
gi es
easonable
p oo
o
0(h
4
)
accu acy
.

'
In
ce ain
cí cuns ances,
howe e
he
compu edin e al
solu ionsmay
la gely
o e es ima e
he
e o
in
he
solu ion
.
The me hod
is
uns able
i
he
ini ial
alue
p oblems
ha e
solu ions
ha
inc ease
apidly
in
absolu e
alue while
he
solu ion
o
he
bounda y
alue
p oblem
emains
almos
s a ione y
.
Example
in
Valenga
(5)
.
I
he
ini ial
alue p oblems
ha e
oscilla o y
solu ions
a
echnique
o
educe
heso
called
"w apping
e ec ",Moo e
(3), mus
be used
.
We
no e
howe e
ha his
is
somewha
unusual
o
well
condi ioned
bounda y
alue
p oblems
.
4
.
Shoo ing
Me hods
o
NonLinea
P oblems
.
We
conside
he
nonlinea
di e en ial
equa ion
o
second
o de
y,'= (X,y,y,

(4
.1)
oge he
wi h
linea
bounda y
condi ions
(3
.2)
We
assume


con inuous
and
Lipschi zian
onAaja,<x
~
b,y?+y'<~

Y

)0,
I

Ó

C

M
on
R
;
a
¡
-
>,
O,b
i
>
0,i=1,2,
a
l
+a2
7
0, b
1
+b
2
7
0, a
l
+b
l
70
.
a
y~
An
i e a i e
scheme
which
is
essen ially
an
in e al
ex ension
o
he
1
classical
shoo ing
echnique
was
de elopd
Valenga
(5)
and
condi ions
o
con e gence
s a ed
.
Example
4
.1
max y(x )y0
.13
,x=0
.5
.
5
.
Conclusions
y''=-1-0
.49y'2
,
y(0)=0,y(1)=0
.
Wi h
s a ing
in e als
o
wid h
o
o de
10-1
and
h=
0 .1
using
a
ou h
o de
in e al
ini ial
alue
me hod
he
algo i hm
con e ges
apidly
p oducing
a e h ee
"s eps"
an
in e al
solu ion
o
max
w(y1)<3.5x10-5
In e alshoo ing
me hods
a e
na u ally
dependen
on
he
pa icula i ies
o
in e al
ini ial
alue
me hods
.
We
we e
able
o
compu e
accu a eboundinj
solu ions o
some
p oblems
whe eas
in
o he cases
e ypessímis ic
bounds
can
be
ob ained
.
One
o
he
causes
o
he
bad
esul s
p oduced
in
hese cases
occu s
in
he classicalme hod
as
well
.
P ecau ions
mus also be
aken
when
he
associa ed
ini ial
alue
p oblems
a e
such ha
he
in luence
o
he
"w apping
e ec "
is
conside able
.
We
no e
ha mo e
es ic ionsa e
imposed
wi h
in e alshoo ing
me hods
han wi h
he
co esponding
classical
me hods
bu
his
a e
all
may
be
he
p ice
o
he
knowledge
o
gua an eed
e o
bounding
solu ions
.
BIBLIOGRAPHY
:
Fox,
L,
and
Valenga
M
.R
"Some expe imen s
wi h
In e al
Me hods
o
2
p
B .V
.P
.
in O .D.E ." o
be
published
(2)
Hansen,
E
.
(ed)
"Topics
in
In e al
Analysis"Cla endon
P ess
.
Moo e,
R
.
E
.
"In e al
Analysis"
P en iceHall
Nickel,
K
.
"The
applica ion
o
In e al
Analysis
o
he
Nume ical
Solu ion
o
Di e en ial
Equa ions"
-
In e ne
Be ich
des
Ins
.
F
.
In o ma ik
69/9
.
Uni e si ae
Ka ls uhe
Valenga,
M
.
R
.
"In e al
Me hods
o
O dina yDi e en ial
Equa ions"
D
.Phil
Thesis,
Ox o d
(1978)
.
A
pa icipagáo
da
au o a
nas VII
Jo nadas
Ma emá icas
oi
subsidiada
pelo
Ins i u o
Nacional
de
In es igagáoCien i ica,Po ugal
.