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),
te 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 folmII
(3)
J(/)=(f -u)r,(t-b)t,..)"*rt-t , te
fa,blwhere 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 partitionaj,
are the roors of the cheby- shev polynomialit
is given an upper delimitation of rhe norm lþ-
¡ll_ , wherer
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 approximantof
the exact solurionof
theproblem
(l)-(2) a
spline function belongingto
the space S,,,,*,¿_l(Â,,) of
2p-derìvative-interpol¡ting spline funcrions, clefined
in
[9], one proves that thel 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 thattf'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 numberof
unknowns:d,4,,,..
Am+2p-t.
aO,at,...'A,t,
:This system has a unique solution
if
and onlyif
the associated homogene- ous system (obtainedfor
cf(q)=0-P({), q=0,p-1,
yk=0,
k=0¡¡
has onlythe 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
eN,
tt)
2,p>
1, m> 2, m+p < n+l.and letA,, 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, tù,
In*t = 1t,,,t,,*1),
jis 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'Definitionl,
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, /€
lRCosticã 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
thatS2,,,+¡-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 theoremwill allow us to
usea
spline function froni Sz,,+zp-r(An)
as an approximant for the solution of the problem ( l)-(2).TueoReu
2,
Suppose thatf
: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,....nBy Theorem 2, there is only one spline function s,. € S2r,+2p_¡(A,,
)
suchthat s,, e
H']'*2|'çL,,,,Y7.
,
Furthermore, we have:
THeoRstr¡
3.([9],
Th. 5 andTh. 6), a)If
geH:'+zp(L,,,y)
thenlls{:,-u lr
¡¡, <
lls(,',*2,,,11, : b)
lf
.f e H[,*2t'{L,,) thun(15) llf^+znt -
'i,,*,r,llrflllt,*znt
..,,,,,,*r,,,llr,for
arryse
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 ¡,,,-rÐ,
rl(Or rr-
. l2r, r ) (rr )-
(e t o,-
r1g, i, )t1 ry_,)] = ok=l '' i
whet'e C¡
-
rlznt+2t'tll,^k=l ,2,...,1
and "tarr-2r'+,r)(4)-
s(rr+2p+t)(b)=0
forj
= 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 - ,
.,,1t82 Costicá lr{usrä1a
,But .r(rrr+2/r+l)(f,, ¡=,t.(,',+14+l)(f,,)=0,
.l=0.rrl 2
(by Con<tition 30 from Defilritionl)sothat
f"'
[.r"*t''
ir)]2dr' =I,o,l.r''
*' r, rr r]2or ==
(
¡¡rrr-ll',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 te [a.
bl.Since
sE 4n+z.p-ton
I¡¡l)1,,r1 it follows
,(rrr+2r)1¡¡ =0 f<lr
anyte
11¡ U/,,*l. By
contirruity s1' r(nr+2pt onR it follows
,(,,,+2p)(Í)=0for
all Ie
.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 onR.
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 eL2[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 ofthe 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 andlrf'-'r - t*'-')l<,,rlll,,ll,
¡=[u-filj
.Since
la-rá''-tr¡
<*lll,,ll
anala
-
tl,::;,tl,l.llo,,llit
fonows thatfor
every te
la, bl there is 19e
{0, 1,..., n-
m+l
} suchthat
i-
r?n-l)tr l=,ilo,,11and
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) =Ofbri
= 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)
wededuce ,
,CoRot-l-RRy
4. If
the exact,solutiony of
the problem(lþ(2) is
inH(''*2pt(La,bl) and J.,,,G S2,r+/,_t(4,,)
is
the spline Junction assocÍatedto
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
exactsolution
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) andll¡,,11 = max{r¡ -t¡_r, i = t, u }.
Pruof.
\{e
have,t2ntQ¡)-s{,2rt11,¡=9, i
=0. 1,2,...,t,.
lBy Rôlle's Theorem
it
follows the existenceof
the points rfl)e (r¡,1¡*¡), i = 0, 1,2,...,n- I
such that112 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.,:,
Thenit will
exist the points r[2'e(rn,rfr'}),
r[2le (r,!),r,,),. t[t' .
f,, such thaty'(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
pointsr[',)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'lFor 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 differenceft) -
s..(/) vanishes:,Finally. we deduce the exjstence of a point
ie
(a,å)
such thatBut then, f'or all ¡
e
lu, bl, we havel rrz n-t\ çt
¡-
.sf2i,- r r 1,,¡ =lË
[r,,r, rl, -
nf u, 1øl] orI
s
<lrrrl ll,u"rl.,13l"ll* ., ,' ' ;
so that
Similarly. there is
[email protected],b)
such thatlvrz n-z
t e
)-
s{z r,-z r ¡, r¡ =[f ¡,,t r,,
(å )-
.s{2r,- 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,, llCostici 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-
tllll,,
jlr.l' . [vt"'.zlr ll, . It followsllvtø'
-
'1øt .for q = 0, 1,2.. ..,2p
-
LProo.f. Since
r(te)-.r,
(/¡) =.t'(f,,)-.r,,(r,,)=0,
it lilllows that there exists atle¿rst onc point tf,,l )
€
lt¡,
l-,)
such that ll-r,r,,r+2r,-:r-
.r{u-z r,-zrll
*
<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 estimationhold:
'll-
.,, - fi2r'-.r.¡;(n -
l)!.lla,,ll''-i
. ll.vi,,,.uzr¡¡,t88 Costicä MustäÌa t0
!l
On a Problenl of B. A Ktupilovskaja I tì9where 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'
3þ
-' a)4 t', J* @-
rl rlll,, ll''-å¡¡r''."'
¡¡,B¡, Definition I, m 22'andthen
lln
-rrll_
isat
least of orderExample,.'
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 values9f
¡he error at the nodes of the uniform partition L,,,: n = 5, 10, 20, 30, 40 are presentedReceived 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