View of Approximation by spline functions of the solution of a bilocal linear problem

REVUE D'ANALYSE _NUMÉRIQUE ET DE THIíORID DE L'APPROXII\,IATION Tome XXVI, No" l_2, 1997,

pp.

137_14g

APPROXIMATION BY SPLINE FL]NCTIONS OF THE SOLUTION OF A BILOCAL LINEAR PROBLEM

COSTICÄ MUSTÃTA

In the last years.the theory of spline fi.rnctions has become an important tool in the numerical solving of some

prôblr-.

for differentiut eqìations (see, for in_

stance,

|1,

L2J, [4]).

In this

paper

we

shall

define a

space

of

spline functions

of

degree 5

which can be used

to

approximate the

ùlution oi

a bilocal linear problem.

Let n

>

_{3 be a}

nafiral

number and let

An:

-æ: t_t1a:

^t0

1tt<

_...

< tr: b 1tn+l: _*

æ be a division of the real axis.

Denote by

^sr (4") _{the set} of all fi.rnctions s : R

-+

R having the following properties:

loseC9(R).

Zo slne

gs, Ik: _ltçt,

to),

k: I,2,...,

n.

3Oslroe

gs,

_tlh*r€

93, I0: ^(t_t, to), Ir*t: ltr,

^tr¡1).

THEoREM

l. If

s

e

S5(A,), then (1)

where Q)

s!)=\,t,ti+

3

l=0

Zooþ-t)1.,

^{r eR,}

k=0

Zoo ^=o

k=0

and

fooro=o

k=0

Proof. Let s e Ss(A,).

If

t > b, then r(4)(t)

:

₀ so that nn

0

= 5!Ð"oQ _k=0 - lt)n- ^{5lX ot(t -rr)*}

^beca*e

f-=o

AMS Subj ect Classifi carion: 6 5D07, 6 5L05, 6 5L 1 0.

138

Consequently

foo =o ^and _iooro=

^o.

^E

*=0 ^k=0

THEoREM 2. a)

If f

^{: R}

+

R verífies ^the conditions

(3) _f(o):

^ur,

Í(b):

^þr,

f"(tk):

^xk,

k: 0,1,"',

n, then there exists a unique splinefunctionsre _'Sr(A) ^{such that}

(4) t.lo): ^{ur, sfb) =} Þ, s|(to): ^)'t, lc:0' r'"''

^n'

b)

If

h: R -+ R verffies ^the conditions

(5)

^h(a)

: ar,

h'(a)

=

þ2, h"(tk)

: _ttt, k:

^0,

^1,..',

^n,

then there exists a uníque splinefunction soe'Ss(^,,) such that

(6)

^so(a) =

ar,

si@)

= Br,

sr"(t¡)

: _lt¡, k: ^0,

^{1,..., n.}

proof.a) _{Using the} representation (1) ^and taking into account conditions ^(4), we ^obüain the system

Ao*

Ara + Ara2

*

Arar

:

a.y,

n-l

Ao+

Aþ

^{+ A2b2+} Atbi

+ _{Zooþ -} ^rr)t =

Ft,

n

^o=o

(7)

^2Az+

6A3tj+r0iot(t¡ -

^ro)t*

^{= ì,¡;} i ⁼ W, no=on

Zot ^{= 0; la¡,t¡, =}

⁰

¡={)

^,t=0

of z

*

5 equations with ¡¿ + 5 unknowns: ^As, A¡, A2,43, ^{as, ctp.,., an'}

The system (7) has aunique solution ifand only ifthe associated homogeneous system

-

(obtained for ø, = Ê r

:

0,

x*:0, ^Ic:0,

^{!,..., n) has only the trivial solution.}

Suppose that

"

^{è Sr1A,)}

^veiifîer

the homogeneous conditions

(4)

(i,e,,

dr

_{= 9r}

:

0, Ài'

:

0,

lc:

0, 1,..., n). Thenwe ^have

¡[,tnl1r)]'za, ⁼

^Io

r(4)(r).(s"'(r))'or = -l:,(Ð(,)'s"'(t)

⁼

-- -äl';,'"'{')'s"'(t)clt = -ä,,olu,' ^(')¿'

⁼

k'

= -i"o[s"(ro)- s"(ro-,)] =

^o,

k=1

where

co:

^s(s)1t¡ I

¡,

^tc

=1r.

Approximation Splino Functions 139

It

follows ^{that s(a)}

:

0,

for

^{all t}

ela,

bl' ^{Since s} e

9,

o1/o.md

ot

^1.'.*.t

*j

s e

C4(R),

^{0 for all t e} R, implying s" ^e

9r'

As

t"(Ð :

0'

k: ^o, 1r.,.,

^{that s"(}

Final

^{e equali}

all ^{I e} R, imPlYing ^that all that ^the homogeneous ^sYS

Assertion b) ^{can be} (1) verifies conditions (6).

tr

CoRoLlenv ^3. Tltere exist ^{the systems} offunctions

g--

{so,s1,,Ss,

E,.",'Sr¡ c

^S5(A')'

6ll:

(us,

\,

^(Is,

^(\,...,

^(Jn)

^c

^S5(Ar)

v er ífy ⁱⁿ g ^t he ^{c on} di ti ^{o n} s

so(a)

: _l,

_so(b)

:

o, so"(f¡)

: _{o' k=l,u'}

sr(a) :

_o,

_s,(å) :

_1,

rr"Qt): o' k=o,n'

So@)

:0,

_s,,(å)

: _o'

_k

=o'n'' sk"(t) :

6or' lc'

i : _o'''

uo(a)

: t, ui@) :

o, un"(t¡)

: _o'

_k

_=o,n'

ur(a)

: _o,

_ul(a)

= t,]r"(l*) : _o'

_k

_=0,;'

lJ*(a):0, Uo'(a):0' k=a'n;

^Uk"U)

=

^Eor'

^{le'i --}

^0'n'

If I ^h:

^R

⁺

^{R verify the} condítions of ^Theorem 2' ^then thefunctions s'and

s r, admit ^the rePresentations

(s) ^{,¡(r) =} so(r)'¡(a)+

^{s1(r) '}

f(b) +åt*tt'

^'

f"(tt')' ^r

^e

^R'

n

(9)r¡,(¿)=uol)'h(a)+u(t)'tt'(o)nZouu(r)'1"(r¿)'reR'

Remarlc ^{1 .} By corollary ³ itfollows ^{that the set s5(4,,) ,r a} (reaI) linear ^space oJ'clhnension ⁿ

r

3 and

I

^{and 6l'l are huo bases rn} 'St(A')'

Soroe prupertier-ùtt

r

^{space S5(An)}

will

^{l¡e piesented in rv'¿rt follows'} Let

(t

- ^t*)*

⁼

Costicá Mustãþ

0 for t<tk t-h for t2t¡

2 3

(10)

w'*,)=

{t:[a't']

'o'o*Ï"'

,î!r' ."í^,),"r'r'

k =r'n

Approximation by Splino Functions 141 TuBoRBtvt 5. a)

If f e

^{W24(A,,) and sre Sr(4,)} verífy conditions ⁽⁴⁾

from

Theorem 2, then

5

(16)

(r7)

and prove that

ll"f' - r''ll, _{= ll'"'} - ,'o'll ror a, ^s

^e

s5(a,)

b)

If

h e

W|(L,)

^and s,,e ,S5(4,,) verify conditions (6)

from

Theorem 2, then

ll'Í', - ^l',11, _{= ll',',} - ^,,o,ll; ror ^{øu s}

^e

^s,(a,)

Proof,In

order to prove (16), we use the identity

lþ'' - l'll,'"= f

[,ro'1,1

- ^,Í)t'l]'

^{at +} _lu

["t"r'l -

¡{+)çt¡fz at ⁺

+2f

lsØ)(tr

- #)r,l] ["Í,r'l - ^{¡ø)çt¡fat}

r =

_[o

[",0,(,) -,1',t,1] [,]',t'l- 7(o)1,¡]a' =

^o

Indeed, integrating byparts, we find

r =

[,tol1,¡

-

^{,Í1)(r)] ,}

_[ ^,'i

^(,1

- r"'(,)]ll -

-f ["t0t,l-'Pt'l] ['i ^{(') -} r"'(t)fat

⁼

= -fc¿(s)[

^{s'i¿ (,0)}

- ^f'(r,

)]

- [r',

^(,0-,)

-

.f" (to-r)]

=

^0,

k--r

where c*1s¡

:

_s{sl1ir¡

_-

_{s}Ðç¡, t e}

_I* k:

t,n ^. çWe trave used the fact that (r(4) -#4)Xa)

:

:6r+) _$+r) (ó):0.)

Therefore,

(18) _ll'n, ^- lull"=ll',., -'t,ll; ^. _ll't., - ^lull",

implying that inequality ( 1 6) holds.

Similarly, in ^the identity

ll,*,

^_

,,rll,',=f

[,,.,(,)_,Í.)(,)]'

^{dt +}

[b["Í-,i,1 _

¡Ø)çr¡f'dt

*

*z[ullÐç,1- ,Í')(')]

['Í'r{,) -

^n{o)ç,¡)a,

we ^have (integrating by parts) r40

But

= l:lr,

Ccsticã Mrlstãla

4

(il) wl,¡(t,),=

{a

u prrro(,1,),s',(tt)

= .¡,,(t¡), ^k = o,;},

(t2) wl,¡,o(a,)'= {r ^. wî,t(A,,),s(ro) ₌ f\o), s(,n) = ¡(ù},

(r3) wf,n,c@),={r . wî*Ø,),g(ro) ₌

_h(to),

_{g,(,0) =}

_¿,(ro)}.

Then we have

Tmonepr _4. a)

If

s e

^Sr(A)

î

Wl¡,o(Lo), _thu,

(t4) lþ"'ll,

=llr"'11,,

ror ^{au g} ewl,,,,(^,,).

b)

If

s e

^Sr(A,,)

ñ

Wl,o,"(A,), rhen

(15)

_lþ,',11, =llr,',11,,

for ^au

^g

ewl,,..(r,),

Proof, We have

= [ulrØ)ç, -,(+)1r¡]4,

₌

o)(

ï

^a,

^- !bl,te)ç,¡]'a, - z[b,(+)çt[rr(Ðçl-

_,(+)1r¡]or

o =

llr(a -,.,11;

t)

ll J^,f,llr,o)(,) -,(o)(,)]0, =

"(o)0)[ s,,,(t)

_,,,,(,)]l:

_

-J:',oU,t=r,,,(r) -

s,,,(r)]or

=

-l:,{s)1r¡1s,,,(r)

-,,,,0)]0,

=

=

-Zi,=,ff,'('){,¡¡r"'(r) - s"'(r)]dr

₌

= -Xi=, ^colr"

^(ro)

-

"" þo)

- ^(s,'þo-r) -

s,,(ro_,))]

=

^o, where

ct = ,(s)(t)lro,,r:1,2,

..,, n.

Ir roflows

llr,,,ll,

-lþ,,,11; > 0, which is equivalent to (ra).

Inequalities (l 5) can be proved by a similar argument.

E

Approximation by Spline Functions 143

142 Costicã Mustãþ

II"u'

- o"ll" =lI".,' -'1,,

_Il,

.

II'f,,

-

o''11,"

6 'l

implying that

(*)

From this equality it follows (17).

tr

CoRor-reny 6.

Ifl h.

Wl

(A)

and sn rå €

^S5(^,) verify conditions (4) and (6)from Theorem 2, then

(1e)

(20)

(2t)

(22) (23) (24)

The Cauchy problems have unique solutions

¡, ^!2,

fespectively ^(see [3], Theorem ^5.15, p, 263), and the function

Qs)

^vQ)

⁼

^vt4)

+\#vr(r) ^with

vzþ)

+ 0, t ^efa,bl,

is the solution of ^the problem (D) ^(see [3]).

Applying Theorom 2b) to the solutions !1,

!2

^{of the problems}

(Ç), (C), it

follows that there exist the functions ty,, sy,e ,Sr(4,) ^such that

snr(a)

= a,

^{s'r, (n)}

= 0,

^s"r,

^(tr) =

^y",

(t¡), k ₌

^0,n,

(**)

_,r',rço¡

= o, ,;r,r(o) ₌ r, st, (tù ₌ y'i (tk), ¡ ₌ o¡'

We call the function s sy,, sy,spline solutions in _^Sr(Â,,) of ^the problems (C, ), (C2), and the function

(26) ^{,r(r) =} ,n(r)+i# ^,r,(t),

^s,,,(u)

^{+ 0, t elo,}

^bl,

is called ^a spline solutíon in ^S5(4,,) of the problem (D)' Tgeonnu 7. Consider theProblem

(c)

Y"

:

_P(t)Y

_{+ q(t), t e'la,}

b) Y(a)

= ú, l'(a):

^Y,

where

p(t)>

0, L

efa,

bl and

p,

q are continuous on la,

bl'

If

y e

\flQt,¡

is the exacÍ solulion ^of

(Q

and sre ,S5(4,,) is its spline solution (cf. ^Theorem 2b)), thenwe have

(zi) llr"-''; ll- = úllo,ll"'

llr(')llr' wherell¡,ll =

^rno*{t¿

- ^{tk-r,k ='u]1'}

Proof, We have

l'(t¡) -c,

^{(tr) = o,}

ⁱ

^{= o,n}

so that, by Rolle's theorem, there exist t!t)

e(t,,t,*r), i=)li

^{such that}

.u"'

(,Ít)) -'";, ('Ítl; ^{= o, i =} o;-

^1'

Applying again ttolle's theorem, it follows the existence

of

4(')

'(t(t),

^+f1ì),

i -,

0,n

¿.such

^that

yþi

(,f))-

"!o(tf

Ð

) = o, ^¡

=l,n -i.

n = [:lJ^)1,¡- "[a)1'l] [#)t'l-

¿{a)1r)]d/

=

^0,

llr"'ll,

⁼

lþf'll; +llrt'r - #'ll;,

lþ"'ll;

⁼

ll'f,,ll; *

lþr'r

- "f.,ll;, ll#'ll,

=llr"'11,,

ll'f"ll, .llr"ll,, llr"' -:i"ll

₌ llr"'11,,

lll"-#'ll,=ll,"'ll,

Proof,, Equalities (19) and (20)

follow

from (18) and

(*) for s: ^0.

^The

remaining inequalities follow from (19) and (20).

tr

Applícation Consider the bilocal linear problem

(D) y" : _p(t).

_{y +}

_q(t),

_t

_ela,

_bl,

y(a):

_{a, y(b) = B.}

If

p,

q are continuous functions on [a, b) and p(t) > 0, t

e

_la, b), then the problem (D) has a unique solutiony ^(see [3], Theorem ^10.1, p. 519),

Consid er the Cauchy problems

(Ct) y" : _{p(t)y + q(t),}

_t

_ela,

_bf,

y(a) : _{a, y'(a):0,}

çCr.)

!" : ^{p(t)y, I ela,}

^b),

y(a) :

_o,

_{y'(a): l.}

144 Costicã Mustãþ T'he inequalities

l,,rÌ

_ d')l <

ziln,ll .,a

lr{iì _ r(,)l< ^{3il^,ll} hold for ¡ = ^O,n

-

^{Zand I = 0,n}

l,respectively,

, ^{,Lo. every t elu, ó]}

^there

^{is an index}

^io

^{e {0, 1,,,.,}

^n_l

}

^{such that}

f ^{-';:'l = 4l¡,ll}

^{so that,} taking into accounr (24),wehave

Approximation by Spline Functions

RemarlcL. From the proof of Corollary

I

ítfollows that the ínequalily

llr'-"',ll* = Jz(o - o)llt,,ll''' llrt"ll,

I

^{9 145}

(2e) holds, too.

The approximative determination of thevalues of the spline solution srof ^the problem (D) on thenodes of ^the dívision An

First observe that the exact soltrtion

/

^{€ W24(^,) of the} problem (D) and its 1.u,,,(r)

- _",,;

(,)l =

lfi ^{{r.,f,) -}

,Í.){,))0,,1

=

,ll',,,, o,l'''

llr,¡r,,

^u,1

-,r'r

]'a,1"' .

<

Jli4lt- ll:l,,rr, ^- ,yt1,¡l'o,l'''

,

< Jr .ll¡,11"' llr,"ll,,

Similarþ, for

every

t ela, ól

there existToe

{0,

^1,..,,

ⁿ _ l)

such that

Pl ₌

^{ll¿,,iI, implying}

lt"(,) - ^,", t,l

⁼

llg ^1r,,,(u)- ,ï ^(r)0,]l

=

= llr"'-''j, ll-

^llo,,ll,

It follows that inequality (27) holds.

tr

coRotrenv

8.

If

y e

wf(A,)

is the exacÍ sorution of the probrem

(e,

then

(28) ll, -'rll.. ^<

^J-z(r'

--o)'ll^,,11',

llr,',11, Proof. For every t e la, b)we have

1,,(r)

- ",(,)l -

llJ, ^{Al -}

s,, (u))aul

= ,, - ^o)

ll/,-",,11*

and

lt'(t) - _"',

^(,)l

=llJr'(u) - ^,,,, ("ùu"l< þ -o)

^.

_lir,,-,,,

_ll-,

From these inequalities _and from (20) we obtain (2g).

tr

spline solution s, e

^Sr(Â,) given by (26)

veriff lr(,) -,,(,)l ⁼

lr,t,l

* \#

^yz(t)

^{-,,,(¿) - B-s tr,(h)}

^{(å) sy,(}

< þ,(,) -,,(r)l

⁺

l=W ^yzl) -#,,,,,,1

""ulTïlr','tåf';iHär?'

^{and rv' are} determined bv the conditions ('t't;'

p _ yt(b)

_

^p

-

^{r,, (å)}

+

o(lla,ll,,,),

yr(b) tr,(b)

showing that

llv(4

-',(,)ll

⁼

r(llo.ll"')

a) The approximatíve determínation of the solutíon sr, on the nodes of the divisíon Ln

Represenúation (9)

yields

n

-

"r(r) ⁼

"oU).

" ^*

_oI-

^ur\). y"r(tt)

₌

, ⁼ uo(t)''*fuo(t)lr(tt). ^y(tt) *

^q4o)1.

vr:: ^sr,(t), i=0,n,

e¡:

: _!tçt) - ^sr,(tr),

ⁱ = ^0,n, one obtains the system

Letting

n

'r,(t,) =

^us(t,)u

+luo(t,)ln(rùko +

v¡) +

q(,ù)=

k=0

=

^us(t¡)a

+fuo(t,¡lt(,*)uo * q!o)l+ o(lln,ll'/'),

k=0

i ₌

^0,n.

146 Costicã Mustãþ _{10 11} Approximation by Spline Functions 147 The approximative values of the spline solutionsr, on the nodes of a, are the

solutions v* of the linear system

v,

=

^uo(r,)a

*lu

oþ,)lp(to)vt ⁺

q(to)f,

_{i =0,n.}

k=0

b)

The approxímatíve determinat

n

^{of the} soluÍion su on the nodes of a,, Using again representation (9), one

obtains

^'v2

trr(t) ₌

ut(t)

+ lu¡(t)' y"z(tt)

₌

and have the exact solutions

!t

⁽

^I

2

I

4

e2t + e-2t

lz (') l"'' -

^u-''1'

Forn:5,

^let

k=0 and

À, :

:

_{/o

:

_0,

_tt:

_{0.2, t2= 0.4,}

lr:

^0.6,

Í.0:

0.8,1s

:

_1}.

Using representation (1), one obtains Table

I

^for the coefficients of sr,

n sy,

= ',t(t) + luoQ). ^{p(tù. yr(t*).}

^Table

^I

k=0 _sy"

0 1

0 0.66',t8202118 0.125',t524863 0.01800170745 -0.17'10186129 0.780465 1 905 -7.970643804 1.163443033 sy,

1

0 2 0.1576268148 0.8446081893 -0.6853940844 -0.601482581 ¹

3,020199906 -5:t77416684 3.139485253

tt=5

Ao A1 A2 A3 d¡

A1

a2 tl3 44 As

Letting

wi:=sy2þ), ¡=o,fl,

e¡

:=

y2(t¡)

- trr(t), i =

^o,n, it follows thatw,are the solutions of the system

w,

:

ur(t,)

* Tp

^{o þ,) p(t o)* o+ o(lla,,} ll',' _).

t=0

Therefore, the approximative values ofsr,(t) can be obtained from the linear system

(30) w,=ur(t,)*fuoQ,)pþ)'*0, _i=0,n.

/<=0

The approximative values of the spline solution s, e

^s5(Â,,) on the nodes of the division Anare given by

(31) ,r(r¡) = ,, _-wn *4w,, i ₌ oi.

A numerícal example. The problem

(D) y": 4y,

/ e [0,

l]

y(0): t, y(7):

ea

has the exact soluti _on y

:

"-zt.

The associated Cauchy problems are

7Cr)

f" :4y,

^{t e [0, 1]}

/(0): l, ^y'(0):0

çCr)

/' :4y,

^{t e} [0,

l]

l0):0, ^y(0):

^I

For the values

ofs,

on the nodes

ofÂr,

^one uses

r"(/,)

= ^sr,

(r,)+ ,r,(t,),;

_{= oJ.}

Table2 contains the values sy(t), _{i =} Q,J, and the errors

Eí:W)-sr(l)|,

_{i =0,5.}

Table 2

Et

0 0.538',1267.10-3 0.t2835s14.rc-'?

0.21215541,rc-1 0,235291('.101

0.E.10-t

"r(',) I 0.6708s$',t',t21 0.4s0612s215 0.3033 15766 0.2t)4249434

0.1 35335284 l¡

0 0.2 0.4 0.ó 0.8 1

148 Costicã Mustäþ l2 _REVUE D'ANALYSE NUMÉRIQUE ET DE THÉORIE DE L'API'ROXIMATION Tome XXVI, Nß 1_2, 1997,

pp.

149_163

II.EFERENCES

1. P. Blaga and G. Micula" Polynoníal natural spline of even degree, Stuclia Univ. "Babeç- Bolyai", Mathematica 38, 2 (1993),3140.

2. P. Blaga, R. Gorenflo and G. Micul4 Evar degree spline teclmiquefor numerical solution oJ delay differential equations, Froie Univorsität Borlin, Preprint No. A-15 (1996), Sorie A-Mathernatik.

3. R. L. Btuden and T. Douglas Fafuæ, Nunøìcal Analysis, Third Blition, PWS-KENT Publiúing Company, Boston, ^1985,

4.

G. Micuta, P. Blaga and M. Micula, On even degree polyomíal splinefunctions with applica- tíons to numerical solution of diffirentiøl equations with retarded argument, Technischo Hochschule Darmstadt, Preprint No. 1771, Fachbereich Matheinatik (1995).

ON p-DERIVATIVE-INTERPOLATING SPLINE FLINCTIONS

RADU MUSTÃTA

Re,coived May 15,

1996

"Tiberiu Popoviciu" Instífiüe olNumericøl Analysß P.O. Box 68

J400 Cluj-Napoca,

I

Rotnania

Following the

idgas frgm [2] ^and [3], \ile define thep-derivative-interpolating spline functions which can be used to approximate

thisolution

of a difierential equation of orderp (

p

e N,

p>

I ) with modified argument, Forp

: _I

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

Let

An:-co=/-l ( ú=t0 (/t <...1tr=b<tn*¡=-+ø

be a partition of an interval

la,bf c

^R.

DEFINITION

I.For n) l,p ) l,m22,m > p

givennaturalnumbers such that

m+ p < n+2,afunction

_s:lR

->

^R satisfyingtheconditi.ons

1)

s

e

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

I

Z)

sl4e

ez,n*p_1,

1r =

ltt _r,tk),

k ₌

1,2,. . . , n, and

3)

^s

lr

lt,,

slr"., € Qn+p_t,Io

= (t¡,to),Io+t ₌ ft*tn*t)

ís called a spline function of ^{degree 2m}

* p - ^L

^{Here g,} denotes the set of

all

polynomials of degree qt most r.

The set of all spli'e flrnctions of degree 2m + p

- ^l

^{is denoted}

^by

^S r,n* o_r(L,) ^. Remark 1. Forp

:

1 one obtains the set

sr,

(Â") of natural polynomial spline functions of even degree 2m considered in [2] ^and [5],

The following representration theorem

will

imply trrat the set ,sr,*r_, (Â,,

)

^is

an(n+

p

+

l)-dimensional subspace

of Cr^ro-r(R).

THnoRBtr¿ 2. Every element

r

^€

sz"*p-r(a,,)

admits the representaÍion

m+p-l

n

(r) _'(t) ⁼ I ^{4Ì *ZqQ} -,ù31'*o-',

j=0 _È=0

AMS Subject Classification: 65D07, 6SLO1