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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) ,Øt@)=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

-

¡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.

(2)

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

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 = 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

(3)

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 ofB, 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 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

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

(4)

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

(5)

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

,) -

(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,,¡ =

[r,,r, rl, -

nf u, 1øl] or

I

s

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

;

so that

Similarly. there is

i.@,b)

such that

lvrz n-z

t e

)-

s{z r,-z 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¡¡,

(6)

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

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