View of The numerical treatment of nonlinear Volterra integral equations of the second kind by the second kind by the discretized collocation method

REVUE'D'AN.aLysE NUÙffRreuE ET DE

r¡rÉonm

DE L'AppRo)ilMATroN

Tome 24, Nos 1-2, 1995, pp.

7f89

THE NUMERICAL TREATMENT OF NONLINEAR

VOLTERRA INTEGRAL ^EQUATIONS OF THE SECOND

KIND BY THE DISCRETIZED COLLOCATION METHOD

I. DANCru (Cluj-Napoca)

I.INTRODUCTION

In

_[5] we have ^described an algorithm for numerical solution of nonlinear Volterra integral equation of the second kind:

(1.1)

v{,) ^-- f(t) *

t

J

0

K(

l,s,

y(s))ds, t

e

I:=[O,f],

in the ^space of polynomial spline functions of degree

mrd

andcontinuity of ^class

d, S9*dØN) (m>-1,Þ-l}

Using the notation and the definitions given

in

_[5], the exact collocation equation (2.3) from [5] ^{can be} written:

(t.za) ^,(,,,¡)-- f(,,,¡)*hþll)["¡l+

^Fo(t,,¡),

where (1.2b)

tt-l

Fn(r,,¡)'

₌

I

t,+!!)1",1

¡'=0

denotes

thelag term and +f/ìL"ì, ¡=l,n

denote the

following

integrals (see

[1], [3])

I

j

0

K(tr,¡,

^t¡

*

rlQ, u¡(t¡

+ rh¡))ar, if

^{0 <}

^{i <} n-l

(1.3) +!i)1",1 '

^{= cj}

J 0

(¡

K(ru,j,

t,

⁺

rlq, ur(t,

+

r,la))aq if i=n Ir."rm)'

2 3

and which is ^defined by:

Q.aa) where Q.4b)

",(r",¡) ^J Í(r,,¡) *

^t

"+fi)la,l * _n(r,,¡),

n-l

1(,,,¡) = Zn+9ìla,l

Nonlinear Volterra Integral Equations

i=0

77

'16 Ioan Danciu

Fr

ll(" ^-

^{d,) I (dt}

^{- d,)ú, I}

⁼

^t,...,ttt

r=lr+l

From [7], we have that an element u ^e

sffJo@¡¡)

^{is well} defined when we know the coefficients

{or,r}r=r-

^{for all}

n:0,I, _{"' '} ^N-l

^(see (2'1) from [5])'

Equation(1.2)represents,foreachn:0,1,"',N-!arecu¡sivesystemwhichwill

give these coefficients.

In the case in which integrals (1,3) ^{can be} evaluated analytically ^the problem of determining the approximative ^solutio

î

^{u €}

SÍÍJoØù

^{and the conver-}

gence and

local

superconvergence properties

of this solution

had already been studied

in _[5].

In this papei

w" will

^{study the case} in which integrals (1.3) occuning in ^the exact collocatión equations cannot be evaluated analytically.

2. TIIE DISCRETIZED COLLOCATION EQUATION

In most applications integrals ^(1,3) occuning inthe exact collocation equations f

r,zl

"a"notbËåvaluateo anJyticatly, ^{and one} is forced to resort to ernploying ìuitâUte quadrature formulas

for their

approximation-

In

the

following

we

õ;;;;

^i'hut

ttt"r"

integrals are approximated

by

quadrature formulas of the

foim

^(see

[1], [2], [3],

[a]):

(z.ta) _+9lAl ^: = fw¡x(tn,j,

pr

^ti

⁺

^dt4,

^{u,(t, + dt4)),}

and

^t=1

Po

(2.lb) +fi)|""], ⁼ )

w¡,¡K(tn,i, ^tn

*

d¡,thn' u¡(t,, + ¿¡,th'))' l=l

whereF^and}Llatetwogivenpositiveintegers.Thesequadraturefornrulasare

".rärrv Ï"t"¡poiato.y

^ottãt,

with

the parameters

{d¡} ^and

{d¡,¡} satisfying, respectively:

0 <

4...dt,

^<

I ^{and 03} d¡1...di,tr.

^<

"¡ (i =1,"',tn)'

The quadrature weights ^are then given by:

Q.2) tf:)1",1 = +fì1"'l-+Íiì1"'1 , i=r,...,*(i=0,...,n),

with {[j[u,]

^and ôÍ1/['/,] civenby ^{(1.3) and (2,1).}

We now use the quadrature formulas to obtain the

firlly

discretized version of the exact collocation equations (1.2), Since the quadrature error terms

will

be disregarded, we generate an approximation A e

SflJoQy)

which has for all

n:0,

1, ...,

N-l

and for all

t ea,

the following form:

(2.3) û(t) =

^ûnþ)

= _r=0 iæ+t(, ^{- t,)' *} f

^a,,,1,

^-,,)o*'

^,

r=l

with

ok)10)

= ,(')10¡ , r=0,15...,d,

i fT t" -

^{d,,,) t (d.} _¡,r

^-

^d

i,,)ds,

^{t =} 1,.'.,r10,

i

⁼

^{\. ..,n.}

i;=l

and the corresponding error terms ^are defined by:

denotes the approxirnation to the lag term.

One can observe that the approximati on û e

ffioçZ*¡,given

^{by the}

^fully

discretized collocation equations (2.4)

will, in

general, be different from the approximation.ø

€ SÍ,!Jo(rr)

^given

by

tire exact collocation equations (1.2).

Denote

by

¿G)

=y(k)':îr(k), k=0,...,m+d

the approximation error

of

the solution y and of its ^derivates up to &-th

respectively by its derivates, Also denote subinterval

on for alln:O,I,...,N-l;

ê[k)

l=\,t,

... ,

m*dwill

depend on the choice in the following theorem:

THEoREM 2.1.

If

thenonlínear Volterra integral equation of thesecond kind (1.1)

r'

^uçm+d+t

^(I)

^and

^K

^{ec'n+d+r (^SxlR)} andifthequadratureformulasQ.l) and (2.2)

s1tísfii:

(2.5) _Je(t, ^{* d,)&} -Zr,q(t,

*r ⁺

^d/+)

^-=

^{o(r¡) ,,}

⁼

^0,...,n-t,

g

l=1

w¡

1

=J

0

and

,r1)¡,¡ '. =

78

(2.8) where

where

Q.sb)

^{h{+18s,, : =}

Ioan Danciu

So, by (2.3) with

n:0,

we have for all re

[0,l]

4 Nonlinear Volterra Integral Equations 79

and

"l

^Fo

e.6) _lo(r,

⁺

^rh)d" -I

w¡,¡<o(tn*

¿¡,th) ₌ o(ür), j

=1,2,...,rn,

6

^/=l

whenever the integrand ís a sfficíently smoothfunctíon, thenfor any choice of ^the collocatíon parameters

{"|¡=mwíth 0<co<..3"^41

and

for ^all

quasi-uniform meshes wíth stfficíently small h> 0, we have:

(2.7)

_llu,*,11. ³

^ô*h'-k ,

^k = o,\. .., s,

withs::min fu*

^{a +}

|

^{ss +}

^{I, srÌ and}

ô¡arefiniteconstanßindependentofh.

Proof.

we

shall prove

it

by induction using the same technique as

in

^the

proof of Theorem 2.2

froml5l.

First, we develop the exact solution y

ín

a s

-

_fo, ¹⁶

_]

^{in Taylor series in the}

neighborhood of the origin, and we obtain for all re _[0,1], that:

(t,

-

^tnbo)ßo

⁼

^¡no-

ft ^{+ 6^-dro,}

Êo : =

(po,r),_ç,

&

^{: =}

(ru("r))r=*,

,'=(4*')¡,r=ffi,

ñs

:

=[ä,r,,

ff(",00, ^o,,th,

^2(d

¡,n)rli')

with

2(t) .[t\),û(t)] _, ^{ro, au}

^{r e}

Js,

)

¡"=r¡'

'o

⁼

(¿6'ì[v])

-,

'J'r=Un

âoi=(â0,¡)¡=w, with

Q o,

i

_"

⁼

^lr* ^r, ff ^(",

^oo,

^0,,,

^{¡'0, 2 (d ¡,t}

^h))

^ru

^('r,,

⁾

By ^(2.1 1) and (2.5) it follows that ^(see [5]) there exist the finite ^constants po and Po such that:

(2.r2) llÊrll,'=Épo,,l <eo+poh-n-d+so,

the estimation which ^together with (2.9a) prove that:

(2.13) _*p

{lalr)

^:

r eos} <ôoh'.

The cstimation,

la(É)1r

)1, ôo,on'-k

çb=a,I,....s) easily

follow

by (2' 13), (2.8), (2.9a).

Suppose now that

if

for

alli:O,1,,..,n-I

(2.14) lui¡'ftl'Ôi,k¡' k,t eoi,k=g,t,.-.,s

hold we shall prove that (2,14) holds forT=n. Therefore we develop the ^exact solutiony in the interval o',, in Taylor ^series

Q.TI)

where

y\,h) ="i_ry,''t6 ^+&(,)

. ¡6t+,t+1,

(z.ea) _{¿("h) =} y{.h) - ^ûo4h)

⁼

*.r.rläß0,,,0*,- _&(,)}

-

^â0,,

^{lú, r} =

^1r2,...,m.

_Sincey is the exact solution ofintegral equation (1.1), then for all

j:1,2,...,m it fulfills

the equation

t("¡h)= f("¡h). ^å00[j?t/] , i

=1,2,...,m, which together with (2.Ib) and (2.2)

(n:0)

gives

(2,t0) t("¡h) " f("¡h).

^¿oô61?tyl

. _untlìlyl

By Q.4), ^{(2.9) and}

Q.l})

^we obtain the system:

80 Ioan Danciu ⁶ Nr¡nline¿¡ Volterr¿ 8l

7

I

m

r=l

y(

r

₎

(t, vØu)- îtl)

(2.rs)

where

y\rn

* xh)

₌

yl.

It ^* l#*o*tn,(t),

rl ,(í=0,1"...,n),

r=Od

&(r)

⁼

-f-- i ,Ø*a*t)çt,

⁺

stç)(t-

")'*'d",

(m+

d) ttn'

for all te(0,11,

So, by (2,3) ^and (2.I5) we have

(2,r6a)

^{ên(t, +}

^th)

⁼

ääþ,,,, ^t,+

^d¡t¡'

^z,(t,

⁺

^{o,ù)of.') ,} if 0<i1n-I,

ñ,,¡

(i.#U, ^{,' t,}

⁺

d¡,th"

z"(tn

+ d¡¡h'))dlî'),,=,r' iî ⁱ

= ^n'

v" (')

_{t{, + ¡{t+} ^ct

^+tläu,,,, *.' ^*

_{4, {.} )

l,

with

z¡(t). [r(r),ûr(l)], ^forall

I eo,and i=0,1,"',N-1,

¿

r=0

tn

(t")

Þ^# ^{r "','}

t'

+ d¡11' z

"(t'

+

øt+Mr)

rl , ^if

⁰

<i <n-1,

where (2.r6b)

)r

Fr,í ¡i-¡n,r-Ìsi

tt*'ßn,,

(ä#U",,, ^{t,+ d¡,t;' zn(t,+} ,,,,''))oir)

r-,¡

^od,ã

, ^{7î i=n,}

Taking

into

account that

y

satisfies integral equation (1,1) and using ^the quadrature formula-s (2.1) and (2.2) we have that:

n-l n-l

(2.17),{,,,,)=.f(t,,¡).|kþy)l,yl+I ^n'n[i)lv]*

ô¡ '= (Fr,"),=u,,(;

⁼ 0,1,...,n), ân:

--(q,,¡)¡=*,

i=0

ⁱ⁼⁰

.n+li)lyl * h{/)lyl, i

= r,2,...,n.

From (2,4), (2.16) ^and (2.17) we obtain the following system:

with

(2

ls) (, ^{-,,. ñ,,,,)ß,} =F^rle,)***'

^u,,,,a,

*nþ; ^{Ê,,,, Ê,I} .

. #^'

^{(h Ê,,,}

^-

^{w) Ê,, + Q, +}

^{*^1n.,,} .|^h\,,,,f,

Qu,¡ :

=- 4,("r) . hi,*,,,1þ,,,, ^{t, *}

^{di14,, 2,,(t,, +}

a¡,,4))4;,(a,,)+

.V-^e)*^

ä#(,n,¡,

,, +

dû¡,

z,(t, +

d,h)B(d,),

and

where

tt

^:

₌ ("j*d)

rn,í i

=("Íl[r])r=

f,,,, ⁱ

= 0,t,...,n.

By (2.5), Q.6'.),(2,14)and (2.18) it follows ^{that (see} [5]) ^{there exist the} finite constants Mr,

i:tr2r,,,,5

such that:

(z.ts)

llp,ll, =

rr,|llÊ,||, ^+(urrt'

⁺ M3hso+t +

Mah\)t h^*o*'

^{+ Ms.}

j,r=l,m

w:

=

("j)

j,r=l,m

Fo

(2.26),

" n(rn,¡) =

hlw

¡,r

l=1 n-1

i=0

^l=7

Volterra lutegral Equations

j=0

n-l

Nonlinea¡

Fl

83

82 Ioan Danciu I

This represents a discrete Gronwall inequality

"t

llp"llr,

(2.20)

llu,ll,

=rr,Ïilp,fl .fi^16+ ^Ms, n= 0),...,N-r,

whereM^t: Mt*

^{Mrhso+I-s * Mohsrs, Thus,} byCorollary ^1.52 from _[3] it follows that therð existinite-constants QnandP, ^{such that:}

(z.zr)

_{llp,ll, =} 9n ⁺

P,tf-'-d-l

for all n=0,1,...,N-l with Mr

I yland l>0

sufficiently small' Now by (2.21), (2.16) ^and (2.14) we obtain that:

(2,22)

_la,(tn

^{+ t4)l<}

^{Ô,,rf ,}

for all

t

e(0,1], where Ôn ut"the positive ^constants.

Deriving relations (2.3) and (2.15) ktimes (1r1,2,...,s) andusing (2.14) and (2.21) we easily obt¿in that:

(2.23) luÍPç,*,h)13Ôn,¡h'-k,

for all

t e(0,1]

wrth Ô,r,¡ the positive constants.

CoRol-¡.RRv 2.2. Let ^tlrc assumptíons ofTheorem ^2.

t

^{ho ld.}

If

^{the quadr} ature

formulas (2.I)

are

of

interpolatory type, wi.th

p"o:

_IL1

:m*d+I,

_{then the}

approxímation ît ^e SÍrlJoØ,) defined by the disuetized collocation ^equation Q.4) leads to an

error

^ê(t) satísfyùtg

(2'24)

_ll¿ll- =

o(tr*d*t),

(as

år0

^and

Nh<vT)

for

every choice of the collocationparameters {c¡}

with0<cf...<c^31,

Proof,It

is know that an

m+d*I-

point inte¡polatory quadrature formula is charactenzed, in ^the terminology of Theorem 2, 1, by min (ss, s1) à m+d+l _' Hence,

wehaves-m*d*L

Now, we consider the approximati

oî

u ^e

SÍÍJoØ)

^{defined by the exact}

collocation equation (1.5) and denote

by

e : = ^u

- ît

the difference between the approximation ¡¿ and the approximation

ît. SffJoØ)

^{defined by} the discretized

collocation equation

(2.4),if

the assumptions gf Theorem 2'1 hold, then one can

,*iiy _f.ou. ^riat

^{the àr¿éi of} this diffeience is',r

:

_min

(m+d+¡ so*l,

s1), thus:

ll"ll-,

⁼

llr-

^¿ll." ₌

llr-yll. *llv-all. t ç¡n+d+t

^{+ Ôh' <} Qh',

where ^rve used the results of Theorem

2.t

^and Theorem 2,2 f¡om [5]. Bu! ^{\rye c¿n} oráu" that the order of e

will

^depend only on the choice of the quadrature formulas

iZ.t¡,

^{u" described in the following theorem'}

THEOREM 2.3. Let ^the assumptions ofTheorcmZ'l hold, then there exísts ^a

finite

^{constant Q such that}

e

ⁱ

:

_7t-

_û

_satísfies

(2.25)

lltll.o = ^Qhs'

, ^{with s'=}

min(so + 1,q),

for ^all

quasi-uniformtneshes wíth sufficiently small

h>

^0'

pro'f,

Lete.^(t')denote the restriction ^{of e (r) to the} subint".vål on, Subtracting (2.4)

froni

(1.2)

äid

using (2.2) we obtain the

following

recufrence relation:

ff(ru,,r,r,,

+ d.¡,¡hn,z,(t, +

d¡,th))".(tn

+ d¡,¡ho) +

*hnli)|",]+ln¡lw¡

ffQ,,i,

^{r, + d¡h¡, z¡(t¡ + d ¡hì))e ¡(t¡ + d¡ní) +}

*>,t tt-l ta[,ìlu¡), i

^{= 1,...,ffi,}

i=0

where ZnQ) e .¡un!), ûnQ)1, for all

t

ec

r(n:0,1,"',N-l)'

gi

_{(Z.t) from [5]} and (2.3) we ^{have that} for all

t

^e(0,11

(2.27)

en(t, +

ún)

=

þ^ryþh,,)' *i(*,, - ã,,,)þh,)o*" ⁿ⁼

^0,...,

^N

₌ ^1'

Ifwedenoteby

î,

^,

=

^(nr,r,

\r,2,"',ïìu,r,) with

^{qn,r:= (or,,}

- ^âr,r)4*'

using tlre notation f¡om the proof of Theorem 2.1 ^then by (2'26) ^{and (2'27) we} obtain the following

relation:

_n-r

(2.2s) ^{(, - ,r' ñ,,,)n,} =fhlñ,,,'1¡ ⁺

^{Ên,¡ '}

4)+

+(hÊ,,,, -w)6, t _hrr,u ^+Zlhrn,i,

Nonlinea¡ Volter¡a Integral Equations ⁸⁵ 84

where

Ioan Danciu ^{10 11}

This equation is analogue to (2.18), Now using the same technique as in the

proof of

^Theorem

2.I or of

^Theorem

^{2.2 from} _[5]

^{one can}

^prove

^that

lln,ll,

= o(nn)

ano lla-ll,

= ^o(hn)

^{and thus bv (2.27)it follows that there exists}

ll"ll-

₌ ^Qhs'

,

^{with s'}

⁼

min{so + I,s}, a finite constant

p

such that:

Renmrkz,4. (i) Theorem 2.3 and Theorent 2.2 ftom [5] imply Theorem 2.1, because we can

write

_llall.

: llv- all*=ll"-i{l- *ll,-all.":llv-rll. *lþll-.

(ii)

If

we take

d:

^{-1 and}

^{m> I}

^the above theorem and corollary are reduced to the theorems given by H, Brunner and P.J. van der Houwen in _{[3], pp.} 260-262,

In the numerical applications it is very irnportant that the convergence order of ^the methods used to be the highest possible. From Theorem2.I

it

follows that the highest convergence order, in the exact collocation method for m and d fixed,

i" t ^:7ntd*1,

which is obtained when ^the quadrature formul¿rs used are such that so*land s, are greater thanm+d*L Also, to reduce the volume of computations it is useflil to employ the simplest possible quadrature formulas and which have highest degreeofprecision. Forinstance, ifweconsider p = Po = Pr ^{and d¡,r =}

d¡'4

^then:

(i)if

2¡r>

m.-rdrl, {aù,=¡

^are the Gauss points for(0,1) and quadrature formulas (2,1) are

of

the Gauis [uadrature formulas, then we ^have s= m+d+l;

(ii)

if2p> m*dtL, {dù,=¡

^{are the Radau} II points for (0,11 and quadrature fornrulas (2.1) are of the Radänduadrature fornrulas, then we have s =

tn*d*|.

In many papers (see

[1], [2],

[3],

[a])

the quadrature formulas used have tto

:

pr

:

rn,

dr:

^crand

d¡,¡: crc¡Ç,1:

_{l,m ),} ^The possibility of employing some quadrature

formulas of this type in our

method

would

lead

us to

some simplifications. These simplifications are useful when they do not

spoil

the convergence order given by Theorern ^{1.1, namely}

s=ntd*l in

Theoretn 2.1. An answer to this problem is given in the following corollary.

CoRou,aRy 2.5.

If

in nonlínear Volterra integral equation of the second

kínd(l.l), _.f e Cm+d+t(f

and

y.çm+d+l(SxlR) andif m> d+I,thenthere

exísts the set of collocatíon parameters {c¡}¡=-t

-

^{such thatfor the} approxùnation

AeSffJo@y) giv"n

^{by the discrete}

collocation

^equations

(2'4) ín

which lro

: _ltl :

_{tn, dr:c,} and

d¡¡:

crc, ^{we have}

(2.2s)

ll¿ll- ^,

=

_llr

-;ll*

⁼

o(t"*o*')'

Proof,,

If

p = _{t, )}

:

_Fr

:

_m and m 2

dt

^1, then it follows \\u:,Zt' ^{> mi d+}-}

y!

bV tfre

uùãä rJ-u*if*i

^{fhere exist the rz Gauss} points for (0,1) ^such that (2'29) fráf¿r.

If

_tr

:

_{m> d*2} then it follows that2¡t 2 m|d+2 and there exist ^{the ,,x Radau}

II

points for ^{(0,11 such that} (2'29) holds.

If the kernel K(t,s,y) can be smooth

^extended

to S'xIR,

^where

,,:{(r,s) :0 ^{< s} (

f

+O}

l^ì

I x l,forsomeô>othentheintegrals +fill"r]

in (1.2) rnay be approximated choosing _P6

:

_P1,

_d¡,t:

dÞ

(2,30) õll),1r,]'

⁼

t i'¡,¡K(t,,i,t,

⁺ d¡hn,un(tn +

d/',))'

with

^I=r

fi¡,¡

^:

=Ïft," -

^d,)

^t(dt - ^d,)ds,

o

i=l

and the corresponding

eÍor

^{terms are} defined by:

(2.3t)

ø!,i,)1",1=

+li)lu,,l-6,!)1",,1 , i ^=t,...,tn.

using quadraílre fornula ^(2.30) in the discretized collocation equation ^we have the following ^equation:

(2.32) t(,,,¡)= f(tu,¡)*n,6!!)l¡,1. i],oO[it¡¡,1, i =r,...,ffi(n=0," ',N-l)j

where the approximations ⁿ

eSfflo@r¡il*.

forms analogous to (2.3) for all

n:

0,1,',',N-1 ^and for all t e _{o n'}

If

we denote

by

ã : = y

-l

^the approximation enor of ^the solution y by the approximatesolution

i

gpd

by ã i _.

^u

-ã

thedifferenceoftheapproxirnation

,'frv

tfr" approximation

i

, theri repeating the above reasonilg ^{one can} prove the following ^theorem:

.IFIEOREM 2.6.Supposetharthegivenfunctionsf e

C"d*l

^{(I) anct K}

.

^çn+d+\

nvergence order a "na..,

1cr,{ I

>

^-0, we have:

r =þfLì(,)ul, ⁱ =0,t,...,n.

' (. r! _),.=*

lr il i rl

i:

i :

Ii

I

il

1f

it

ii

l

I

;l

it ti il il

86 Ioan Danciu 12

llzll.'

= lly

-¡ll_

⁼

^o(h'

₎

and

llãll-' ⁼

ll, - rll*

=

o(t"'), withs;: {mld*1,"t};

b)

if

ihe.quadratureformulas (2.1a) and

(2.30)

are of interpolatory ^type,

with¡t"1:4*d*1,

^then

llãll.'

⁼

lly-;ll*

⁼

o(t'^*d*t);

c) tf

m>

d+1, then there ^{exists the set} of collocatíon

paratn"t"rt

{"¡} ¡=t*

such ^that

þr

^the approxímation

I eSffJo@*)

^{gív.n by the} disuete collocatíon equatíons (2.32) in which

llr:

^tÍr,

dj:

"j

^and

^d¡,t: "r",

^{we have} llãll-' ⁼

lly-zll."

⁼

o(t **o*t).

Remark 2.7: (í) The results of the above theorem fot d

:

_{-1 and m}

> I

is reduced to the results given in _[3].

(ii) other possibilities for discretization

of

+9)1""1 can be found in _{[3 ]} ,

3. LOCÄL ST]PERCONVERGENCE

We now deal with the question of the attainable order of superconvergence

þn ^{Z*) in}

approxirnarions

AeSffJo@*)

^defined

by

the

fully

discretized collocation equation (2.4) attd, respectively,

in

approximations

i eSffJr@*)

defined by (2.32), As

in

_[5], we state the results for the linear integral equation

(3,1)

v{,) = f(t)

⁺ )dr

the modification for the ^{general case} being straightforward.

It is

again assumed that the underþing ^mesh sequence is quasi-unifonn. Wc have ttre following theorem:

THEoREM

3.L If

m> d+2

and A,ieS$Jo(Z¡¡),

denote, respectívely, ^the collocations approximations determined by

(2.4)

and (2.32), the collocatíott parameters

l"iì rt *ç

^{, with 0 < ct < ... <}

^{cm: I}

^{are chosen such that:'}

(3.2)

t

_o ^:=

_J skfl("-"r)4"

₌ 0, for _{k =} 0,\...,

p-\

0 i=l

Jo*0

_, ^where

d+l<p<m,

and

ff

^{and K have} continuous

de

vates ofstfficíently high order on their respective domaíns, then theþllowíng estimations hold:

(3.3)

,ng;lt(t")- ^{û(,)l= o(h")}

and (3.4)

where ^c¿ :

:

_min (tn-lp, so*I,

s)

and

!

^{is the} solution of linear Volterra integ,al equation (3.1).

Proof. We shallprove that formula (3,3) holds, theproofof ^(3.a) is analogous.

The collocation equationfor

îtesffJoQfi

(equations (z,4)),holds only ar tlre collocation points

X(l{).

k can be written in the following form:

(3.5)

û(t)=

f(t)+ _Jr(r,s)

t

u(s)as-6þ),

t

er,

0

where

ô

denotes a suitable function, subsequently called the defect function. This firnction has the form:

u(ç,,

n"øg)

ío,tÍ:)f*1,

^{ror all}

t,,¡ ex(N),

r3' ^Nonlinea¡ Volterra Integral Equations 87

lm

,^ä;ÞQ)-r(t,)l= ^o(h")

(3.6) +

i=0

Subtraction of ^(3.5) from (3,1) yields a second-kind integral equation for

the '

error function:

(3.7) ^¿(r) ₌ 6(r) +

[ xQ,la(^s)ds,

r

e/,

un

trs)v{

K(

t

J

0

,s

, teI:=[o,r]

,

0

whose solution is given by:

(3.8)

_¿Q)

_=ô(r)n

^t

_n(

J

0

trs

)6(")os, rer,

where rR(fÐ ^denotes the resolvent kernel for

K(ts).

Substituting æ1,,

in

(3.8), n

:

_{1,2,...,N it follows:}

88 Ioan Danciu L4

15

(3. l3a) and

Nonlinea¡ Volterra Integral Equations

REFEREN C ES

89

(3.e) a(t)=6(t,)+

lnJn(r,,,s)6 (s)os=

,0

. n-l

¹

= 6(r, ₎

* I _Í=0

^k

!

^{nQ,,t, +}

^*16Q,

⁺

^út

⁾

^&.

If

^each integral from equati"on n (3.9) is approximated with an interpolator m-pornt quadrature formula based on the abscissas _{t¡,7\ ¡=7^for all i

:

0, 1, , , ,,¡ú-

I

we are led

to:

^ ^{n-l (} m _I

(3.10) êQ)=61r,¡+)alìan(r,,,,¡)6(r,,r) *8,,,1,

¡=o [Ër ⁾

where

{

^;, tepresents the error of the interpolator quadrature formulaused, and år, I

-

1,2,.'.'.,m are the weight of ^{these formulas.}

By the hypothesis

c^=

Iand (3,2) we have

ltr,,l=C,

^hm+P andby (3,6) rve

g.t lô(t.r)l<

ehso+t+Phst

(Ci,Q, P

are the positive constants). Using these estiriraiioir'd in equation (3.9) we obtain that

laçt,,¡l<

ctt" , ^{with a = min{m* p,so+}

^l,s1},

Remark3.2, Theorem3.l canbeprovedusing Theorem2.3 stated above and Theorem 3.1 from [5].

By

Theorem 3.1 one observes that the local superconvergence

on Z*

^is

closely connected with the choice of collocation parameters {cr} ^(see

Ul,l2l,l3),

[4]), ^the relationbetween theirnumber ofthe coefficients ofthe approximate solution determined frorn the smooth conditions and

with

the choice

of

the quadrature formulas (2.1), respeciively (2,30).

If

the parameter

c*:1,

^{then the} number

ofp

cannot exceed m-1, ^and the convergence order cannot exceed the value 2m-l. The convergence ordera.:2nt-1 ^{canbe obtained} whenwe choos em>

drL, ktlr=ç -*

Radau II points for (0,1] ^{and the} quadrature formulas used are such that

so*l

and s, are greater than2m-l ^(see

[],

[3], [5]),

If in ^the quadrature formulas

(2.I)

and (2.30) we consider _Fo

:

pr

:

_{m and d,}

: _{c, d¡l:} crcrfori,l:

|,...,ffi a¡d then we obtain an algorithm for which the local su¡íerionveigence order is given in the following theorem.

THEoREM 3.2. Let the assumption of Theorem 3.1 hold.

If in

quadrature formulas

(2.I)

_{and Q.30)} we consider,po-

I

^Fr

^:

^{tn and}

^d¡:

_"j,

^d¡,¡:

^{crc¡ for}

j,l:

1,...,ffi, then

theþllowíng

estimations hold:

(3.r1) _^a-xly!)- a(r,)l = o(n^*r)

(3,r2\ trúw _-q

_1y0,,)-

^r(,,)l

=

o(n *t¡

_(as

_{å\0, Nh<yr),}

tr4N

where y is the solution of linear Volterra integral equatíon ⁽³

.l).

Proof,

Il

^po

: l\ :

^{m and}

^d,: _"i, ^d¡.t:

^{c,c, for}

^j,l :

|,...,ffi, then by (3.2)

follows

that the convergence ordér

foi

qúädrañre formulas (2,1) and

lz.lo¡ ii

r0

:

_st

:

m*p, i.e. there exists the positive constiants

Cþ

Czsuch that:

ltfl t",ll= ^øh*+p,i

^{= 0,.}

." (lu])f",Jl= ^{crn^. o),}

(3.13b)

lnn,,l<crn*+P,

i=o,...,n

for

allj :

_1,...,ffi (z

:0,1,...,N-l).

From (3.6) and (3,l3) we eet lôþ,,r¡f ³ C3hn+P, and using this estimarion in equation (3.10) we obtain that:

la!,)l<

ctt"'+ P, n =0,1,. ^. .,N

-

^l.

coRor.L¡.Rv 3.3.Let the assumptíon

of

Theorem 3.2

hold,

tlrcn

íf

the collocatíon pararneters

{"1¡=f;are

^{the zeros of} P^_1(2s-l) -

p^es-r)

_{(i.e the}

Radau II poíntsþt'(0,11), then in Theorem 3.2we

havep:

rn-|, í.e.:

'"øla(t,)l ⁼ o(t?^a),

(as

å\0, Nh<yr)

tnezN

^q'lz(t,)l

⁼

o(t?^u),

(as

å\0, Nh<vr).

tnezN

l. J. G. Blom, H. Brunner, The nunterícal solutiott of nonlinear llohena integral equalions of the second kìnd by collocøtíon and íterated collocatíon melhods, u¡r SIAM.

j.

SCt. Sfa1..

COMPUT., 8(J), (1987), pp. 806-830.

2.H.

Volterra_type

3.

H.

olJ;"r*Í"nor,;

4. H. Brunner, S. P. Norsett, ,Superconvergence ollocation melhodsþr Yolta.ra andAbel equations of rlte second-,kur,C, Numer. Metb.36 (1981), pp.347-358.

5. I. Danciu, The nunerical treatuent of,nonlineør Yolterra íntegral equatíons of the second kínd by

the exact collocatíon method, i¡t.' Revue d'Analyse Numérique'et de Théorie de l'.4.pproximation, Ton:e 24, I-2 (1995), pp.59-73.

6. G. Micul4 Funclíí spline ;i ^aplicalii (Romaniau), Edittua Tehnicã, Bucharest, 1978.

7. M. Micula" G. Micul4 Sur la résolution nunérique des équaliorts íntégrales du type de Voltena de seconde espèce à l'aide desþnctìons splínes, StudiaUniv.Babeç-Bolyai Math., 18 (1973), pp.65-68.

Inst¡tutul de Calcul str. Republícíi nr. 37 OJiciul PoStal

I

C.P. 68

3400 Cluj-Napocø Rontânía