A mechanical algorithm for solving the Volterra integral equation

AmechanicalalgorithmforsolvingtheVolterraintegralequationqWeimingWang

SchoolofMathematicsandInformationScience,WenzhouUniversity,Wenzhou325035,PRChina

Abstract

Inthispaper,byusingthetheoriesandmethodsofintegralequationandcomputeralgebra,areliablealgorithmforsolvingtheVolterraintegralequationisestablished,andanewMaplealgorithmmainprocisestablished,too.Someexamplesarepre-sentedtoillustratetheimplementationsofthealgorithm.TheresultsoftheexamplesindicatethatthealgorithmofTaylorpolynomialmethodissimpleandeffective,andcanprovideanaccuracyapproximatesolutionorexactsolutionoftheVolterraintegralequationandintegro-differentialequation.Thiswillbeusefulforsolvingasystemofintegralequationorintegro-differentialequation.Ó2005ElsevierInc.Allrightsreserved.

Keywords:Volterraintegralequation;Taylorpolynomial;Algorithm;Mechanization;Maple

1.Introduction

InrecentyearstherehasbeenagrowinginterestintheVolterraintegralequation.Volterraintegralequationarisesinmanyphysicalapplications,

Foundationitem:TheprojectissupportedbytheNaturalScienceFoundationofChina(GrantNo.10371090)andNaturalScienceFoundationofZhejiang(GrantNo.M103043).E-mailaddress:weimingwang2003@163.com0096-3003/$-seefrontmatterÓ2005ElsevierInc.Allrightsreserved.doi:10.1016/j.amc.2005.02.056

q1324W.Wang/Appl.Math.Comput.172(2006)1323–1341

e.g.,potentialtheoryandDirichletproblems,electrostatics,mathematicalproblemsofradiativeequilibrium,theparticletransportproblemsofastro-physicsandreactortheory,andradiativeheattransferproblems[1–10].

SomevalidmethodsforsolvingVolterraintegralequationhavebeendevel-opedinrecentyears,suchasquadraturemethod,single-termWalshseriesmethod[7],Langrangeinterpolation[8]andmixedinterpolationcollocationmethods[9],AdomainÕsdecompositionmethod[10],etc.Besides,thenumericalsolutionofTaylor-seriesexpansionmethod[1–6]forsolvingVolterraintegralequationhadbeendeveloped.

However,theTaylorpolynomialmethodalsorequiresahugesizeofcalcu-lations.Itiswellknownthattherapiddevelopmentofcomputerscienceandcomputeralgebrasystemhasaprofoundeffectontheconceptandthemethodsofmathematicalresearches[11–15].Itisanewdevelopmentorientationinthefieldofmathematicsandcomputertoconductsciencecalculationbycomputer.Foralongtime,themathematiciansandcomputerscientistshavedreamedofreplacinghumanbrainwithcomputertoconductsymbolicoperationandanykindofmathematicalprocesses,leadingthemathematicstoamechanizingwayandmakingthecomputeritselfbecomemoreintelligent,i.e.,mathematicsmechanization[11,14–22].

Thesubject,mathematicsmechanization,triestodealwithmathematicsinaconstructiveandalgorithmicmannersothatthereasoningsbecomemechani-cal,automated,andasmuchaspossibletobeintelligence-lacking,withthere-sultoflesseningthepainstakingheavybrain-labor[11].Mathematicsmechanizationisoneoftheimportantmethodsofmathematicsstudies.Thebasisofmathematicsmechanizationarealgorithmestablishingandprogram-mingtechniques[15–22].

Maple,aninternationalmathematicalsoftware,isanexchangedcomputeralgebrasystemwithgreatabilityofsymbolicoperation,numericalcalculation,copingwithgraphics,etc.Itspowerfulfunctionslibraryanduniqueinteriorprogramminglanguageprovidescientificcalculationandprogrammingwithfriendlyplatform[15–22].

TheobjectiveofthispaperistoestablishapromisingalgorithmthatcanbeeasilyprogrammedinMapleforsolvingtheVolterraintegralequation.2.Basicmethods

LetusfirstrecallthebasicprinciplesoftheTaylorpolynomialmethodforsolvingtheVolterraintegralequation.AlthoughtheTaylorpolynomialmethodisaclassicalmethod,butitisthekeyforourproblems,sowegivethemindetails.ConsiderthesecondkindofVolterraintegralequationoftheform

Zs

xðsÞÀkkðs;tÞxðtÞdt¼yðsÞ;06s61;ð2:1Þ

0

W.Wang/Appl.Math.Comput.172(2006)1323–13411325

wheretheparameterkandthefunctionkandyaregiven,andxisthesolutiontobedetermined.Weassumethatsystem(2.1)hasauniquesolutionandthatthekernelk(s,t)=k(sÀt)withkcontinuousinI=[0,1]anddecreasingassÀtincreasesfromzeroorthatthekernelk(s,t)isoneoftheformaðs;tÞKðsÀtÞwitha(s,t)continuousfors,t2IandKweaklysingular,i.e.,

Àa

Kðs;tÞ¼OðjsÀtjÞ,0ByusingtheTaylorpolynomialmethod,aTaylorseriesapproximationcanbemadeforthesolutionx(s)in(2.1)

xðsÞ¼

1Xi¼0

aisi¼a0þa1sþa2s2þÁÁÁþansnþRnðsÞ;

ð2:2Þ

whereRn(s)istheremainder,ai(i=0,1,...,n)arethecoefficientstobe

determined.

NeglectingtotheremainderRn(s),i.e.,converting(2.2)intotheformofTay-lorpolynomial

xðsÞ¼a0þa1sþa2s2þÁÁÁþansn.

ð2:3Þ

Next,weusetheTaylorpolynomialexpansionsfory(s)(inpoints=0)andk(s,t)(inpointt=0).SubstitutingtheseresultsinEq.(2.1),theintegralpartin(2.1)willbetransformedintoamorereadilysolvableintegral,whichcanbesolvedmoreeasilythantheoriginal.Then,wecantransformtheintegralequa-tion(2.1)intoapolynomialinthevariableswithn+1unknownparametersai(i=0,1,...,n).Constructingaequationofallthecoefficientsofthepolynomialofsi(i=0,1,...,n),wecangetanalgebraicsystemswithn+1unknownparametersai(i=0,1,...,n).

Easytoknowthat,inpracticethedifficultpartissolvingforthecoefficientsai(i=0,1,...,n).ThismethodwillleadustoaclosedformsolutionfortheequationonlyintherarecasethatweareabletorecognizetheresultingpowerseriesastheTaylorseriesofsomeknownfunction.Thedisadvantageofthistechniqueisthatitisnotaswidelyapplicable,duetothedifficultyofsolvingforthecoefficients.Andtheadvantageofthistechniqueisthatweareabletoworkwithgeneralsolutionsandequationsinvolvingunspecifiedparameters,whereaswithanumericalmethodeveryquantitymustbespecifiedasanumber.

Solvingthealgebraicsystems,wecandeterminethecoefficientsai(i=0,1,...,n).Thenthesolutionof(2.1)isreadilyobtainedinaTaylorpoly-nomialform.Clearly,theexactsolutioninaclosedformisreadilyobtainabledependingontheresultingpowerseries.Thiscompletesthetechniques.

Itiswellknownthat,ifthevalueofnisbigger,thecomplexityofsolvingthealgebraicsystemscanbeveryhighinpractice.Inthefollowingsection,wewillgiveamechanicalalgorithmforsolving(2.1).

1326W.Wang/Appl.Math.Comput.172(2006)1323–1341

3.AmechanicalalgorithmforsolvingVolterraintegralequation

TaylorpolynomialmethodcanbewelladaptedtocalculatetheapproximateanalyticsolutionoftheintegralequationbyapplyingMaple.Thewholepro-cessoftheTaylorpolynomialmethodcouldbeprogrammedinMaple.NowifwewanttosolvetheintegralequationbasedonTaylorpolynomialmethod,everythingwehavetodoisjusttoinputinformationabouttheequation,thentheprogramwillgiveouttheapproximateanalyticsolutionoftheproblem.Themainalgorithmofmainprocisasfollows:

mainproc:¼proc(expr,N)localExpr,parme,lambda,k,xfunction,yfunction,bounds,Xfunction,Func,TaylorT,TaylorS,eqn,key,i,result;result:¼0;

Expr:¼Union(expr);

parme:¼parmeint(Expr);lambda:¼parme[1];k:¼parme[2];

xfunction:¼parme[3];yfunction:¼parme[4];bounds:¼parme[5];

xfunction:¼sum(a[i]*s^i,i=0..N+1);Xfunction:¼unapply(xfunction,s);

TaylorT:¼convert(taylor(k,s=0,N+1),polynom);

TaylorS:¼convert(taylor(TaylorT,t=0,N+1),polynom);Func:¼sort(collect(Xfunction(s)-convert(taylor(yfunc-tion,s=0,N+1),polynom)-lambda*int(col-lect(expand(TaylorS*Xfunction(t)),t),t=0..s),s),s);

lprint(ÔTheequationcanbeconvertedintothefollowingalgebraicequations:Ô);

eqn:¼seq(coeff(Func,s,i)=0,i=0..N):print(eqn);

key:¼solve(eqn,seq(a[i],i=0..N));

lprint(ÔSolvethisalgebraicequations,wecanget:Ô);print(key);

forifrom1toN+1do

result:¼result+rhs(key[i])*s^op(lhs(key[i]));od;

lprint(ÔThenthesolutionoftheequationisasfollows:Ô);print(parme[3]=result);endproc:

W.Wang/Appl.Math.Comput.172(2006)1323–13411327

Forconvenience,inthemainproceduremainproc,wealwaysletsrepre-senttheindependentvariable,andsettheparametersasfollowing:expr:theintegralequationtobesolved,

N:thedegreeoftheTaylorpolynomialsolution.Forexample,tothefollowingequation

Zs

xðsÞ¼essinsþesÀtxðtÞdt.

0

Forsolvingthisequation,everythingwehavetodoisjusttoinputinformationabouttheequation

expr:¼x(s)=exp(s)*sin(s)+int(exp(s-t)*x(t),t=0..s);mainproc(expr,10);

Ontheotherhand,therearetwosub-proceduressuchasUnionandparm-eintintheproceduremainproc.TheuseoftheprocedureUnionistrans-forming(2.1)intothefollowingnormalform:

Zs

xðsÞÀkkðs;tÞxðtÞdtÀyðsÞ¼0ð3:1Þ

0

andthemainuseoftheprocedureparmeintistoobtaintheinformationofEq.(3.1)astheform

Àk;kðs;tÞ;xðsÞ;yðsÞ;½0;s󰀃.

Thealgorithmparmeintisasfollows:parmeint:=proc(expr)localhexpr,temp,i,k,xfunction,yfunction,lambda,TEMP,bounds,x_int;

hexpr[1]:¼lhs(expr);hexpr[2]:¼rhs(expr);temp:¼[];TEMP:¼[];

yfunction:¼0;

forifrom1to2do

ifhexpr[i]<>0then

iftype(hexpr[i],Ô+Õ)andepxr[i]<>0thentemp:¼[op(temp),op(hexpr[i])];else

temp:¼[op(temp),hexpr[i]];fi;fi;od;

1328W.Wang/Appl.Math.Comput.172(2006)1323–1341

forifrom1tonops(temp)doifhas(temp[i],ÔintÕ)thenk:¼temp[i];

elifhas(temp[i],x)thenxfunction:¼temp[i];else

yfunction:¼yfunction+temp[i]fi;od;

iftype(k,Ô*Õ)then

lambda:¼convert([seq(op(i,k),i=1..nops(k)-1)],Ô*Õ);k:¼op(nops(k),k);else

lambda:¼1;k:¼k;fi;

temp:¼[op(op(1,k))];

bounds:¼[op(op(2,op(2,k)))];forifrom1tonops(temp)doifhas(temp[i],x)thenx_int:¼temp[i];else

TEMP:¼[op(TEMP),temp[i]];fi;od;

k:¼convert(TEMP,Ô*Õ);

return(Àlambda,k,xfunction,Àyfunction,bounds);end:

Inthefollowingsection,someexamplesarepresentedtoillustratetheefficiencyandaccuracyofthemechanizedprocess.4.Examples

Inthissection,togiveaclearmechanizedprocessofthealgorithm,wehavechoseneightdifferentexamples,fivesecondkindofVolterraintegralequa-tion—threelinearandtwononlinear,twofirstkindofVolterraintegralequa-tion—onelinearandonenonlinear,andoneintegro-differentialequation.Webeginourillustrationwithasimplelinearexample.Example1.Considerthesecondkindofequation

Zs

xðsÞ¼cossÀessinsþesxðtÞdt.

0

ð4:1Þ

W.Wang/Appl.Math.Comput.172(2006)1323–13411329

InMaple,whatwehavetodoisjustinputtingthefollowingcommand:expr1:¼x(s)=cos(s)Àexp(s)*sin(s)int(exp(s)*x(t),t=0..s);mainproc(expr1,10);

Thenwecangetthefollowingresultsimmediately.

Theequationcanbeconvertedintothefollowingalgebraicequations:8

a0À1¼0;>>>>a1Àa0þ1¼0;>>>>a1Àa0þ3¼0;a2À1>22>>>>aÀ1aþa3À1aþ1¼0;À1>3221203>>11111>¼0;a4À4a3À3a2À4a1À6a0À24>>>311111>¼0;a5À5a4À4a3À6a2À12a1À24a0À30>><1111117

¼0;a6À6a5À5a4À8a3À18a2À48a1À120a0À720111111>aÀ1aÀ10a4À24a3À72a2À240a1À720a0À630¼0;a7À1>6665>>111111>aÀ1aÀ12a5À30a4À96a3À360a2À1440a1À5040a0À40;1¼0;a8À1>>8776320>>11111>>a8À1a7À14a6À36a5À120a4À480a3À2160a2À10;1aa9À1>980801>>>>À40;1aþ22;1¼0;>3200680>>111111>>a9À1a8À16a7À42a6À144a5À600a4À2880a3À15;1aa10À10>91202>:11

À80;1aÀ3621aþ1;209¼0.6401;8800;600Solvethisalgebraicequations,wecanget󰀁

111

a10¼À;a9¼0;a8¼;a7¼0;a6¼À;

3;628;80040;320720

󰀂

11

a5¼0;a4¼;a3¼0;a2¼À;a1¼0;a0¼1.

242Thenthesolutionoftheequationisasfollows:

111611

xðsÞ¼1Às2þs4Àsþs8Às10.

22472040;3203;628;800Asamatteroffact,theexactsolutionofsystem(4.1)is

111611

sþs8Às10þÁÁÁxðsÞ¼coss¼1Às2þs4À

22472040;3203;628;800EasytoknowthattheTaylorpolynomialsolutionobtainedbythealgo-rithmmainprocisthe10-orderapproximatesolutionoftheexactsolution

coss.Andweknowthat10isthevalueofthesecondparameterinproceduremainproc.Infact,ifweinputthevalueofthesecondparameterwithN,thenwecangettheN-orderapproximatesolutionoftheexactsolutioncoss.

1330W.Wang/Appl.Math.Comput.172(2006)1323–1341

Example2.ConsiderthesecondkindofVolterraintegralequation

Zs

esÀtxðtÞdt.xðsÞ¼sinsþ2

0

ð4:2Þ

Inputthefollowingcommands:

expr2:¼x(s)=sin(s)+2*int(exp(sÀt)*x(t),t=0...s);

mainproc(expr2,10);

Wecanget

Theequationcanbeconvertedintothefollowingalgebraicequations:8

a0¼0;>>>>>>a1À2a0À1¼0;>>>>>a2Àa1Àa0¼0;>>>>>>a3À2a2À1a1À1a0þ1¼0;>3336>>>>11>a4À1aÀ1aÀ12a1À12a0¼0;>2362>>>>11111>a5À2aÀ10a3À30a2À60a1À60a0À120¼0;>54>><21111a6À1aÀ15a4À60a3À180a2À360a1À360a0¼0;35>1111111>>a7À2aÀ21a5À105a4À420a3À1260a2À2520a1À2520a0þ5040¼0;>76>>>1111>a7À28a6À168a5À840a4À3360a3À10;1a2À20;1a1À20;1a¼0;a8À1>>40801601600>>>>1111>aÀ36a7À252a6À1512a5À7560a4À30;1aÀ90;1aa9À2>9824037202>>>>>À1811a1À1811a0À3621¼0;>>;440;440;880>>>>111>a10À1aÀ45a8À360a7À2520a6À15;1aÀ75;1aÀ3021aÀ9071a>5912056004;4003;2002>>>:11

À1;814aÀ1;814a¼0.;4001;4000Solvethisalgebraicequations,wecanget󰀁

524973

;a6¼;a0¼0;a1¼1;a2¼1;a3¼;a4¼;a5¼

63120360

󰀂

4374139371181a7¼;a8¼;a9¼;a10¼.

50401260362;880362;880Thenthesolutionoftheequationisasfollows:

524957364377418

sþsþsþsxðsÞ¼sþs2þs3þs4þ

631203605040126039379118110

sþs.þ

362;880362;880

W.Wang/Appl.Math.Comput.172(2006)1323–13411331

Easytoprovethattheexactsolutionofsystem(4.2)is112

xðsÞ¼e3sÀcossþsins

555524957364377418

¼sþs2þs3þs4þsþsþsþs

631203605040126039379118110

sþsþÁÁÁþ

362;880362;880Example3.SolvethesecondkindofVolterraintegralequation

Zs

1þs22

xðsÞ¼1þsþxðtÞdt.2

01þtWiththefollowingcommands:

expr3:¼x(s)=1+s^2+int((1+s^2)/(1+t^2)*x(t),t=0..s);mainproc(expr3,10);

Wecangetthesolutionof(4.3)asfollows.

Theequationcanbeconvertedintothefollowingalgebraicequations:8

a0À1¼0;>>>>>a1Àa0¼0;>>>>>>aÀ1¼0;a2À1>21>>>>aÀ2a¼0;a3À1>3230>>>>aÀ1a¼0;a4À1>4341>>>22>a4À15a2þ15a0¼0;a5À1>5>>>11>aÀ12a3þ12a1¼0;a6À1>>65>>73535351111>a1¼0;a8À8a7À24a5þ24a3À24>>>>2222>aÀ63a6þ63a4À63a2þ63a0¼0;a9À1>98>>>11111>a9À40a7þ40a5À40a3þ40a1¼0;a10À10>>>>222221>a2À99a8À99a4À99a0þ99a6À11a10¼0;a11þ99>>>>111111>>a3À60a5À60a9þ60a7À60a1À12a11¼0;a12þ60>>>>2221222>a13þ143a8À143a2À143a10À13a12À143a6þ143a4þ143a0¼0;>>>>1111111>a7À14a13þ84a1þ84a5À84a3þ84a9À84a11¼0;a14À84>>>:22222221

a6þ195a10À195a8À195a12À195a4þ195a2À195a0À15a14¼0.a15þ195ð4:3Þ

1332W.Wang/Appl.Math.Comput.172(2006)1323–1341

Solvethisalgebraicequations,wecanget

󰀁

371373143

;a7¼;a0¼1;a1¼1;a2¼;a3¼;a4¼;a5¼;a6¼

2624407205040

a8¼

19731337;a9¼;a10¼;a11¼;

13;440362;880518;40013;305;600

1915761

;a13¼;a14¼;

68;428;8006;227;020;80029;059;430;400

󰀂

211

a15¼.

1;307;674;368;000a12¼

Thenthesolutionoftheequationisasfollows:

3713731643719

sþsþs8xðsÞ¼1þsþs2þs3þs4þs5þ

262440720504013;44073133719

s9þs10þs11þs12þ

362;880518;40013;305;60068;428;800

15761

s13þs14þ

6;227;020;80029;059;430;400

211

s15.þ

1;307;674;368;000Easytoprovethattheexactsolutionofsystem(4.3)isxðsÞ¼esð1þs2Þ

3713731643719

sþsþs8¼1þsþs2þs3þs4þs5þ

262440720504013;44073133719

s9þs10þs11þs12þ

362;880518;40013;305;60068;428;800

15761

s13þs14þ

6;227;020;80029;059;430;400

211

s15þÁÁÁþ

1;307;674;368;000

TheaboveexamplesareallthesecondoflinearVolterraintegralequation.Thefollowingtwoexamplesaresimplenonlinearequation.Wecanseethecomplexityofthecomputationsofsolvingthenonlinearintegralequation.Example4.ConsiderthesecondkindofnonlinearVolterraintegralequation

Zs

xðsÞ¼1þeÀtx2ðtÞdt.ð4:4Þ

0

W.Wang/Appl.Math.Comput.172(2006)1323–13411333

InMaple,wejustneedinputthecommands:

expr4:¼x(s)=1+int(exp(Àt)*(x(t))^2,t=0..s);mainproc(expr4,10);

Thenwecangettheresultsimmediately.Theequationcanbeconvertedintothefollowingalgebraicequations:

8

a0À1¼0;>>>>>>a1Àa2>0¼0;>>>>>>a2¼0;a2Àa0a1þ1>20>>>>>>aaÀ1a2À2aaÀ1a2¼0;a3þ2>3016030231>>>>>12>aaÀ1aaþ24a0À1aaþ1aaþ1a2¼0;a4À1>40121220320241>>>>>12112>a0þ15a0a1À2aaÀ1aaÀ2aaÀ1a2À10a1þ2aaþ2aa¼0;a5À120>51350250452503512>>>>1211>>aaþ720a0À72a0a1À11aaÀ1aaþ1aaÀ1aaþ18a0a2a6þ1>313612603304323>>>>1211>þ1a2>2þ36a1À3a0a5À3a1a4¼0;>6>>>>11121>a0a5À2a0a6þ420a0a1À2a1a5þ21a1a2À5040a2a7þ2>0þ7a1a4þ21a0a3777>>>>>11212>þ2aaÀ2aaÀ84a0a2À1aaÀ1aaÀ1a2À14a2À168a1¼0;>72372471370473>>>>>11111121>>þ1a1a5À1a1a6À1a2a3þ1a2a4À1a2a5À1a3a4þ1aa>448444406>>>>>111212>À1aaþ24a1a3þ24a0a4þ1a2þ48a2þ960a1¼0;>40783>>>>>1212>a2þ22;1aaÀ18a3À216a2À1a2a9À3621>;88006800194>>>>>1111122>À6480a2>1þ540a1a2À108a1a3þ27a1a4À9a1a5þ9a1a6À9a1a7>>>>>111>þ27a2a3À1aaþ2aaÀ2aaþ2aaÀ2aaÀ3240a0a2þ540a0a3>924925926934935>>>>>11>À108a0a4þ27a0a5À1aaþ2aaÀ2aa¼0;>906907908>>>>>11211212>a2À2011aaÀ1aaþ60a3þ1200a2a10þ3;628>2þ10a4þ50;400a1;8000;60001509>>>>>11111>À3600a1a2þ600a1a3À120a1a4þ30a1a5À10a1a6þ1a1a7À1aa>5518>>>>>1111>À120a2a3þ30a2a4À10a2a5þ1aaÀ1aaÀ10a3a4þ1aa>526527535>>>>>11111>À1aaÀ1aaþ25200a0a2À3600a0a3þ600a0a4À120a0a5þ30a0a6>536545>>>:1À10a0a7þ1aa¼0.508

1334W.Wang/Appl.Math.Comput.172(2006)1323–1341

Solvethisalgebraicequations,wecanget

󰀁

a1;a1;a111

0¼1¼2¼2;a3¼6;a4¼24

;

a15¼

120;a1720;a1

6¼7¼5040

;a18¼40;320;a¼1362;880;a1

󰀂

910¼3;628;800.

Thenthesolutionoftheequationisasfollows:

xðsÞ¼1þsþ1s2þ1s3þ1s412624þ

120

s5

þ1s6þ1715040sþ40;320s8720þ

1s9þ1

3;628;800

s10362;880.

Easytoprovethattheexactsolutionofsystem(4.4)isasxðsÞ¼es¼1þsþ1s2þ1s3þ1

s4

2624

þ15161120sþ720sþ7

5040sþ

140;320s8þ1

362;880

s9þ

1

3;628;800

s10þÁÁÁ

Example5.Considerthefollowingsystem:

s

x2

ðsÞ¼e2s

Àes

þ1þZ

eÀtxðtÞ2

dt.

0

Onejustneed

expr5:¼(x(s))^2=exp(2*s)-exp(s)+1+int(exp(Àt)*(x(t))^2,t=0..s);mainproc(expr5,10);

Thenwecangetthemorecomplicatedsolvingprocessasfollows.

ð4:5Þ

W.Wang/Appl.Math.Comput.172(2006)1323–13411335

Theequationcanbeconvertedintothefollowingalgebraicequations:8a2>0À1¼0;>>>>>2a0a1þa20À3¼0;>>>125>>2a0a2þa2>1À2a0þa0a1À2¼0;>>>>2a1a2þ2a0a3þ2aaþ1a2À2aaþ1a2À3¼0;>30260301312>>>1111212117>2a0a4þ2a1a3þa2>2þ2a0a3À2a0a2þ2a1a2À24a0À4a1þ4a0a1À24¼0;>>>12121>2a0a5þ120a0þ10a1þ1a2þ2aaÀ2aaÀ2aaÀ15a0a1þ1aa>52504503512502>>>>þ2aaþ2a1a4þ2a2a3À11¼0;>51340>>>1211121>2a0a6þ2a1a5þa2>3þ2a2a4À720a0þ3a2a3À3a1a3À36a1þ6a1a2>>>1113>>þ1aaþ1aaÀ18a0a2À1aaþ1aaÀ1a2þ72a0a1À144¼0;>31460330430562>>12>>2a0a7þ2a1a6þ2a3a4þ2a2a5À2a2a3þ2a1a5þ168a1þ1a0a4þ1aa>777713>>>11211>À420a0a1þ2aaþ1a2À2aaþ2aaþ14a2À21a0a3À21a1a2>72473705706>>>1143>À2aaþ84a0a2þ5040a2>0À1680¼0;714>>>111112121221>2a0a8À40320a2>0þ8a1a4þ8a0a5þ96a1a2À960a1À48a2À8a3þa4À24a1a3><

À1aaþ1aaþ2a1a7þ1aaÀ1aaþ1aaþ1aaþ2a3a5415416823424425434

>1111>þ2880a0a1À480a0a2þ96a0a3À24a0a4À1a0a6þ1aa>4407>>>257>þ2a2a6À40320¼0;>>>>111112121212>>2a0a9þ362880a2>0À27a1a4À27a0a5À540a1a2þ6480a1þ216a2þ18a3þ9a4>>11>>þ108a1a3þ1aaÀ2aaþ2aaþ2a1a8À27a2a3þ1aa>915916917924>>>11>À2aaÀ2aaþ2aaþ2a3a6þ2a4a5À22680a0a1þ3240a0a2>925934935>>>11>À540a0a3þ108a0a4þ1aaÀ2aaþ2aaþ2aa>906907908926>>>>þ2a2a7À1319¼0;>;440>>>>1111112>2a0a10À3;628a2þ120a1a4þ120a0a5þ3600a1a2À50;1a2À1200a2>2À60a3;80004001>>>1211111>>À10a4þa2>5À600a1a3À30a1a5þ10a1a6À5a1a7þ5a1a8>>>1111>þ2a1a9þ120a2a3À30a2a4þ10a2a5þ10a3a4À1aaþ1aa>535536>>>1>þ2a3a7þ1aaþ2a4a6þ2011aaÀ25;1aaþ3600a0a3>545;6000120002>>>111>>À600a0a4À30a0a6þ10a0a7À1aaþ1aaÀ1aa>508509526>>:þ2aaþ1aaÀ41¼0.

28527145;152Solvethisalgebraicequations,wecanget

󰀁

111111

a0¼1;a1¼1;a2¼;a3¼;a4¼;a5¼;a6¼;a7¼;

26241207205040

󰀂

111

a8¼;a9¼;a10¼.

40;320362;8803;628;800

1336W.Wang/Appl.Math.Comput.172(2006)1323–1341

Thenthesolutionoftheequationisasfollows:

1111516171

sþsþsþs8xðsÞ¼1þsþs2þs3þs4þ

2624120720504040;32011

s9þs10.þ

362;8803;628;800TheexactsolutionofEq.(4.5)isx(s)=es.Easytoseetheaboveresultis10-orderapproximatesolutionof(4.5).

ItiswellknownthattheTaylorpolynomialmethodhaswideapplicationginmanyareas.Here,wechosetwofirstkindofVolterraintegralequation,onelinear,anothernonlinear.

Example6.ConsiderthefirstkindofVolterraintegralequation

Zs

esÀtxðtÞdt¼coss.

0

ð4:6Þ

InMaple,inputthefollowingcommands:

expr6:¼int(exp(s-t)*x(t),t=0..s)=cos(s);mainproc(expr6,10);

Thenwecanobtain

Theequationcanbeconvertedintothefollowingalgebraicequations:8

a0À1¼0;>>>>>a1þa0¼0;>>>>>aþ1aþ1¼0;a2þ1>21202>>>>a2þ1a1þ1a¼0;a3þ1>3660>>>1111>>a3þ12a2þ24a1þ24a0À24¼0;a4þ1>4>>1111>>aþ20a3þ60a2þ120a1þ120a0¼0;a5þ1>54>>>1111111>111111

aþ42a5þ210a4þ840a3þ2520a2þ5040a1þ5040a0¼0;a7þ176>>>1111>aþ56a6þ336a5þ1680a4þ6720a3þ20;1aþ40;1aþ40;1aa8þ1>87160232013200>>>>>À40;1¼0;>320>>>>11111111>a9þ9a8þ72a7þ504a6þ3024a5þ15;120a4þ60;480a3þ181;440a2þ362;880a1>>>>>>þ3621a¼0;>;8800>>>11111>>a9þ90a8þ720a7þ5040a6þ30;1a5þ1511a4þ6041a3þ1;814aa10þ10>240;200;800;4002>>:111þ3;628aþ3;628aþ3;628¼0.;8001;8000;800W.Wang/Appl.Math.Comput.172(2006)1323–13411337

Solvethisalgebraicequations,wecanget

󰀁

111

a0¼1;a1¼À1;a2¼À;a3¼;a4¼;

262411

;a6¼À;12072011a7¼;a8¼;

504040;320a5¼À

󰀂

11

;a10¼Àa9¼À.

362;8803;628;800

Thenthesolutionoftheequationisasfollows:

111

xðsÞ¼1ÀsÀs2þs3þs4

26241516171sÀsþsþs8À120720504040;320

À

11

s9Às10.

362;8803;628;800

Theexactsolutionofsystem(4.6)isas

11

xðsÞ¼cossÀsins¼1ÀsÀs2þs3

26

11516171

sÀsþsþs8þs4À

24120720504040;320

À

11

s9Às10þÁÁÁ

362;8803;628;800

Example7.SolvethenonlinearfirstkindofVolterraintegralequation:

Z

0s

eÀ2tx3ðtÞdt¼esÀ1.

ð4:7Þ

WiththefollowingcommandsinMaple:

expr7:¼int(exp(À2*t)*(x(t))^3,t=0..s)=exp(s)À1;

mainproc(expr7,8);

Wecangetthecomplexresultsimmediately.

1338W.Wang/Appl.Math.Comput.172(2006)1323–1341

Theequationcanbeconvertedintothefollowingalgebraicequations:8

1Àa3>0¼0;>>>1>>À3a2aþa3>0¼0;2201>>>22321>Àa2>0a2Àa0a1À3a0þ2a0a1þ6¼0;>>>1>>þ3a2a2À3a2a3À1a3À3a2a1þ3a0a2À3a0a1a2þ1a3001010¼0;>242442223>>>23>a2aÀ6a2aþ2a3À6aa2À15a0þ12aaaÀ3aa2À3a2aÀ3>504502515015012502512>>>6621>þ4a2>0a1À5a0a1a3þ5a0a3þ120¼0;>5>>>132122122312>Àa2>0a3À3a1þa0a2À2a0a5þa0a4À3a0a1Àa0a2a3þ45a0À2a1a2>>>2212221>À2a0a1a2þa2>1a2þ3a0a1Àa0a1a4þ2a0a1a3À2a1a3þ3a0a2þ720¼0;>>>43>>a3À6aaaÀ6aaaÀ6a2aÀ2aa2À6aaaÀ315a0À1>72702471237047017015>>>>þ12aaaÀ3a2aþ6a2aÀ3aa2þ6aa2þ4a2aÀ2a2a>7023714713703712703702>>>43>À3a2aÀ6a2aþ12aaaþ21a1þ8aaaÀ6aa2þ6a2a>>70671270147012702705<

421

À12aaaþ35a0a1þ5040¼0;7013>>3>aaaþ3aaaÀ3aaaÀ3aaaÀ1aaaÀ3aaaÀ3aaa>2015202420234034201240252014>>>1313>>þa0a1a3þ315a0À3a0a1a6À12a1þ1a3þ3a1a2a3À3aaa>24424124>>>1123212121232>þ10a0a2>1þ2a0a2þ4a0a3À4a0a3À30a0a1þ10a0a2À4a0a5>>>321212323232>þ3a2>0a6À8a0a7þ2a0a4þ2a1a2À4a1a3þ4a1a4À8a1a5>4>>>>À3aa2À3aa2À3a2aþ40;1¼0;>412813823320>>>>>aaaÀ4aaaÀ4aaaÀ2aaaþ8aaaþ4aaaÀ2>302630153024303590233034>>>82>À2aaaþ45a0a1a2þ4aaaþ8aaaÀ4aaaÀ2835a3>03017302590149013>>>43>>þ4aaaþ135a1À2a3À4aaaþ4aaaÀ2aaaÀ2aaa>3016923123312431253134>>>4224282424222>À135a0a2>1À9a0a2À3a0a3þ45a0a3þ945a0a1À135a0a2þ9a0a5>>>>>À2a2a6þ2a2a7À1a2a8À2a2a4À2a2a2þ4a2a3À2a2a4þ2a200001111a5>33399933>>:À1a2aþ4aa2þ2aa2þ2a2aÀ1a2aÀ1aa2À1aa2þ1¼0.

316

912

313

323

324

323

304

362;880

Solvethisalgebraicequations,wecanget󰀁

11111

a0¼1;a1¼1;a2¼;a3¼;a4¼;a5¼;a6¼;

2624120720

󰀂

11a7¼;a8¼.

504040;320Thenthesolutionoftheequationisasfollows:

1111516171

sþsþsþs8.xðsÞ¼1þsþs2þs3þs4þ

2624120720504040;320

W.Wang/Appl.Math.Comput.172(2006)1323–13411339

Easytoprovethattheexactsolutionofsystem(4.7)isas

1111516

sþsxðsÞ¼es¼1þsþs2þs3þs4þ

2624120720

171sþs8þÁÁÁþ

504040;320Thefollowingexampleisanintegro-differentialequation.Example8.Considerthefollowingintegro-differentialequation:

Zs

00sxðsÞ¼1þseÀesÀtxðtÞdt.

0

ð4:8Þ

InMaple,weneedinput:

expr8:¼diff(x(s),s$2)=1+s*exp(s)Àint(exp(s-t)*x(t),t=0..s);

mainproc(expr8,8);

Thenwecangettheresults.

Theequationcanbeconvertedintothefollowingalgebraicequations:8

a0¼0;>>>>a1þa0À1¼0;>>>>>a1þ1aÀ1¼0;a2þ1>220>>>>a3þ1a2þ1a1þ1a0À1¼0;>3662>>>41224246111111

a0À24¼0;a5þ5a4þ20a3þ60a2þ120a1þ120>>>>111111>aþ30a4þ120a3þ360a2þ720a1þ720a0À120¼0;a6þ1>65>>>1111111>aþ42a5þ210a4þ840a3þ2520a2þ5040a1þ5040a0À720¼0;a7þ1>76>>>1111>a8þ1aþ56a6þ336a5þ1680a4þ6720a3þ20;1a>871602>>:1þ40;1aþ40;1aÀ5040¼0.32013200Solvethisalgebraicequations,wecanget󰀁

11111

a0¼0;a1¼1;a2¼;a3¼;a4¼;a5¼;a6¼;

2624120720

󰀂

11a7¼;a8¼.

504040;320Thenthesolutionoftheequationisasfollows:

1111516171

sþsþsþs8.xðsÞ¼sþs2þs3þs4þ

2624120720504040;320

1340W.Wang/Appl.Math.Comput.172(2006)1323–1341

Easytoprovethattheexactsolutionofsystem(4.8)isas111151617

sþsþsxðsÞ¼esÀ1¼sþs2þs3þs4þ

26241207205040

1

s8þÁÁÁþ

40;320

5.Conclusionandremarks

MechanizationofsolvingtheVolterraintegralequationbyusingtheTaylorpolynomialmethodhaveproposedinthisstudy.TheproceduremainprocgivesnotonlytheapproximateanalyticsolutionsofVolterraintegralequation,butalsofortheintegro-differentialequation.Theresultsoftheexamplesindi-catethatthealgorithmofTaylorpolynomialmethodisverysimpleandeffec-tiveformostofVolterraintegralequationandintegro-differentialequation.Butwemustrealizethatcomputerisnotomnipotent.Oneside,fortherea-sonoffastefficiency,wemustestablishsimplealgorithm.Ontheotherhand,forhighaccuracy,wemustusecomplexalgorithm.Itisacoupleofcontradictions.

Easytoseethat,inthemainproceduremainproc,forthesakeofreading,weestablishedtheformofreadableoutputtingfortheresultsbyusingthecom-mandslprintandprint(see,fordetails,Section3).So,thewholeprocesses,solvingtheintegralequationweobtainedbymainproc,arejustlikewedoitwithourhandsandpapers.Theformofreadableoutputtingmaybeagoodmethodforcomputer-aidedprovingorsolving.References

[1]A.M.Wazwaz,Twomethodsforsolvingintegralequations,Appl.Math.Comput.77(1996)79–89.

[2]Y.Ren,B.Zhang,H.Qiao,AsimpleTaylor-seriesexpansionmethodforaclassofsecondkindintegralequations,J.Comp.Appl.Math.110(1999)15–24.

[3]K.Maleknejad,Y.Mahmoudi,NumericalsolutionoflinearFredholmintegralequationbyusinghybridTaylorandblock-pulsefunctions,Appl.Math.Comput.149(2004)799–806.[4]K.Maleknejad,N.Aghazadeh,NumericalsolutionsofVolterraintegralequationsofthesecondkindwithconvolutionkernelbyusingTaylor-seriesexpansionmethod,Appl.Math.Comput.161(2005)915–922.

[5]S.Yalcinbas,M.Sezer,Theapproximatesolutionofhigh-orderlinearVolterra–Fredholmintegro-differentialequationsintermsofTaylorpolynomials,Appl.Math.Comput.112(2000)291–308.

[6]S.Yalcinbas,TaylorpolynomialsolutionsofnonlinearVolterra–Fredholmintegralequations,Appl.Math.Comput.127(2002)195–206.

[7]B.Sepehrian,M.Razzaghi,Single-termWalshseriesmethodfortheVolterraintegro-differentialequations,Engi.Anal.Boun.Elem.28(2004)1315–1319.

W.Wang/Appl.Math.Comput.172(2006)1323–13411341

[8]M.T.Rashed,Numericalsolutionoffunctionaldifferential,integralandintegro-differentialequations,Appl.Numer.Math.156(2004)485–492.

[9]H.Brunner,A.Makroglou,R.K.Miller,MixedinterpolationcollocationmethodsforfirstandsecondVolterraintegro-differentialequationswithperiodicsolution,Appl.Numer.Math.23(1997)381–402.

[10]E.Deeba,S.A.Khuri,S.Xie,Analgorithmforsolvinganonlinearintegro-differential

equation,Appl.Math.Comput.115(2000)123–131.

[11]W.Wu,Basicprinciplesofmechanicaltheoremprovinginelementarygeometries,J.Auto.

Reas.2(1986)221–252.

[12]A.Friendman,J.Glimm,J.Lavery,Themathematicalandcomputationalsciencesin

emergingmanufacturingtechnologiesandmanagementpractices,SIAM62–63(1992).

[13]J.G.Glimm,MathematicalSciences,Technology,andEconomicCompetitiveness,National

Academypress,Washington,DC,1992.

[14]Z.Lu,B.He,Y.Luo,ANewAlgorithmforIsolatingIntervalsforRealRootsofa

PolynomialSystemswithApplication,SciencePress,Peking,China,2004(inChinese).

[15]W.Wang,ComputerAlgebraSystemandSymbolicComputation,GansuSci-Techpress,

Lanzhou,China,2004(inChinese).

[16]W.Chen,Z.Lu,AnalgorithmforAdomiandecompositionmethod,Appl.Math.Comput.

159(2004)221–235.

[17]W.Wang,X.Lian,Anewalgorithmforsymbolicintegralwithapplication,Appl.Math.

Comput.162(2005)949–968.

[18]W.Wang,C.Lin,Anewalgorithmforintegraloftrigonometricfunctionswithmechanization,

Appl.Math.Comput.164(2005)71–82.

[19]W.Wang,AnewalgorithmforsolvingDAEswithmechanization,Appl.Math.Comput.,in

press,doi:10.1016/j.amc.2004.08.010.

[20]W.Wang,Analgorithmforsolvingnonlinearsingularperturbationproblemswith

mechanization,Appl.Math.Comput.,inpress,doi:10.1016/j.amc.2004.11.005.

[21]Z.Li,W.Wang,MechanizationforsolvingSPPbyreducingordermethod,Appl.Math.

Comput.,inpress,doi:10.1016/j.amc.2004.11.006.

[22]W.Wang,Computationsofmulti-resultantwithmechanization,Appl.Math.Comput.,in

press,doi:10.1016/j.amc.2004.11.034.