View of On a problem of B.A. Karpilovskaja

REVUE D'ANAI,YSE NUMÉRIQUE E'I' DF] 'T'HÉORIE DE I,'APPROXII!1A'I'TON 'l-omc 28, N" 2, I999, pp. 179-189

ON A ^PROBLEM OF B. A. KARPILOVSKAJA

. ÇOSTICÄ MUSTÀTA

In [7] ^{one consider the} following problem:

(l)

y(2P,U)-gr(¡)

y(2rt-ttç,!-...-ezr(r) y(t)=.Í'(r),

t

e _la,bl

(2) ,Ø[email protected])=ykt,(b)=0,

_{q =} 0,

l,

2,...,

p- I',

^::'

, ' )'

where

pe

N,

pl L ln

the same paper one determines an approxirnate solution of the folm

II

(3)

J(/)=(f -u)r,(t-b)t,..)"*rt-t , te

_fa,bl

where the coefïicients

ck, k= 1,2,...,n

are determined from the system of equations:

(4) yQp)ç)-qt\¡)r(ztt-t)Q)-...-ezpQi)vft)= !'(t,),

ⁱ = 1,2,...,n., where t¡, i =1,

¡

are the nodes of a partition

(5)

L',, := tt <

tt 1r,

of the interv al _{[a, b],}

b

"In

^{the case} when the nodes of the partition

aj,

are the roors of the cheby- shev polynomial

it

is given an upper delimitation of rhe norm _lþ

-

¡ll_ , ^where

r

is the exact. solution of the problem ( I )-(2). ^{From this} delimitation it f-ollows that the'order of approximation of the exacr solution by the functions

Í

^{given by (3)}

i* o/b¿l \n l

In

the fbllowing, taking as ân approximant

of

the exact solurion

of

the

problem

(l)-(2) _a

spline function belonging

to

the space S,,,,*,¿_l(Â,,

) ^of

2p-derìvative-interpol¡ting spline funcrions, clefined

in

_[9], one proves that the

l 99l AMS Classilìcation: 65D07: 6.5LI0.

On a Problent of B, A. Karpilovskaja l8l wherc

it¡.

_k

_=0,h

_{arc the nodes of rhe partition L,, and a(4\,}

and

y¡,

_{k =} 0, n,

lre

given numbers.

þ(q', q=o,p-l

(10)

Then there exisls a unique spline functio,.t

se

S2nt+zp_t(L,,) such that

tf'to)=s(e), q=0,p1, sf)þ)=B(ri), q=o,pi,

{i"'er) =yr, ^k=o,n.

(q+I)l

i!

(11) Q=0,

P-l

at t'f

=0'

i =0,

ln-l

k=0

Proof.

If

s¡ is of the form (6),

fulfilling

the conditions (7), then, imposing the conditions (10), we find the system:

',rt2$ø-t (q +

llr _

/r'":Ao*it'o-a(ql'

^{Ç =0'}

P-l

i=0 m+2p-q-l

3

0

A,t+,th+>

¡Un-t¡¡Znr+2p-q- =B(ø),

k=0

having

2p+n+ I

+m equations and

'

^.: th:same number

of

unknowns:

d,4,,,..

Am+2p-t.

aO,at,...'A,t,

_:

This system has a unique solution

if

and only

if

the associated homogene- ous system (obtained

for

cf(q)

=0-P({), q=0,p-1,

^yk

=0,

^k

=0¡¡

^{has only}

the null solution..

Let's show that,

if

s€ S2u,12r-¡ verifies rtø)(a) =

rkt)(b)=0,

_{Q =0,}

p-l;

r(2\l (tk) ₌

0,

^& = 0,n then ^.ç ₌ 0 or IR .

Integrating by parts we obtain

It' l "rn

+2 n) 1r)]'4, =

Ï,-,r'

^s( t't+z t'+ i t

(t)'

^{*( m+2 p- j-t )} _{7rll l;} + Jr,,

L

j=o

*

(- | ¡rrr-r Ir" .rt2r,+21-t ⁾ (f _¡ ^. 5(21l+l ⁾ (l ₎ d¡.

J 1,,

180

(6) where (7)

"(*)

order of appr'oximation is at least

DEF'INITION L Leî ¡n,n,

p

e

N,

tt

)

2,

p>

1, m> 2, m+p < n+l.and let

A,, i=

* =t-l

^{1 o} =

t¡ <fl

^<

"'

1t,r =

b(

fr+l =

l-

be u partitktn o.f the interval la, bl.

A.function s:lR -+

R

søtisf ing the conditions 10 *ç ç2n+p-r(R) ;

20 slr^ e i'4.r,+p-t,'

I¡= _ltr-t,

t¡), k = ^{1.2,..., n;}

,o

sll,, e 4,+p..t, Ig= lt-r, ^t

ù,

^In*t = ^1t,,,

t,,*1),

^j

is called a naturcl sþline function ^{of degree 2m 4 p}

-

^{1 .}

Here ,tJ (r e N) stands for the set of polynomials of degree at most r.

Denoting

b!

Sz,r+p-t(Ar) the set of all functions verifying the conditions l0'30 frorn'Definition

l,

one sees that each s€ Sir¡+p_l(4,,) admits a represen-

tation of the

form

¹

,{r) =

"'f'

_a,ti

_+faolt

-r*¡2*''*r'-t, /€

^lR

Costicã Mustã¡a 2

i=0

)

^{n¡r1 =}

^{0, i}

^{= 0,1 ,2,}

A=0

0, ifrlt*,

Í-tk, if t>tk

and

^'(=o

(8) (t-rr)* -

(see Theorem 2 from [9]).

,nt-l

,tefa,bl.

Taking into account the representation (7) and the conditions (8), it follows that each se Si,n*,,-¡(Ar) depends

on

n+ p +

I fiee

parameteres,

so

that

S2,,,+¡-l(4,,) is a vector space of dimension n + p +

I

with respect to the usual (pointwise) of addition and multiplication by scalar of real functions.

The

following theorem

will allow us to

use

a

spline function froni Sz,,+zp-r(An

)

^{as an} approximant for ^the solution of the problem ( l)-(2).

TueoReu

2,

Suppose that

f

^{:lR. -+ IR} veriJies tlrc conditktns:

¡fu\@)- çxktt,

q =0,1,2,.,.,p

-l

(9) fkt)@)=p('/), q=0,1,2,...,p-l

.f'2Pt(tt)=

yr.

^/< = 0,1,2,....n

By Theorem 2, there is only one spline function s,. € S2r,+2p_¡(A,,

)

^such

that s,, e

H']'*2|'çL,,,,Y7.

,

Furthermore, we have:

THeoRstr¡

3.([9],

Th. 5 andTh. 6), a)

If

ge

H:'+zp(L,,,y)

then

lls{:,-u lr

¡¡, <

lls(,',*2,,,11, ^: b)

lf

^{.f e} H[,*2t'{L,,) thun

(15) _llf^+znt -

'i,,*,r,llrflllt,*znt

..,,,,,,*r,,,llr,

for

^arry

^se

Szm+2p-t(L,,) (lIere s¡ is, given bv Theòrem 2).

Proo.f. To prove ( 14) we shall use the

identity

, :,

n+2pl _r(lr+2p)

where the last term is null. Indeed, integrating by parts, we find [u, ^r\1"

*"

çr ¡18( "'*2 ^{t' t}

{ù-

^{s{:"* 2 r' )} 1r ¡

] dl =

=

(-

¹ _¡,,,-r

Ð,

^{rl(Or rr}

^-

^{. l2r, r ) (rr )}

^-

^{(e t o,}

-

^{r1g, i,} )t1 ry_,)] = ^o

k=l '' ⁱ

whet'e C¡

-

rlznt+2t'tll,^

k=l ,2,...,1

and "tarr-2r'+,r)(4)

-

s(rr+2p+t)(b)

=0

_for

j

₌ 0,

l,

..., tlt

-

^{2 (by Condition 30} from Definition I ).

It follows

-5 On a Problcln of_{B, A,} Karpilovskaja

o <

llet,,*:nr ll2

-

llrÍ:,-'r'¡¡e ^'

l'8:l'

(t4)

implying the relarion (14).

To prove ( I 5) we shall use the identity

llsr,*znr

-.f<,,.rr,ll',=llsr,rzrrt -,,i',r,,Illl+llr¡'t*tr, -.¡t*,tzt,\li

⁺

+2 [t',lst,, ^{nz t, t ( t )}

-

^{sltt +2 t',} {

ùflrt _;,

^{rr, ) ( t} ¡ - ¡ ^{(,,, + z, ¡,,} tr I

] a,

where, by integrating by parts. the last tenn is again null. To show this one uses

theequalities - ,

^.,,1

t82 _{Costicá lr{usrä1a}

,But .r(rrr+2/r+l)(f,, ¡=,t.(,',+14+l)(f,,)=0,

.l=0.rrl 2

^(by Con<tition 30 from Defilrition

l)sothat

f"'

[.r"*t''

^{ir)]2dr' =}

_I,o,l.r''

^{*' r, rr r]2or =}

=

(

¡¡rrr-l

l',rrz,,r+zn-tt(¡¡.5{2i,-r)(¡¡ ^{d¡ =}

=

(:t)"'-r

þ.r*1,,^'* ,"t2r'+r)1¡) d/ =

= f-l),,r-l

f

^{c* 1.r,tr,f t ¡)}

^-

^{sen\ {.t u- r )) = 0,}

k=l

where c^

-

,ç(2,r+1r'-l'(¡)lr, _k

.=

li

^(by condition 20 fiom Defìnition l).

Tlrerefore, .ç(r,+2¿)(/) = 0,

fbr

all t

e _[a.

bl.

Since

sE 4n+z.p-t

on

I¡¡l)

1,,r1 it follows

,(rrr+2r)1¡¡ =

0 f<lr

any

te

^11¡ _U/,,*l

. By

contirruity s1' r(nr+2pt on

R it follows

,(,,,+2p)(Í)=0

for

all I

e

.l? (see the condition l0 from Definition r ). Then s e ,!4n+tp_t on R . impries

"(2¡;)a

4r-t on lR. But s(2p)(t*)=0,

ft

=O,tt (n>rn)

implies .ç(21,)(¡)=0 for all ¡

e

lR and, consequently, se ,4p_t on

R.

4"

r(ø)¡4)=r('/)(b)

=0,

^q

=0, l, ...,p- I

we infer that

.r=0 orlR.

But then all the coefficients of ,ç are nulr, so that the homogeneous _sy.stem associated to

(l l)

^has only the null solurion.

Re¡nark

l.

By Theorem 2, tJ'y is the e.xac:Í sorutiott ctf the dffirerúiar equ u_

tir¡¡ts

(l)

with condition (2), _{the:n there} is onr¡- ort,

¡unciit,,i

,,."è szr,+zp_r(a,,) verifi,ittg _the conditions (2).

.

Let

( I

2¡

IÌ'l'+2 r' ç¡4, øl¡:=

{s ^{, k,'t:)}

:

^{lQ, *',(rrr+2 rr-tt} absolurely conrinuous onlu, b) ànd gtttt+2pt e

L2[a,il].

andlet Y=(ar0).q,(l),,.,,o(/,-t),0(0),0(r,,...,0(p-l),y0,y1,...,y,,)e pr+2r+l

6.u

fixed vector.

Denote

(l.l)

H':'*7t'(L,,'

x):={.ge ¡1'l'*?r'¡¡u.bl:

ger\1¡r)=yr,

'

_k

_-

0. l. 2,.... n: s(q) ¡.r.t. tix(qt . g(q) (h) = fJ(q) ⁽

ü

=

f)ttrt

ø = ^O, ¡, _ _f

]

I

lri" -r,,.1ìl.tllo,,ll,

i = o, 1,2,..,, ⁿ

-

^2:

Applying again Rôile's Theorem

fot

vep+t) one obtains the existence of

the points ₁₂₎

e(tftt,r,fJl),

_{i,= 0.}

_l,

_{2,. .., n} _2süch _thar

,(2Þ2) (rÍ',)-.s{.2r,*zr

(rjr,) =0,

ⁱ = 0, t, 2,...,

^ _ _Z and

l,l"-r,til1=¡ll¡,,11,

^{i = o,}

l,

2,...,

I _-

3.

A k-times applications of Rôlle's Theorem, yields the poinrs rjÅ)e

(rft-tr,r,fj¡',), ¡=(nJ, _{k= l,}

?,,..,,r._

^|

such that

,(2rt+k\

(rjt))

^_

"1zr*k) (rÍr) ₎ = 0

7 On a Problem of B.:A Karpilovskaja

lrÍ-'

-r,,Íll<(k+r¡lla,,ll, i=0, n-k

^{and k}

₌ t,2,..;,

n

-

^t.

Fork= m-lweobtain

,(n+2p-t) (rl,,tr)-.r(a,+2r,-tr (rf,r-D) = O, ¡ =

çl¡

_ m + I and

lrf'-'r ^- t*'-')l<,,rlll,,ll,

^¡

=[u-filj

^.

Since

la-rá''-tr¡

<*lll,,ll

ana

la

-

tl,::;,tl,l.llo,,ll

it

fonows that

for

every t

e

_{la, bl} there is 19

e

_{0, 1,..., n

-

^m

^+l

} ^such

that

ⁱ

-

^r?n-l)tr l=,ilo,,11

and

and

(according to (17)).

I'

lrrnt+2 r,-t) 1¡¡- r{,rr+2r,-r)(¡1 ₌

lf;,, l/,,*zt,t {u)-

^{s.{1,*2 r,r 1, ¡}

.

(

li,,

^o,

)

t

(r,,,,(!enr,,, {,¡-

^s{r,,+22) 1¡,¡)' o, )+ ⁼

<

v[, liÇi

^{.lþ,r, r z u t}

^-

^{sl:,*z a r} _ll, ^{< .Æ'} ¡a,, ¡] ll-v (,,, * 2,, _r

¡,

t84 Cosricä Mustãta

(.r(rr+2r,+lr ,^tm+)P+i)

){r)

⁼

(r,r.,

^r)+¡)

-

Jy+2p*r,){ø) _=O

fbri

₌ 0, 1,2. ..., m

-

^{2, and}

(r(,rr+zr,-l)1r¡

-

r(;n+2rttr,))1,

=c¡(s),,

k = ^1,,t (constants depending on s). In conclusion

( l6)

llsr,,+zlr

-

.¡rn+zntll2 =

llsr,,*u rr

- li,,.rr,ll'+

ll.s!,,*zrr

_ ,¡t^*zt,tll2 which imply ^(15),

Remark 2. Bv (15), _we obtain.for s = 0

(11) ,,llf,u,tr,,r-s!,-3lr¡¡, <lll(,:zrtl¡.,

Returning to the problem

(l)-(2)

we

deduce ,

^,

CoRot-l-RRy

4. If

the exact,solution

y of

the problem

(lþ(2) _is

_in

H(''*2pt(La,bl) and J.,,,G S2,r+/,_t(4,,)

is

the spline Junction assocÍated

to

v, veriJ.ving the samë boundq4, ¿snditions as y, then the following evaluaÍit¡n

(18) ,; ,

ll.vr,',*zar.s{l,rnzrrll,

<ll.o1,,li;,,¡¡r.

_:,

: ^,i

^,;;,,,

holds.

TueoRepr

5. ff v ^{is the}

^exact

^solution

^oJ'

^the

^probtem (l )-(2),

ye

.flott+2¡t)(¡a,b1¡and t¡.€ S2nr+,¡i_t(4,,)

is

the approximunt .spline funuion, then the .fol low ing ine q uol itie s :

..^., \ ' , l[rt''*lu-t1-çl+,3r';-irl[-.,

(re)

<,[^Qr-

1¡.;.(n

-/

⁺

r¡lll,,l[-å

,]l.vt,,.z,r¡¡, holds, frn' I -12,3,...,m) and

ll¡,,11 = max{r¡ -t¡_r, i = ^{t, u} }.

Pruof.

\{e

have

,t2ntQ¡)-s{,2rt11,¡=9, i

=0. 1,2,...,t,.

^l

By Rôlle's Theorem

it

follows the existence

of

the points rfl)e (r¡,1¡*¡), i ₌ 0, 1,2,...,n

- ^I

^{such that}

112 Èt) (rj' ⁾

)

-

^r1.rr.,, (,Í,, ₎ = O, i ₌ 0, ^I

,2,...,

^rt

-

^|

Furthermolr. we have

6

On a Prohlenl of B. A. Karpilovskaja IriT

4)

-'{(r{,r')=o,

implying

..,,'(r¡ ) -.ti, {re ) = .v'(rf,r ⁾

-tí,

(,å") =)'(f,,) -s{ ^(r,,,)

t,0.,:,

Then

it will

exist the points r[2'e

(rn,rfr'}),

r[2le (r,!),r,,),

. t[t' .

f,, such that

y'(t0)

_{-si(16 ) =}

r"(r,!t')-'í (rá'')=.v'(rÍ') )-'í ^(rÍt')=

,,'(r

,) -

^{sí (r,,} ) = o,

In ^general, for every q'e,12,..:,

^p

- 1) there are the

^points

r[',)e

(ro,rår-',), 4r,.(tf-t,,rÍu-',),

..,tt't_)te(,,;:rt,.r,,),,

Q,

<t[f, <l'tt...

9

to.tÍî' .

þ:!)- '!'

^(tl;';,)

^:

^{-'',lf o'(1,,¡}

^-

^{siÍr)(r,, ¡ = 0.}

"'=y(

^q'l

For r7 = ¡r

-

^{I we} deduce the existence of p

+

I distinct points to

. ^tl'-t'

^{< tfn-tt}

. "'.

^l,Y-1')

.

^t,,

at which the

þ -

^{1) -} derivative of the difference

ft) ^-

^{s..(/) vanishes:,}

Finally. we deduce the exjstence of a point

ie

^(a,

^å)

^{such that}

But then, f'or all ¡

e

_{lu, bl,} we have

l ^{rrz n-t\ çt}

¡-

^{.sf2i,- r r 1,,¡ =}

lË

[r,,r, rl, ^-

^{nf u, 1øl] or}

I

s

<lrrrl ll,u"rl.,13l"ll* ., ,' '

^;

so that

Similarly. there is

[email protected],b)

^{such that}

lvrz ^n-z

t e

)-

^{s{z r,-z r ¡, r¡ =}

[f ^{¡,,t r,,}

^{(å )}

^-

^.s{2

r,- ^r r

¡ø I ] ^oa

|

<

v(r/,-

) _{,.Ítr- ,_ l(r,} -ÐJ-*qr-l)!

^A,,

"'-)!. ,t,*ztù

_,

3 (h

-

^a) _llvrzr'-t ^I -.slzr'-ll _{l¡_}

I86

so that

Here fiom we deduce

llrl,','2¡,-r)

-.r{l+ìr-',ll_ .J, _lln,[å

1v,,,-:r,,1¡,

Similarly, for every t

e

_[a,D] there exists _d,

e

_{0, I,,.., n

-

^{m +}

2l

such that lr

-

rll"-',1 ^< (m -rrlln,, _ll

Costici Mustãta 8

I.urr,+2r,-2r1r)-,ç{,rr'3p-z¡(rl=ll,;',,,[,r,{,,,'2ri-r)1ø)*s{,,,*2r,-rr1ut]l

= lt

-

t1"'-''

¡

.ll-r,rrrr+:r'-rr -.rr,r+zrr*irll_ <

< J

^ Ør-

^t

llll,,

^{jlr.l' .} [vt"'.zlr ll, ^. It follows

llvtø'

-

'1øt .for q = 0, 1,2.. ..,2p

-

^L

Proo.f. Since

r(te)-.r,

^(/¡) =.t'(f,,)-.r,,(r,,)

=0,

^{it lilllows that there exists at}

le¿rst onc point tf,,l )

€

lt¡,

^l-,

)

such that ll-r,r,,r+2r,-:r

-

.r{u-z r,-zr

ll

*

^<

^{J *}

^@,- l)llA" _ll ^{r*1 .}

[.r,r,,,*

z r,,

¡¡,

ln general we finrl

llvt,r,+zr,-rr -.r{1,,.:r'-rrll_

<J*ç,,,-

l)...(ni

-t

⁺ ttlln,,lll-1 .¡|n,,,-rr,¡¡, tbr all / = 2,3, 4,..., m.

Remark3. For

l=

^myve.frncl

(20)

ll.r,rzr',-r1:r,ll_

<J*er*t)!lln,,ll'l-li.llrr,-:ut¡¡r.

In the following,we shall give estimations fbr the norms

(2t)

ll.n'u'

-rlf 'll-, c¡=0,1,2,...,2p - t,,.

^:

necessary for the numerical treattnent of the problem ( I )-(2).

CoRolr-nRy 6. If' the exuct ,solutirsn !- of the problem

(t)-(2)

lselongs to H'l'*2t'([a, bD and s.,,. G S2,,,*2/,-¡ (Â,,

)

^{ls the} ctssot'iatetl spline solurion, then tlte .fttllot+'ing estimation

hold:

^'

ll-

.,, - fi2r'-.r.¡;(n -

^l)!.lla,,

ll''-i

^. ll.vi,,,.uzr¡¡,

t88 Costicä MustäÌa t0

!l

_{On a} Problenl of B. A Ktupilovskaja _{I tì9}

where from

ll.rtzrr,

-

^s(2/-2) _{ll_} ^{3 (b}

-

^{6¡,2 J}

* Ø,-

^{I ) tll^,,} ll "'-å ^.

fl.r,.r,,-,:r

r

¡¡ ^. Continuing in'this nìànn..,

*e

obtain'

, ,,

;llvrzr,-ri -.sr.zrhrrll' ,g

(b'-o¡t J*Ø- rlrlll,,lll'-å

.||vt,-zlr¡¡.,

REI.'ERENCES

t'or I =0,1,2,

,2p-l

llJJ-P'Aubin, A.Cellina. Dffirential Int.lusion;;. Set-Valued Mctps tuttl viubitit¡. Theon., Springer-Verlag. I 984.

l2l o' Aramã. D. Ripianu, on the polylocal probleln tbr difterential equations with constant co- elïicients (I), (ll) (romanian),Studii çi t'ercetãri ,rriitrtiÍit.c - ecad. R.p.R., Filiala Cluj Vlll

( I 9s7).

[3] o-Aramã, D. Ripianu, euclques recherche actuelles concernant l,équation de ch. de la Vallée-Poussin rélative au problem polylocal dans la rhéorie <trl; équations ditÈre'tielles, Mathentaticu (Cluj¡ ^g (3f ) / (1966). l9_28.

l4l ^M' Biernacki. Sur un problénrc d'interpolation relatif aux équations différentielle linéaires.

Atttt. de Socié10 po,o,rrri,t¿, de Mqrhetntuiclue 20 (1947).

[5f ^P. _" Blaga' G. Micula, Polynornial natural spline functions of even degree. .!r'rlir

¡

^IJnit,.

Babes-Br¡lvai", Mathenwrica XXXVlll, 2 ⁽ I993), 3l_40.

[6] ^Ch' de la Vallée Poussìn' Sur l'équation difterentielle du second ordre. Déterninarion d,une integrale par deux valeurs assignées. Extension aux équations d'order n, Journ. Math. ^pures

er Appl. (9) 8 (1929).

t7l ^B' E' Karpilovskaja. The convergence of a nrethod of interpolation tbr clifþrential equations (russian). U.M.N,r. VIII. J ( l9-53) I I l_t l8

f 8f C' Micula. P. Blaga' M. Micula, on et'en degree polvnontiul .rpline.litnuiotts rtÍt¡ ap¡tlica- líons to numerical solution of differential equation.r u,irh reiar(te¿ argunrcqt, Technische Hochschule Darmstadr, preprint No, r77l 099-5), Fachbereich Mathemãrik.

[9] R. Musta¡ä, On P'derivative-interpólating splinc t'unctions. Rettue d'Attt¿:. Nuttt. et de Th. cte

l'Apprlx. XXVI I -2 ^{( t997). t49_163.}

f l0ì C. Mustäta, A. Muresan, R. Mustalä, The apploximation by spline t'unctions ol'the .solution of

a singularpenurbed bilocal problem, Ret'ne d'An¿tl, Nut¡t. ¿tt'tle't'h. tle I'Ap¡trtu.27 (lgglj) _?.

297--308

I I

lf

I. Pävãloiu, Lttro(luüion in tlrc theon' oJ ttppntinrution of'the equ(ttions ;;olutir¡ns, Ed. Dacia, Cluj-Napoca.

f l2ì S. A. Pruess, Solving Linear Boundary Value Problems by Apploximating the CoetTicie¡ts.

M ut h. o.l' C om¡t utat i o n 27 ( I 23) ( I 973), 55 I --56 I ^.

If 3l D. Ripianu, Intervalles d'interpolation pour des équarions différentielles linéaires, Ma¡¡rc.

tnatica (Clufl ₁₄ (37) _{2 ( I} 972), 363-368.

ll4l

^{D' Ripianu.} Sut' certaincs classes d'équations rlifférentielles interpolatoire dans un intcrv¿rlle donnée, Ret'ue d'Anul. Nunt. et de Th. de I'Approx..3 (1974) 2,215_22?.

Therefore

lþ(ø) - ^r{r) _{ll_} ³

(b,

r1t? n-,t

.,[*f*-

_{I )} tlll,,

ll,,-]

^. _[vr,*:r,i ¡¡r,

for q=0;1,2,...,2P-l

which ends the proof.

Remark 4, For _Q = ^{0 one obtains}

ll.,,r.,

ll'

³

þ

^{-' a)4 t', J}

* _@-

^rl rlll,, ll''-å

¡¡r''."'

^¡¡,

B¡, Definition I, m 22'andthen

lln

-rrll_

^is

^at

^{least of order}

Example,.'

Considpr _!þe prgh!eq

(Pl) ^{y14)(r)= (ta} +14tt +49t2

+3zt

_-12)et

, rc

[0, U

.v(0)=.y'(0)=v(l)=¡'(1.),=0 ,

^:,j;

Problem

(Pl)

is a probiem of Karpilovs[aja type for a fourth order differ- ential equation which is studied also in

[21.

In Table

I

the maxirpum values

9f

^¡he error at the nodes of the uniform partition L,,,: n = ^{5, 10, 20, 30, 40 are presented}

Received Mars 03, I998 ^"'1. Popoviriu" Institut of Nunrcric,al Analvri.r P.O. Bo.u 68

3400 Cluj-Napotct I

Ronaniu Table I

maxirnunl values of the error at the nodes of A 5

t0

',

²⁰

-10

40

r,l

0:tn00380786 0.000002307?

Q.0m000il62 0.0000000 I 98 û.0000û00055