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 (mokhtary.payam@gmail.com, mokhtary@sut.ac.ir). ‡DepartmentofMathematics,K.N.ToosiUniversityofTechnology,Tehran,Iran (ghoreishif@kntu.ac.ir). 289 ETNA Kent State University http://etna.math.kent.edu 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 ETNA Kent State University http://etna.math.kent.edu 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 √µx t √µx 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 ETNA Kent State University http://etna.math.kent.edu 292 P.MOKHTARYANDF.GHOREISHI 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) ETNA Kent State University http://etna.math.kent.edu 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. { } ETNA Kent State University http://etna.math.kent.edu 294 P.MOKHTARYANDF.GHOREISHI 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 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 √µx 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 ! ETNA Kent State University http://etna.math.kent.edu 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

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