View of On p-derivative-interpolating spline functions

148 Costicã Mustãþ t2 REVUE D'ANALYSE NUMÉRIQUE ET DE THÉORID DE L'APPROXIM.TIION Tome XXVI, Nß 1-2, 1997,

pp.

149_163

I{EFERENCES

l.

^p.

Blaga

^{I splíne of} even degree, stutlia univ. "Baboç- Bolyai

2. ^p.

Blaga,

spline techniquefor numerical solution ^oJ

cletayBorlin,ProprintNo.A-15(1996)'Serio

A-Mathernatik'

3. R.

t.

^{Bruden and} T. Douglas Fa;xæ, Numøical Analysis, Third Edition, PWS-KENT Publiúing

4 G i'{:{:"::i"

Received May 15,

1996

"Tíbefiu Popoviciu" Institute of Numerical Analysß P.O. Box 68

3400 Clui-NaPoca,

I

Romania

ON p-DERTVATIVE-INTERPOLATING SPLINE FI-INCTIONS

RADU MUSTÄTA

Following the ideas from [2] and [3], we defîne thep-derivative-interpolating spline functions which can be used to approximate the solution of a differendal equation of orderp

(p eN,p

^>

l)

with modified argument,

Forp:

1 one obtains the spline functions considered in _[2] and [3],

Let

Ân

:-æ = t_l 1

^a

=

^{t0 <}

\ ^<...1

^tn

⁼

^b

^<

^tn*1

⁼

^+æ

be a partition of an interval

la,b] c

^R.

DeFrNIrIohl L

For n) l,p ) r,m)-2,m 2 p

givennaturalnumbers such

lhat m+ p < n+Z,afunction

s:.R

-+

R satisfyíngthecondítions 1)

s

e

,z'n*t-2(R),

2)

slre

I Q2,n*p-t,Io =

lto-r,tk),k = 1,2,.,.,n,

^and

3)

^s

lr,, ^,1r,.,

. en+p-t,Io = (t-t,ts),

Ir¡1

=

_ltn,tn*t)

is called a spline function of ^{degree 2m} +

p - ^|.

^{Here p,} denotes the set of

all

polynomials of degree at most r.

The set of all spline functions of degree 2m + p

-

¹ is denoted

by

^,S2,,*r_,

(^,

₎ .

-

^-

Remark 1.

Forp:

I one obtains the set

^t

,(^,)ofnaturalpolynomial

^spline functions of even degree 2n considered in [2] ^and t3l.

The following representation theorem

will

imply that the set ^,S2,,+p_r (4"

)

^is

an (n +

p +

l)-dimensional subspace

of C',*o-'(R)

^.

Tgeonpu

2. Every element

s

€ Sz,r*p_l(A,,,) admits the representation

m+p-l

ⁿ

(r)

^s(r)

⁼ I ^{4t' *ZakQ - ,ù?Ï*o-',}

i=0 k=0

AMS Subject Classification: 65D07, 65L05

Dorivativs- Spline Functions ¹⁵¹

I J

150

where

a)

(s)

Radu Mustãþ

Proof. Taking into account condition 3) from Definition

l, it

follows that

s('*e)(/):0 ^for

^all

t e1,*,,giving

nil

0

= i

^{(zm +}

^p -r)'.'(^ *

^p

- ^t)a¡(t -'k)î't ⁼ *}ooft - ^t¡)m-|

⁼

k=0

^k=0

n (nt-r \ ¿'-t ( n

,jl.r,_l_r

= M; "olïc!,-{m-i-\il= u2?t)i ci-rlZoo,í

),"-'-',

oã t;- ) i=o

^\¿=o

where

M::(2m+ P-l)".(m+ P-l).

, The above equalities imply that

å ^{aotl =}

^0,

^for i ⁼

^0,1,

"',ffi - l'

^tr

Tneonnv

^3,

Let (l eN,03q< p-l

^and

let f ^:l?+

'R

beafunclion

v er ifyin g th e c on diti ons

í

f@0,\ = vlf), I =0,1,',., P -1,

(3) ll')i,-i ^{= yf), ft =}

^o,

^r,...,n.

Then there ^{exists a uníque} splinefunction

s,

€ Sz',*p-r(À

,)

^{such that}

["!r)1ro¡

= yf), I=0,1,..,,p-1,

^..

(4) lú't,rl =vf),k=,,r,"',n'

Proof.

Since the spline function

s,

admits representation (1),

it

follows

,f)(,) ='.1^^ +

^{Ao, rtk}

^. ä##.ao(t ^-

^{t o)ù''*}

^rt-t

for

q:0, ' l,

.,., P.

Imposing õonditions ^{(4) to} this function, ^{one obtains the system}

Zrorto =0, i ⁼

^0,1,

"',ffi-\'

yf),q=o,Pl

(2m+p-r or(t¡-rr)'i" = y(i'), i ^=o,n

2n - ^t)l

withþ ⁺ n+l ^+m)

^equations

^{andm+ p+ n+ l}

^unknowns:

lo, "',A,,*p-r,

Agr,,,r(In.

"

Thó systern (5) has aunique solution if and only if.the associated homogeneous system

(rf) =0,q=0,1,.,.,n-rrrllù -0,i=0,t,...,n)

hasonlythebivialsolution'

Dànoting by s a function

in

Sz,*o=r (À,

) ^verifiiig

the homogeneous condi- tions (4) 1i.e,, ^s(u)(lo)

:

0,Q

=

^0,

p - l;r(')(t,,) = 0,k = W),and

integratingby parts, we get

Jt ["t,,.o1

qrr]'a,

= !- ^(-Ð,

^{r,'+p+r) (r)} "{

"*r-r-r) _{1r ¡1"'} +

+(-rf'-t

J', r{z'*r-t) _{1¡ ¡r(r..1)}

(/)dt,

Taking into ^account Definition ^1, we further obtain

J'[,('*o1,,¡f'a,--(-r)*-'Ðrl;',,t'**n-r)1r¡,(z*r)1r;or=

=

(-r)^-t

I'l-, r(r*1)1r¡or =

(-,)'-tÉ"o[r(o)1,0¡- r(o)1,0-,¡]=

o, wlrere

cr - ,(2'*t-')(r)lr-, k = l,n

lt

^L

Itfollows

that ,('n*o)(r)

=

^O

for all t

^{elto,t,,f ,}

Since

s

e Q,rrp-t on the intervals 1o and /61, ^we find ,1tu1"ú"+l)(l)

:

0, for all

t e

IoU 1,,*,, too, ^{so that} by the continuity we ^get

"(rz+r)(l) = 0, for

all r

e 1ì' The equality s('ÉP)(Ð

:

0, for

all I

^{e R,} implies that t(P/

e4n-7

^{on R. Since}

,(r')(lo)= ⁰ for k=0,1,...,t1,

^because

m3n+l

and

l< p<n-nt+7,it f"il;#

"(r) : ₀

<¡n R. Now, using ^the condition, u(n)(lo)

-

^0,t7

=

^{0, p}

J,

^we

obtains=0on,Rand,consequently,allthecoefficientslo,'.',Am*p-l,av"',an in (l)

^are

^null.

Taking into account the linear independence

of

the flinctions

{t, ,, . .

.,t''*o-' ,(, -

^{,o)1''* o-t}

^{,... ,(t} * ,)1^.l-tl],

it follows that the system (5) has

aunique solution, o

An immediate consequence of Theorem 3 is the following

COROU,ARy 4. There exisls a uníque subset

of

^{n +}

^{p +} I

splinefunctions {ss,s1,.'., ^{Jo-rr 50,} Sr,...,

S,} c

Szr,*p-r(Â,)

s a tisfying ^{the con clitiott s}

n

k--0

n

Zt¡|=o,i=o,m-l

r52

(6)

Rarhr Mustãfa

)=ô¡qi 8,i ⁼ o,p-l )=o; j=o,P-l,k=1,n,

4 5

is null. Indeed,

Therefore,

Then the inequalitY (17)

Splino Functions ¹⁵³

o <

lls{'.ot -

",,,.,,111

= lolrr,.^(r)- s(,.r)qr¡]'ar

₌

: f [r1'.,,1r¡]'.rr - f [J" 'rir¡]'o -

^zlo

r("*o)p)[s(".'){r¡ -

rt'-'o)1r¡]ar

and,respectively,

r,¡

(7)

^Js[o'(to)

^=o' Q=o'P-r'k=r'n

fs[o)i,,¡ ⁼ o"; k,i = o,n

^.

IÊt -f .

C(n)(R) and

^S :

"tr)1r)

-+,sr,,*o-,(Â,,)be ^{the spline operator defined}

by,s(fl :

_sr,

Obvi-ously, the f,rnctions ^{J0, ,tl, .} .,, s

p-b

^S0, S1,.,., S,, defured by (6) and

(7) form a basis

of

the space

,Sr,*r-,(^,),

Therefore,

tlt^

^the representation .p-ln

(B) _"r(r) ⁼ |'oUY@(¡o)

⁺

Z ^soQ)¡@Q)

^'

4=0

^k=0

In order to study the properties of the space

^Sr,*r-r(Â"), we consider ^the notations

W{.p(L,),= {g:la,bl-+ R,r@*r-t) is

abs, cont. on

Io, k = þ

anrJ gbn*ù e

\la,bl|

Wi'.p(lo,r]):= {s :la,bf

^-+

R,g1n*rt) is

abs. cont. on[ø,å]

and

gtu"ù

e

\1a,fl\

wi;o(L,,)'= {, ew{-p(d,,),g@Qo) = f@þo),k = ou}

(r2) wîî,'(A,)'= {r ewi'}p(a,,),g(o)(ro) = ¡k) (ro),q=o,p-l

THEoREM 5.

If s

e ^Sr,,*o_,(A

^)ìWi:,'(L,),

^{th.n the inequalíty}

(13) llJ'.''ll,

⁼

llr''.''ll,

holdsþr

every

g eWii,P(L,,).

Proof, Observe that the last term of ^the relations

'Í)(',

'l')Qo

I

I

i

, = f r('.r)1r¡lr{".n)(r)-

,('.n)1r¡]or ₌

= Ï(-r)'r('.r.i)1r¡[r(n+r-;)1r¡ -

r(m+r-r)(tl]l;=:

.

j=o

*(-l)''-'f ,(z'.r1)(r)lr{r.r)1r) -

"r,'t)1r)] ^or

6o¡ "(2,r+r,-r)þ)

= rt,"rr)(ó) = 0, j ₌ 0,*-Z,and

s(2'*p-1)(r)

= c¡, k =li,

(e)

(11)

r = (-r)*-tÐ.t

,,Q^+r-r)7tt[sk.t){r)- ,(r*t)1r¡]ar

= k=l

--

(-r)*-'Ðr"olrþ)Qù-

^,(o)(,0)

- (s(o){,*-,)- ,þ)00-'))] =

^o

It follows

o

= llJ'.''ll, - lltt'".''llr,

which is equivalent ^to (13)' o Remark2. From ^the proof of Theorem ⁵ it is clear that

(r4) llt'''.''ll, ⁼ il'f

.'il,

₊

ll/{'.or - _'y'.''ll,

for /

^e Wì'*o(L,),imPlYing

(r5) il'f.'ll, =ll/'"."11,

(16) llrr".ø -'f.''ll, ₌ llr''.''ll,'

THEoREM6.

Izf I eW;'*o(L,)

^and

s¡

€ Sz,,*p-r(A,,) btgtvubyTheoretn3'

llrt".,r - "f.,,11, .

llrt".a - r'".''ll,

holds

for all s

eSz,,*p-r(Âu)

t54 ^{Radu Mustãþ} Proof. ^{The last} term of the identity

is null. Incleecl,

obt¿ins

Therefore,

implying(17). o

I. APPLICATIONS

In the following we ^{shall use the} spline functions in the solutions of Cauchy problems for differential ^equations

oforder p(p > t).

Consider ^{the Cauchy} problem

6 7 Dorivative-interpolating Spline Functions 155

llrr,.or - ^{fr.^ll',=} _f frt".,r(r)-

s!,.o)1r)]'ar +

*f [rt".,, - ^{,t,-o)]'o t +}

^zlb fr{,.o)1r¡

- ,!'.,,(r)][r!".,)1r¡ - tt'.o)ir¡]or

and suppose that the conditions enstuing the existence and uniqueness of the solution y of this problern are fulfilled ^(see

[

^]),

From Theorem 3 ^one obtains

TnBoRBvt 7 .

If

y is the exact solutíon of the problem (P), then there exists a uníque splínefunction sy

€

Sz,n* p-r(A,) vertfying ^lhe condilions

a

lrÍ,"0r) ⁼ yb)(ò= nq,

^q

_=o,p-t

(18) {;,

^/-\

\ /

['f,(r-) =yþ)Q), ^k=0,n.

Consider the notations

,, = Ï[s{".n)1r¡- 'Í.,,0)][rP.,)(r) - ¡{'.òçt¡þt

⁼

= Ï(-r)'["('r+r+r)(r) - sf*t*i\çt)]["Í.'-41,

_¡

-

yØ+n-i)(rl]ll=:

.

j=o

*(-

_Ð''-'

l"' [s(2".n-r) ^1r¡

- ^rf'n

^o-ù

çtl]l'f ^."

^(,)

- rtør)

ir ¡]or

and,

since

(s('*r*.r)

- rf'.e.r))1ø) = (s('*r*i) - ,@*r*i'Xu) =

^0,

i ⁼

^0,m

a

and

,(2,,+n-t)(r)- ,f'*r-r)çt) =.¿(r)

(consrants)forall

t

e

I¡,k _{= t,n,one}

tv\) ⁼

^vo;

^{v(q(t)) =} lr,

(1e) \__.,

_{

[#,t+l ⁼ f(tr,twr*),

Using Corollary 4, we obtain

Turonnu

^8.

If

_{ss,rr, ,. ., so-r, S0, Sl, . .

.,

^Sr} _, the basis

of

S2,,*p-r(^,),

,r

gíven by Corollary 4, then the spline function ^s

y

^{e S 2,n* p_1(L}

^) given by Theorem

7 admits the representation

(20) ^,r(r) = f

^,oQ)*o

+f so@¡(,r,ro,ro).

q=0

k=0

We call the spline function s, given by (20) ^the approximøte spline. solutíon of the problem (P).

Tsnon¡u

^9.

If y

e ÍTi*Pla, bf ^{is the exact solution of the problem} (P) and s, rs its a¡tproximate splíne solution, then

(zt)

llr(''.o-'l - ,t".'',ll- < .[^(r,-

^l).,,

_{(* -} ,+ t)ll^,||'iil ,'".0'll, t ₌

2,3,.,.,n;ll¡,,11

= ^mu*{r, -t¡-t,i = þ}.

Proof.

Since _-v(')(1,¡

-

r{,")(r,)

=

^0,

i ₌ O¡,

Ay an application

of

Rôlle's theoremwe obtain the existence of points

t!t) u(t,,t,,r),i = 0,n- l,

^{such that}

,tr.t)(r{'))_ rþ.')(rf')) = 0, f

₌ ^{õ, r¡}

* _l.

Applying agailr ltôlle's thec:em, it follou,s that there exists

/"'-r)

e

(t!*-'l

,

r!:';'))

_,

i=0,n-m+ l,suehthat

k=0,n

k =

^0,n.

4 ⁼ (-1)"-'i+(')(["i'Qo)- t@)u)f- ['f'{'--') - ^{l@Qo-,)]) =}

^o

k=0

ll,{, "r) - l^.

^rll,'o=

_ll,(,.,)

^{1, )}

^-,?'

^{* u) (,,11;}

^- _il,f

^.

^ù

^Q)

^{- /,'. ò}

^Øll'o,

S2r,+ p-tto aPProximate with inodified argument

(r

/(')(r) =

l'(,,y(¿),

y(q(r))), t

^{ela, b)}

yk\@)=fttn,Q=qp-1

<p:la, bf

-+

_{la, bl}

)- r!',oon')(,t' 'l; ^{= o, r}

⁼

^br, -;;i

9 Spline Functions

We obtain, in ^general,

r:21

^{3, ..,, w, E}

Remark 3. For

r:

^{m orre obtains the evaluation}

Similarly,

llr'-'',ll-

^{< @}

-o)'llv"-'';,

_ll-

lþ(,,.o-,1

- s!,-r'rll_

=

J^(^- ^r)...(, -r +r)lll,,ll'

ållr,'.,,11,

157

156 Radu Mrrstãla ⁸

,l*+ø)(t@-r)

þ,qi

-'Ío)l ₌

¿il¿"¡¡,

¡ = 0,,, -r.

It

follows that

for

every

/ elø,bf

there exists ^Ío such ttrat

þ- ^{1f'-t'l <}

orlln"ll It is obviot¡s that

and, consequentlY,

The last inequality from ^above follows

fron

(16).

Therefore,

(n) ll,t'r-,y,ll-

⁼

J^(*-r¡r¡¡a,¡¡'-llþ''."'ll,

CoRolreRY

^{I0. The inequalilY}

e3) llr-,,11. <çr-o)o Jm(nr-r)rllr,ll''-i¡¡rr'.'r¡¡,

holdsfor every

y eWl'*ela,b].

Proof. ^{Vy'e have}

lr(,) -.,;(,)l ⁼

lJ,

t, f, ^-

s', (a))oøl

= ^þ

-',llþ'-'',ll-

^<

þ -o)llv'-'',ll- lr(,.r-,)(r) -

^{r(,no r-1)(tl}

.

lJ*, 1r{".,)(u) - rt'.,,(r)þrl

l l

t,

.l ri

1l

. lü,,*lt lft, çra'.,)(u)- s!".')1,¡)'u,l'

=

= ¡r,,"4ft -,,(tþ'*o)çu¡ - sf;'.')çu,)'o,lj

=

= .Fli{l[ ^(y('.')

^(il)

^- s!'.') iø)' o,l:

=Jãll^,,

^il;

il/'".n'

^ll,

llr{".'-'r -,lr'.'-"ll* ₌ J'¡a,¡¡ållr''."'11,

^.

Sinrilarly,forevery

t elo,ó]wecanfurdaninclexiostrcfrtfratlr-{j'-'ll<ir-f¡+,ll

so that

lr(^.

u,)

(t) _-r('.rz)

_i, ¡l

.

lJ;,, (/(,.

^{n -r)} _@)

-r!".,'r ir¡þul .

<

llr{".o-'l -,y.'-',11.þ - _4'-"1

₌

J*(*- r¡¡a"¡¡"*þ''.''ll,

for

all t ela,ó],

^implYing

llt{*.r-zl -

^þt+P-z)

_{ll- =} tt, ^- r¡lla,ll1.ållr,''.,,11,

and, finally,

ll, -',ll- ^<

^(å

^- ò'llr"'- "fr'll-

Now (23) follows

homQL).

o

CoRorleRY ll.

^{The relatíon}

(24) ÉlËjr'*'-'Íi'll. ^{= 0,k =} P,P*1,,p*m-2

hold.sfor every

y eWi'*ela,bf

^.

Proof.It

follows immediately from (21).

Now we shall ^show how the above results can be applied

to

obtain ^the approximate spline solution (20) of ^the problem (P).

Denote

Splino Functions 159

Radu Mustãþ

l0

¹¹

158

Qs)

p-l ⁿ

14,,:=

ry(/¡):= >ro(t,)y(o)(¿o)* Iso0,),r(t¡,!r,yx), ⁱ

^{= 0,n}

9=0 ^t=0

p-l ⁿ

w, := ^s,,(e(r,))'=

I

r,(.p(/,))/(o)(ro) +

Iso(e(r'))/(ro' !*,tr)'

q=0 ^l=0

(26)

p-l ⁿ

*i ₌ 2,

^{n(t)ma +}

>

^sfr

^(rr)/(t¡,

^{w r,,w}

^¡)

^{+ E t}

Ço

^&=o

p-l

ⁿ

ø,

= [so

^{(<dr, ))zro}

ⁿ I

^so

^(,(

^{t ¡))}

^f

^{(t r, w ¡, w}

^¡)

^{+ E}

^¡,

Ço

^,k=o

i=0rn

where and let

e,:= e(t,) --

y(t)- ^{rr(t,), i}

^-- 0,1,".,n

",t= ^u(,o}))

⁼

y(q(t,))-

'r(e(r,)), i

=

0,r,.'.,n

denote the deviation of ^the approximate spline solutions, from ^{the exact} solutiony of ^the problem (P), on ^the knots t, and <p(t), i

:

_0,

_{I, "''}

n'

Wehave

!¡=W¡*e¡

n

õf(tt ,\*,\*)

ôyr

ek

+ >sr(r;)

õf( 4,Er,wr) _{ê¡; i =}

_0,n

E¡

= Zsr(r¿)

Øn

k=0 k=0

E,= f;^ro(q(,,)) W+ù"¿ ⁺ is¿(e(r¿)) ^{e¡' i = 0¡}

/ç= 0

Supposingthatthederivativesoff(t,u,v)withrespecttou,vafeboundedonD,

there exist

Mu

Nr> 0 ^such that Yr ^wk

*e*

for

fr:

^{0, 1, ..., n} and the system (25) can be written in ^the following form

of(tr,Er,qr)

3 M,,l*+\3 ^{N,, k =}

^o,n

Øt

P-l /\ ⁿ

,r¡ = ïsq(1,Þ(q)(/0) * I s¡(t¡)f(+,'tvk

⁺

ek,îk +

ek),

i

= ^{0,1, n} _and, taking into account Remark 3, we deduce that

4 ⁼ o(lb,lr-å),r, = o[iln,il' ;),

consequently E,

-+ 0,

E,

-+

^o,

for

llÀ"ll

+

^o'

Now, neglecting the quantities

E¡, Ei, i ₌

^O,n,we obtain the following system of 2n + 2 equations

wi=

8=o

^k=o

n-l ⁿ

'f

"o1o{r,))r(ø)1r¡)

+ | s¡(r(r,))f

^(tt

,w¡ * e¡,w¡

^{+ ao)'}

q=o

^k=0

i

= 0,1,...,tt

with

w,

and w,, i

:

0, 1, ..., txas unknowns'

If

the derivatives of the function

f

^(t,u,v),

f

^:D

^c

^{R3 -+}

^{n;(o c}

^{[4, ó] x 'R2)}

with respectto

u,

vare continuous, ^then

f

^{(to,w* +}

^e¡,fr¡

^+Ak)=

f(tr,wr,wo)+ ^k+

af(tr,Er,nr)

_p-l

u

4=O

t

p-1 q=o

,l

I

ir

õ-y* k)

,o(t

i)na * I ^so(,,)/(t

^{¡, wr<,}

^{*r), i = o,'i}

k=o

,t_

sn(e(r,))nro +

| so((r,))f(to,w¡,frt), i ⁼

^0,n

Å=0

\4tí

where (27)

Wi

min(wo, ^wo

+

^eo)

_{< €*} <

max(w¡, wr

+

^eo)

min(w*,

wr

+

e) I r1 I

max(wr,wo

+

^eo) _{rvith 2n l- 2 unknowns: w0, wys} ",tYl'urfrùrî-vr, ^{' '} ', ^ì-r;

Consider ^the notations One obtains the ^sYstem

Derivativo-intorpolating Spliue Functions ¹⁶¹

12 13

160 Radu Mustãþ

"o(ro) ^rr-r(ro)

^@,1

= rnl,,(e(,,))l *,'rils-(e(,,))1,, = t"Ì

since

llis

^a continuous application of ^the compact convex

o

into itself, it follows that ^equatio n(29) ^{has at least one solution} W* e {ù'

Let IFJ ^{be the} following diagonal matrix 0

[']

⁼

"r-1(q(r6))

'o-t(e(r,))

o-')'

ro (,p(,0 )) 0

ôf

(tr,wo,frs)

õwo

0

f(t*wu,wn)

0

[r]

^{= õw,,}

^af(t,

,WOrW0)

so(t,₎

s'(rr)

awo

[s]

⁼

so(a(to)) ^s,(o0o))

⁰ o¡(tn,wn,wn)

0 _Aw

n

Then

,i

'W = (tu¡,w¡t...

tv)r,fro,frrr'..,frr)'

f ^{(w) =}

^{(.f (t o,r's, il6 ), .' ., f'(t}

r,' r,fr

^n),

^f

^(t

^o,'0,'o

^)'

"'' ^f

^{(t n' * n'}

^{fr n))''}

using

these notations, the system

(27)

^{can be written}

in

the

following

^form

# ^=tsllrl

rseoRBrr¡

" ^ull#ll=

is unique ^and can befòund'bY

g

<

I,thenthesolution

^{W* e} d) ofequationQ9) the iterative _Process

(28) We obtain Qe)

sr =lsfm+[s]¡(r)

ql ₌ u(W),

W&).=

U(Wt*r11, k = t,2,3,...,

- where W@)

. (t

^ís arbitrariþ ^chosen. Furthermore, thefollowing evalualion holds where H:Çù ^.->

a,H(W)=

_[s]ru

+lsl\@)

and the domain ç)

c

À2"*2will ^be

ll*. -",',ll . ftrrr"' ^- "(''ll

specifiedbelow:

Let

u ₌

mao{lmoltQ'=g,',"',r

-r}

andlri> ⁰ suchthat

lf('o'*u'tt)l ^< r'

proof.It

is an immediate consequence of the Banach's fxed point ^theoretn' o

k = ^o,n.The

^set

^O c

^,R2'*2 is definedbY

Remartc4

we *.

ll#ll . iltslllllt"lll

^<

1 ir

^and

^{onlv rr} lltrlll' Ñ

(30)

n

₌

I, ^e

^{R2"*2 ;W = (ro, w 1,'.', w n,fr}

¡,frr,'..,fr u)',

V,l= ,f1,,(',)l+ ^ruilsoP,

)1,

,

₌

q=o &=0

If

the norm of a matrix

ll)

⁼

(or),,,=,ø*, it

defined by

162 Radu Mustáþ 74 15 Derivative-intorpolating Spline Functions ¹⁶³

wx=(iil,uÍlå,

\ ,=l j=l )

(,1 12 ,tl'

I

¡r"r¡ ⁼ [;( )'.åt ^{)'Ju = 12,,+2M.,}

REFERENCES

thÞn

l. [L Akça ærd Gh. Micula, Munø'ical soltttion ^of tlífierenlial equations of n'h order with deviating argument lry splinefuncÍbns, Bul. St. Udv. Baia Mare, Seria B, Matenaticã-Inlbrr¡aticã

vIJ,

l-2

^{(1991), 4',7-54.}

2.P.Blaga, R. Goreltflo and Gh. Micula, Even degree spline techniquefor nunrerical solution of rtelay diferential eqtatíons, F¡oio Universität Berlin, Proprint No. A-15 (1996), Serie A-Mathematik.

3. Gh. Micula, P. Blaga and M. Micula, On even degree polynotniøl spline frnctions ^with applicatíons to nunterical soltttion of differential equations with relarded ørgilmenl, Tàrnisclro Hochschulo Darmstadt, Proprint No. 1771 (1995), Fachboroich Mathematik.

4. A. I. Rus, Princíples and Applicatíons of Fixed PointT'heory, Ed. Dacia, Cluj-Napoca, 1979 (in Romanian).

5. Ma Tsoy-Wo, Classical Analysis on Nonned ^{Spaces, 'Wodcl} Scientihc, Singapore-New ^Jcrsoy- Lonclon-Hong Kong, 1994.

where Mo: _{mar {M t,} N _L} .It follows that llt"]il

.

Ñ

^{if and only}

^if

^{M 0 <}

1

,lltslll

Received May 15, 1996 Facully of Mathenatics and Cornputer ^Science

" BabeS-B olyai

"

Un iversity

l,

^M. Kogãlniceantt St.

3400 Clui-Napoca Rotnania 2. A NUMERICAL EXAMPLE

Consider the Cauchy problern

y"'(t)

₌

{o) =t +;

"(i).),t

t),

t € 0,

rl

(P)

v'(o) = t y"(o) =

^1.

Its exact solution is

t(t): _"t'

Table ¹ contains the calculated values of the spline approximating function ^on indicated points as well as the absolute errors (in the case Ìn -- 3, p

:3, n:

^4).

Table

I

þ(")l

0

0.13ó.10-6 0,2025.10-5 0.?552.10-5 0.16234.rc-4 0.23442.rc-4 0.20762-rc-1 0.852. 10-6 0.44204.n-4 0.94542.110-4 0.104048.10-3

rr(r)

1.1 05170755 1.221400733 1.349851256 1.491808464 1.648697829 1.822098038 2.013153559 2.225585132 2.45969't653 2.718385876 x

0

1lt0

2 /10

3 /70

4tt0

5/10 6 t10

7 /10 8/10 9

ll0

1