Efficient algorithms for solving nonlinear Volterra integro differential equations with two point boundary conditions

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Internat. J. Math. & Math. Sci. VOL. 21 NO. 4 (1998) 755-760

EFFICIENT ALGORITHMS FOR

SOLVINGNONLINEAR

VOLTERRA

INTEGRO-DIFFERENTIAL

EQUATIONS

WITH

TWO

POINT BOUNDARY CONDITIONS

LE. GAREY andR.E. SHAW" Univrsxty ofNewBrunswick P.O.Box5050,SaintJohn,N.B.

Canada, E2L 4L5

755

(Received August19, 1996andinrevisedform December 2,1996)

ABSTRACT.

In

thispaperacollection of efficient algorithms are describedforsolvinganalgebraic system

withasymmetric Toeplitz coecientmatrix. Systemsof this form arise when approximating thesolutionof boundary value Volterra integro-differential equationswith finite differencemethods.Inthenonlinearcase,

aniterativeprocedureisrequired andisincorporatedintothe algorithmspresented.Numericalexamples

illustrate theresults.

KEY WORDS AND PIIRASES. NonlinearVolterra integro-differential equation, symmetric Toeplitz matrix,algorithmefficiency.

AMS CLASSIFICATION CODE. 65R,45. 1. INTRODUCTION

The form ofthesystemtobe solvedin thispaperis

AY

F(Y)

(I I)

whereF(Y)is a nonlinear function in anunknownvectorYandAis asymmetric Toeplitzmatrix.In [1,4]

circulant matricesand synunetric bandmatrices arediscussedin connection withsolvingalinearsystem. In

[7]

there is an application ofafast algorithmto asystem ofthe form(1.1). In [5]this algorithm

involves thesolutionofasecondorder Volterraintegro-differential equation. Thebackgroundfor that workandthework considered hereisbased on the Sherman-Morrison formula [2,p.113]. Considertwo

squarematricesAandBandtwocolumnvectors u andv relatedby A B nvT

If theinverseofBexistsandvTB’*u, 1, then

A"t

B

"

₊

B’muvTB

"

_I/(I-

vTB’u)

Inthispaperanapplication involving thenumerical solution ofasecond order boundary value problemof the Volterra type will be discussed.Inparticular, theproblemmodelwillbenonlinear inthe unknown and the computer implementationwillemploytheSherman-Morrison formula.Twonumerical

exampleswillbegiventocomparethismethodwith an efficientformof theLUmethod andavariantof the methoddescribedin

[5]

and

[7].

2. THE DISCRETE SOLUTION

Considera nonlinearintegro-differentialequation of the form

y)(x)

f(x,y(x),z(x)),O<x a (2.1)

Supported by NSERCof Canada

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where

z(x)

"

K(x,t,y(t))dt

subject to the boundary conditions y(0)=b and y(a)=b Let

Rl={(x,t,y):0za,

lYl<(R)} and

R2={(x,y,z):0xa,lyl<(R),lzl<(R)}. For equation

(2.1)

defined for points in R and R2, the following

conditions areassumed:

I) f and

K

areuniformlycontinuousineach variable

ii) for the function f and for all(x,y,z), (x,y,z)and(x,y,z)in

Rz,

]ffx,

y,z)-f(x,y,z)l

ly-

YI

lffx,

y,z) ffx,y,z)l

lz-

zl

iii) for the function

K

andforall(x,t,y)and

(x,t,)

in

R,

IK(x,t,y) K(x,t,y)l s

l=Jly-

YI

and

iv) the functions

fr’

fz

and K

r are continuous and satisfy

I(x,t,y)<

0for all(x,t,y)6R and(x,y,z)6R2.

f(x,y,z)>0,

fz(x,y,z)a0

and

A proofof the uniqueness ofa solutionfor problems satisfying the abovecanbe foundinShaw

[6].

To develop a discrete solution for

(2.1)

let

IN={X.

x--ih, I=0(1)N,

Nh=a}

be a partition of I=[0,a]. Ageneral k-step method ofsolution isgivenby

where

giYa*i

h2

[i f(Xa*i’Yt*i’ga*i) nf0(l)lq-k

i-O i-O

-.

h

%x(..,j,y?,

>

,,

"-o

0

j-O

(2.2)

The coefficients

{wj}

denote the weights for thechoiceofaquadrature rule.I,naddition, there are special rules requiredin connection withstarting values. These rules haveweights

w,

msmax{k,s}

ands isrelated

to theorderofthe method.Notethat whenk=-2 thetwoboundaryconditionsprovide the required auxiliary

conditionsand the(N+1)x(N+1)system can thenbe solved.Inthelinearcaseonlyonesolveisrequiredbut inthe nonlinear caseaniterativeschememustbeemployed.

Forthe remainderofthissectionweconsideramodelof the form

ya)(x)

y(x) + g(x,y(x),z(x))

withboundary values y(0)=b and y(a)fb. Applying atwo-step method to (2.3) gives rise to an

(IN+ l)x(N+ 1)system.Furthermore,if themethod for solvingthe differentialequationisdenotedby (p,o)

with

then a particular method, known as Numerov’s method, has

(ao,

a_l,a

2)=(1,

-2,1) and

[3o,[3 ,13

z)=(1/12,10/12,1/12).

ThequadratureruleQselectedaspartof thisparticular methodisthe fourth

order

Newton-Gregory

rule. Conditionsfor theconvergenceoftwopart methods ((p,o),Q)canbe found

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NONLINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS 757

involving Yoand_Ysto thefight handside togivean

(N-1)x(N-1)

system. This systemcanbe expressedas

(j-h2B)V

h2(BG(Y)

+ S(Y)) R (2.4)

where

,scr)--go]

h

(.i-

1,

Denoting(J-h

2B)

byCwe have

2 0

o

C C

Of particularinterest isthatCisadiagonallydominantsymmetric Toeplitzmatrix.

3. ALGORITHM DEVELOPMENT

Tosolve system

(2.4)

aniterationschememustbeemployed.Using{i}todenote the

=

iterategives

(J-hZB)Y

i’* _h

Z(BG(Y

i)

₊ _s(yi)) ₊ _R _(3.1)

There areseveralwaystosolve system(3.1). For example,by efficiently usingLUdecompositiononthe Toeplitz tridiagonal symmetric system the operationcountisabout 9n,where n is the sizeof thesystem.

For repeatedapplicationswiththe same coefficientmatrixCand different fight hand sidesthecountis

5n.The formof theLUmethod usedcanbe described asfollows:

2)

3)

4)

5)

assign valuestoelements ofC:

co=l-hZ/12

c=-2.0-10h2/12

c=Co

compute superdiagonal ofU

for

j=2,N-uj=(u,c,-coS/Uj.,

performforward substitutiontosolve

Lx

(brepresents thefighthandsideof(3.1))

and thesubdiagonal ofLisgeneratedfi’omthe superdiagonal ofU

Xl=bl

for

j=2,N-xj--bj-xjq

where

=co/ujq

performbackward substitutiontosolveUy=x ys.--xs./us.

for

jfN-2,1,-j=(Xj’CoYj+

I)/Uj

if

IY

+ >tolerance,repeat from step3

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Workingwith

C,

notethat

{c,I>{CO{ +lcxl,

_%=5 and the divisionby -c givesa matrixwhich is tridiagonal symmetric and diagonallydominant.Thenonzero elementsineach roware given by(-1,

.,

-1)

with

.=2+(10h2/12).

Denotingthis matrixbyDgives

Dyi.

_I 012(BGCY

i)

scxr

i))

_R) _(3.2)

o_o

Let u=(-g)e]and

v--e,

wheree=(1,O O)

T.

PerturbDsuch that

13

isgivenby

Thenitiseasily seen that

Dffib-

uv

’.

Method2uses anLUdecomposition of f) from[4]asfollows:

Foreaseof notation, let

K

=h2(BG(yi)+s(Yi))+R

represent the siterateof the righthandsideofequation (3.2). Let

aj=(’J-gJZ’2)d

where

d=g’(1-’a’+))

",

m=N-1.The stepsinthe methodaregivenby

1) factor

b

into itsLUdecomposition anddeterminethe_aj,j=

1(1)N-2) solve the system Z’= KwhereZt=

()

3) evaluate

Y’

(j)

where

,+’

-

9

,

J=

(I)N-4) if

IY

/’

-lil

>tolerance,repeat from step2

Therearejustafew calculationsforstep and approximately4ncalculations for step2.The remaining steps require about 5noperationsforatotalof9n.For repeated applicationswith the same

13

the operationcount isabout 6n.

Asa thirdapproach, Method3 utilizesthemethod givenin[2,p.113]. Toobtain anapproximate solutionthe stepsare asfollows:

1)

z) 3) 4) )

factor15

imo itsLU decomposition solve

13

W u

solve

13

Z

Y

calculate theconstant

13--vZ/(1-vTW)

andsetY

Z+I3W

if

IY

/’

-1

>tolerance,repeatfromstep3

As before, step essentially requires just afew operations and step 2 requires4n operations. Since

1,0 0)

3,

step4requires2noperations. The remaining steps giveatotal of9noperationsthatreduces

to6nwithrepeatediterations.

To close this section, we willlookattheconvergenceof thethreemethods. Forthe nonlinear

problems, each methodisiterativein nature.Forthe directLU method,theapplicationof the methodis

equivalenttosolving

Y+’=

A"

(hX(BG(Y

)

+

S(Y))

+ R).

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NONLINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS 759

fromwhich

IF-.,[I

<

hI/[HIBNII

_(3.3)

whereLis aLipschitzconstantandconvergencetakesplaceprovided h2LI,IIB|<I.NotethatAisa continuousfunctionof hwhichtendstoJash-,0. The matrix-Aismonotone [3,p.360] and henceis invertible.Theinverseofa monotone matrixhas nonnegative elements.Since-Jand-Aaremonotoneand

(-A)-(-J)0we seethat

(-J")-(-Aa)0

[3,Theorem7.5,p.362].Itfollows,therefore,that

0s(-A’l)s(-J).

From [3]itisalsoknownthat

IFI

N2/8 and therefore

Il

isbounded.

Theothertwomethodsaresomewhat similar soonlythelast one will beanalyzed.Thereare two

partstotheiterationprocess.

In

thefirstpart, anapproximationisobtained.

By

addinganappropriate correction, thenextiterate

Y

is givenby

Y/

=b

"’

K +

=

15

"v

b -’ K%.

(3.4)

Again,exactvaluesfor thesolutionof the discreteproblemare substituted and the resultissubtracted from(3.4)togive

Ei+’

15

-’

hZB[G(Y)-G(Y)+S(Y)-S(Y)]

+

I

-1

v

x15

"h2B(CK’Y’)-G(Y)+S(Y)-S(Y))u.

(3.5)

Inparticular,v

Tb

"*

h2B(G(Y)-G(Y)+S(Y)-S(Y))

isjusta constantbecauseof thedefinitionofv.

Hence,

equation

(3.5)

canbewrittenas

E

i+’

(I

-*₊

x

I

-1nv

1

-1

)h2B(G(yi).G(Y)+S(yi).S(y))

hZA-’B(G(y).G(Y)+S(y).S(y)).

By

takingnormsand assuming

F

satisfiesaLipschitzconditionwehave thesameresultas

(3.3).

4. NUMERICAL RESULTS

Two Volterra integro-differential equations oftheform(2.1)areusedtoillustratethemethods described.Method istheefficientform oftheLUmethod, Method2 is a variantofthe fast algorithm given

in[4]and

[6]

and Method3 isthe implementation of the Sherman-Morrison formula[see, 2].The iteration countfor all methods is givenalongwiththeaverage CPUtimeinseconds. AlLprogramswerenan on aSUN SPARC20 indoubleprecision arithmetic.

Example1

y//

y -x

xS

/ 2 _x ((x-l)* + I) + at, y(0)=l, y(1)=l+e 4

exactsolution:y= x +

stepsize(h)

10.01

n

"ll

1[

Method

II

Metho

d

2

][

Method

i,

"1

il

II#i, a,o

llavscP

avgCPUtime, .avgC.PUtime

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llx

f

Example2: 3.//

er

+ 2 + + (x+t)y dt, y(0)=0, y(l)=l

30

exactsolution: 3" x

Anotherapproachtosolving system

(3.1)

isgiveninthepaperbyYahandChung

[7].

Their modification

reduces the operationcountforstep 3from5n to 4n+2twhere representsthenumber of operationsin the correction termapproximation(tsn).The actualvalue of dependsonthedegreeof the diagonaldominance

ofthe coefficientmatrixC. Todescribetheiralgorithm,one canreplacethe approximationtothecorrection terminstep3of Method 2by

yi.1 _Z

Let

b=t

"]. Thevectorpisgivenby pfCu,b b’,0 0)T_where_the_{number of}_{components in}_p_is_dependent

on

[.1

andatolerance%.Inparticular,% ischosen such that

tog(ll 2)

t() >

togClbl)

Forthe matrixCas given in

(3.1),

as the step size h used in thek-stepmethod andquadraturerulein

(2.2)

decreases

I.]

approaches2and bapproaches1. Thisgivesavalue of >Nand the method ofYanand Chung [7],as pointedoutin theirpaper,will notwork.

CHEN,

M.,

Onthe solution ofcirculant linearsystems,

SIAMJ.

Numer.Anal. 24(668-683),987.

HAGER,

W.,

Appfied NumericalLinearAlgebra,PrenticeHall, 1988.

HENRICI,

P.,

DiscreteVariable MethodsinOrdinary

Differential

Equations, JohnW’dey,1962.

ROJO, O., Anewmethod for solving symmetriccirculanttfidiagonalsystemsoflinearequations,

Computers

Math. Applic.20(1990),61-67.

SHAW,

R.E.,

GAREY,

L.E. andGLADWIN,

C.J.,

Afast algorithm for solving second order Volterraintegro-diffcrential equationswithtwo-point boundaryconditions,

Congressus

Numerantium 106(1995), 193-203.

SHAW,

R.E.,

Numerical Solutions for Two-Point Boundary Value Problems, Ph.D. Thesis, University ofNewBrunswick,1994.

YAN,

W.M.andCHUNG,

K.L.,

Afast algorithm for solving special tfidiagonal systems, Computing