Research Article A Jacobi-Collocation Method for Second Kind Volterra Integral Equations with a Smooth Kernel

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1 Abstract ad Appled Aalyss Volume 014, Artcle D , 10 pages Research Artcle A Jacob-Collocato Method for Secod Kd Volterra tegral Equatos wth a Smooth Kerel Hogfeg Guo, Haotao Ca, ad X Zhag School of Mathematcs ad Quattatve Ecoomcs, Shadog Uversty of Face ad Ecoomcs, Ja, Shadog 50014, Cha Correspodece should be addressed to X Zhag; zhag80@sdufe.edu.c Receved 8 May 014; Revsed July 014; Accepted July 014; Publshed 17 July 014 Academc Edtor: Jua J. Neto Copyrght 014 Hogfeg Guo et al. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. The purpose of ths paper s to provde a Jacob-collocato method for solvg secod kd Volterra tegral equatos wth a smooth kerel. Ths method leads to a fully dscrete tegral operator. Frst, t s show that the fully dscrete tegral operator s stable both L ad weghted L orms. The, the proposed approach s proved to arrve at a optmal (the most possble) coverget order both orms. Oe umercal eample demostrates the effcecy ad accuracy of the proposed method. 1. troducto ths paper, we provde a Jacob-collocato approach for solvg the secod kd Volterra tegral equato of the form u () + k (, t) u (t) dt = f (), := [, 1], (1) where the kerel fucto k ad the put fucto f are gve smooth fuctos about ther varables ad u s the ukow fucto to be determed. Foreaseofaalyss,wewllwrte(1) toaoperator form. By troducg the tegral operator K by (KV)() := k (, t) V (t) dt,, () (1)sreformulatedas ( + K) u=f. (3) t s well kow that there are may umercal methods for solvg secod kd Volterra tegral equatos such as the Ruge-Kutta method ad the collocato method based o pecewse polyomals; see, for eample, Bruer [1] ad refereces there. For more formato of the progress o the study of the problem, we refer the readers to [ 8]. Recetly, a few works touched the spectral appromato to Volterra tegral equatos. [9], Elagar ad Kazem provded a ovel Chebyshev spectral method for solvg olear Volterra-Hammerste tegral equatos. The, ths method was vestgated by Fujwara [10] forsolvg the frst kd Fredholm tegral equato uder multpleprecsoarthmetc.nevertheless,otheoretcalresultswere provded to justfy the hgh accuracy. [11], Tag et al. developed a ovel Legedre-collocato method for solvg (3). spred by the work of [11], Che ad Tag [5, 1] obtaed the spectral Jacob-collocato method for solvg the secod kd Volterra tegral equatos wth geeral weakly sgular kerels k(, t)( t) μ for <μ<0.[13], a spectral ad pseudospectral Jacob-Galerk approach was preseted for solvg (3). [14], We ad Che cosdered a spectral Jacob-collocato method for solvg Volterra type tegrodfferetalequato. [15], Ca cosdered a Jacobcollocato method for solvg Fredholm tegral equatos of secod kd wth weakly sgular kerels. Ufortuately, all these papers [5, 11 14] gve thecovergece aalyss but suffer from the stablty aalyss. Because of lack of the stablty aalyss, the appromate solutodoes ot atta the most possble covergece order. Moreover, all of those papers do ot aswer that the appromate equato has a uque soluto. Hece, ths paper, we wll provde a Jacob-collocato method for solvg (3), whch eteds the Legedre spectral method developed [11]. Ths spectral method leads to a fully dscrete lear system. We are gog to show that the fully dscrete tegral operator s stable; that

2 Abstract ad Appled Aalyss s, the appromate equato has a uque soluto, ad the, preset the optmal (the most possble) coverget order of theappromatesolutobasedothestabltyaalyss.we orgaze ths paper as follows. Secto, as demostrated [13], we revew a spectral Jacob-collocato method for solvg (3). Secto 3, a few mportat results are preseted to aalyze the Jacob-collocato approach. Sectos 4 ad 5, we aalyze the Jacob-collocato method, cludg the stablty of the appromate equato ad the coverget order of the appromate soluto, both L ad weghted L orms, respectvely. Secto 6, oe umercal eample s preseted to show the effcecy ad accuracy of ths method. The problem uder study deserves more vestgatos futureworks.moreover,webelevethatthesemaalytcal approaches are useful to vestgate the problem. For related termologes ad applcatos of semaalytcal approaches, please refer to [16 18].. A Spectral Jacob-Collocato Method ths secto, we are gog to revew the spectral Jacobcollocato method for solvg (3). To ths ed, we troduce several de sets: N := {1,,...,,...}, N 0 := N {0}ad Z := {0,1,,...,}.Welet () := (1 ) α (1 + ) β for α, β > be a weght fucto ad the use the otato L () to be the set of all square tegrable fuctos assocated wth the weght fucto, equpped wth the orm V := ( (t)v 1/ (t)dt). (4) For N, we deote the pots by α,β, Z to be the set of +1Jacob-Gauss pots correspodg to the Jacob weght fucto.bytroducg π () :=( α,β 0 )( α,β 1 ) ( α,β ), (5) we defe the Lagrage fudametal terpolato polyomal L α,β, Z by L α,β () := π () ( α,β )π ( α,β ),. (6) Let P be the set of all polyomals of degree ot more tha ;clearly, P = spa {L α,β : Z }. (7) We use the oto C() to deote the set of all cotuous fuctoso, equpped wth the orm V := ma V (). (8) For s, we defe a lear fuctoal δ s o C() such that, for ay V C(), δ s, V :=V (s). (9) The collocato method for solvg (3)s to seek a vector u := [a : Z ] T such that satsfes δ α,β j u () := Z a L α,β (), (10), ( + K) u = δ α,β,f, j Z. (11) j The above equato ca be rewrtte as a + a j j Z α,β k( α,β,t)l α,β j (t) dt = f ( α,β ), Z. (1) For V C(), we defe the terpolatg operator L α,β : C() P by (L α,β V)(α,β )=V( α,β ), Z. (13) t t well kow that L α,β V s wrtte as the form (L α,β V) () = V ( )L α,β (),. Z (14) Usg these otatos we ca reformulate (1)to a operator form ( + L α,β K) u = L α,β f. (15) The dffculty solvg the lear system (1) stocompute the tegral term (1), accurately. ths paper, we adopt the umercal tegrato rule proposed [11] toovercome ths dffculty. For ths purpose, we troduce a smple lear trasformato t=g(, τ) := +1 τ+, (16) whch trasfers the tegral operator K to the followg form: (KV)() = +1 k (, g (, τ)) V (g (, τ)) dτ. (17) The, by usg N+1-pot Legedre-Gauss quadrature formula relatve to the Legedre weght w, Z N,weca obta the dscrete tegral operator K N as follows: (K N V) () := +1 w k(,g(, 0,0 )) V (g (, 0,0 )). Z N (18) Thus, usg those otatos, a fully dscrete spectral Jacob-collocato method for solvg (3) s to seek a vector ũ := [ a : Z ] T such that u () := Z a L α,β (),, (19)

3 Abstract ad Appled Aalyss 3 satsfyg ( + L α,β K N) u = L α,β f. (0) t s easy to show that the operator equato (0) hasthe followg form: a + α,β +1 a j w l k( α,β,g( α,β j Z l Z N =f( α,β ), Z., 0,0 l )) L α,β j (g ( α,β, 0,0 l )) (1) [11], for the case N =,basedothegrowall equalty, Tag et al. aalyze the covergece of a spectral Jacob-collocato method for solvg (3) bothc() ad weghted L w0,0() spaces. However, the stablty aalyss of the spectral method s ot gve. Moreover, we observe that the covergece order of the appromate soluto the space L w0,0() s ot optmal. Hece, the purpose of ths paper s to llustrate that for suffcetly large ad N, theoperator + L α,β K N :P P has a uformly bouded verso both C() ad L () spaces, respectvely. Moreover, we also show that the appromate soluto u attas at the most possble coverget order. 3. Some Prelmares ad Useful Results ths secto, we wll troduce some techcal results, whch cotrbute to aalyze the stablty ad covergece o the spectral Jacob-collocato method for solvg (3). To ths ed, for N 0,weusetheotatoD to deote the th dfferetal operator o the varable. Forr N, we troduce the ouformly weghted Sobolev space H r () by H r () := {V : D V L w α+,β+ (), Z r}. () t follows from [19] that there ests a postve costat γ 1 depedet of such that, for V H r () ad Z r, D (V Lα,β V) γ w α+,β+ 1 Dr V w α+r,β+r r, (3) whch mples that D (Lα,β V) γ w α+,β+ 1 Dr V w + α+r,β+r D V. (4) w α+,β+ Moreover, we have the followg. Lemma 1. Suppose that < α 1,β 1 <1.ftheparameters α, β satsfy the et codtos: 1 +α 1 <α< 3 +α 1, 1 +β 1 <β< 3 +β 1, (5) the there ests a postve costat γ depedet of such that, for V H r () C(), V Lα 1,β 1 V γ Dr V w α+r,β+r r. (6) Proof. Ths s a cosequece of Theorem 3.4 ad (3.13)-(3.14) [0]. For r N, Z r, the bomal coeffcets are gve by C r := r (r) (r +1). (7) We use the otato C r () to deote the set of all fuctos whose rth dervatve s cotuous o, edowed wth the usual orm V r := Z r D V. (8) For r 1,r N 0,theotatoC r 1,r ( ) s used to deote the set of all fuctos such that, for V C r 1,r ( ), D r 1 Dr y V s cotuous o.let Dr 1 Dr y V 1 := ma (,y) Dr Dr y V (, y). (9) Net we cosder the dfferece betwee KV ad K N V. Lemma. Assume that the kerel fucto k C 0,m ( ) for m N.ftwoparametersα ad β satsfy the codtos 1 <α,β<3, α+β 1, (30) the there ests a postve costat γ 3 depedet of N such that whe V C m (), KV K NV γ 3 ( Z m D V w α+,β+ )N m. (31) Proof. Frst of all, by settg b (, τ) := k (, g (, τ)) V (g (, τ)), (L 0,0 N b) (, τ) := b(, 0,0 )L 0,0 (τ),,τ, Z N the tegral operator K N s wrtte as (K N V) () = +1 (3) (L 0,0 N b) (, τ) dτ,. (33) addto, usg the hypothess that k C 0,m ( ) ad V C m () mples that b C 0,m ( ). Thus, we wrte the dfferece betwee KV ad K N V as follows: (KV)() (K N V) () = +1 (w α/, β/ (τ)) (w α/,β/ (τ) (b (, τ) (L 0,0 N b) (, τ))) dτ. (34) Employg Cauchy-Schwartz equalty to the rght had sdeoftheaboveequatoadtheusgtheresult(6)wth

4 4 Abstract ad Appled Aalyss α 1 := 0, β 1 := 0, := N ad r:=mproduce that there ests a postve costatξ depedet of N, (KV)() (K NV)() ξg() 1 w α, βn m, (35) where G() s gve by to G () := ( 1+ ) w α+m,β+m (τ) ((D m τ b) (, τ)) dτ,. (36) t remas to estmate G(). Adrectcomputatoleads G () =( 1+ ) m+ w α+m,β+m (τ) ( C m (D gv)(g(, τ))(dm g k)(, g(, τ))) dτ. Z m (37) Makg use of a lear trasform t := g(, τ) to the rght had sde of the above equato produces G () =( 1+ ) 1 α β (1+t) α+m ( t) β+m ( C m (D t V)(t)(Dm t k)(, t)) dt. Z m (38) Usg the dscrete Cauchy Schwartz equalty to the rghthadsdeoftheaboveequatoobtas G () ma Z m D t k ( (C ) )( 1+ 1 α β Z ) m Z m ( (1+t) α+m ( t) β+m (D t V) (t) dt), wherecombgthefactthatα+β 1leads to (39) G () ma Z m D t k ( (C ) )( D t V ). w α+,β+ Z m Z m (40) Substtutg the above estmate o G() to the rght had sde of (35) yelds the desred cocluso (31) wthγ 3 beg gve by γ 3 := ma Z m D t k (ξ 1 w (C α, β m ) ) Z m Usg Lemma, we ca obta the followg. 1/. (41) Corollary 3. Suppose that the codtos of Lemma hold, the for V P,thefollowgtwoestmateshold: KV K NV γ 3 V ( N )m, (4) KV K NV γ 3 1 V ( N )m. (43) Proof. We observe that f (4) holds, the by usg the fact V 1 V, V P, (44) wecaeaslyobtatheresult(43). Thus, we oly requre to estmate (4). fact, by usg the verse equalty relatve to two orms weghted wth dfferet Jacob weght fuctos Theorem 3.31 [19], there ests a postve costat ξ depedet of such that, for V P ad Z m, D t V ξ w α+,β+ D t V w. (45) α+,β+ By the above equalty, we ca obta that D t V (ξ) V w α+,β+, Z m Z m where combg (31) yelds the desred cocluso (4). 4. The Stablty ad Covergece Aalyss uder the L Norm (46) ths secto, we wll establsh that, for suffcetly large ad N, theoperator + L α,β K N : P P has a uformly bouded verso the space C() ad the show that the appromate soluto u arrves at the most possble coverget order uder the L orm. To ths ed, we frst gve some otatos. For r N 0 ad ] (0, 1], the otato H r,] () susedtodeotethespaceoffuctos whose rth dervatve s Hölder cotuous o wth epoet ].Theormofthespacesdefedby V r,] := V r + sup,y, =y (Dr V) () (Dr y V)(y) y ]. (47) Lemma 4. Supposethatthekerelfuctok C 1,0 ( ); the the operator K s a bouded lear operator from C() to H 0,1 (); that s, for V C(), KV 0,1 (4 k + D1 k ) V. (48)

5 Abstract ad Appled Aalyss 5 Moreover, for <α,β<1,theoperatork :L () C() s also a lear bouded operator; that s, for V L (), KV k 1 1/ V w α, β. (49) Proof. t s easly proved that the operator K s a lear operator from the space C() to the space H 0,1 () or from L () to C(). Net we llustrate that (48) holds. By the defto of the orm, KV V ma k (, t) dt, (50) whch mples that KV k V. (51) O the other had, for all 1,,bytroducg 1 1 := (k ( 1,t) k(,t))v(t) dt, we ca obta that 1 := k(,t)v(t) dt, (5) (KV) ( 1 ) (KV) ( ) = 1 +, (53) whereusgthetragleequaltyyeldsthat (KV) ( 1) (KV) ( ) 1 +. (54) The left thg s to gve a estmato of 1 ad.frst, employg Lagrage mdvalue dfferetal theorem to 1 yelds that 1 D1 k V 1. (55) A drect estmato for produces that k V 1. (56) Thus, substtutg the estmates (55)-(56)totherghthad sde of (54)leadsthat (KV) ( 1) (KV) ( ) (57) ( k + D1 k ) V 1, where (51) yelds the desred cocluso (48). the followg we show that the result (49) holds. Notcg, KV = ma k (, t) V (t) dt = ma (w α/, β/ (t) k (, t))(w α/,β/ (t) V (t))dt. (58) Usg Cauchy-Schwartz equalty to the rght had sde of the equato above yelds KV ma (w α, β (t) k (, t)) V wα,β, (59) whch mples that KV k 1 w α, β V wα,β. (60) Ths complete the proof of (49). The et result cocers o the boud of the orm KV L α,β KV for V C(). For ths purpose, we troduce the result o the Lebesgue costat correspodg to the Lagrage terpolato polyomals assocated wth the zeros of the Jacob polyomals, whch comes from Lemma 3.4 [5]: Lα,β = ma V =1 Lα,β V = { O (log ), < α, β 1 {, { O ( (1/)+ma{α,β} ), otherwse. (61) Further, we also requre to make use of aother result of Ragoz, comg from [1, ], whch states that, for ay V H r,] (), there est a polyomal q P ad a postve costat ς 1 such that V q ς 1 r ] V r,]. (6) A combato of (61) ad(6) leads to that there ests a postve costat ς such that V Lα,β V { r ] log, <α,β 1 ς V r,] {, { (1/)+ma{α,β} r ], otherwse. (63) Lemma 5. Supposethatthekerelfuctok C 1,0 ( ).The there ests a postve costat ς 3 depedet of such that whe V C(), KV { log, <α,β 1 ς 3 V {, { (1/)+ma{α,β}, otherwse. (64) Proof. t follows from Lemma 4 that KV H 0,1 () for V C(), where combg (63) obtas that there ests a postve costat ξ depedet of : KV { log, <α,β 1 ξ KV 0,1 {, { (1/)+ma{α,β}, otherwse. (65)

6 6 Abstract ad Appled Aalyss Substtutg the estmate (48) totherghthadsdeofthe above equato yelds the desred cocluso wth ς 3 beg gve by ς 3 := ξ (4 k + D1 k ). (66) We use the otato [] to deote the largest teger ot more tha. Moreover, by Theorem 3.10 [3], f the kerel fucto k s a smooth fucto, the operator +K :C() C() has a bouded verso; that s, for ay V P,there ests a postve costat ρ such that ( + K) V ρ V. (67) Theorem 6. Suppose that k C 1,m ( ), /<α,β<1/.f we choose N as follows: N N m := [ 1+(1/m)+(m{α,β}/m) log 1/m ] + 1, (68) the there ests a postve teger 0 such that whe 0 ad for V P, ( + Lα,β K N)V ρ V, (69) where ρ appears (67). Proof. t follows from the hypothess that / < α, β < 1/ that (1/)+ma{α,β} tedstozeroas teds to.hece,usg (64) there ests a postve teger 1 such that 1, V ρ 4 V. (70) O the other had, usg (61) wth the hypothess that / < α, β < 1/ yelds that there ests a postve costat ξ 1 such that, for V P, Lα,β K NV ξ 1 KV K NV (1/)+ma{α,β}, (71) where combg (43) ad(68) produces that there ests a postve costat ξ, Lα,β K NV ξ V log. (7) Smlarly as before, by the fact that log teds to 0 as teds to, there ests a postve teger such that, for, Lα,β K NV ρ 4 V. (73) Hece, whe 0 := ma{ 1, },combgthesethree estmates (67), (70), ad (73) yelds that ( + Lα,β K N)V V + KV Lα,β K NV ρ V, provg the desred cocluso (69). KV (74) Theorem 6 esures that, for suffcet large, the operator equato (0) hasauquesoluto u. The et result cosders the coverget order of the appromate soluto u L orm. Theorem 7. Suppose that the kerel fucto k C m,m ( ), f C m (),ad/ < α, β < 1/.fwechooseN as (68), the there est a postve costat η ad a postve teger 0 such that, for 0, u u η u m (1/)+ma{α,β} m. (75) Proof. We frst otce that t follows from the hypothess that k C m,m ( ) ad f C m () that (3) hasauquesoluto u C m (), whch mples that u H m,1 (). Byusgthe tragle equalty, u u u Lα,β u + Lα,β u u. (76) Upo the estmato (63) wthv := u, weolyrequreto estmate the secod term the rght had sde of the above equato. fact, employg L α,β to both sdes of (3) yelds that L α,β u+lα,β Ku=Lα,β f. (77) A drect computato of the above equato ad (0)cofrms that ( + L α,β K N)( u L α,β u) = L α,β Ku Lα,β K NL α,β u. (78) By Theorem 6, there ests a postve teger 0 such that 0, u L α,β u ρ Lα,β Ku Lα,β K NL α,β u, (79) where combg (61) leads that there ests a postve costat ξ 1 such that u L α,β u ξ 1 Ku K NL α,β u (1/)+ma{α,β}. (80) To obta the estmato of the rght had sde of equato (80), we let Clearly, 1 := Ku KLα,β u, (81) := KLα,β u K NL α,β u. Ku K NL α,β u 1 +. (8) t remas to estmate 1 ad, respectvely. Frst, usg the hypothess that /<α,β<1/ad the result (49) wth V := u L α,β u produces that there ests a postve costat ξ depedet of such that 1 ξ u Lα,β u, (83)

7 Abstract ad Appled Aalyss 7 wherecombgtheresult(3) wth := 0, r := m, V := u yelds that there ests a postve costat ξ 3 depedet of such that 1 ξ 3 Dm u w α+m,β+m m. (84) Hece, a combato of (84)ad the followg equalty Dl u 1 w α+l,β+l Dl u, l N, (85) produces that there ests a postve costat ξ 4 such that produces +h w λ +h 1,λ (t) dt (1 t) λ 1 dt h1+λ 1, (9) 1+λ 1 whch esures the desred cocluso. a smlar approach as the above case, clearly, the result (89)holdsforthecasethatλ 1 0whle λ <0. At last, whe the codtos λ 1 <0ad λ <0hold, usg the et equato 1 ξ 4 u m m. (86) O the other had, usg the results (4) ad(31) leadsthat there ests a postve costat ξ 5 such that ca produce (1 t) λ 1 (1+t) λ = (1 t)λ 1 + (1+t) λ (1 t) λ 1 + (1+t) λ (93) ξ 5 ( Z m D u w α+,β+ )N m, (87) where combg (68) ad(85) yelds that there ests a postve costat ξ 6, ξ 6 u m m (1/) ma{α,β} log. (88) A combato of the above estmato ad (80), (8), ad (86) yelds the desred result. Theorem 7 llustrates that the appromate soluto obtaed by the proposed method arrves at the most possble coverget order. 5. The Stablty ad Covergece Aalyss uder the L Norm As demostrated the prevous secto, ths secto we are gog to prove that, for suffcetly large ad N, the operator + L α,β K N : P P has a uformly bouded verso the L () space ad the show that the appromate soluto arrves at the optmal coverget order. To ths ed, we frst gve a few results. Lemma 8. Suppose that < λ 1,λ <1ad h [0,1].f, + h, the oe has that +h w λ 1,λ (t) dt ma {h, h1+m{λ 1,λ } }. (89) 1+m {λ 1,λ } Proof. We wll prove that the result (89) holds the followg four cases: (1) λ 1 0 ad λ 0; () λ 1 0 but λ < 0; (3) λ 1 <0whle λ 0;(4)λ 1 <0ad λ <0. Frstly, we otce that, for λ 1 0, λ 0, (1 t) λ 1 (1+t) λ, (90) whch cofrms the desred cocluso. f the codtos λ 1 <0ad λ 0hold, the usg (1+t) λ (91) (1 t) λ 1 (1+t) λ (1 t) λ 1 + (1+t) λ. (94) Thus,agausgthesamemethodasbeforeyeldsthe desred result. Net we esure that the operator K :L () H 0,κ () s a bouded lear operator wth certa postve costat κ. Lemma 9. Suppose that < α, β < 1, k C 1,0 ( );the K s a bouded lear operator from L () to H 0,κ () wth κ:= m{1/, (1/) + m{ α/, β/}};thats,thereestsa postve costat ζ 1 such that, for V L (), KV 0,κ ζ 1 V. (95) Proof. By the estmato (49) Lemma5, there ests a postve costat ξ 1 such that, for V L (), KV ξ 1 V. (96) O the other had, for 1,, wthout loss of geeralty, we assume that 1.Bytroducg we have 1 := (k ( 1,t) k(,t))v(t) dt, := k( 1,t)V(t) dt, 1 (97) (KV) ( 1 ) (KV) ( ) = 1 +. (98) Hece, t remas to estmate 1 ad,respectvely.forths purpose, by reformulatg 1 ad as follows 1 := (w α/, β/ (t) (k ( 1,t) k(,t))) (w α/,β/ (t) V (t))dt, := (w α/, β/ (t) k( 1,t))(w α/,β/ (t) V (t))dt 1 (99)

8 8 Abstract ad Appled Aalyss ad the employg Cauchy-Schwarz equalty to 1 ad, respectvely, we ca obta that ( 1 ( w α, β (t) (k( 1,t) k(,t)) dt) 1 ( (t) V (t) dt), w α, β (t) k ( 1,t)dt)( (t) V (t) dt). 1 (100) Usg the hypothess that k C 1,0 ( ) ad the Lagrage mdvalue dfferetal theorem yelds that 1 D1 k 1 w α, β V 1. (101) A drect estmato for produces that k V 1 w α, β (t) dt. (10) f the codto 1 >1holds, the we have k V 1 w α, β 1 κ, (103) otherwse, usg (10), where combg (89) wthλ 1 := α, λ := β, := 1 ad +h := leadstothatthere ests a postve costat ξ such that ξ V ma { 1, 1 1+m{ α, β} }. (104) A combato of (98) (104) ad the tragle equalty yelds that there ests a postve costat ξ 3 such that (KV) ( 1) (KV) ( ) ξ 3 V 1 κ, (105) where (96) draws the desred cocluso. The et result cocers o dfferece betwee KV ad L α,β KV for V L ().Forthspurpose,wewllmakeuse of the et result proposed [5]. For ay V C(), there ests a postve costat ζ depedet of : Lα,β V ζ V. (106) A combato of (61) ad(106) leads to that there ests apostvecostatζ 3 such that, for V H r,] (), V Lα,β V ζ 3 V r,] r ]. (107) Aga usg Theorem 3.10 [3], we kow that 0 s the uque egevalue of Volterra tegral operator K; cosequetly, the operator + K : L () L () has a bouded verso; that s, for ay V P, there ests a postve costat such that ( + K)V V. (108) Theorem 10. Suppose that k C 1,m ( ) ad / < α, β < 1, α + β 1.foechoosesN as follows: N N m := [log 1/m ] + 1, (109) the there ests a postve teger 0 such that 0 ad for V P, ( + Lα,β K N)V w α,β V wα,β, (110) where appears (108). Proof. Ths proof s smlar to that of Theorem 6. By Lemma 9, forv P,wehaveKV H 0,κ (), where combg (95) ad(107) obtas that there ests a postve costat ξ 1, KV ξ 1 V κ. (111) Hece,bythefactthatlm κ =0, there ests a postve teger 1 such that, for 1, KV 4 V wα,β. (11) O the otherhad,usg (106) obtas that there ests a postve costat ξ such that Lα,β K NV w ξ α,β KV K NV. (113) By the hypothess that / < α,β 1,adα+β 1, a combato of (4) ad(109) yelds that there ests a postve costat ξ 3 such that KV K NV ξ 3 V log. (114) Substtutg the above estmato to the rght had sde of (113)producesthat Lα,β K NV w ξ α,β ξ 3 V log. (115) Aga usg the same fact that lm log = 0,there ests a postve teger such that for, Lα,β K NV w α,β 4 V wα,β. (116) Whe 0 := ma{ 1, },thesethreeestmates(108), (11), ad (116)yeldthat ( + Lα,β K N)V w α,β w wα,β, (117) whch fers our result. Ths above result shows that (0)hasauquesoluto u the space L (). Net result cosders the appromate order of the soluto u. Theorem 11. Suppose that the kerel fucto k C m,m ( ), f C m (),ad/<α,β<1,α+β 1.foechoosesN as (109), the there est a postve costat θ ad a postve teger 0 such that, for 0, u u θ( Z m D u w α+,β+ ) m. (118)

9 Abstract ad Appled Aalyss 9 Table 1: The umercal result based o the collocato odes 0,0, Z u u 1.40e 3.16e 4 4.8e e 8.60e e 1 u u w0,0 1.85e 4.10e e e e e 1 Table : The umercal result based o the collocato odes 1/4,1/3, Z u u 1.16e.61e e e 8.14e e 13 u u w1/4,1/3 1.5e 3.38e 4 4.5e e 8.53e e 1 Proof. The proof of Theorem 11 s smlar as that of Theorem 7. tfollowsfromtheorem 7 that u C m (), whch mples that u H m (). Byusgthetragle equalty, u u w α,β u Lα,β u + Lα,β u u. (119) Upo the estmato (3)wth:=0,r:=mad V := u,we oly eed to estmate L α,β u u.employgtheresult (78), (106), ad Theorem 7, there est a postve costat ξ 1 ad a postve teger 0 such that 0, u L α,β u ξ 1 Ku K NL α,β u (10) To obta the estmato of the rght had sde of (10), we let Clearly, 1 := Ku KLα,β u, (11) := KLα,β u K NL α,β u. Ku K NL α,β u 1 +. (1) Upo the estmato (84), we oly requre to estmate. fact, a combato of (87) ad(109) yelds that there ests apostvecostatξ : ξ ( D u ) m log. (13) w α+,β+ Z m A combato of (84) ad(10) (13) yelds the desred result. Theorem 11 llustrates that the proposed method preserves the optmal order of covergece. 6. Oe Numercal Eample ths secto, we are gog to preset oe umercal eample to demostrate the effcecy of the spectral Jacobcollocato method for solvg (3). each eample, we use two spectral collocato approaches assocated wth the weght fucto w 0,0 ad w 1/4,1/3, respectvely. Here, we compute the Gauss-Jacob quadrature rule odes ad weghts by Theorems 3.4 ad 3.6 dscussed [19]. All computer programs are compled by Matlab laguage. Eample. Cosder the secod kd Volterra tegral equato (1)wth k (, t) =e t, f() =e + e(+) e (+). (14) + The correspodg eact soluto s gve by u() = e. As epected, the errors show a epoetal decay, sce ths semlog represetato the error varatos are essetally lear versus the degrees of the polyomal. From the theoretcal results we observe that the umercal errors should decay wth a epoetal rate, ad we also fd that the errors show a epoetal decay (Tables 1 ad ). Coflct of terests The authors declare that there s o coflct of terests regardg the publcato of ths paper. Ackowledgmets Hogfeg Guo s supported by Natural Scece Foudato of Cha Grat , Scece ad Techology Developmet Project of Shadog Provce Grat 01GGB0141, ad Postgraduate Educato ovato Project of Shadog Uversty of Face ad Ecoomcs Grat SCY1308. Haotao Ca s supported by Natural Scece Foudato of Shadog Provce Grat ZR010AQ001. X Zhag s supported by Natural Scece Foudato of Shadog Provce Grat ZR010AQ01. Refereces [1] H. Bruer, Collocato Methods for Volterra tegral ad Related Fuctoal Equatos Methods, Cambrdge Uversty Press, Cambrdge, UK, 004. [] A. H. Bhrawy, A Jacob-Gauss-Lobatto collocato method for solvg geeralzed Ftzhugh-Nagumo equato wth tmedepedet coeffcets, Appled Mathematcs ad Computato, vol., pp , 013. [3] H. Majda, Composte quadrature rules for a class of weakly sgular Volterra tegral equatos wth ocompact kerels, Appled Numercal Mathematcs,vol.83,pp.1 11,014. [4] Z. Che ad W. Jag, The eact soluto of a class of Volterra tegral equato wth weakly sgular kerel, Appled

10 10 Abstract ad Appled Aalyss Mathematcs ad Computato, vol.17,o.18,pp , 011. [5] Y. Che ad T. Tag, Spectral methods for weakly sgular Volterra tegral equatos wth smooth solutos, Joural of Computatoal ad Appled Mathematcs, vol.33,o.4,pp , 009. [6] S. Pshb, F. Ghoresh, ad M. Hadzadeh, The sem-eplct Volterra tegral algebrac equatos wth weakly sgular kerels: the umercal treatmets, Joural of Computatoal ad Appled Mathematcs,vol.45,pp.11 13,013. [7] G. Zaker ad M. Navab, Sc collocato appromato of o-smooth soluto of a olear weakly sgular Volterra tegral equato, Joural of Computatoal Physcs, vol.9, o. 18, pp , 010. [8] Y. Jag ad J. Ma, Spectral collocato methods for Volterrategro dfferetal equatos wth ocompact kerels, Joural of Computatoal ad Appled Mathematcs, vol.44,pp , 013. [9] G. N. Elagar ad M. Kazem, Chebyshev spectral soluto of olear Volterra-Hammerste tegral equatos, Joural of Computatoal ad Appled Mathematcs, vol.76,o.1-,pp , [10] H. Fujwara, Hgh-accurate umercal method for tegral equatos of the frst kd uder multpleprecso arthmetc, Preprt, RMS, Kyoto Uversty, 006. [11] T. Tag, X. Xu, ad J. Cheg, O spectral methods for Volterra tegral equatos ad the covergece aalyss, Joural of Computatoal Mathematcs,vol.6,o.6,pp ,008. [1] Y. Che ad T. Tag, Covergece aalyss of the Jacob spectral-collocato methods for Volterra tegral equatos wth a weakly sgular kerel, Mathematcs of Computato, vol.79,o.69,pp ,010. [13]Z.Xe,X.L,adT.Tag, Covergeceaalyssofspectral Galerk methods for Volterra type tegral equatos, Joural of Scetfc Computg, vol. 53, o., pp , 01. [14] Y. We ad Y. Che, Legedre spectral collocato methods for patograph Volterra delay-tegro-dfferetal equatos, JouralofScetfcComputg, vol. 53, o. 3, pp , 01. [15] H. T. Ca, A Jacob-collocato method for solvg secod kd Fredholm tegral equatos wth weakly sgular kerels, Scece Cha: Mathematcs,014. [16]M.R.Hajmohammad,S.S.Nourazar,adA.H.Maesh, Sem-aalytcal treatmets of cojugate heat trasfer, Joural of Mechacal Egeerg Scece, vol.7,o.3,pp , 01. [17] M. R. Hajmohammad ad S. S. Nourazar, O the soluto of characterstc value problems arsg lear stablty aalyss; sem aalytcal approach, Appled Mathematcs ad Computato,vol.39,pp.16 13,014. [18] M. R. Hajmohammad, R. Mohammad, ad S. S. Nourazar, Cojugate forced covecto heat trasfer from a heated flat plate of fte thckess ad temperature-depedet thermal coductvty, Heat Trasfer Ege,vol.35,pp ,014. [19]J.She,T.Tag,adL.Wag,Spectral Methods: Algorthms, Aalyss ad Applcatos, vol.41ofsprger Seres Computatoal Mathematcs, Sprger, New York, NY, USA, 011. [0] G. Mastroa ad M. G. Russo, Lagrage terpolato weghted Besov spaces, Costructve Appromato,vol.15,o., pp , [1] D. L. Ragoz, Polyomal appromato o compact mafolds ad homogeeous spaces, Trasactos of the Amerca Mathematcal Socety,vol.150,pp.41 53,1970. [] D. L. Ragoz, Costructve polyomal appromato o spheres ad projectve spaces, Trasactos of the Amerca Mathematcal Socety,vol.16,pp ,1971. [3] R. Kress, Lear tegral Equatos, Sprger, New York, NY, USA, 1st edto, 1989.

11 Advaces Operatos Research Volume 014 Advaces Decso Sceces Volume 014 Joural of Appled Mathematcs Algebra Volume 014 Joural of Probablty ad Statstcs Volume 014 The Scetfc World Joural Volume 014 teratoal Joural of Dfferetal Equatos Volume 014 Volume 014 Submt your mauscrpts at teratoal Joural of Advaces Combatorcs Mathematcal Physcs Volume 014 Joural of Comple Aalyss Volume 014 teratoal Joural of Mathematcs ad Mathematcal Sceces Mathematcal Problems Egeerg Joural of Mathematcs Volume Volume 014 Volume Volume 014 Dscrete Mathematcs Joural of Volume Dscrete Dyamcs Nature ad Socety Joural of Fucto Spaces Abstract ad Appled Aalyss Volume Volume Volume 014 teratoal Joural of Joural of Stochastc Aalyss Optmzato Volume 014 Volume 014