Generalized Jacobi spectral-galerkin method for nonlinear Volterra integral equations with weakly singular

Transcription

1 JOURAL OF MAHEMACAL SUDY J. Mah. Sudy, Vol. x, o. x (1x), pp Geeralized Jacobi specral-galerki mehod for oliear Volerra iegral equaios wih weakly sigular kerels Jie She 1,, Chagao Sheg 1 ad Zhogqig Wag 3 1 School of Mahemaical Scieces, Xiame Uiversiy, Xiame, Fuia 3615, P.R. Chia. Deparme of Mahemaics, Purdue Uiversiy, Wes Lafayee, , USA. 3 School of Sciece, Uiversiy of Shaghai for Sciece ad echology, Shaghai, 93, P.R. Chia. Absrac. We propose a geeralized Jacobi specral-galerki mehod for he oliear Volerra iegral equaios (VEs) wih weakly sigular kerels. We esablish he exisece ad uiqueess of he umerical soluio, ad characerize he covergece of he proposed mehod uder reasoable assumpios o he olieariy. We also prese umerical resuls which are cosise wih he heoreical predicios. Key Words: Geeralized Jacobi specral-galerki mehod, oliear Volerra iegral equaios wih weakly sigular kerels, covergece aalysis. AMS Subec Classificaios: 45D5, 65L6, 41A3, 65L7 1 roducio his paper is cocered wih he umerical soluios of he oliear Volerra iegral equaios wih weakly sigular kerels: y() = f ()+Vy() := f ()+ ( s) µ K(,s)G(s,y(s))ds, := [,], (1.1) where < µ < 1, K C(D) wih D := {(,s) : s }, f C() ad G is a coiuous fucio. rece years, here has bee a icreasig ieres i sudyig VEs. he mai difficulies for dealig wih weakly sigular VEs are: (i) he iegral operaor is o-local; (ii) Correspodig auhor. address: she7@purdue.edu (J. She), csheg@su.xmu.edu.c (C. Sheg), zqwag@uss.edu.c (Z. Wag) hp:// 1 c 1x Global-Sciece Press

2 J. She, C. Sheg ad Z. Wag / J. Mah. Sudy, x (1x), pp he soluios are usually sigular ear =. Bruer [4] ad Lubich [17] ivesigaed he smoohess properies of he exac soluios of VEs wih weakly sigular kerels. Various umerical approaches, usig he piecewise polyomial collocaio mehods ad he Ruge-Kua mehods, have bee proposed for approximaig VEs wih weakly sigular kerels [4, 5, 1, 3]. However, hese umerical mehods do o paricularly deal wih he above wo difficulies. Specral mehods are capable of providig exceedigly accurae umerical resuls wih relaively less degree of freedoms, ad have bee widely used for scieific compuaio, see, e.g., [1,,6,13,14,19,]. Sice he specral mehods are global mehods, so hey could be beer suied for o-local problems. Recely, may kids of specral collocaio mehods are proposed for solvig VEs wih smooh kerels. Li, ag ad Xu [16] iroduced a ime parallel mehod wih specral-subdomai ehaceme for VEs; Sheg, Wag ad Guo [1] preseed a mulisep specral collocaio mehod for oliear VEs; Wag ad Sheg [] also proposed a mulisep specral collocaio mehod for oliear VEs wih delays. o solve VEs wih weakly sigular kerels, may aemps have bee made o overcome he difficulies caused by he sigulariies of he soluios. Che ad ag [9,1] proposed specral collocaio mehods for weakly sigular VEs; Huag, ag ad Zhag [15] sudied he supergeomeric covergece of specral collocaio mehods for weakly sigular Volerra/Fredholm iegral equaios. hese mehods usually use orhogoal polyomials as basis fucios. Aoher approach for solvig weakly sigular VEs is o use he o polyomial sigular fucios (which reflec he sigulariies of he exac soluios) as basis fucios. For example, Bruer [3] employed a o polyomial splie collocaio mehod for VEs wih weakly sigular kerels; Cao, Herdma ad Xu [7] preseed a o polyomial sigulariy preservig collocaio mehod for VEs wih weakly sigular kerels. his paper, we develop a o polyomial specral-galerki mehod for VEs wih weakly sigular kerels. More precisely, we cosruc a specral-galerki mehod for weakly sigular VEs (1.1), usig he geeralized Jacobi fucios as basis fucios. his kid of basis fucios have bee used by Zayerouri & Kariadakis [4] ad Che, She & Wag [8] for approximaig fracioal differeial equaios. he mai sraegies ad coribuios are as follows. We propose a geeralized Jacobi specral-galerki mehod for oliear VEs wih weakly sigular kerels. he basis fucios ca be ued o mach he sigulariies of he uderlyig soluios, ad lead o a efficie implemeaio. he exisig works (cf. [9, 1]) We approximae he problem (1.1) direcly wihou ay variable rasformaios, as opposed o he approach i [9, 1] where a specral-collecio mehod is cosruced for he rasformed VEs.

3 J. She, C. Sheg ad Z. Wag / J. Mah. Sudy, x (1x), pp he res of his paper is orgaized as follows. Secio, we prese he geeralized Jacobi specral-galerki mehod for oliear VEs (1.1). Some useful lemmas for he covergece aalysis are provided i Secio 3. he exisece, uiqueess ad covergece of he geeralized Jacobi specral-galerki mehod are give i Secio 4. We prese i Secio 5 umerical experimes, which cofirm he heoreical expecaios. Some cocludig remarks are give i he fial secio. he geeralized Jacobi specral-galerki mehod his secio, we shall propose a specral-galerki mehod usig geeralized Jacobi fucios as basis fucios for problem (1.1). o his ed, we firs iroduce he shifed Jacobi polyomials ad he shifed geeralized Jacobi fucios o he ierval..1 he shifed Jacobi polyomials o. For α,β > 1, le J (α,β) (x), x Λ := ( 1,1) be he sadard Jacobi polyomial of degree, ad deoe he weigh fucio χ (α,β) (x) = (1 x) α (1+x) β. he se of Jacobi polyomials is a complee L χ (α,β) (Λ)-orhogoal sysem, i.e., Λ J (α,β) m (x)j (α,β) (x)χ (α,β) (x)dx = γ (α,β) m δ m,, (.1) where δ m, is he Kroecker fucio, ad γm α,β = α+β+1 Γ(α+1)Γ(β+1), Γ(α+ β+) m =, α+β+1 Γ(m+α+1)Γ(m+ β+1), (m+α+ β+1) m!γ(m+α+ β+1) m 1. (.) paricular, J (α,β) (x) =1. he shifed Jacobi polyomial of degree is defied by Clearly, he se of { J (α,β) fucio χ (α,β) J (α,β) () = J (α,β) ( 1),. (.3) ()} is a complee L ()-orhogoal sysem wih he weigh χ (α,β) () = ( ) α β. fac, by (.1) ad (.3) we kow ha J (α,β) m () J (α,β) ()χ (α,β) ()d = ( ) α+β+1γ (α,β) m δ m,. (.4) For ay ieger, we deoe by {x (α,β),ω (α,β) } = he odes ad he correspodig Chrisoffel umbers of he sadard Jacobi-Gauss ierpolaio o he ierval Λ. Le

4 4 J. She, C. Sheg ad Z. Wag / J. Mah. Sudy, x (1x), pp P () be he se of polyomials of degree a mos o he ierval, ad (α,β) shifed Jacobi-Gauss quadraure odes o he ierval, be he (α,β) = (x(α,β) +1),. (.5) Due o he propery of he sadard Jacobi-Gauss quadraure, i follows ha for ay φ() P +1 (), φ()χ (α,β) ()d = ( ) 1 α+β+1 φ( 1 (x+1))χ(α,β) (x)dx = ( ) α+β+1 φ( (x(α,β) +1))ω (α,β) (.6) = = ( ) α+β+1 φ( (α,β) )ω (α,β). = By (.4) ad (.6), we furher obai ha for ay m+ +1, = J (α,β) m ( (α,β) ) J (α,β) ( (α,β) )ω (α,β) = γ (α,β) m δ m,. (.7). he shifed geeralized Jacobi fucios o. he shifed geeralized Jacobi fucio of degree is defied by (cf. [8]) P (α,β) (α,β) () := β J (), α,β > 1,. (.8) Le F (β) () be he fiie-dimesioal fracioal polyomial space (cf. [8]) F (β) () := {β ψ() : ψ() P ()} =spa{p (α,β) : }. (.9) Due o (.4) ad (.8), i is clear ha he se of {P (α,β) orhogoal sysem wih he weigh fucio χ (α, β) P (α,β) m ()P (α,β) ()χ (α, β) ()d = = (α,β) β J m J (α,β) m () J (α,β) () J (α,β) ()} is a complee L ()- χ (α, β) (), amely, ()χ (α, β) ()d ()χ (α,β) ()d = ( ) α+β+1γ (α,β) m δ m,. Because of (.6), i follows ha for ay ϕ() = β φ() wih φ() P +1 (), (.1) ϕ()χ (α, β) ()d = φ()χ (α,β) ()d = ( ) α+β+1 = = ( ) α+β+1 = ( (α,β) ) β ϕ( (α,β) )ω (α,β). φ( (α,β) )ω (α,β) (.11)

5 J. She, C. Sheg ad Z. Wag / J. Mah. Sudy, x (1x), pp ex, le (u,v) χ (α, β) ad v χ (α, β) be he ier produc ad orm of space L (), re- χ (α, β) specively. We also iroduce he followig discree ier produc ad orm o he ierval, u,v (α, β) χ = ( ) α+β+1 = v,χ (α, β) = v,v 1 χ (α, β) haks o (.11), for ay φ,ψ F (β) (), (φ,ψ) χ (α, β) = φ,ψ χ (α, β). ( (α,β) ) β u( (α,β), φ χ (α, β) )v( (α,β) )ω (α,β), (.1) = φ (α, β),χ. (.13).3 he geeralized Jacobi specral-galerki mehod for problem (1.1) o describe he specral-galerki scheme for problem (1.1), we firs rasform he iegral ierval [,] o usig he rasformaio: he he equaio (1.1) becomes s = τ, τ. (.14) y() = f ()+Vy() = f ()+ ( ) 1 µ ( τ) µ K (, τ ) ( τ G,y( τ )) dτ. (.15) he geeralized Jacobi specral-galerki scheme is o seek Y() F (1 µ) (), such ha (Y,ϕ) χ = ( f,ϕ) χ +(VY,ϕ) χ, ϕ F (1 µ) (). (.16) We ow describe a umerical implemeaio for (.16). o his ed, we se Y() = m= y m P ( µ,1 µ) m (). (.17) Subsiuig (.17) io (.16) ad akig ϕ = P ( µ,1 µ) (), we obai ha for, m= Se y m (P ( µ,1 µ) m,p ( µ,1 µ) ) χ = ( f,p ( µ,1 µ) ) χ y = (y,,y ), A = (a m ),m, a m = (P ( µ,1 µ) m f = ( f,p ( µ,1 µ),p ( µ,1 µ) v (y) = (VY,P ( µ,1 µ) ) χ = ( ) χ, f = ( f,, f ), +(VY,P ( µ,1 µ) ) µγ ( µ,1 µ) m δ m,, ) χ, v(y) = (v,,v ). ) χ. (.18) (.19)

6 6 J. She, C. Sheg ad Z. Wag / J. Mah. Sudy, x (1x), pp he, he sysem (.18) becomes Ay = f+v(y). (.) acual compuaio, we use he quadraure formula (.1) o approximae he erms f ad v, amely, f f,p ( µ,1 µ) χ ( ) µ = = ( ( µ,1 µ) ) µ f ( ( µ,1 µ) )P ( µ,1 µ) ( ( µ,1 µ) )ω ( µ,1 µ), (.1) ad v (y) µ 3 3µ i,= G( ( µ,1 µ) i ( ( µ,1 µ) i ( µ,) ) µ 1 K( ( µ,1 µ) i /,Y( ( µ,1 µ) i, ( µ,1 µ) i ( µ,) ( µ,) /) /))P ( µ,1 µ) ( ( µ,1 µ) i )ω ( µ,1 µ) i ω ( µ,). (.) his is a (oliear) implici scheme, which ca be solved, for isace, by he ewo ieraive mehod. 3 Some useful lemmas his secio, we prese some useful lemmas. For his purpose, we firs recall he defiiios of he fracioal iegrals ad fracioal derivaives i he sese of Riema-Liouville (see, e.g., [11, 18]). Defiiio 3.1. (Fracioal iegrals ad derivaives). For ρ R +, he lef ad righ fracioal iegrals are respecively defied as axu(x) ρ = 1 x Γ(ρ) a u(y) (x y) 1 ρ dy, x > a; x ρ b u(x) = 1 b Γ(ρ) x u(y) dy, x < b, (3.1) (y x) 1 ρ where Γ( ) is he usual Gamma fucio. For s [k 1,k) wih k, he lef-sided Riema-Liouville fracioal derivaive of order s is defied by ad s xu(x) = 1 Γ(k s) d k x dx k a u(y) dy, x (a,b), (3.) (x y) s k+1 ad he righ-sided Riema-Liouville fracioal derivaive of order s is defied by xdb s ( 1)k u(x) = Γ(k s) d k b dx k x u(y) dy, x (a,b). (3.3) (y x) s k+1

7 J. She, C. Sheg ad Z. Wag / J. Mah. Sudy, x (1x), pp is clear ha for ay k, ad k x = D k, xd k b = ( 1)k D k, where D k := dk dx k. hus, we ca defie he Riema-Liouville fracioal derivaives as adxu(x) s = D k ax k s u(x), xdb s = ( 1)k D k x k s u(x). Accordig o heorem.14 of [11], we have ha for ay absoluely iegrable fucio u ad real s, adxa s xu(x) s = u(x), xdbx s b s u(x) = u(x), x (a,b). (3.4) ad ex, le F (β) (Λ) := {(1+x)β ψ(x) : ψ(x) P (Λ)} =spa{(1+x) β J (α,β) (x) : }, B r α,β (Λ) := {u(x) : u L χ (α, β) (Λ), 1 D β+l x u L χ (α+β+l,l) (Λ) for l r}, r. Deoe by c a geeric posiive cosa idepede of, ad he soluios of y() ad Y(). Accordig o heorem 4.3 of [8], we have Lemma 3.1. Le α > 1, β >, for ay u B α,β r (Λ) wih ieger r, we have where π (α,β) π (α,β) u u L χ (α, β) (Λ) c (β+r) 1 Dx β+r u L (Λ), (3.5) (α+β+r,r) is he sadard L (Λ)-orhogoal proecio upo F (β) χ (α, β) (Λ), defied by Λ Similarly, we defie ( π (α,β) u(x) u(x))ψ(x)χ(α, β) (x)dx =, ψ F (β) (Λ). (3.6) B r α,β () := {v() : v L χ (α, β) H r α,β () := {v() : v L χ (α, β) Deoe by π (α,β) he L χ (α, β) (), D β+l (), D β+l b χ v L () χ (α+β+l,l) for l r}, r, () for l r}, r. v L χ (α, β) ()-orhogoal proecio upo F (β) (), (π (α,β) v v,φ) χ (α, β) By Lemma 3.1, we obai he followig resuls. =, φ F (β) (). (3.7)

8 8 J. She, C. Sheg ad Z. Wag / J. Mah. Sudy, x (1x), pp Lemma 3.. Le α > 1, β >. For ay v Bα,β r () wih ieger r, we have π (α,β) paricular, if v Hα,β r (), he Proof. Se u(x) := v() π (α,β) v v χ (α, β) v v χ (α, β) c β (β+r) D β+r c r (β+r) D β+r. Sice π(α,β) = v() = (x+1) (x+1) i he variable x, ad hece by he defiiios (3.6) ad (3.7), π (α,β) v (α+β+r,r) χ. (3.8) v (α, β) χ. (3.9) ad π(α,β) u(x) belog o F (β) (Λ) v() = = (x+1) π(α,β) u(x). (3.1) he above wih (3.5) ad (3.) yields ( π (α,β) ) α β+1 v v = χ (α, β) Λ c α β+1 (β+r) c β (β+r) ( π (α,β) u(x) u(x)) (1 x) α (1+x) β dx ( 1Dx β+r u(x) ) (1 x) α+β+r (1+x) r dx Λ (D β+r v() ) ( ) α+β+r r d. (3.11) his leads o he resul (3.8). Furhermore, (D β+r v() ) ( ) α+β+r r d ( (D )(β+r) β+r v() ) ( ) α β d. his leads o he resul (3.9). 4 Exisece, uiqueess ad error esimae his secio, we firs verify he exisece ad uiqueess of he soluio of (.16), ad he we aalyze ad characerize he covergece of scheme (.16) uder reasoable assumpios o he olieariy. heorem 4.1. Assume ha K(,s) C(D) ad G saisfies he followig Lipschiz codiio: he, for sufficiely small such ha he equaio (.16) possesses a uique soluio. G(s,y 1 ) G(s,y ) γ y 1 y, γ. (4.1) c µ β <1, (4.)

9 J. She, C. Sheg ad Z. Wag / J. Mah. Sudy, x (1x), pp Proof. We firs prove he exisece. Cosider he followig ieraio process: (Y (m),ϕ) χ = ( f,ϕ) χ +(VY (m 1),ϕ) χ, ϕ F (1 µ) (). (4.3) Accordig o he defiiio (3.7) of he proecio operaor π ( µ,1 µ), we kow from (4.3) ha Y (m) = π ( µ,1 µ) ( f +VY (m 1) ). (4.4) ex, le Ỹ (m) =Y (m) Y (m 1). he, by (4.4) we ge Ỹ (m) = π ( µ,1 µ) ( VY (m 1) VY (m )). his, alog wih he proecio heorem, implies Ỹ (m) χ = π ( µ,1 µ) ( VY (m 1) VY (m )) χ VY (m 1) VY (m ). χ Furher, by (4.5), (4.1), (.1), he Cauchy-Schwarz iequaliy ad usig he raformaio = s+ (τ+1)( s), we ge ha Ỹ (m) χ ( ( s) µ K(,s) ( G(s,Y (m 1) (s)) G(s,Y (m ) (s)) ) ( ) ds) µ µ 1 d [ c ( s) µ ds ( s) µ( Ỹ (m 1) (s) ) ds ]( ) µ µ 1 d c ( ) µ ( s) µ( Ỹ (m 1) (s) ) dsd c (Ỹ(m 1) (s) ) ( ) µ ( s) µ dds s c (Ỹ(m 1) (s) ) 1 (1 τ) µ (1+τ) µ ( s) 1 µ dτds 1 c (Ỹ(m 1) (s) ) ( s) 1 µ ds c µ (Ỹ(m 1) (s) ) ( s) µ s µ 1 ds = c µ Ỹ (m 1) χ. hus, if c µ β < 1, he Ỹ (m) χ as m. his implies he exisece of soluio of (.16). is easy o prove he uiqueess of soluio of (.16). heorem 4.. Assume ha K(,s) C(D), y B µ,1 µ r () wih ieger r, G fulfills he Lipschiz codiio (4.1) ad is sufficiely small saisfyig he codiio (4.). he, here holds y Y χ paricular, if y H µ,1 µ r (), he y Y χ c µ 1 µ r 1 D µ+r+1 c r µ r 1 D µ+r+1 (4.5) (4.6) y (1 µ+r,r) χ. (4.7) y χ. (4.8)

10 1 J. She, C. Sheg ad Z. Wag / J. Mah. Sudy, x (1x), pp Proof. By (.16) we kow Y = π ( µ,1 µ) ( ) f +VY. (4.9) Subracig (4.9) from (1.1) yields Usig (1.1) agai, we obai y Y = f π ( µ,1 µ) f π ( µ,1 µ) f = y π ( µ,1 µ) A combiaio of he previous wo equaliies leads o y Y = y π ( µ,1 µ) his, ogeher wih he proecio heorem, gives f +Vy π ( µ,1 µ) VY. y+π ( µ,1 µ) Vy Vy. y+π ( µ,1 µ) ( ) Vy VY. (4.1) y Y χ y π ( µ,1 µ) y π ( µ,1 µ) y χ y χ + π ( µ,1 µ) ( ) Vy VY χ + Vy VY χ. (4.11) ex, by a argume similar o (4.6) we deduce ha Vy VY χ c µ y Y. χ (4.1) Moreover, by (3.8) we ge ha for y B µ,1 µ r () wih ieger r, y π ( µ,1 µ) y χ c µ 1 µ r 1 D µ+r+1 y (1 µ+r,r) χ. (4.13) herefore, by (4.11) - (4.13) ad (4.), we obai he desired resul (4.7). Fially, by (3.9) ad a similar argume, we derive he resul (4.8). Remark 4.1. he Lipschiz codiio (4.1) appears o be ecessary for our covergece aalysis. However, some umerical experimes below show ha, eve if he Lipschiz codiio is o saisfied, he scheme is sill coverge. 5 umerical Resuls his secio, we prese some umerical resuls o illusrae he efficiecy of he geeralized Jacobi specral-galerki mehod.

11 J. She, C. Sheg ad Z. Wag / J. Mah. Sudy, x (1x), pp Liear problem Cosider firs he liear VE wih weakly sigular kerel (cf. [9]): y() = f () ( s).35 y(s)ds, [,6], (5.1) where f () = B(4.6,.65) ad B(, ) is he Bea fucio defied by 1 ( 1 ) x+y 1γ B(x,y) = x 1 (1 ) y 1 (y 1,x 1) d =. he exac soluio is y() = 3.6. Figure 5.1, we lis he discree L ()-errors χ ad he maximum errors of (5.1), hey idicae he algebraic covergece. fac, a direc compuaio shows ha y B µ,1 µ r () wih µ =.35 ad r = 6. Hece, accordig o (4.7), we ca expec a covergece rae for he L ()-orm o be of he order r µ+1=6.65. χ ()-orm ploed i Figure 5.1 is abou 8.3. he observed covergece rae for he L χ able 5.1 below, we compare he maximum errors of our algorihm wih ha of he collocaio mehod suggesed i [9] (see able 1 of [9]). We observe ha our mehod provides more accurae umerical resuls. Figure 5.1: he umerical errors of (5.1). Figure 5.: he umerical errors of (5.). able 5.1: A compariso of L ()-errors for (5.1) Ref. [9] e e e- 1.95e-3 Our mehod e e e e Ref. [9] 4.486e e e e-5 Our mehod.8175e e-7.795e e-8

12 1 J. She, C. Sheg ad Z. Wag / J. Mah. Sudy, x (1x), pp Cosider ex he liear VE wih weakly sigular kerel: y() = f ()+ ( s) 1/3 y(s)ds, [,1]. (5.) We choose f such ha he soluio y of (5.) is give by y()= /3 cos(). ca be verified readily ha y B µ,1 µ () wih µ = 1/3, so heorem 4. predics ha he errors of he geeralized Jacobi specral approximaio will decrease faser ha ay algebraic rae. Figure 5., we lis he discree L ()-errors ad he maximum errors of (5.). χ We observe ha he umerical errors decay expoeially as icreases. 5. oliear problem Cosider firs he oliear VE wih weakly sigular kerel: y() = f ()+ ( s) 1/ exp ( s 1/ y(s)/ ) ds, [,1]. (5.3) We choose f such ha he soluio y of (5.3) is give by y() = 1/ l(+e). Clearly, he exac soluio y B 1/,1/ (). However, he Lipschiz codiio (4.1) is o saisfied for problem (5.3). Figure 5.3, we plo he discree L ()-errors wih µ=1/ ad he maximum errors of (5.3). is show ha he umerical errors decay expoeially as icreases. his χ meas ha our algorihm is sill valid for problem (5.3), eve if he Lipschiz codiio (4.1) is o saisfied. Figure 5.3: he umerical errors of (5.3). Figure 5.4: he umerical errors of (5.4). Cosider ex he oliear VE wih weakly sigular kerels: y() = exp() / ( s) 1/ exp( s)y (s)ds, [,1], (5.4) wih he exac soluio y() = exp(). Clearly, he exac soluio y B 1/,1/ (). Moreover, he Lipschiz codiio (4.1) is o saisfied for problem (5.4). Figure 5.4, we lis he discree L ()-errors wih µ = 1/ ad he maximum χ errors of (5.4). hey also idicae ha he umerical errors decay expoeially as icreases.

13 J. She, C. Sheg ad Z. Wag / J. Mah. Sudy, x (1x), pp Cocludig Remarks his paper, we proposed a geeralized Jacobi specral-galerki mehod for he oliear VEs wih weakly sigular kerel. his mehod ca be implemeed efficiely. We showed he exisece ad uiqueess of he umerical soluio ad proved is covergece rae uder reasoable assumpios o he olieariy. umerical experimes demosrae ha he proposed mehod are very effecive for dealig wih liear ad oliear VEs. Ackowledgmes he work of J.S. is suppored i par by SF of Chia gras ad ; ad he work of Z.W. is suppored i par by SF of Chia gra ad he Research Fud for Docoral Program of Higher Educaio of Chia (gra ). Refereces [1] C. Berardi ad Y. Maday, Specral Mehod, i Hadbook of umerical Aalysis, par 5, edied by P. G. Ciarle ad J. L. Lios, orh-hollad, [] J. P. Boyd, Chebyshev ad Fourier Specral Mehods, Spriger-Verlag, Berli, [3] H. Bruer, opolyomial splie collocaio for Volerra equaios wih weakly sigular kerels, SAM J. umer. Aal., (1983), [4] H. Bruer, Collocaio mehods for Volerra egral ad Relaed Fucioal Differeial Equaios, Cambridge Uiversiy Press, Cambridge, 4. [5] H. Bruer, A. Pedas ad G. Vaiikko, he piecewise polyomial collocaio mehod for oliear weakly sigular Volerra equaios, Mah. Comp., 7 (1999), [6] C. Cauo, M. Y. Hussaii, A. Quareroi ad. A. Zag, Specral Mehods: Fudameals i Sigle Domais, Spriger-Verlag, Berli, 6. [7] Y. Cao,. Herdma ad Y. Xu, A hybrid collocaio mehod for Volerra iegral equaios wih weakly sigular kerels, SAM J. umer. Aal., 41 (3), [8] S. Che, J. She ad L. Wag, Geeralized Jacobi Fucios ad heir Applicaios o Fracioal Differeial Equaios, submied. [9] Y. Che ad. ag, Specral mehods for weakly sigular Volerra iegral equaios wih smooh soluios, J. Compu. Appl. Mah., 33 (9), [1] Y. Che ad. ag, Covergece aalysis of he Jacobi specral-collocaio mehods for Volerra iegral equaios wih a weakly sigular kerel. Mah. Comp., 79 (1), [11] K. Diehelm. he Aalysis of Fracioal Differeial Equaios, Lecure oes i Mah., Vol. 4. Spriger, Berli, 1. [1]. Diogo ad P. Lima, Supercovergece of collocaio mehods for a class of weakly sigular Volerra iegral equaios. J. Compu. Appl. Mah., 18 (8), [13] D. Fuaro, Polyomial Approximaios of Differeial Equaios, Spriger-Verlag, 199. [14] B. Guo, Specral Mehods ad heir Applicaios, World Scieific, Sigapore, 1998.

14 14 J. She, C. Sheg ad Z. Wag / J. Mah. Sudy, x (1x), pp [15] C. Huag,. ag ad Z. Zhag, Supergeomeric covergece of specral collocaio mehods for weakly sigular Volerra ad Fredholm iegral equaios wih smooh soluios, J. Compu. Mah., 9 (11), [16] X. Li,. ag ad C. Xu, Parallel i ime algorihm wih specral-subdomai ehaceme for Volerra iegral equaios, SAM J. umer. Aal., 51 (13), [17] C. Lubich, Ruge-Kua heory for Volerra ad Abel iegral equaios of he secod kid, Mah. Comp., 41 (1983), [18]. Podluby, Fracioal Differeial Equaios, volume 198 of Mahemaics i Sciece ad Egieerig. Academic Press c., Sa Diego, CA, [19] J. She ad. ag, Specral ad High-order Mehods wih Applicaios, Sciece Press, Beiig, 6. [] J. She,. ag ad L. Wag, Specral Mehods: Algorihms, Aalysis ad Applicaios, Spriger Series i Compuaioal Mahemaics, Vol. 41, Spriger-Verlag, Berli, Heidelberg, 11. [1] C. Sheg, Z. Wag ad B. Guo, A mulisep Legedre-Gauss specral collocaio mehod for oliear Volerra iegral equaios, SAM J. umer. Aal., 5 (14), [] Z. Wag ad C. Sheg, A hp-specral collocaio mehod for oliear Volerra iegral equaios wih vaishig variable delays, Mah. Comp., i press. [3] Q. Wu, O graded meshes for weakly sigular Volerra iegral equaios wih oscillaory rigoomeric kerels, J. Compu. Appl. Mah., 63 (14), [4] M. Zayerouri ad G.E. Kariadakis. Fracioal Surm-Liouville eige-problems: heory ad umerical approximaio. J. Compu. Phys., 5: , 13.