Computable error bounds with improved applicability conditions for collocation methods

Internat. J. Math. & Math. Sci. VOL. 21 NO. 2 (1998) 375-379

375

COMPUTABLE ERROR BOUNDS WITH IMPROVED APPLICABILITY

CONDITIONS

FOR

COLLOCATION

METHODS

A.H.AHMED VisitingProfessor

InstitutodeMatematicaPuraeAplicada,

IMPA

Estrada

Dona

Castorina110 Jardim Botanico

CEP22460-320 RiodeJaneiro ILl,

BRAZIL

(ReceivedJuly 10, 1995and inrevisedform February 5, 1997)

ABSTRACT. This paper is concernedwith errorbounds for numerical solutionoflinear ordinary differential equation usingcollocation method. It isshown thatifthe differential operatoris split in

different operator forms thenthe applicabilityconditions forthe computable errorboundswhich are

based on the collocation matrices could beimproved.

KEY

WORDSAND PHRASES: Error bounds,collocation,differentialequations.

1991AMSSUBJECT CLASSIFICATION CODES: G17.

1. INTRODUCTION

Thispaperextends previous work ofCruickshank

&

Wright

1]

and Ahmed

[2]

oncomputable error boundsfor collocationsolutionof ordinarydifferential equations. In

[1

computableerror boundsforthe

solutionwiththeglobal collocationmethod were described intermsofmatricesrelatedtothe highest

derivativesof thesolution. Itisshown laterin[2]thatifthe boundswererelated directlytothe matrices involvedinthesolutionandnot viathe highest derivatives then significant improvementinthe closeness of these bounds could beachieved.

However,

despitethisimprovement, the conditions of applicability

which were the main drawback of

[1]

haveturned out tobe the same. That is for many practical problemsaninordinateamountof workisnecessarytoproduceanystricterrorbounds ofthistype. The aimofthispaperistoconsider thisprobleminordertoimprove the applicability ofthesebounds. The workinthispaperisdevelopedfromAhmed’s thesis

[3].

2.

THE

NEW SPLITTING OF

THE DIFFERENTIAL

OPERATOR

Before investigating the remit more preciselywe introducethe following assumptions and notations. Weconfider anm-thorder differentialequation oftheform

m-1

zm(t)

+

E

P.i(t)zO)(t)

F(t)

(2.1)

I=o

with associatedhomogeneousboundaryconditions. Thetheoryin

1]

and[2] dealswiththis equationin the operatorform

(D"’

T)z

F,

(2.2)

where

(D’m)()

(--)()

Butasindicated in Kantorvich

&

Akilov[4]and Anselone[5],the theory

canbe appliedto ageneral equation of the form

(D-

T)x

/, (2.3)

where

D

isinvertible. Clearly differentchoicesmaybe treatedtodealwiththeproblem ofapplicability.

A.H.AHIVD

beknown explicitly. Secondlytheprocedurefor calculating the projectionnorm orboundonitneedsto be available. Thirdlyallthe assumptions required bythetheoryin[4]and[5]should besatisfied.

Toavoid the difficulty of knowing theinverseof

D

explicitly, perhapsthe simplestextensioncould betodefine thedifferential operator

D

by

Dz

D’z

+ A_D’-z +

.--+ A0z

(2.4)

where the A’s are some parameters to be chosen to g/ve the highest possible applicability with a

reasonableamountof work. Since

D

is a lineardifferential operatorwih constantcoefficientsits inverse if" itexistscanbefound analytically.

The norm of the project

operaor

associated with

D,

which will bedenoted by

"

was shown in Ahrned[6]

o

be asymptotically thesameas thenormof theusualinterpolatingprojection operator if, for the globalcollocationmethod,the pointsare moreusually Tchebychev. Itwasalso shownherethat bounds of" the norm ofthisprojection operator canbe calculated in terms of the usual interpolating projection. These resultswillovercome thesecondproblem. Forsatisfactionofthe conditions required by the theoryin[4]and[5],using the analysisin

[6]

this willbestraightforwardif weprovethaTO

-

is a

compac

operator. That isguaranteed byKolomogorov and Fomin

[?]

since TO

-

is an integral operator.

Nowreplacing

D

byDand following thesameanalysisin

Ill

and

[2]

wereachsmilarexpressions fortheerrorboundswithslight computationalmodifications in certain terms. This isshowninthenext section.

3. EXPRESSIONS FOR THEERROR BOUNDS WITH THE NEW SPLITTING OF

THE

DIFFERENTIAL OPERATOR

Ifwerefertotheboundspresentedin

[1

and

[2],

then theboundswiththe newsplittingwillhave exactlythesameexpressionswiththe following modificationsincomputationofthe followingterms:

1. Theintegraloperator

K

whichstands for

TD

"-

in

[1]

and

[2]

willbereplaced by

TD-;

2. The projection operator

Oq

willbereplacedby

Oq;

3. The collocation matrix

Q)

which relates to the highest derivative of the solution and is denoted in

[1]

by

W,

willbereplacedby thematrix

Q

where

Q

Q + A_Q,"

-:

+

+

AoQ,q

and

Q

relatestothekthderivativeof the solution.

4. The subscriptnrefers tothe number ofpartitionsandq referstothenumber of collocation points. For globalcase n 1, then the expression of the error bounds withthis new splitting and

notation willbe

and

d

(D

T)

-1

II

<-

P.

6

(3.1)

d 0,1, 2,

II(D-

T)

-1

-<

₁

-/k

(3.2)

COMPUTABLE ERRORBOUNDS 37 7

Sincethemain

purpose

ofthispaperis tostudythe problem of applicability we arenotgoingtocompare

thecloseness of thesenewbounds. However,wedon’t expect to haveany significantchanges,especially

withthe bounds using matrices related directlytothe solutionastheywill notbeaffected bythenew

splitting. Butfor the applicability conditions we expect significantchangessinceallmodifiedtermsare involved.

4. NUMERICALAPPLICATIONS

Inournumericalapplication we will consider the simple second ordercase

"()

+

()’() +

o()()

(),(

+/-

1)

o

withDtaking the simples form

Dx

x"

+

oZ,

that is, the coefficient of

x’

istakenzero. Obviously, ifp(t)

-

0, then the inclusion of that parameteris

expectedtogive better results. Generalizationtohigher order equations and morecomplicated

D

is

straightforward buta bittedious.

The testproblemswillbe thesame onesconsidered in[1]and[2]. Problem

(1)

z"

+

a(1

+

t2)x

y

x(

+ t)

0

Problem

(2)

x"

x y

z(

+ 1)

O Problem

(3)

x"

2x

5-

(+)

=0

Problem(4)

x"-+-

2zxt+3

_(t+3)

[2x

y

x(

=[=

l)

0.

The parameterc=is included tovary the smoothness of theproblem. Problem2wig_{be neglected because} it isa trivialcasewiththeaboveD. A, isfirstlychosenthe one point best approximation ofp,

().

That

is,A,

1/2{maxlpi(s)[ -+-

min,

lp,(s)l}.

In

table

(1)

wepresent bounds for

IITD

-

and

(TD’)

-

II

and the norms of thematricesrelated

to

D"

and

D

forn 1,q 5 andn 1,q 20.

In

comparing those boundswenoticethe following: (i)There are someodd valuesinproblem

(1)

(valueswith

*)

wherethe new splitting giveslarger

values. Theeasiestpractical way of avoiding these nearly singularcases is to considerother values of nearby andtochoose thebest ofthem,as shownlater.

(ii) For problem(3)hugereductions wereachieved. Thatisbecause

,

isnegative and hence

TD

-is wellbehaved, the functionp0(t)= does not varymuch and can bewell approximated by a constantand_Pi

(t)

0.

(iii) For

problem

(4)

all values are reduced but the reductions arenotasinproblem

(3)

since here /h

(t)

:{-0. Obviously,if

D

includes anapproximation ofpithensimilarreductionsareexpected.

(iv) For mostof the cases thereductions increase witha, which indicatesbetter resultswithless smoothproblems.

Incomparing the norms of the collocation matrices relatedto

D

and

D"=,

Q

andW,_wrespectively,

we notethe followingintable

(1).

(v)For

problem

(1)

Q

<

W,

wforeveryvalue ofa 1,2 where

[[TD-

was shownin(i)tobe

relatively large. Theexplanationofthismaybeduetosingularity of

D

ata 2,18which willaffect the values of

W,

_wmorethan

Q.

(vi)For problem (3)

Q

<<

W,

_wasexpectedfrom(ii).

(vii) For problem

(4)

Q

isalmost similar to

W,.

This isprobablyduetothedominateof the derivativeterm.

378

Wenotehere the following:

(viii) For problem

(1)

asexpected from(i)and

(v)

we seegood improvementswitha 0.5 and

a Ibut noimprovementwith a 2. Thiscasewillbereconsidered.

(ix) Forproblem(3) hugeimprovementswereachieved, especiallywithc 100asexpected from (ii)and(vi).

(x) Forproblem(4),therearealsogoodimprovementsasexpected from(iii)and(vii).

Intable(3)thebounds for

[[TD-I[[

areconsidered withother values of

A

for problem(1),a 2.

We seethat the boundstake their minimumaround

A

1. The applicabilityistestedintable

(4)

with

A

0.8, 1, 1.28. We observe that the best applicability ocrs for

A

1 where

[ITD-I[[

takesits minimum whichsupportsourmethodindealingwiththe choice of

A

values.

Problem .5 .75 1.5 2 3 100 150 3 .5 -0.04514 -.09028 2 -.8056 100 9.028 4 .5 -0.15625 -0.3125 2 -0.625 100 -312.5 Table(I)

Comparison of bounds on imcgraloperators andthe normsof the rchtcdmatrices

0.1812 0.2500 1.3258 1.0122 0.6494 0.5000 1.9318 1.3572 2.4064 1.000 13.4358 5.678 8.0191 50.000 526.872 112.6927

0.0045 0.03125 1.0025 1.0042 0.0088 0.0625 1.0048 1.0027 0.0171 0.1250 1.0089 1.0060 0.2058 6.250 1.475 1.2439

0.4755 0.6250 1.4935 1.5017 0.9074 1.250 2.0520 2.0659 1.6665 2.500 3.3172 3.337 17.888 125.0 32.1639 34124

W2o 1.3257 1.9318 13.4934 26.2084 1.01409 1.0283 1.0546 1.4831 1.5515 2.2038 3.7425 148.5813 1.1136 1.3543 5.6359 7.3665 1.0045 1.0144 1.0278 1.245 1.552 2.2043 3.7439 148.7529 Table(2)

Comparison of applicabilitycxmditions

(The smallest number ofcollocationpoints required for applicability)

Problem A .5 0.75 1.5 2 3 100 150 3

.5 -0 04514 -.09028 2 -.18056 100 -9.028 4 .5 -0.15625 -0.3125 2 -0.625 100 -312.5 original modified 19 4 82 35

>120 > 120 >120 > 120

2 2 2 > 120 33 >120 >120 >120 2 2 2 22 17 97 > 120 >120 8 4 22 19

> 120 >120 > 120 > 120

2 2 2 > 120 2 2 2 10 6 22 74 >120 A1 original modified 3 2 36 18

> 120 > 120 > 120 > 120

COMPUTABLEERROR BOUNDS 37 9

Table(3)

Boundon

TD-]I

withothervalues ofAnear 3for problem(l),a 2

[2.75

12.5

l:25

12

1"75 1"5 1"25 0.75

ii!D-*ll

3.393

19.1538 i.981’_611.234211.038510.9.6571.9401

0.9365

b.9459

Table(4)

The applicability for values of)with small bounds

of[[TD-*l[

for problem(1),a 2

A Po i A,

A2

.5 >120 95 >120 61 >120 85

>120

48

1.25

>10

87 >120 49

5. CONCLUSION

Itisshownin thispaperthatextensionof the principal part ofthe differentialoperatortobe a linear combination with constantparameters ofall the derivatives can be one practical solution to theproblem of applicability of the error boundsderived in and

[2].

Thenumericalresults have shown that significant improvementsinthe applicabilityare achieved with this new extension. More investigationinthe method of choosing the parametersor inlooking for other principal part extensions may be needed for further improvements. Oneoption could beto considerthe equationinthe form

(I

D-’

T)

x

D"-

I,

and toinvestigate for satisfaction of thetheoryandlook for practical bounds for

m"-

Tll.

REFERENCES

[I]

CKUICKSI-IENK,

D.M. andWRIGHT,

K.,

Computable error bounds for polynomialcollocation

methods, SIAJ. Nm.Anal.I$_(1978), 134-151.

[2]

AHMED, AH.,

Computable error bound forcollocationmethods,Intrnat. J.Math.

&

Math.Sci.

[3]

AHMD,

A.H., Collocationalgorithms and erroranalysis for approximate solutions of ordinary

differentialequations, Ph.D. thesis, University ofNewcastle

Upon

Tynq (198 I).

[4]

KANTOROVICH,

L.V. and

AKILOV,

G.P.,

Fnctional AnalysisinNormed

Spaces, Pergaman

Press,

NewYork

(I

96).

[5]

ANSELONE, P.M.,

Collectiely

Compact Operator

Approximation Theory, Prentice Hall, Englewood Cliffs,NJ

(I

97

I).

[6]

D, A.H,

,

Asymptotic properties of collocation projection norms,

ComFuter

Math. Applic. (990),-0o