CONVERGENCE ANALYSIS OF THE OPERATIONAL TAU METHOD FOR ABEL-TYPE ...

17 Pages·2014·0.24 MB·English

Electronic Transactions on Numerical Analysis. ETNA
Volume 41, pp. 289-305, 2014. Kent State University
Copyright2014, Kent State University. http://etna.math.kent.edu
ISSN 1068-9613.
CONVERGENCEANALYSISOFTHEOPERATIONALTAUMETHODFOR
ABEL-TYPEVOLTERRAINTEGRALEQUATIONS
∗
P.MOKHTARY†AND F.GHOREISHI‡
Abstract.Inthispaper,aspectralTaumethodbasedonJacobibasisfunctionsisproposedanditsstabilityand
convergencepropertiesareconsideredforobtaininganapproximatesolutionofAbel-typeintegralequations. This
workisorganizedintwoparts. First,wepresentastableoperationalTaumethodbasedonJacobibasisfunctions
thatprovidesanefficientapproximatesolutionfortheAbel-typeintegralequationsbyusingareducedsetofmatrix
operations. WealsoprovidearigorouserroranalysisfortheproposedmethodintheweightedL2-anduniform
norms under more general regularity assumptions on the exact solution. It is shown that the proposed method
converges,butsincethesolutionsoftheseequationshaveasingularityneartheorigin,alossintheconvergence
orderoftheTaumethodisexpected.Toovercomethisdrawbackwethenproposearegularizationprocess,inwhich
theoriginalequationischangedintoanewequationwhichpossessesasmoothsolution, byapplyingasuitable
variabletransformationsuchthatthespectralTaumethodcanbeappliedconveniently. Wealsoprovethatafter
thisregularizationtechnique,thenumericalsolutionofthenewequationbasedontheoperationalTaumethodhas
exponentialrateofconvergence. Somestandardexamplesareprovidedtoconfirmthereliabilityoftheproposed
method.
Keywords.OperationalTaumethod,Abel-typeVolterraintegralequations
AMSsubjectclassifications.45E10,41A25
1. Introduction. This paper deals with the numerical treatment of the following class
ofAbel-typeVolterraintegralequations
x K(x,t)
(1.1) u(x)=f(x)+λ u(t)dt, x I =[0,1],
√µx t ∈
Z0 −
where f(x) and K(x,t) are a given continuous function and a sufficiently smooth kernel
function, respectively, u(x) is the unknown function, and µ N with µ 2 (N is the col-
∈ ≥
lectionsofallnaturalnumbers). ThisequationisaparticularcaseoflinearVolterraweakly
singular integral equations of the second kind. Several numerical methods have been pro-
posedfor(1.1);see[2,3,5,6,8,9,11].
From the well-known existence and uniqueness theorems, it can be concluded that in
Abel-type integralequations wemustexpect somederivatives ofthesolutiontohave adis-
continuityattheorigin; see[3,8]. Asdiscussedin[3,8],thefirstderivativeofthesolution
1
u(x)behaveslikeu′(x) x−µ andu′(x) / C(I).
∼ ∈
In order to approximate the solution of (1.1) we propose a strategy mainly consisting
in two steps. In the first step, we introduce the operational Tau method based upon Jacobi
basisfunctionsto(1.1). Thisstrategyisanapplicationofthematrix-vectorproductapproach
in spectral approximation. The main characteristic behind this approach is that it reduces
such problems to those of solving a system of linear algebraic equations. In this step, we
alsoinvestigateconvergenceandstabilitybehaviorofthenumericalsolutionof(1.1). Wecan
deduceconvergenceoftheTaumethodatthispoint,butduetothefactthatthesolutionsof
theseequationsusuallyhaveaweaksingularityattheorigin,thismethodleadstoverypoor
numericalresults. Thus,itisnecessarytointroducearegularizationprocedurethatallowsus
∗ReceivedFebruary26,2013.AcceptedMay21,2014.PublishedonlineonOctober2,2014.Recommendedby
A.Wathen.
†DepartmentofMathematics,SahandUniversityofTechnology,Tabriz,Iran
([email protected], [email protected]).
‡DepartmentofMathematics,K.N.ToosiUniversityofTechnology,Tehran,Iran
([email protected]).
289
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290 P.MOKHTARYANDF.GHOREISHI
toimprovethesmoothnessofthegivenfunctionsandthentoapproximatethesolutionwitha
satisfactoryorderofconvergence.In[8],TaoTangintroducedanddiscussedasimplevariable
transformationfortheregularizationofsolutionsofAbel-typeVolterraintegralequationsof
the second kind, such that the resulting equation possesses a smooth solution. Then, in the
secondstepwetransform(1.1)byusingthiscoordinatetransformationandweprovethatafter
this regularization technique the numerical solution of the new equation by the operational
Taumethodhasexponentialrateofconvergence.
Theorganizationofthispaperisasfollows: inSection2, weintroducetheoperational
Tau method and its application to the Abel-type integral equation (1.1). In addition, we
analyzeconvergenceandstabilitybehaviorofthenumericalsolutionof(1.1).InSection3,by
applyingasuitableregularizationprocess,anoperationalschemeforobtainingthenumerical
solution of the transformed equation is considered and an exponential rate of convergence
for the proposed regularized scheme is proved. In Section 4, some numerical examples are
consideredwhichconfirmourtheoreticalpredictions.
2. Operational Tau method for Abel integral equations. In this section, we present
a numerical solution of (1.1) by using the operational Tau method based on Jacobi basis
functions.
2.1. Numerical treatment. Let Vα,β := [vα,β(x),vα,β(x),...,vα,β(x),...]T with pa-
0 1 N
rameters α,β ( 1,1) be an arbitrary Jacobi polynomial bases with respect to the inner
∈ −
product
(vα,β,vα,β) = vα,β(x)vα,β(x)wα,β(x)dx,
i j α,β i j
ZI
wherewα,β(x)=(2 2x)α(2x)β istheJacobiweightfunction;see[4,10]. Clearly,wecan
writeVα,β :=Vα,βX−, whereVα,β isaninfinitelynon-singularlowertriangularcoefficient
matrixwithdegree(vα,β(x)) i,fori=0,1,2,...,andX =[1,x,x2,...,xN,...]T.
i ≤
Supposethatf(x)isagivenpolynomialandconsider
Nf
(2.1) f(x)= f vα,β(x)=fVα,β =fVα,βX, f =(f ,f ,f ,...,f ,0,...).
i i 0 1 2 Nf
i=0
X
Iff(x)isnotpolynomial,thenitcanbeapproximatedbypolynomialstoanydegreeofaccu-
racybyinterpolationoranyothersuitablemethod. Letu(x)= ∞i=0aiviα,β(x)=aVα,βX
be the Jacobi series expansion of the exact solution of (1.1), where a = (a ,a ,...) and
0 1
P
uα,β(x)isaTauapproximationofdegreeN foru(x)as
N
(2.2) uα,β(x)= ∞ a vα,β(x)=a Vα,βX, a =(a ,a ,...,a ,0,...).
N i i N N 0 1 N
i=0
X
Wedefine
x K(x,t)
(2.3) L(u(x))=u(x) λ u(t)dt
− √µx t
Z0 −
andassumethat
K(x,t)= ∞ ∞ k vα,β(x)vα,β(t),
ij i j
i=0j=0
XX
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OPERATIONALTAUMETHODFORABEL-TYPEINTEGRALEQUATIONS 291
whichcanberearrangedas
K(x,t)= ∞ ∞ k˜ xitj.
ij
i=0j=0
XX
Now,thefollowingtheoremshowshowtoreplace(1.1)byamatrixformulationoftheoper-
ationalTaumethod.
THEOREM 2.1. Let vα,β(x) be the orthogonal Jacobi polynomials with respect to the
j
weightfunctionwα,β(x)inI andlettheapproximatesolutionuα,β(x)begivenbytherela-
N
tion(2.2). Thenwehave
L(uα,β(x))=a id λVα,βΠα,β Vα,β,
N N −
whereidistheinfiniteidentitymatrix, (cid:0) (cid:1)
Πα,β = ∞ ∞ k˜ j i,j,
ij
B C
i=0j=0
XX
Bj is the infinite diagonal matrix with diagonal entries Bsjs = B(j +s+1,µ−µ1), for all
j,s=0,1,2,... (hereB(a,b)denotestheBetafunction),and
Ci,j = clk ∞l,k=0 = (cid:0)xi+jv+αl,+β1,−vαµ1,,βvkα,β(cid:1)α,β.
k k α,β
(cid:0) (cid:1)
(cid:0) (cid:1)
Proof. Fromrelations(2.2)and(2.3)wecanwrite
x K(x,t)
(2.4) L(uα,β(x))=a Vα,β λVα,β X dt ,
N N − √µx t t
(cid:18) Z0 − (cid:19)
whereX =[1,t,t2,...,tN,...]T. Thus,theintegraltermof(2.4)canbewrittenas
t
(2.5) x K(x,t) X dt= ∞ ∞ k˜ xi x tj+l dt ∞ .
t ij
õx t õx t
Z0 − i=0j=0 (cid:20)Z0 − (cid:21)l=0
XX
Byusingtherelation
x tj+l µ 1
dt=xj+l+1−µ1B(j+l+1, − ),
√µx t µ
Z0 −
wecanrewrite(2.5)as
x K(x,t) Xtdt= ∞ ∞ k˜ij xi+j+l+1−µ1B(j+l+1,µ−1) ∞
√µx t µ
Z0 − i=0j=0 (cid:20) (cid:21)l=0
XX
(2.6) = ∞ ∞ k˜ij j xi+j+l+1−µ1 ∞ ,
B
i=0j=0 (cid:20) (cid:21)l=0
XX
Substitutingxi+j+l+1−µ1,i,j,l=0,1,2,...,byitsorthogonalJacobiprojectionpolynomial
ofdegreeN weget(see[4,10])
(2.7) xi+j+l+1−µ1 ∞ = ∞ clkvkα,β(x) ∞ =Ci,jVα,β,
(cid:20) (cid:21)l=0 (cid:20)k=0 (cid:21)l=0
X
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where
Cij = clk ∞l,k=0 = (cid:0)xi+jv+αl,+β1,−vαµ1,,βvkα,β(cid:1)α,β, i,j =0,1,2,... .
k k α,β
(cid:0) (cid:1)
(cid:0) (cid:1)
Bysubstituting(2.7)into(2.6)weobtain
(2.8) x K(x,t) X dt= ∞ ∞ k˜ j i,j Vα,β =Πα,βVα,β,
t ij
Z0 √µx−t i=0j=0 B C !
XX
where Πα,β = ∞i=0 ∞j=0k˜ijBjCi,j, and inserting (2.8) into (2.4) we can conclude the
result.
P P
We are now ready to obtain the following algebraic form of the operational Tau dis-
cretizationof(1.1)basedonJacobipolynomials. Accordingtotheproposedmethod,follow-
ingTheorem2.1andrelation(2.1),weobtain
(2.9) a id λVα,βΠα,β Vα,β =fVα,β.
N
−
(cid:0) (cid:1)
If we let α,β = id λVα,βΠα,β, then, because of the orthogonality of vα,β(x)
A − { k }∞k=0
(see[4,10]),projecting(2.9)onto vα,β(x) N yields
{ k }k=0
a α,β =f ,
NAi i
where α,β isthei-thcolumnof α,β. Bysetting
Ai A
˜α,β =[ α,β, α,β,..., α,β], f˜ =[f ,f ,...,f ],
AN A0 A1 AN N 0 1 N
weobtaina ˜α,β =f˜ ,whichgivesustheunknownvector(a ,a ,...,a ).
NAN N 0 1 N
2.2. Stabilityandconvergence analysis. Inthissection, weprovideasuitablestabil-
ityanderroranalysiswhichtheoreticallyjustifiesstabilityandconvergenceoftheproposed
methodforthenumericalsolutioninthespecialcaseof(1.1)whenK(x,t)=1,i.e.,
x u(t)
(2.10) u(x)=f(x)+λ dt, x I.
√µx t ∈
Z0 −
Inoursubsequentanalysis,somepreliminaryresultsareneeded. Throughoutthepaper,
C willdenoteagenericpositiveconstantthatisindependentofN. LetL2 (I)bethespace
α,β
offunctionswhosesquareisLebesqueintegrableinIrelativetotheweightfunctionwα,β(x).
ThelatterisaBanachspacewiththenorm(see[4])
v 2 =(v,v) = v2(x)wα,β(x)dx, v L2 (I).
k kα,β α,β ∀ ∈ α,β
ZI
Bk (I)denotesthenon-uniformJacobi-Sobolevspaceofallfunctionsv(x)onI suchthat
α,β
v(x) and its weak derivatives of order s are in L2 (I) for 0 s k. We define the
α+s,β+s ≤ ≤
norm(see[10])
k
v 2 = v(s) 2 .
k kk,α,β α+s,β+s
s=1
X(cid:13) (cid:13)
(cid:13) (cid:13)
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OPERATIONALTAUMETHODFORABEL-TYPEINTEGRALEQUATIONS 293
It is trivial that v = v and v(k) v . The Ho¨lder space
0,α,β α,β α+k,β+k k,α,β
k k k k k k ≤ k k
Ck,γ(I)wherek 0isaninteger,consistsofthosefunctionsonI havingcontinuousderiva-
≥
tives up to order k and such that the kth derivative is Ho¨lder continuous with exponent γ
where0<γ 1. IftheHo¨lderconstant
≤
v(x) v(t)
v = sup | − |,
| |0,γ x tγ
x=t I
6 ∈ | − |
is finite, then the function v is said to be Ho¨lder continuous with exponent γ in I. If the
functionvanditsderivativesuptoorderkareboundedonI,thentheHo¨lderspaceCk,γ(I)
canbeassignedthenorm
v = v +max Dηv ,
k,γ k 0,γ
k k k k η=k| |
| |
where η ranges over multi-indices and v = maxsup Dηv . When γ = 0, Ck,0(I)
k
k k η kx I| |
denotesthespaceoffunctionswithkcontinuousde|ri|v≤ativ∈esonI,alsodenotedbyCk(I)and
withnorm . ; see[1]. Itcanbeeasilyseenthatthefunctionv(x) = xs with0 < s 1
k
kk ≤
definedonI belongstothespaceC0,γ(I)for0 < γ s.Wefurtherdefinealinearweakly
≤
singularintegraloperator
x K˜(x,t)
(v)= v(t)dt, x,t I,
K √µx t ∈
Z0 −
whereK˜(x,t)issufficientlysmoothkernelfunctionandµisanaturalnumberwithµ 2. It
≥
canbeproventhatforanycontinuousfunctionv,thereexistapositiveconstantC suchthat
1
(2.11) (v) C0,γ(I), (v) C v , γ 0,1 ,
0,γ
K ∈ kK k ≤ k k∞ ∈ − µ
(cid:16) (cid:17)
where . istheusualuniformnorm. Theproofof(2.11)canbefoundin[5].
kk∞
Let bethespaceofallalgebraicpolynomialsofdegreeuptoN. Now,weintroduce
N
the orthoPgonal projection α,β : L2 (I) which is a mapping such that for any
PN α,β → PN
v L2 (I),
∈ α,β
(v α,βv,φ) =0, φ .
−PN α,β ∀ ∈PN
ConcerningthetruncationerrorofaJacobiseries,thefollowingestimateshold(see[1,10])
(2.12) v α,βv CNk s v(s) , v Bs (I), s k, k 0,
k −PN kk,α,β ≤ − α+s,β+s ∈ α,β ≥ ≥
(cid:13) (cid:13)
(cid:13) (cid:13)
v α,βv C 1+σ (N) N (k+γ) v ,
(2.13) −PN ≤ p − k kk,γ
∞
(cid:13) (cid:13) (cid:0) (cid:1) v Ck,γ(I), k 0, γ [0,1],
(cid:13) (cid:13) ∈ ≥ ∈
where
O(log(N)) 1<α,β 1,
σ (N)= − ≤−2
p (O(Ns+21) otherwise,
withs=max α,β ,isthewellknownLebesgueconstant.
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InouranalysisweshallapplytheHardyandGronwallinequalities(see[5,7]):
LEMMA 2.2 (GeneralizedHardy’sinequality[7]). Formeasurablefunctionsg 0,the
≥
followinggeneralizedHardy’sinequality
1/q 1/p
b b
(λg)(t)qw (t)dt C g(t)pw (t)dt ,
1 2
Za | | ! ≤ Za | | !
holdsifandonlyif
as<ut<pb Ztbw1(t)dt!1/q(cid:18)Zatw21−p′(t)(cid:19)1/p′ <∞, p′ = p−p 1,
for1<p q < . Here,λisanoperatoroftheform
≤ ∞
t
(λg)(t)= k(t,s)g(s)ds,
Za
withgivenkernelk(t,s)andweightfunctionsw (t),w (t)for a<b .
1 2
−∞≤ ≤∞
LEMMA 2.3 (Gronwall inequality [5]). Assume that v(x) is a non-negative, locally
integrablefunctiondefinedonI whichsatisfies
x
v(x) b(x)+B (x t)mtnv(t)dt, t I,
≤ − ∈
Z0
whereb(x) 0andB 0. ThenthereexistaconstantC suchthat
≥ ≥
x
v(x) b(x)+C (x t)mtnb(t)dt, t I.
≤ − ∈
Z0
Now,westateandprovethemainresultsofthissectionregardingthestabilityanderror
analysisoftheproposedmethodforthenumericalsolutionof(2.10).
THEOREM 2.4 (Stability). Letuα,β(x)betheTauapproximation(2.2)totheexactso-
N
lutionu(x)oftheAbelintegralequation(2.10). Assumethatthefunctionf(x)iscontinuous
and λ <1. Alsosupposeu˜ andf˜ C(I)aretheerrorsofuα,β andf,respectively.
| | ∈PN ∈ N
Then,wehave
u˜ C f˜ .
k kα,β ≤ α,β
Proof. Weknowthatuα,β anduα,β +u˜sati(cid:13)(cid:13)sfy(cid:13)(cid:13)thefollowingequations
N N
x uα,β(t)
(2.14) uα,β(x)= α,βf(x)+λ α,β N dt,
N PN PN õx t
Z0 −
x uα,β(t)+u˜(t)
(2.15) uα,β(x)+u˜(x)= α,β f(x)+f˜(x) +λ α,β N dt.
N PN PN õx t
Z0 (cid:0) − (cid:1)
(cid:0) (cid:1)
Subtracting(2.15)form(2.14)weget
x u˜(t)
u˜(x)= α,βf˜(x)+λ α,β dt
PN PN õx t
Z0 −
andthen
x u˜(t)
(2.16) u˜ α,βf˜ + λ α,β dt .
k kα,β ≤ PN α,β | | PN √µx t
(cid:13) Z0 − (cid:13)α,β
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13)
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OPERATIONALTAUMETHODFORABEL-TYPEINTEGRALEQUATIONS 295
Since α,β isanorthogonalprojection, α,β =1andfrom(2.16)weobtain
PN kPN kα,β
x u˜(t)
u˜ f˜ + λ dt .
α,β α,β
k k ≤k k | | √µx t
(cid:13)Z0 − (cid:13)α,β
(cid:13) (cid:13)
Using Hardy’s inequality (i.e., Lemma 2.2) in t(cid:13)(cid:13)he integral term(cid:13)(cid:13)of the (2.13), we obtain the
desiredresult.
Inthefollowingtheoremweprovideanerroranalysiswhichtheoreticallyjustifiescon-
vergenceoftheproposedschemeforthenumericalsolutionof(2.10)intheL2 anduniform
norms.
THEOREM 2.5 (Convergence). Letuα,β(x)betheTauapproximation(2.2)totheexact
N
solutionu(x)oftheAbelintegralequation(2.10). Ifu(x) Ck,γ(I) Bm (I)withk 0,
∈ ∩ α,β ≥
γ (0,1],andm 0,thenwehave
∈ ≥
Clog(N)N (γ+k) u for 1<α,β 1,
eα,β(u) − k kk,γ − ≤−2
(cid:13) N (cid:13)∞ ≤CN21+s−γ−kkukk,γ forγ+k > 12, s=max{α,β}<0,
(cid:13) (cid:13)
aswellas, 
eαN,β(u) α,β ≤C N−mkukm,α,β +log(N)N−γ1 eαN,β(u) ,
∞
(cid:13) (cid:13) (cid:16) (cid:13) (cid:13) (cid:17)
forγ (0,(cid:13)1 1),(cid:13) 1<α,β 1,and (cid:13) (cid:13)
1 ∈ − µ − ≤−2
eαN,β(u) α,β ≤C N−mkukm,α,β +N12+s−γ1 eαN,β(u) ,
∞
(cid:13) (cid:13) (cid:16) (cid:13) (cid:13) (cid:17)
forγ (1(cid:13)+s,1 (cid:13)1),s = max α,β < 0,whereeα,β(u(cid:13)) = u(x)(cid:13) uα,β(x)isdefined
1 ∈ 2 − µ { } N − N
aserrorfunction.
Proof. Consideringequation(2.10),accordingtotheproposedmethodwehave
x uα,β(t)
(2.17) uα,β(x)= α,βf +λ α,β N dt.
N PN PN õx t
Z0 −
Subtracting(2.10)from(2.17),wehave
x u(t) x uα,β(t)
(2.18) eα,β(u)=eα,β(f)+λ dt α,β N dt ,
N PN Z0 √µx−t −PN Z0 √µx−t !
whereeα,β(v)=v α,βv. Bysomesimplemanipulationswecanrewrite(2.18)asfollows
PN −PN
x u(t)
eα,β(u)=eα,β(f)+λ dt
N PN Z0 √µx−t
x u(t) x eα,β(u) x eα,β(u)
α,β dt+ N dt eα,β N dt .
−PN Z0 √µx−t Z0 √µx−t − PN Z0 √µx−t !
From(2.10)wehaveλ x u(t) dt=u(x) f(x)andcanrewritetheaboverelationas
0 √µx t −
−
R
x eα,β(u) x eα,β(u)
eα,β(u)=eα,β(u)+λ N dt eα,β N dt ,
N PN Z0 √µx−t − PN Z0 √µx−t !
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296 P.MOKHTARYANDF.GHOREISHI
whichyields
x eα,β(u) x eα,β(u)
(2.19) eα,β(u) eα,β(u) λeα,β N dt + λ | N |dt.
(cid:12) N (cid:12)≤(cid:12)(cid:12) PN − PN Z0 √µx−t (cid:12)(cid:12) | |Z0 √µx−t
(cid:12) (cid:12) (cid:12) (cid:12)
UsingGronwa(cid:12)ll’sinequ(cid:12)ality(cid:12) (Lemma2.3)in(2.19),wecan(cid:12)write
(cid:12) (cid:12)
x eα,β(u)
(2.20) eα,β(u) C eα,β(u) + eα,β N dt .
(cid:13) N (cid:13)∞ ≤ (cid:18)(cid:13) PN (cid:13)∞ (cid:13)(cid:13) PN Z0 √µx−t (cid:13)(cid:13)∞(cid:19)
Byapplyingthe(cid:13)relation(cid:13)s(2.11)an(cid:13)d(2.13)i(cid:13)n(2.20(cid:13)),weobtain (cid:13)
(cid:13) (cid:13)
x eα,β(u)
eαN,β(u) ≤C 1+σp(N) N−(γ+k)kukk,γ +N−γ1 √µNx tdt
∞ (cid:18) (cid:13)Z0 − (cid:13)0,γ1(cid:19)
(cid:13) (cid:13) (cid:0) (cid:1) (cid:13) (cid:13)
(cid:13) (cid:13) ≤C 1+σp(N) N−(γ+k)kukk,γ +N−γ1(cid:13)(cid:13)eαN,β(u) , (cid:13)(cid:13)
∞
(cid:0) (cid:1)(cid:16) (cid:13) (cid:13) (cid:17)
where k 0,γ [0,1] and γ (0,1 1). The first result o(cid:13)f the the(cid:13)orem regarding the
≥ ∈ 1 ∈ − µ
errorestimateintheuniformnormcanbeobtainedunderthefollowingconditions
1 1
0<γ <1 1<α,β ,
1
− µ − ≤−2
(2.21)
1 1
+s<γ <1 s=max α,β <0.
1
2 − µ { }
Now,wederiveasuitableerrorboundfortheproposedschemeintheL2-norm. Tothis
end,byapplyingagainGronwall’sinequality(Lemma2.3)in(2.19),wewrite
x eα,β(u)
eα,β(u) C eα,β(u) + eα,β N dt
N α,β ≤ (cid:18) PN α,β (cid:13) PN Z0 √µx−t (cid:13)α,β(cid:19)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) x eα,β(u) (cid:13)
C eα,β(u) +(cid:13)eα,β N dt(cid:13) .
≤ (cid:18)(cid:13) PN (cid:13)α,β (cid:13)(cid:13) PN Z0 √µx−t (cid:13)(cid:13)∞(cid:19)
Usingrelations(2.11)and(2.13)in(cid:13)theinteg(cid:13)ralterm(cid:13)oftheaboveequatio(cid:13)nyields
(cid:13) (cid:13)
(2.22) eαN,β(u) α,β ≤C(cid:18) eαP,Nβ(u) α,β + 1+σp(N) N−γ1 eαN,β(u) ∞(cid:19),
(cid:13) (cid:13) (cid:13) (cid:13) (cid:0) (cid:1) (cid:13) (cid:13)
whereγ sati(cid:13)sfies(2.21(cid:13)). (cid:13) (cid:13) (cid:13) (cid:13)
1
Finally,thesecondresultofthetheoremcanbededucedbyadoptingtherelation(2.12)
in(2.22).
In general, the exact solution u(x) of the Abel integral equation (2.10) behaves like
x1−µ1. Thus, u(x) ∈ C0,γ(I) with 0 < γ ≤ 1− µ1. In this case, in Theorem 2.5 we have
m=1,k =0andγ (0,1 1]. Asaresultwecandeduceconvergenceoftheoperational
∈ − µ
Tau method from Theorem 2.5 by choosing suitable values of α,β. But due to the low
smoothness of the exact solution, this scheme leads to a low order numerical method. To
overcomethisdrawback,inthenextsectionweproposearegularizationprocessinwhichthe
originalequation(1.1)willbechangedintoanewequationwhichpossessesasmoothsolution
byapplyingavariabletransformationintroducedbyTaoTangin[8]. Alsoforequationswith
highorderofsmoothnessintheexactsolutionwecandeducetheconvergenceoftheproposed
methodwithlargervaluesofm,k,andγ andobtainahigherrateofconvergence. Itistrivial
thatinthiscase,wedonotrequiretheregularizationprocess.
ETNA
Kent State University
http://etna.math.kent.edu
OPERATIONALTAUMETHODFORABEL-TYPEINTEGRALEQUATIONS 297
3. OperationalTaumethodforregularizedAbelintegralequations. Itiswellknown
that the spectral Tau method is an efficient tool for solving the differential equations with
smoothsolutions. InordertomakeitefficientfortheAbelintegralequation(1.1),theorigi-
nalequationwillbechangedintoanewintegralequationwhichpossessesasmoothsolution,
byapplyingasuitablevariabletransformation. Ourapproachtochoosingthepropertransfor-
mationisbasedupontheimpressivepaper[8]. Considerequation(1.1). Weknowthatnear
1
x = 0thefirstderivativeofthesolutionu(x)behaveslikeu′(x) x−µ. Toovercomethis
≃
difficulty,weapplythevariabletransformation
x=zµ, z = √µx, t=wµ, w = √µt,
andchangetheAbelintegralequation(1.1)asfollows
z K¯(z,w)
(3.1) u¯(z)=f¯(z)+λ u¯(w)dw, z I,
√µz w ∈
Z0 −
where
µwµ 1K(zµ,wµ)
f¯(z)=f(zµ), K¯(z,w)= − ,
µ µj=−01zµ−1−jwj
q
andu¯(z) = u(zµ)isthesmoothsolutionofequationP(3.1). Sincetheexactsolutionof(1.1)
canbewrittenasu(x) = u¯(z),wecandefineu˜α,β(x)=u¯α,β(z),x,z I,astheapproxi-
N N ∈
matesolutionofproblem(1.1).
3.1. Numerical treatment. Assume that u¯α,β(z) is the spectral Tau approximation of
N
degreeN for(3.1)as
N
(3.2) u¯α,β(z)= b vα,β(z)=b Vα,βZ =b Vα,β,
N j j N N
j=0
X
whereb =[b ,b ,...,b ,0,...],Z =[1,z,z2,...,zN,...]T. Letusdefine
N 0 1 N
z K¯(z,w)
(3.3) L¯(u¯(z))=u¯(z) λ u¯(w)dw,
− √µz w
Z0 −
andassumethat
K(zµ,wµ)= ∞ ∞ k¯ vα,β(z)vα,β(w),
ij i j
i=0j=0
XX
whichwecanrearrangeas
K(zµ,wµ)= ∞ ∞ k˜¯ ziwj.
ij
i=0j=0
XX
Now,wefindtheunknownvectorb in(3.2)usingtheoperationalTaumethodbasedon
N
theJacobipolynomialsaccordingtothefollowingtheorem.
THEOREM3.1. Letvα,β(z)betheorthogonalJacobipolynomialswithrespecttoweight
j
functionwα,β(z)inI. Assumethattheapproximatesolutionu¯α,β(z)isgivenby(3.2). Then
N
wehave
1
L¯(u¯α,β(z))=b id λVα,βΠ¯α,β Vα,β − Vα,β,
N N −
(cid:16) (cid:16) (cid:17) (cid:17)
ETNA
Kent State University
http://etna.math.kent.edu
298 P.MOKHTARYANDF.GHOREISHI
whereidistheinfiniteidentitymatrix,
Π¯α,β = ∞ ∞ k˜¯ i,j,
ij
T
i=0j=0
XX
and i,j fori,j =0,1,2,...,istheinfinitediagonalmatrixwithdiagonalentries
T
0, s i+j+µ 2,
≤ −
i,j =
Tss  πcsc(π)Γ(1+s−(i+µ−1))
 Γ(1µ)Γ(2+s−(i+µµ)) , s≥i+j+µ−1.
µ µ

Proof. Fromtherelations(3.3)and(3.2)wecanwrite
z K¯(z,w)
L¯(u¯α,β(z))=b Vα,β λVα,β Wdw
N N − √µz w
(cid:16) Z0 − (cid:17)
z µwµ 1K(zµ,wµ)
(3.4) =b Vα,β λVα,β − Wdw ,
N − √µzµ wµ
(cid:16) Z0 − (cid:17)
whereW =[1,w,w2,...,wN,...]T. Thus,theintegraltermof(3.4)canbewrittenas
(3.5) z µwµ−1K(zµ,wµ) Wdw = ∞ ∞ k˜¯ zi z µwj+l+µ−1dw ∞ .
√µzµ wµ ij √µzµ wµ
Z0 − i=0j=0 (cid:20)Z0 − (cid:21)l=0
XX
Usingtherelation
z µwj+l+µ 1 πcsc(π)Γ(1+ j+l)
− dw =zj+l+µ 1 µ µ ,
Z0 √µzµ−wµ − (cid:16) Γ(µ1)Γ(2+ j+µl−1) (cid:17)
wecanrewrite(3.5)as
z µwµ−1K(zµ,wµ) Wdw = ∞ ∞ k˜¯ zi+j+l+µ 1πcsc(πµ)Γ(1+ j+µl) ∞
Z0 √µzµ−wµ i=0j=0 ij(cid:20) − Γ(µ1)Γ(2+ j+µl−1) (cid:21)l=0
XX
(3.6) = ∞ ∞ k˜¯ij i,j Z =Π¯α,βZ =Π¯α,β Vα,β −1Vα,β,
T
(cid:18)i=0j=0 (cid:19)
XX (cid:0) (cid:1)
where i,j istheinfinitediagonalmatrixwiththeaforementioneddiagonalentries,and
T
Π¯α,β = ∞ ∞ k˜¯ i,j, Z =[1,z,z2,...,zN,...]T.
ij
T
i=0j=0
XX
Substitutingtherelation(3.6)into(3.4)wededucetheresult.
Now,considerthetransformedequation(3.1). Assumethat
Nf¯
f¯(z)= f¯vα,β(z)=f¯Vα,β,
i i
i=0
X