SPECTRAL-COLLOCATION METHOD FOR FRACTIONAL FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

Transcription

1 J. Korean Math. Soc. 5 o. pp. 3 SPECTRAL-COLLOCATIO METHOD FOR FRACTIOAL FREDHOLM ITEGRO-DIFFERETIAL EQUATIOS Yn Yang Yanpng Chen and Yunqng Huang Abstract. We propose and analyze a spectral Jacob-collocaton approxmaton for fractonal order ntegro-dfferental equatons of Fredholm- Volterra type. The fractonal dervatve s descrbed n the Caputo sense. We provde a rgorous error analyss for the collecton method whch shows that the errors of the approxmate soluton decay exponentally n L norm and weghted L -norm. The numercal examples are gven to llustrate the theoretcal results.. Introducton Many phenomena n engneerng physcs chemstry and other scences can be descrbed very successfully by models usng mathematcal tools from fractonal calculus.e. the theory of dervatves and ntegrals of fractonal nonnteger order. Ths allows one to descrbe physcal phenomena more accurately. Moreover fractonal calculus s appled to model the frequency dependent dampng behavor of many vscoelastc materals economcs and dynamcs of nterfaces between nanopartcles and substrates. Recently several numercal methods to solve fractonal dfferental equatons FDEs and fractonal ntegro-dfferental equatons FIDEs have been proposed. In ths artcle we are concerned wth the numercal study of the followng fractonal Fredholm ntegro-dfferental equaton: D γ yt = yt + y = y t t τyτdτ + T < γ < t [ T ] t ςyςdς + ft Receved May 3; Revsed September 3. Mathematcs Subject Classfcaton. Prmary 65R 5J5 65. Key words and phrases. spectral Jacob-collocaton method fractonal order Fredholm ntegro-dfferental equatons Caputo dervatve. Ths wor was supported by SFC Project Chna Postdoctoral Scence Foundaton Grant 3M53789 Program for Changjang Scholars and Innovatve Research Team n Unversty IRT79 Project of Scentfc Research Fund of Hunan Provncal Scence and Technology Department 3RS57 and the Research Foundaton of Hunan Provncal Educaton Department 3B6. 3 c The Korean Mathematcal Socety

2 Y. YAG Y. CHE AD Y. HUAG where the source functon f and the ernel functon are gven the functon yt s the unnown one and y R. Here the gven functons f are assumed to be suffcently smooth on ther respectve domans I and τ t T. In the equaton D γ denotes the fractonal dervatve of order γ defned as a Caputo dervatve. Dfferental and ntegral equatons nvolvng dervatves of non-nteger order have shown to be adequate models for varous phenomena arsng n dampng laws dffuson processes models of earthquae [5] flud-dynamcs traffc model [6] mathematcal physcs and engneerng [9] flud and contnuum mechancs [5] chemstry acoustcs and psychology []. Let Γ denote the Gamma functon. For any postve nteger n and n < γ < n the Caputo dervatve D γ ft s defned as follows: D γ ft = Γn γ t a f n τ dτ t [a b]. t τ γ n+ Also the Remann-Louvlle fractonal ntegral I γ of order γ s defned as 3 I γ ft = Γγ We note that t a t τ γ fτdτ. n I γ D γ ft = ft f a t!. From the fractonal ntegro-dfferental equaton can be descrbed as 5 D γ yt = yt + yt = Γγ t t t τyτdτ + T < γ < t [ T ] t τ γ D γ yτdτ + y. t ςyςdς + ft Several methods have been ntroduced to solve FDEs n analytcal and numercal frames. Analytcal methods nclude varous transformaton technques [8] operatonal calculus methods [] the Adoman decomposton method [38] and the teratve and seres-based method [37]. A small number of algorthms for the numercal soluton of FDEs have been suggested [] and most of them are fnte dfference methods whch are generally lmted to low dmensons and are of lmted accuracy. As we now fractonal dervatves are global they are defned by an ntegral over the whole nterval [ T ] and therefore global methods such as spectral methods are perhaps better suted for FDEs. Standard spectral methods possess an nfnte order of accuracy for the equatons wth regular solutons whle falng for many complcated problems wth sngular solutons. So t s relevant to be nterested n how to enlarge the adaptablty of spectral methods and

3 SPECTRAL-COLLOCATIO METHODS FOR FFIDES 5 construct certan smple approxmaton schemes wthout a loss of accuracy for more complcated problems. Spectral methods have been proposed to solve fractonal dfferental equatons such as the Legendre collocaton method [ 36] Legendre wavelets method [3 3] homotopy perturbaton method [] and Jacob-Gauss-Lobatto collocaton method []. The authors n [ 3 39] constructed an effcent spectral method for the numercal approxmaton of fractonal ntegro-dfferental equatons based on tau and pseudo-spectral methods. Moreover Bhrawy et al. [7] ntroduced a quadrature shfted Legendre tau method based on the Gauss-Lobatto nterpolaton for solvng mult-order FDEs wth varable coeffcents and n [6] shfted Legendre spectral methods have been developed for solvng fractonal-order mult-pont boundary value problems. In [35] truncated Legendre seres together wth the Legendre operatonal matrx of fractonal dervatves are used for the numercal ntegraton of fractonal dfferental equatons. In [8] the authors derved a new explct formula for the ntegral of shfted Chebyshev polynomals of any degree for any fractonal-order. The shfted Chebyshev operatonal matrx [5] and shfted Jacob operatonal matrx [] of fractonal dervatves have been developed whch are appled together wth the spectral tau method for numercal soluton of general lnear mult-term fractonal dfferental equatons. However very few theoretcal results were provded to justfy the hgh accuracy numercally obtaned. Recently Chen and Tang [ ] developed a novel spectral spectral Jacob-collocaton method to solve second nd Volterra ntegral equatons wth a wealy sngular ernel and provded a rgorous error analyss whch theoretcally justfes the spectral rate of convergence. Inspred by the wor of [] we extend the approach to fractonal order ntegro-dfferental equatons and provde a rgorous convergence analyss for the Jacob-collocaton method whch ndcates that the proposed method converges exponentally provded that the data n the gven FIDE are smooth. Ths paper s organzed as follows. In Secton we outlne the spectral approach for. Some lemmas useful for establshng the convergence result wll be provded n Secton 3. The convergence analyss wll be carred out n Secton and Secton 5 contans numercal results whch wll be used to verfy the theoretcal result obtaned n Secton.. Jacob-collocaton method Let ω αβ x = x α + x β be a weght functon n the usual sense for α β >. The set of Jacob polynomals {Jn αβ x} n= forms a complete L ω -orthogonal system where L αβ ω s a weghted space defned αβ by L ω = {v : v s measurable and v αβ ωαβ < }

4 6 Y. YAG Y. CHE AD Y. HUAG equpped wth the norm v ω αβ = vx ω αβ xdx and the nner product u v ω αβ = uxvxω αβ xdx u v L ωαβ. For a gven we denote by {θ } the Legendre ponts and by {ω } the correspondng Legendre weghts.e. Jacob weghts {ω }. Then the Legendre-Gauss ntegraton formula s 6 fxdx fθ ω where ω = ω x. Smlarly we denote by { θ } the Jacob-Gauss ponts and by {ω αβ } the correspondng Jacob weghts. Then the Jacob-Gauss ntegraton formula s 7 fxω αβ xdx f θ ω αβ where ω αβ = ω αβ x. For a gven postve nteger we denote the collocaton ponts by {x αβ } = whch s the set of + Jacob-Gauss ponts correspondng to the weght ω αβ x. Let P denote the space of all polynomals of degree not exceedng. For any v C[ ] we can defne the Lagrange nterpolatng polynomal v P satsfyng I αβ I αβ vxαβ = vx αβ. The Lagrange nterpolatng polynomal can be wrtten n the form I αβ vx = vx αβ F x where F x s the Lagrange nterpolaton bass functon assocated wth {x αβ } =. For the sae of applyng the theory of orthogonal polynomals we use the change of varables to transfer the ntegraton nterval [ T ] to a fxed nterval I =: [ ] t = T + x x τ = T + s s = t T = τ T

5 and let SPECTRAL-COLLOCATIO METHODS FOR FFIDES 7 ς = T + ξ ξ = ς T ux = y T + x Dγ ux = D γ y T + x gx = f T + x K x s= T T + x T + s K x ξ= T T + x T + ξ. The fractonal ntegro-dfferental equaton n one dmenson or 5 s of the form 8a D γ ux = ux + K x susds + K x ξuξdξ + gx < γ < x I ux = γ T x 8b x s γ D γ usds + u. Γγ Let µ = γ. Set the collocaton ponts {x µ µ } = as the set of + Jacob-Gauss ponts assocated wth ω µ µ x. Assume that Eq. 8 holds at x µ µ : 9a 9b D γ ux µ µ = ux µ µ + + ux µ µ = T Γγ µ µ K x µ µ susds K x µ µ ξuξdξ + gx µ µ γ µ µ x µ µ s µ D γ usds + u. The man dffculty n obtanng a hgh order of accuracy s to compute the ntegral term n 9. In partcular for small values of x µ µ there s lttle nformaton avalable for us. To overcome ths dffculty we wll transfer the ntegraton nterval [ x µ µ ] for a fxed to a fxed nterval [ ] and then mae use of some approprate quadrature rule. More precsely we frst mae a smple lnear transformaton: sx θ = + x θ + x θ. Then 9 becomes a D γ ux µ µ = ux µ µ + + x µ µ + K x µ µ K x µ µ ξuξdξ + gx µ µ sx µ µ θusx µ µ θdθ

6 8 Y. YAG Y. CHE AD Y. HUAG b ux µ µ = Γγ T + x µ µ γ θ µ D γ usx µ µ θds + u. ext usng a + -pont Gauss quadrature formula relatve to the Legendre weghts {ω }.e. Jacob weghts {ω } the ntegraton term n a can be approxmated by 3 K x µ µ K x µ µ K x µ µ ξuξdξ sx µ µ θusx µ µ θdθ sx µ µ θ usx µ µ θ ω K x µ µ θ uθ ω where {θ } s the set of Jacob-Gauss ponts correspondng to the set of Jacob weghts {ω }.e. {θ } s Legendre-Gauss ponts. θ µ D γ usx µ µ θds D γ usx µ µ θ ω µ where { θ } s the set of Jacob-Gauss ponts correspondng to the weghts {ω µ }. We use u ux µ µ u γ Dγ ux µ µ and 5 Ux = u j F j x U γ x = u γ j F jx j= where F j j =... s the Lagrange nterpolaton bass functons assocated wth {x µ µ } = whch s the set of + Jacob-Gauss ponts. Combnng the above equaton and a yelds 6a u γ = u + + x µ µ + K x µ µ θ K x µ µ j= j= sx µ µ θ u j F j θ ω j= + gx µ µ u j F j sx µ µ θ ω

7 6b = u + + x µ µ + j= u = Γγ = Γγ u j SPECTRAL-COLLOCATIO METHODS FOR FFIDES 9 u j K x µ µ sx µ µ θ F j sx µ µ θ ω j= K x µ µ θ F j θ ω + gx µ µ T + x µ µ T + x µ µ γ γ j= j= u γ j u γ j F jsx µ µ θ ω µ + u F j sx µ µ θ ω µ + u. We can get the values of {u } = and {uγ } = by solvng the system of lnear equatons 6 and obtan the expressons of Ux and U γ x accordngly. 3. Some useful lemmas In ths secton we wll provde some elementary lemmas whch are mportant for the dervaton of the man results n the subsequent secton. Lemma 3. see [9]. Assume that an +-pont Gauss quadrature formula relatve to the Jacob weght s used to ntegrate the product uϕ where u H m I wth I for some m and ϕ P. Then there exsts a constant C ndependent of such that 7 uxϕxdx u ϕ C m u H m I ϕ L ω αβ ω αβ I where 8 u H m ωαβ I = u ϕ = m j=mnm+ ux j ϕx j ω j. j= u j L ω αβ I Lemma 3. see [9 ]. Assume that u H m I and denote by I αβ ω αβ u ts nterpolaton polynomal assocated wth the + Jacob-Gauss ponts {x j } j= namely I αβ u = ux F x. = /

8 Y. YAG Y. CHE AD Y. HUAG Then the followng estmates hold: 9a u I αβ 9b u L ω αβ I C m u H m ωαβ I u I αβ u L I where ω c = ω { C m log u H m C γ m u H m ω c I ω c I α β γ = maxα β otherwse denotes the Chebyshev weght functon. Lemma 3.3 see [6]. Assume that {F j x} j= are the -th degree Lagrange bass polynomals assocated wth the Gauss ponts of the Jacob polynomals. Then I αβ L I max F j x = x [] j= { Olog < α β O γ+ γ = maxα β otherwse. Lemma 3. Gronwall nequalty see [7] Lemma 7... Suppose L < µ < and u and v are a non-negatve locally ntegrable functons defned on [ ] satsfyng ux vx + L Then there exsts a constant C = Cµ such that ux vx + CL x τ µ uτdτ. x τ µ vτdτ for x <. If a nonnegatve ntegrable functon Ex satsfes Ex L where Jx s an ntegrable functon then Esds + Jx < x E L C J L E L p ω αβ C J L p ω αβ q. Lemma 3.5 see [3 3]. For a nonnegatve nteger r and κ there exsts a constant C rκ > such that for any functon v C rκ [ ] there exsts a polynomal functon T v P such that v T v L I C rκ r+κ v rκ where rκ s the standard norm n C rκ [ ] T s a lnear operator from C rκ [ ] nto P as stated n [3 3].

9 SPECTRAL-COLLOCATIO METHODS FOR FFIDES Lemma 3.6 see []. Let κ and let M be defned by Mvx = x τ µ Kx τvτdτ. Then for any functon v C[ ] there exsts a postve constant C such that Mvx Mvx x x C max vx x [] under the assumpton that < κ < µ for any x x [ ] and x x. Ths mples that Mv κ C max x [] vx < κ < µ. Lemma 3.7 see [7]. For every bounded functon v there exsts a constant C ndependent of v such that sup vx j F j x L ω αβ I C j= max vx x [] where F j x j =... are the Lagrange nterpolaton bass functons assocated wth the Jacob collocaton ponts {x j } j=. Lemma 3.8 see []. For all measurable functons f the followng generalzed Hardy s nequalty /q b /p b T fx uxdx q fx p vxdx a holds f and only f /q b utdt sup a<x<b x a a /p v p tdt < p = p p for the case < p q <. Here T s an operator of the form T F x = a x tftdt wth x t a gven ernel u v are nonnegatve weght functons and a < b.. Convergence analyss Ths secton s devoted to provde a convergence analyss for the numercal scheme. The goal s to show that the rate of convergence s exponental.e. that spectral accuracy can be obtaned for the proposed approxmatons. Frstly we wll carry out our convergence analyss n the functon space L I.

10 Y. YAG Y. CHE AD Y. HUAG Theorem.. Let ux be the exact soluton of the fractonal ntegro-dfferental equaton 8 whch s assumed to be suffcently smooth. Assume that Ux and U γ x are obtaned by usng the spectral collocaton scheme 6 together wth a polynomal nterpolaton 5. If γ assocated wth fractonal order < γ < and µ = γ u H m+ ω I then µ µ 3 U γ D γ u L I C γ K m u L I + U C m log K u L I + U < γ < < γ U u L I C γ K m u L I + U < γ < C m log K u L I + U < γ provded that s suffcently large where C s a constant ndependent of but depends on the bound of the functon Kx s and the ndex µ 5 K = max K x sx θ H m x [] ω I + max K x ξ H m x [] ω 6 U = D γ u H m ω c I + u H m ω c I. Proof. Snce U γ x = j= uγ j F jx P we have θ µ U γ sds = the numercal scheme 6 can be wrtten as 7a 7b where u γ = u + + x µ µ + u = Γγ I = + x µ µ I = K x µ µ U γ sx µ µ θ ω µ I sx µ µ θusx µ µ θdθ K x µ µ ξuξdξ + I + I + gx µ µ T + x µ µ γ θ µ U γ sds + u + x µ µ K x µ µ K x µ µ K x µ µ θ Uθ ω sx µ µ θ Usx µ µ θ ω sx µ µ θusx µ µ θdθ K x µ µ ξuξdξ.

11 SPECTRAL-COLLOCATIO METHODS FOR FFIDES 3 Let e and e γ denote the error functons ex = Ux ux e γ x = U γ x D γ ux. Usng the ntegraton error estmates for the Jacob-Gauss quadrature stated n Lemma 3. we have 8 9 I x C m max K x sx θ H m x [] ω I U L I C m max K x sx θ H m x [] I u L I + e L I ω I x C m max K x ξ H m x [] ω I U L I C m max K x ξ H m x [] I u L I + e L I. ω It follows from 9 and a that 3a 3b u γ = u + u = Γγ µ µ K x µ µ susds + + I + I + gx µ µ T + x µ µ γ µ µ K x µ µ ξuξdξ x µ µ s µ U γ sds + u. Multplyng by F x both sdes of 3 and summng from to yeld 3 U γ x = Ux + I µ µ + I µ µ + I µ µ Ux = I µ µ where + I µ µ It follows from 8 that 3 K x susds + I µ µ K x susds + I µ µ g + J x + J x T + x µ µ γ x Γγ T + x µ µ γ x Γγ = K x µ µ sesds K x µ µ sesds x s µ D γ usds x s µ e γ sds J x = I F x J x = I F x. U γ x = Ux + I µ µ D γ u u gx = +u

12 Y. YAG Y. CHE AD Y. HUAG Then we have 33a Ux = I µ µ e γ x = ex + 33b where ex = Γγ + I µ µ K x sesds + K x sesds + I µ µ g + J x + J x ux + I µ µ Γγ T + x µ µ γ x x s µ e γ sds. Kx sesds + J x + J x + J 3 x + J x + J 5 x T + x µ µ J 3 x = I µ µ D γ ux D γ ux J x = I µ µ ux ux J 5 x = I µ µ J 6 x = Γγ Kx sesds γ I µ µ T + x µ µ γ x s µ e γ sds + J x + J 6 x Kx sesds x s µ e γ sds Due to 33b and usng the Drchlet s formula whch states that τ Φτ sdsdτ = provded the ntegral exsts we obtan 3 e γ x =ex + Γγ T + x µ µ s γ s Φτ sdτ ds x s µ e γ sds. Kx sds x τ µ e γ τdτ τ + Kx s J s + J 6 s ds + J x + J x + J 3 x + J x + J 5 x Denote D := {x s : s x x [ ]}. We have s Kx sds max Kx s Γγ Γγ xs D M and then 3 gves 35 τ e γ x M x τ µ e γ τ dτ + ex

13 SPECTRAL-COLLOCATIO METHODS FOR FFIDES 5 + Kx s J s + J 6 s ds + J x + J x + J 3 x + J x + J 5 x. It follows from the Gronwall nequalty Lemma 3. that 36 e γ x L I C ex L I + 6 J L I It follows from 33b that 37 ex L I C e γ x L I + J x L I Then 38 e γ x L I C 39 ex L I C = =6 6 J L I = 6 J L I. = Usng Lemma 3.3 the estmates 8 and 37 we have J L I { C µ max I < µ < C log max I µ < C µ m max x [] K x sx θ H m ω u L I + e γ x L I + =6 J x L I C m log max x [] K x sx θ H m ω u L I + e γ x L I + =6 J x L I J L I { C µ max I < µ < C log max I µ < C µ m max x [] K x sx θ H m ω u L I + e γ x L I + =6 J x L I C m log max x [] K x sx θ H m ω u L I + e γ x L I + =6 J x L I.. < µ < µ <. < µ < µ <.

14 6 Y. YAG Y. CHE AD Y. HUAG Due to Lemma 3. { C µ m D γ u H m ω J 3 L I c I < µ < C m log D γ u H m ω c I µ < { C µ m u H m J L ω I c I < µ < C m log u H m ω c I µ <. By vrtue of Lemma 3.9b wth m = 3 J 5 L I { C µ e L I < µ < C e L I { µ < C µ e γ L I + J 3 L I + J 5 L I < µ < C e γ L I + J 3 L I + J 5 L I µ <. We now estmate the term J 5 x. It follows from Lemma 3.5 and Lemma 3.6 wth Kx τ = Γγ that J 6 L I= I µ µ IMe γ L I = I µ µ IMe γ T Me γ L I + I µ µ L I C Me γ κ { C µ κ e γ L I < µ < C κ log e γ L I µ < where n the last step we have used Lemma 3.6 under the followng assumpton { µ < κ < µ when < µ < < κ < µ when µ < provded that s suffcently large. Combnng 3 and gves U γ x u γ x L I C K µ m u L I+ D γ u H m ω c C m log K u L I+ D γ u H m ω c I + u H m I + u H m ω c I < µ < ω c I µ <. Ux ux L I C K µ m u L I+ D γ u H m ω c C m log K u L I+ D γ u H m ω c I + u H m Usng γ = µ we have the desred estmate 3 and. I + u H m ω c I < µ < ω c I µ <. ext we wll derve the error estmates n the functon space L ω µ µ I.

15 SPECTRAL-COLLOCATIO METHODS FOR FFIDES 7 Theorem.. If the hypotheses gven n Theorem. hold then 5 U γ x u γ x L ω µ µ I C V m + γ κ V + γ κ U < γ < C V m + κ log V + κ log U < γ 6 Ux ux L ω µ µ I C V m + γ κ V + γ κ U < γ < C V m + κ log V + κ log U < γ for any κ γ provded that s suffcently large and C s a constant ndependent of where V = K u L I + D γ u H ω c I + u H ω c I V = K u L I U = D γ u H m ω c I + u H m ω c I. Proof. By usng the generalzaton of the Gronwall nequalty Lemma 3. and the Hardy nequalty Lemma 3.8 t follows from 33 that 7 e γ L ω µ µ I C and 8 e L ω µ µ I C ow usng Lemma 3.7 we have 9 5 J L ω µ µ I C max I x x [] 6 = 6 = C m max K x sx θ H m x [] J L ω µ µ I C max I x x [] C m max K x ξ H m x [] J L ω µ µ I J L ω µ µ I. u ω I L I + e L I. u ω I L I + e L I. By the convergence result n Theorem. m = we have e L I C D γ u H ω c I + u H ω c I + u L I.

16 8 Y. YAG Y. CHE AD Y. HUAG So that 5 = J L ω µ µ I C m K D γ u H Due to Lemma 3.9a 5 J 3 L 3 ω µ µ I C m D γ u H m ω c I J L ω µ µ I C m u H m ω c I. By vrtue of Lemma 3.9a wth m = J 5 L ω µ µ I C Kx sesds 53 C e L ω µ µ I. Fnally t follows from Lemma 3.5 and Lemma 3.7 that 5 J 6 L ω µ µ I = I µ µ IMe γ L ω µ µ I = I µ µ IMe γ T e γ L ω µ µ I ω c I + u H ω c I + u L I H ω µ µ I I µ µ Me γ T e γ L ω µ µ I + Me γ T e γ L ω C Me γ T e γ L I C κ Me γ κ C κ e γ L I where n the last step we used Lemma 3.6 for any κ µ. convergence result n Theorem. we obtan that µ µ I. By the 55 6 C K µ m κ u L J L ω µ µ I I + U < µ < = C m κ log K u L I + U µ < for suffcently large and for any κ µ. The desred estmates 5 and 6 are obtaned by combnng and γ = µ. 5. Algorthm mplementaton and numercal results Wrtng U = u u... u T and U γ = uγ uγ... uγ T we obtan the followng equatons of the matrx form from 6: 56 U γ = E + A + BU + G U = U + CU γ

17 SPECTRAL-COLLOCATIO METHODS FOR FFIDES 9 where E s the dentty matrx A j = + x µ µ K x µ µ sx µ µ θ F j sx µ µ θ ω B j = C j = Γγ K x µ µ θ F j θ ω T + x µ µ γ G = gx µ µ gx µ µ... gx µ µ T U = u... T. F j sx µ µ θ ω µ Example. Consder the followng fractonal ntegro-dfferental equaton 57 D.75 yt = 6t.5 Γ3.5 t t e t yt + y =. t e t τyτdτ + t ςyςdς The correspondng exact soluton s gven by yt = t 3 D.75 yt = 6t.5 Γ3.5. Fgure presents the approxmate and exact solutons on the left-hand sde and presents the approxmate and exact dervatves on the rght-hand sde whch are found n excellent agreement. In Fgure the numercal errors are plotted for n both L and L ω µ µ norms. As expected an exponental rate of convergence s observed for the problem whch confrmed our theoretcal predctons. Example. Our last example s about a nonlnear problem n one-dmenson. Consder the followng fractonal ntegro-dfferental equaton 58 D.5 yt = ftyt + gt + + t y = t y τdτ + yςdς wth ft = t + t 3 3 t + t ln + t gt = arcsn h t t 3. π + t The exact soluton s yt = ln + t. Ths s a nonlnear problem. The numercal scheme 6 leads to a nonlnear system for {u } = and a proper solver for the nonlnear system e.g. ewton method should be used. The numercal results can be seen from Fgure 3.

18 Y. YAG Y. CHE AD Y. HUAG.. Exact soluton Approxmate soluton.5 Exact soluton Approxmate soluton Fgure. Example : Comparson between approxmate soluton and exact soluton yt left approxmate fracton dervatve and exact dervatve D.75 yt rght 3 L L ω 3 L L ω Fgure. Example : The errors of numercal and exact soluton yt left and the errors of numercal and exact soluton D.75 yt rght versus the number of collocaton ponts n L and L ω norms. These results ndcate that the spectral accuracy s obtaned for ths problem although the gven functons ft and gt are not very smooth. Example 3. Consder the followng fractonal ntegro-dfferental equaton 59 D α yt = + t yt + t + t y = + e ςt ς yςdς t when α = the exact soluton of 59 s yt = e t. e τt τ yτdτ

19 SPECTRAL-COLLOCATIO METHODS FOR FFIDES.7.6 Exact soluton Approxmate soluton 3 L L ω Fgure 3. Example : Comparson between approxmate soluton and exact soluton of yt left. The errors of numercal and exact soluton yt versus the number of collocaton ponts n L and L ω norms rght Approxmate solutonα=.5 Approxmate solutonα=.5 Approxmate solutonα=.75 Exact soluton α= Approxmate soluton α= 6 5 Approxmate dervatve Exact dervatve Fgure. Example 3: Approxmaton solutons wth dfferent α and exact soluton of yt wth α = left. Comparson between approxmate soluton and exact soluton of y t. In the only case of α = we now the exact soluton. We have reported the obtaned numercal results for = and α = n Fgure. We can see that as α approaches the numercal solutons converges to the analytcal soluton yt = e t.e. n the lmt the soluton of fractonal ntegro dfferental equatons approaches to that of the nteger order ntegro dfferental equatons. In Fgure 5 we plot the resultng errors versus the number of the steps. Ths fgure shows the exponental rate of convergence predcted by the proposed method.

20 Y. YAG Y. CHE AD Y. HUAG L L ω L L ω Fgure 5. Example 3: The errors of numercal and exact soluton ytleft and the errors of numercal and exact soluton y t rght versus the number of collocaton ponts n L and L ω norms. References [] P. Agrawal and P. Kumar Comparson of fve numercal schemes for fractonal dfferental equatons Advances n fractonal calculus 3 6 Sprnger Dordrecht 7. [] W. M. Ahmad and R. EL-Khazal Fractonal-order dynamcal models of love Chaos Soltons Fractals 33 7 no [3] P. Baratella and A. Ors A new approach to the numercal soluton of wealy sngular Volterra ntegral equatons J. Comput. Appl. Math. 63 no. 8. [] A. H. Bhrawy and M. A. Alghamd A shfted Jacob-Gauss-Lobatto collocaton method for solvng nonlnear fractonal Langevn equaton nvolvng two fractonal orders n dfferent ntervals Bound. Value Probl. no [5] A. H. Bhrawy and A. S. Alof The operatonal matrx of fractonal ntegraton for shfted Chebyshev polynomals Appl. Math. Lett. 6 3 no [6] A. H. Bhrawy and M. Alshomran A shfted Legendre spectral method for fractonalorder mult-pont boundary value problems Advan Dffer Eqs. 8. [7] A. H. Bhrawy A. S. Alof and S. S. Ezz-Elden A quadrature tau method for fractonal dfferental equatons wth varable coeffcents Appl. Math. Lett. no [8] A. H. Bhrawy M. M. Tharwat and A. Yldrm A new formula for fractonal ntegrals of Chebyshev polynomals: Applcaton for solvng mult-term fractonal dfferental equatons Appl. Math. Model no [9] C. Canuto M. Y. Hussan and A. Quarteron Spectral Methods Fundamentals n sngle domans Sprnger-Verlag Berln 6. [] Y. Chen and T. Tang Convergence analyss of the Jacob spectral-collocaton methods for Volterra ntegral equaton wth a wealy sngular ernel Math. Comp. 79 no [] D. Colton and R. Kress Inverse Acoustc and Electromagnetc Scatterng Theory Sprnger-Verlag Hedelberg nd Edton 998. [] E. H. Doha A. H. Bhrawy and S. S. Ezz-Elden Effcent Chebyshev spectral methods for solvng mult-term fractonal orders dfferental equatons Appl. Math. Model. 35 no

21 SPECTRAL-COLLOCATIO METHODS FOR FFIDES 3 [3] A Chebyshev spectral method based on operatonal matrx for ntal and boundary value problems of fractonal order Comput. Math. Appl. 6 no [] A new Jacob operatonal matrx: an applcaton for solvng fractonal dfferental equatons Appl. Math. Model. 36 no [5] J. H. He onlnear oscllaton wth fractonal dervatve and ts applcatons In: Internatonal Conference on Vbratng Engneerng Dalan Chna [6] Some applcatons of nonlnear fractonal dfferental equatons and therr approxmatons Bull.Sc. Technol [7] D. Henry Geometrc Theory of Semlnear Parabolc Equatons Sprnger-Verlag 989. [8] F. Huang and F. Lu The tme fractonal dffuson equaton and the advecton-dsperson equaton AZIAM J [9] H. Jafar and S. A. Yousef Applcaton of Legendre wavelets for solvng fractonal dfferental equatons Comput. Math. Appl. 6 no [] M. M. Khader and A. S. Hendy An effcent numercal scheme for solvng fractonal optmal control problems Int. J. onlnear Sc. no [] M. M. Khader. H. Swelam and A. M. S. Mahdy An effcent numercal method for solvng the fractonal dffuson equaton J. Appl. Math. Bonf. no.. [] A. Kufner and L. E. Persson Weghted Inequaltes of Hardy Type World Scentfc ew Yor 3. [3] Y. L. L Haar wavelet operatonal matrx of fractonal order ntegraton and ts applcatons n solvng the fractonal order dfferental equatons Appl. Math. Comput. 6 no [] Y. Lucho and R. Gorenflo The ntal value problem for some fractonal dfferental equatons wth the Caputo dervatves Preprnt seres A8-98 Fachbrech Mathemat und Informat Frec Unverstat Berln 998. [5] F. Manard Fractonal calculus: some basc problems n contnuum and statstcal mechancs Fractals and fractonal calculus n contnuum mechancs Udne CISM Courses and Lectures 378 Sprnger Venna 997. [6] G. Mastroann and D. Occorsto Optmal systems of nodes for Lagrange nterpolaton on bounded ntervals: a survey J. Comput. Appl. Math. 3 no [7] P. eva Mean convergence of Lagrange nterpolaton. III Trans. Amer. Math. Soc no [8] A. Pedas and E. Tamme Pecewse polynomal collocaton for lnear boundary value problems of fractonal dfferental equatons J. Comput. Appl. Math. 36 no [9] I. Podlubny Fractonal Dfferental Equatons Academc Press ew Yor 999. [3] D. L. Ragozn Polynomal approxmaton on compact manfolds and homogeneous spaces Trans. Amer. Math. Soc [3] Constructve polynomal approxmaton on spheres and projectve spaces Trans. Amer. Math. Soc [3] E. A. Rawashdeh Legendre wavelets method for fractonal ntegro-dfferental equatons Appl. Math. Sc. 5 no [33] umercal soluton of fractonal ntegro-dfferental equatons by collocaton method Appl. Math. Comput no. 6. [3] M. Rehman and R. A. Khan The Legendre wavelet method for solvng fractonal dfferental equatons Commun. onlnear Sc. umer. Smul. 6 no [35] A. Saadatmand and M. Dehghan A new operatonal matrx for solvng fractonal-order dfferental equatons Comput. Math. Appl. 59 no [36] A Legendre collocaton method for fractonal ntegro-dfferental equatons J. Vb. Control 7 no

22 Y. YAG Y. CHE AD Y. HUAG [37] G. Samo A. A. Klbas and O. I. Marchev Fractonal ntegrals and dervatves: theory and applcatons Gordon & Breach Yverdon 993. [38]. T. Shawagfeh Analytcal approxmate solutons for nonlnear fractonal dfferental equatons Appl. Math. Comput. 3 no [39]. H. Swelam and M. M. Khader A Chebyshev pseudo-spectral method for solvng fractonal-order ntegro-dfferental equatons AZIAM J. 5 no []. H. Swelam M. M. Khader and R. F. Al-Bar Homotopy perturbaton method for lnear and nonlnear system of fractonal ntegro-dfferental equatons Int. J. Comput. Math. umer. Smul. 8 no [] Y. We and Y. Chen Convergence analyss of the spectral methods for wealy sngular Volterra ntegro-dfferental equatons wth smooth solutons Adv. Appl. Math. Mech. no.. Yn Yang Hunan Key Laboratory for Computaton and Smulaton n Scence and Engneerng Xangtan Unversty Xangtan 5 P. R. Chna E-mal address: yangynxtu@xtu.edu.cn Yanpng Chen School of Mathematcal Scences South Chna ormal Unversty Guangzhou 563 P. R. Chna E-mal address: yanpngchen@scnu.edu.cn Yunqng Huang Hunan Key Laboratory for Computaton and Smulaton n Scence and Engneerng Xangtan Unversty Xangtan 5 P. R. Chna E-mal address: huangyq@xtu.edu.cn