ETNA OPERATIONAL MÜNTZ-GALERKIN APPROXIMATION FOR ABEL-HAMMERSTEIN ...

18 Pages·2016·0.33 MB·English

ElectronicTransactionsonNumericalAnalysis.
ETNA
Volume45,pp.183–200,2016.
KentStateUniversity
Copyright(cid:13)c 2016,KentStateUniversity.
http://etna.math.kent.edu
ISSN1068–9613.
OPERATIONALMÜNTZ-GALERKINAPPROXIMATIONFOR
ABEL-HAMMERSTEININTEGRALEQUATIONSOFTHESECONDKIND∗
P.MOKHTARY†
Abstract.SincesolutionsofAbelintegralequationsexhibitsingularities,existingspectralmethodsforthese
equationssufferfrominstabilityandlowaccuracy.Moreover,fornonlinearproblems,solvingtheresultingcomplex
nonlinearalgebraicsystemsnumericallyrequireshighcomputationalcosts.Toovercomethesedrawbacks,inthis
paperweproposeanoperationalGalerkinstrategyforsolvingAbel-Hammersteinintegralequationsofthesecond
kindwhichappliesMüntz-Legendrepolynomialsasnaturalbasisfunctionstodiscretizetheproblemandtoobtain
asparsenonlinearsystemwithupper-triangularstructurethatcanbesolveddirectly.Itisshownthatourapproach
yieldsawell-posedandeasy-to-implementapproximationtechniquewithahighorderofaccuracyregardlessofthe
singularitiesoftheexactsolution.Thenumericalresultsconfirmthesuperiorityandeffectivenessoftheproposed
scheme.
Key words. Abel-Hammerstein integral equations, Galerkin method, Müntz-Legendre polynomials, well-
posedness
AMSsubjectclassifications.45E10,41A25
1. Introduction. Abel’sintegralequation,oneoftheveryfirstintegralequationsseri-
ouslystudied,andthecorrespondingintegraloperatorhaveneverceasedtoinspiremathe-
maticianstoinvestigateandtogeneralizethem. Abelwasledtohisequationbyaproblemof
mechanics,thetautochroneproblem[16]. However,hisequationandsomevariantsofithave
foundapplicationsinsuchdiversefieldsasstereologyofsphericalparticles[3,18],inversion
ofseismictraveltimes[4],cyclicvoltametry[5],waterwavescatteringbytwosurface-piercing
barriers[12],percolationofwater[15],astrophysics[19],theoryofsuperfluidity[22],heat
transferbetweensolidsandgasesundernonlinearboundaryconditions[25],propagationof
shock-wavesingasfieldstubes,subsolutionsofanonlineardiffusionproblem[29],etc.
SeveralanalyticalandnumericalmethodshavebeenproposedforsolvingAbelintegral
equationssuchasrationalinterpolation[2],non-polynomialsplinecollocation[6],Adomian
decomposition[8],productintegration[9],homotopyperturbation[20],trapezoidaldiscretiza-
tion[13],waveformrelaxation[11],JacobispectralcollocationandTaumethods[14,21,27],
Taylorexpansion[17],Laplacetransformation[19],fractionallinearmultistepmethods[23],
Runge-Kuttamethods[24],modifiedTaumethods[26],LegendreandChebyshevwavelets
[31,33],etc.Almostallaforementionedmethodshavebeenappliedtothelinearcase,andvery
fewapproachesintheliteratureutilizeapproximationsforanumericalsolutionofnonlinear
Abelintegralequations. Thus,providingasuitablenumericalstrategywhichapproximates
solutionsofnonlinearAbelintegralequationscanbeworthwhileandnewinthearea. Inthis
paper,weconsideraHammerstein-typeofnonlinearityintheAbelintegralequation.
TheGalerkinmethodisoneofthemostpopularandpowerfulprojectiontechniquefor
solving integral equations. It belongs to the class of weighted residual methods, in which
approximationsaresoughtbyforcingtheresidualtobezerobutonlyinanapproximatesense.
Thismethodhastwomainadvantages. First,itreducesthegivenlinear(nonlinear)problem
tothatofsolvingalinear(nonlinear)systemofalgebraicequations. Second,ithasexcellent
errorestimateswiththeso-called"exponential-likeconvergence"beingthefastestpossible
whenthesolutionisinfinitelysmooth;see[10,30].
∗ReceivedNovember3,2015.AcceptedMarch1,2016.PublishedonlineonMay6,2016.Recommendedby
R.Ramlau.
†DepartmentofMathematics,FacultyofBasicSciences,SahandUniversityofTechnology,Tabriz,Iran
([email protected], [email protected]).
183
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Inthispaper,weintroduceanumericalschemebasedontheGalerkinmethodtoapproxi-
matethesolutionsofthefollowingAbel-Hammersteinintegralequationofthesecondkind
x
(cid:90) K(x,t)
(1.1) u(x)=f(x)+ √ G(t,u(t))dt, x∈Λ=[0,1],
x−t
0
usingsomesimplematrixoperations. Throughoutthepaper,C willdenoteagenericpositive
i
constant. Thefollowingtheoremstatestheexistenceanduniquenessresultfor(1.1).
THEOREM1.1([7]). Letthefollowingconditionsbefulfilled:
• f(x)isacontinuousfunctiononΛ,
• K(x,t)isacontinuousfunctiononD :={(x,t)|0≤t≤x≤1}withK(x,x)(cid:54)=0,
and
• G:Λ×R→RisaLipschitz-continuousfunctionwithrespecttothesecondvariable,
i.e.,
|G(t,u (t))−G(t,u (t))|≤C |u (t)−u (t)|.
1 2 1 1 2
Thenequation(1.1)hasauniquecontinuoussolutionu(x)onΛ. Here,Rdenotesthesetof
allrealnumbers.
However,fromwell-knownexistenceanduniquenesstheorems[7,21,26,27],wecan
concludethat(1.1)typicallyhasasolutionwhosefirstderivativeisunboundedattheorigin
andbehaveslike
1
(1.2) u(cid:48)(x)(cid:39) √ ·
x
After providing criteria for the existence and uniqueness of solutions of (1.1) in the
previoustheorem,weinvestigatethewell-posednessoftheproblem. Itiswellknownthat
linearAbelintegralequationsofthesecondkindarewell-posedinthesensethatsolutions
dependcontinuouslyonthedata. Inthefollowingtheoremweprovethatthisalsoholdstrue
for(1.1).
THEOREM1.2. Considertheequation(1.1)andletu˜(x)bethesolutionoftheperturbed
problem
x
(cid:90) K˜(x,t)
(1.3) u˜(x)=f˜(x)+ √ G(t,u˜(t))dt.
x−t
0
IftheassumptionsofTheorem1.1holdfor(1.3),thenwehave
(1.4) (cid:107)u(x)−u˜(x)(cid:107) ≤C (cid:107)f −f˜(cid:107) +C (cid:107)K−K˜(cid:107) ,
∞ 3 ∞ 4 ∞
where(cid:107).(cid:107) denotesthewell-knownuniformnorm.
∞
Proof. Subtracting(1.1)from(1.3)yields
x x
(cid:90) K(x,t) (cid:90) K˜(x,t)
u(x)−u˜(x)=f(x)−f˜(x)+ √ G(t,u(t))dt− √ G(t,u˜(t))dt
x−t x−t
0 0
x
(cid:90) K(x,t)
=f(x)−f˜(x)+ √ (G(t,u(t))−G(t,u˜(t)))dt
x−t
0
(cid:16) (cid:17)
(cid:90)x K(x,t)−K˜(x,t)
+ √ G(t,u˜(t))dt.
x−t
0
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OPERATIONALMÜNTZ-GALERKINAPPROXIMATION 185
SinceK(x,t)andG(x,t)arecontinuousfunctions,wecanwrite
x
(cid:90) |G(t,u(t))−G(t,u˜(t))|
|u(x)−u˜(x)|≤|f(x)−f˜(x)|+(cid:107)K(x,t)(cid:107) √ dt
∞
x−t
0
x
(cid:90) |K(x,t)−K˜(x,t)|
+(cid:107)G(t,u˜(t))(cid:107) √ dt.
∞
x−t
0
SinceG(t,u(t))satisfiestheLipschitzconditionwithrespecttothesecondvariable,weobtain
x
(cid:90) |u(t)−u˜(t)|
|u(x)−u˜(x)|≤|f(x)−f˜(x)|+C (cid:107)K(x,t)(cid:107) √ dt
2 ∞
x−t
0
x
(cid:90) |K(x,t)−K˜(x,t)|
+(cid:107)G(t,u˜(t))(cid:107) √ dt.
∞
x−t
0
Finally,usingtheGronwallinequality(see[26,Lemma4.2]),wecandeducethat
(cid:107)u(x)−u˜(x)(cid:107) ≤C (cid:107)f(x)−f˜(x)(cid:107) +C (cid:107)K−K˜(cid:107) ,
∞ 3 ∞ 4 ∞
whichgivesthedesiredresult(1.4).
FromTheorems1.1and1.2,wecandeducetheexistenceanduniquenessofsolutions
of(1.1)andalsothewell-posednessoftheproblembyassumingLipschitz-continuityofthe
nonlinearfunctionG(t,u(t)). However,animplementationofspectralmethodscanleadto
some difficulties from the numerical point of view such as a complex nonlinear algebraic
systemthathastobesolved,highcomputationalcosts,andtherebypossiblylowaccuracy. In
thispaper,weshowthatifG(t,u(t))isasmoothfunction,thenwecanobtainitsGalerkin
solutionbysolvingawell-posedupper-triangularnonlinearalgebraicsystemusingforward
substitution.
ItiswellknownthattheGalerkinmethodisanefficienttooltosolveintegralequations
with smooth solutions. However, from (1.2) we can conclude that for (1.1) this approach
leadstoanumericalmethodwithpoorconvergencebehaviour. Moreover,aswementioned
above, for nonlinear problems, its implementation produces a numerical scheme with a
highcomputationalcost. Thereforetomakeitapplicablefor(1.1),wemustprospectfora
well-conditioned,easytouse,andhighlyaccuratetechnique. Tothisend,inthispaperwe
employsuitablebasisfunctionswhichhavegoodapproximationpropertiesforfunctionswith
singularitiesattheboundaries. HereweconsiderMüntz-Legendrepolynomials[28,32]as
naturalbasisfunctionstodefinetheGalerkinsolutionof(1.1). Thesefunctionsareofnon-
polynomialnatureandaremutuallyorthogonalwithrespecttotheweightfunctionw(x)=1.
WewillshowthatifwerepresenttheGalerkinsolutionof(1.1)asalinearcombinationofthese
polynomials,wecanproduceanumericalschemewithasatisfactoryorderofconvergence,
andtheunknowncoefficientsoftheapproximatesolutioncanbecharacterizedbyawell-posed
upper-triangularsystemofnonlinearalgebraicequations. Wewillsolveitdirectlybyapplying
thewell-knownforwardsubstitutionmethod.
Thereminderofthispaperisorganizedasfollows: inSection2,wedescribetheMüntz-
Legendrepolynomials, whicharethebasisforoursubsequentdevelopment. InSection3,
weexplaintheapplicationoftheoperationalGalerkinmethodbasedontheMüntz-Legendre
polynomials to obtain an approximate solution of (1.1). Here we also give results on the
numericalsolvabilityandacomplexityanalysisoftheresultingsystemofnonlinearalgebraic
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equations. InSection4,weapplytheproposedmethodfromSection3toseveralexamplesand
presentsomecrucialnumericalcharacteristicssuchasnumericalerrors,orderofconvergence,
and comparison results with other existing numerical techniques and thereby confirm the
efficiencyandeffectivenessoftheproposedmethod. Section5,isdevotedtoourconclusions.
2. Müntz-Legendrepolynomials. Inthissection,wegivethedefinitionandthebasic
propertiesoftheMüntz-Legendrepolynomialswhichareusedinthesequel. Allthedetailsin
thissectionaswellasfurtheronescanbefoundin[28,32].
Then-thMüntz-Legendrepolynomialisdefinedas
L (x):= 1 (cid:90) n(cid:89)−1t+ξk+1 xt dt,
n 2πi t−ξ t−ξ
k n
k=0
D
wherethesimplecontourDsurroundsallthezerosofthedenominatorintheaboveintegrand
∞
and0=ξ <ξ <...<ξ <...,withξ →∞and (cid:80) ξ−1 =∞. Thesepolynomialsare
0 1 n n k
k=1
mutuallyorthogonalwithrespecttotheweightfunctionw(x)=1suchthatwehave
(cid:90) δ
L (x)L (x)dx= n,m , n≥m,
n m 2ξ +1
n
D
whereδ istheKroneckersymbol. ItcanbeshownthattheMüntz-Legendrepolynomials
n,m
havethefollowingrepresentation
n
(cid:88)
Ln(x):= ρk,nxξk =Span{1,xξ1,xξ2,...,xξn},
k=0
n−1
(2.1) (cid:81) (ξ +ξ +1)
k j
ρ = j=0 , forn∈N ,
k,n n 0
(cid:81)
(ξ −ξ )
k j
j=0,j(cid:54)=k
andtheysatisfytherecursionformula
1
(cid:90)
L (x)=L (x)−(ξ +ξ +1)xξn t−ξn−1L (t)dt, x∈(0,1].
n n−1 n n−1 n−1
x
Here,N isasusualthesetofnonnegativeintegers.
0
3. TheMüntz-Galerkinapproach. Inthissection,wedevelopanoperationalGalerkin
methodwhichappliestheMüntz-Legendrepolynomialsasbasisfunctionstoobtainanapprox-
imatesolutionto(1.1). Wepresentourresultsintwoparts. First,usingasequenceofmatrix
operations,weobtainasuitablenonlinearalgebraicformoftheMüntz-Galerkindiscretization
of(1.1). Second,weexplainhowtheuniquesolutionisfound. Hereweshowthattheresulting
nonlinearsystemfromthefirststepcanberepresentedbyanupper-triangularstructureand
thisenablesustocomputethesolutiondirectly.
3.1. Outlineofthemethod. Inthissection,weapplytheMüntz-Galerkinapproachto
convert(1.1)intoasuitablenonlinearalgebraicform. Tothisend,wesupposethat
i
(3.1) ξ = , i=0,1,...,
i 2
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OPERATIONALMÜNTZ-GALERKINAPPROXIMATION 187
anddefineu (x)asMüntz-Galerkinapproximationoftheexactsolutionu(x)inthefollowing
N
way:
∞
(cid:88)
(3.2) u (x):= u L (x)=uL=uLX,
N i i
i=0
where u := [u ,u ,...,u ,0,...] and L := [L (x),L (x),...,L (x),...]T is a Müntz-
0 1 N 0 1 N
Legendrepolynomialbasis. Fromtherelations(1.2),(2.1),and(3.1),weconcludethatour
approximatesolution(3.2)hasthesameasymptoticbehaviorastheexactones,whichisthe
crucialfactforachievingasatisfactoryaccuracyintheGalerkinmethod. Clearly,wecanwrite
L := LX whereL = {L }∞ isaninfinitelynon-singularlower-triangularcoefficient
i,j i,j=0
matrixwithdegreedeg(Li(x)) ≤ ξi i = 0,1,...,andX := [1,xξ1,xξ2,...,xξN,...]T are
thestandardMüntzbasisfunctions. Assumethat
Nf
(cid:88)
f(x)(cid:39) f xξi =f X =fL−1L, f :=[f ,f ,...,f ,0,...]T,
i 0 1 Nf
i=0
(cid:88)NG
(3.3) G(t,u(t))(cid:39) s (t)up(t).
p
p=0
Substituting(3.2)into(1.1)andusing(3.3),ourMüntz-Galerkinmethodistoseeku (x)
N
suchthatwehave
u (x)=fX+(cid:88)NG (cid:90)x K(√x,t)sp(t)up (t)dt
N x−t N
p=0
0
(3.4) =fX+(cid:90)x K(√x,t)s0(t)dt+(cid:88)NG (cid:90)x K(√x,t)sp(t)up (t)dt.
x−t x−t N
p=1
0 0
LetusdefineK (x,t)=K(x,t)s (t)forp≥1,andassumethat
p p
∞ ∞
(cid:88)(cid:88)
(3.5) K (x,t)= kp xitj, p≥1.
p i,j
i=0j=0
Byinserting(3.5)into(3.4),weobtain
(3.6) u (x)=fX+(cid:90)x K(√x,t)s0(t)dt+(cid:88)∞ (cid:88)∞ (cid:88)NG (cid:90)x k√ip,jxitjup (t)dt.
N x−t x−t N
0 i=0j=0p=10
Now,weintendtoderiveamatrixrepresentationfortheintegraltermsin(3.6). Forthis,
wefirstgivethefollowinglemmawhichtransformup (t)intoasuitablematrixform. Our
N
strategy is to extend the proof in [14, Theorem 2.1] to the case of Müntz-Legendre basis
functions.
LEMMA3.1. SupposethatuN(x)isgivenby(3.2). Thenwehave
up (x)=uLQp−1X,
N
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whereQisthefollowinginfiniteupper-triangularmatrix
 
uL uL uL ···
0 1 2
 0 uL0 uL1 ···
Q:= 0 0 uL0 ···,
 ... ... ... ...
withL ={L }∞ , j =0,1,... .
j i,j i=0
Proof. Weproceedusingmathematicalinductiononp. Forp=1thelemmaisvalid. We
assumethatitholdsforpandtransittop+1asfollows:
up+1(x)=up (x)×u (x)
(3.7) N N N
=(cid:0)uLQp−1X(cid:1)×(uLX)=uLQp−1(X×(uLX)).
Now,itissufficienttoshowthat
(3.8) X×(uLX)=QX.
Tothisend,wecanwrite
(cid:32) ∞ ∞ (cid:33) (cid:40) ∞ ∞ (cid:41)∞
X×(uLX)=X× (cid:88)(cid:88)urLr,sx2s = (cid:88)(cid:88)urLr,sxs+2i
s=0r=0 s=0r=0 i=0
(cid:88)∞ (cid:32)(cid:88)∞ (cid:33) j∞ (cid:88)∞ (cid:32)(cid:88)∞ (cid:33) j∞
= urLr,j−i x2 = urLr,j−i x2
   
j=0 r=0 j=i r=0
i=0 i=0
=QX,
whichproves(3.8). Clearly,inserting(3.8)into(3.7)givesthedesiredresult.
NowapplyingLemma3.1inthethirdtermontheright-handsideof(3.6)yields
(cid:88)∞ (cid:88)∞ (cid:88)NG (cid:90)x k√ip,jxitjup (t)dt=(cid:88)∞ (cid:88)∞ (cid:88)NG (cid:90)x k√ip,jxitjuLQp−1X dt
x−t N x−t t
i=0j=0p=10 i=0j=0p=10
(3.9) =uL(cid:88)NG Qp−1(cid:88)∞ (cid:88)∞ kp xi(cid:90)x √tj+m2 dt∞ ,
i,j x−t
 
p=1 i=0j=0 0 m=0
whereXt :=[1,tξ1,tξ2,...,tξN,...]T. Usingtherelation[27]
(cid:90)x √tj+m2 dt∞ =(cid:26)√πΓ(1+j+ m2) x1+22j+m(cid:27)∞ ,
x−t Γ(3+m +j)
  2 m=0
0 m=0
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OPERATIONALMÜNTZ-GALERKINAPPROXIMATION 189
wecanrewrite(3.9)as
(cid:88)∞ (cid:88)∞ (cid:88)NG (cid:90)x k√ip,jxitjup (t)dt
x−t N
i=0j=0p=10
 √ 
=uL(cid:88)NG Qp−1(cid:88)∞ (cid:88)∞ kip,j(cid:26) πΓΓ((31+m++j+j)m2) x1+2(i+2j)+m(cid:27)∞ 
p=1 i=0j=0 2 m=0
(3.10)  
=uL(cid:88)NG Qp−1(cid:88)∞ (cid:88)∞ kip,j(cid:110)Amj x1+2(i+2j)+m(cid:111)∞ 
m=0
p=1 i=0j=0
(cid:32)(cid:88)NG (cid:33)
=uL Qp−1A X
p
p=1
=uLQ X,
p
where
√
πΓ(1+j+ m) (cid:88)NG
Am = 2 , Q = Qp−1A ,
j Γ(3+m +j) p p
2 p=1
andA isaninfiniteupper-triangularmatrixwithnonzeroentries
p
l
(cid:88)
(3.11) (A ) := kp As , s,l=0,1,2,... .
p s,2l+s+1 j,l−j l−j
j=0
Itcaneasilybeseenthatbyproceedinginthesamemannerasin(3.9)–(3.11),wefind
that
x x
(cid:90) K(x,t)s (t) (cid:90) K(x,t)
(3.12) √ 0 dt=s √ X dt=s AX,
0 t 0
x−t x−t
0 0
where the infinite upper-triangular matrix A is obtained from (3.11) with k in place
j,l−j
ofkp .
j,l−j
Now,togetthealgebraicformoftheMüntz-Galerkindiscretizationof(1.1),itissufficient
thatweinserttherelations(3.2),(3.10),and(3.12)into(3.6). Thuswehave
(cid:0) (cid:1)
(3.13) uL(id−Q )X = f +s A X,
p 0
whereidistheinfinite-dimensionalidentitymatrix. UsingX =L−1L,wecanrewrite(3.13)
as
(3.14) uL(id−Q )L−1L=(cid:0)f +s A(cid:1)L−1L.
p 0
Becauseoftheorthogonalityof{L (x)}∞ ,projecting(3.14)onto{L (x)}N yields
k k=0 k k=0
(3.15) uNLN(cid:16)idN −QN(cid:17)(cid:0)LN(cid:1)−1 =(cid:16)fN +s NAN(cid:17)(cid:0)LN(cid:1)−1,
p 0
whereuN =[u ,u ,...,u ], fN =[f ,f ,...,f ], s N =[s0,s1,...,sN],andLN,idN,
0 1 N 0 1 N 0 0 0 0
QN,(cid:0)LN(cid:1)−1,AN aretheprincipalsubmatricesoforderN+1ofthematricesL,id,Q ,L−1,
p p
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190 P.MOKHTARY
andA,respectively. Sincetheproductofupper-triangularmatricesgivesanupper-triangular
matrix,thematrixQ definedin(3.11)hasanupper-triangularstructure. Consequently,the
p
nonlinearalgebraicsystem(3.15)hasanupper-triangularstructure,whosesolutionyieldsthe
unknowncomponentsofthevectoruN. Inthenextsectionwepresentmoredetailsregarding
thecomplexityanalysisanduniquesolvabilityofthissystem.
REMARK3.2. Clearly,forG(t,u(t))=up(t),thesystem(3.15)canberepresentedas
(3.16) uNLN(cid:16)idN −(cid:0)Qp−1(cid:1)NAN(cid:17)(cid:0)LN(cid:1)−1 =fN(cid:0)LN(cid:1)−1, p(cid:54)=1,
(3.17) uNLN(cid:16)idN −AN(cid:17)(cid:0)LN(cid:1)−1 =fN(cid:0)LN(cid:1)−1, p=1(linearcase),
where(cid:0)Qp−1(cid:1)N istheprincipalsubmatrixoforderN +1fromQp−1.
3.2. Numericalsolvabilityandcomplexityanalysis. Themainobjectofthissectionis
toprovideanexistenceanduniquenesstheoremforthenonlinearalgebraicsystem(3.16)and
itscomplexityanalysis. Inotherwords,weexplainhowwecancomputetheunknownvector
ubysolving(3.16)byawell-posedtechnique. Furthermore,weextendthefollowinganalysis
alsotothesystem(3.15).
Multiplyingbothsidesof(3.16)byLN yields
(3.18) uNLN(cid:16)idN −(cid:0)Qp−1(cid:1)NAN(cid:17)=fN.
Bydefining
(3.19) u=uNLN =[u ,u ,...,u ],
0 1 N
wecanrewrite(3.18)as
u(cid:16)idN −(cid:0)Qp−1(cid:1)NAN(cid:17)=fN.
FromthestructureofthematrixQinLemma3.1,wehave
uL0 uL1 uL2 ··· u0 u1 u2 ···
Q:= 00 uL00 uuLL10 ······=00 u00 uu10 ······.
 ... ... ... ...  ... ... ... ...
From[14],weobservethat(cid:0)Qp−1(cid:1)N hasthefollowingupper-triangularToeplitzstruc-
ture:
(cid:0)Qp−1(cid:1)N
(cid:16) (cid:17)p−1 (cid:16) (cid:17)p−2 (cid:16) (cid:17)p−3(cid:18) (cid:16) (cid:17)2 (cid:19) 
u (p−1) u u 1(p−1) u (p−2) u +2u u ···
 0 0 1 2 0 1 0 2 
 (cid:16) (cid:17)p−1 (cid:16) (cid:17)p−2 
 
 0 u0 (p−1) u0 u1 ··· 
 
= (cid:16) (cid:17)p−1 
 0 0 u ··· 
 0 
 . . . . 
 .. .. .. .. 
 
 (cid:16) (cid:17)p−1
0 ··· 0 u
0
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OPERATIONALMÜNTZ-GALERKINAPPROXIMATION 191
Qp−1(u ) Qp−1(u ,u ) Qp−1(u ,u ,u ) ··· Qp−1(u ,u ,...,u ) 
0,0 0 0,1 0 1 0,2 0 1 2 0,N 0 1 N
 0 Qp−1(u ) Qp−1(u ,u ) ··· Qp−1 (u ,u ,...,u )
 0,0 0 0,1 0 1 0,N−1 0 1 N−1 
= . . . . ,
 . . . . 
 . . . ··· . 
0 0 0 ··· Qp−1(u )
0,0 0
whereQp−1(u ,u ,...,u ),i,j =0,1,...,N,arenonlinearfunctionsoftheelementsu ,
i,j 0 1 j 0
u ,...,u . Thus,fromthestructureoftheupper-triangularmatrixAdefinedby(3.11),we
1 j
obtainthat
 
0 A 0 A ··· ···
0,1 0,3
0 0 A1,2 0 A1,4 ··· 
 
(3.20) (cid:0)Qp−1(cid:1)NAN =(cid:0)Qp−1(cid:1)N0 0 0 A2,3 ... ... 
... ... ... ... ... ... 
 
0 0 0 ··· 0 AN−1,N
0 0 0 0 0 0
0 Qp0,−01(u0)A0,1 Qp0,−11(u0,u1)A1,2 Qp0,−01(u0)A0,3+Qp0,−21(u0,u1,u2)A2,3 ··· 
0 0 Qp0,−01(u0)A1,2 Qp0,−11(u0,u1)A2,3 ··· 
=0.. 0.. 0.. Qp0,−01(u..0)A2,3 ··..· .
. . . . . 
 
0 0 0 0 Qp0,−01(u0)AN−1,N
0 0 0 0 0
Nowwiththeaboverelation,wecanwrite
(3.21) uQp−1A=[0,F (u ),F (u ,u ),...,F (u ,u ,...,u )],
1 0 2 0 1 N 0 1 N−1
whereF (u ,u ,...,u ),fori=1,2,...,N,arenonlinearfunctionsoftheelementsu ,
i 0 1 i−1 0
u ,...,u . Substituting(3.21)into(3.16)yields
1 i−1
u0 f0  0 
u...1=f...1+ F1(...u0) ,
uN fN FN(u0,u1,...,uN−1)
andtherefore
u =f ,
0 0
u =f +F (u ),
1 1 1 0
(3.22)
.
.
.
u =f +F (u ,u ,...,u ),
N N N 0 1 N−1
givestheunknowncomponentsofthevectoru. Finally,(3.19)yieldstheunknownvectoruN
intheGalerkinrepresentation(3.2). Theanalysisaboveissummarizedinthenexttheorem.
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KentStateUniversity
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192 P.MOKHTARY
THEOREM3.3. Thenonlinearsystem(3.18)hasauniquesolutionuN whichisobtained
by(3.22)and(3.19).
Obviously, this analysis can be extended similarly to the system (3.15) yielding the
followingresult:
THEOREM3.4. Thenonlinearsystem(3.15)hasauniquesolutionuN.
Proof. ByproceedinginasimilarmannerasinSection3.2,wecanrewrite(3.15)as
(cid:16) (cid:17)
u idN −QN =fN,
p
wherefN =fN +s NAN. From(3.11),(3.20),theupper-triangularstructureofthematrix
0
(cid:0)Qp−1(cid:1)NAN,andsinceasummationofupper-triangularmatricesyieldsanupper-triangular
p
matrix,wesimilarlyobtainthat
(3.23) uQN =[0,F˜ (u ),F˜ (u ,u ),...,F˜ (u ,u ,...,u )],
p 1 0 2 0 1 N 0 1 N
whereF˜(u ,u ,...,u ),fori=1,2,...,N,arenonlinearfunctionsoftheelementsu ,
i 0 1 i−1 0
u ,...,u . Byadoptingthesamestrategyasin(3.21)–(3.22),wecanconcludetheresult.
1 i−1
4. Numerical results. In thissection, we apply theproposed methodof the previous
sectiontoapproximatethesolutionof(1.1)andprovidesomeexamples. Thecomputations
wereperformedusingMathematicawithdefaultprecision. Inthetablesbelow,thelabels"Nu-
mericalerrors"and"Order"alwaysrefertotheL2-normoftheerrorfunctionandtheorderof
convergence,respectively. Wealsocompareourresultswiththoseobtainedbyseveralexisting
methodsforthisclassofproblemsandtherebyconfirmthesuperiorityandeffectivenessofour
scheme. Wealsoillustratethewell-posednessoftheproblemandthusnumericallyverifythe
theoreticalresultofTheorem1.2. InallexampleswehaveN =N =N.
G f
EXAMPLE4.1. Considerthefollowingintegralequations:
√ √ x
√ 4 x3 12 x5 (cid:90) 1+x+t
(4.1) u(x)= x− − + √ u2(t)dt,
3 5 x−t
0
x
πx √ (cid:90) u(t)
(4.2) u(x)= + x− √ dt,
2 x−t
0
x
√ 3πx2 (cid:90) u3(t)
(4.3) u(x)= x+ − √ dt,
8 x−t
0
√
whichallhavethesameexactsolutionu(x)= x.
ApplyingthemethoddescribedintheprevioussectionwithN =1,theMüntz-Galerkin
approximationoftheseproblemsisgivenby
u (x)=u1L1X1,
1
where
(cid:20) 1 0(cid:21) √
u1 =[u ,u ], L1 = , X1 =[1, x]T.
0 1 −2 3