Research Article Cubic Spline Collocation Method for Fractional Differential Equations

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1 Hndaw Publshng orporaon Journal of Appled Mahemacs Volume 3, Arcle ID 8645, pages hp://dx.do.org/.55/3/8645 Research Arcle ubc Splne ollocaon Mehod for Fraconal Dfferenal Equaons Shu-Png Yang and A-Guo Xao Deparmen of Mahemacs, Huzhou Unversy, Guangdong 567, hna School of Mahemacs and ompuaonal Scence, Hunan Key Laboraory for ompuaon and Smulaon n Scence and Engneerng, Xangan Unversy, Hunan 45, hna orrespondence should be addressed o A-Guo Xao; xag@xu.edu.cn Receved February 3; Revsed 7 Aprl 3; Acceped 7 May 3 Academc Edor: Alan Mranvlle opyrgh 3 S.-P. Yang and A.-G. Xao. Ths s an open access arcle dsrbued under he reave ommons Arbuon Lcense, whch perms unresrced use, dsrbuon, and reproducon n any medum, provded he orgnal work s properly ced. We dscuss he cubc splne collocaon mehod wh wo parameers for solvng he nal value problems (IVPs of fraconal dfferenal equaons (FDEs. Some resuls of he local runcaon error, he convergence, and he sably of hs mehod for IVPs of FDEs are obaned. Some numercal examples verfy our heorecal resuls.. Inroducon Inherecenpasyears,heuseoffraconaldfferenal equaons (FDEs has ganed consderable populary n some felds (e.g., nonlnear oscllaon of earhquake [], flud-dynamc raffc model [], maeral vscoelasc heory and physcs [3 6], ec.. Based on hese requremens, he numercal approach has become very mporan o solve FDEs and analyze he expermenal daa whch are descrbed n a fraconal way. The numercal approaches o FDEs have been recenly suded by numerous auhors [6 5]. However, he sae of ar s far less advanced for general fraconal order dfferenal equaons. In he recen years, splne collocaon mehods ncludng wavele mehods have been successfully appled o some nal value problems (IVPs and nal-boundary valueproblemsoffdes[6 3]. Pedas and Tamme [6, 8] dscussed splne collocaon mehods for some classes of IVPs of lnear mulerm fraconal dfferenal equaons and obaned he correspondng convergence resuls. L e al. [9] used he hgher-order pecewse nerpolaon for he fraconal negral and fraconal dervaves, proposed a hgherorder algorhm based on he Smpson mehod for FDEs, and gave he error and sably resuls. L []appledhecubcbsplne wavele collocaon mehod o FDEs by he approach ha such problems are convered no a sysem of algebrac equaons whch s suable for compuer programmng. I s worh noce ha he splne collocaon mehods for he IVPs of FDEs are ofen acheved by her solvng of he equvalen IVPs of negral equaons wh weakly sngular kernels. In hs paper, he cubc splne collocaon mehod s desgned o solve drecly he IVPs of general FDEs. And he correspondng heorecal resuls of he local runcaon error, he convergence, and he sably of he cubc splne collocaon mehod for he IVPs of general FDEs are gven. Ths paper s organzed as follows. In Secon, we propose he splne cubc collocaon mehod for solvng he IVPs of FDEs. In Secon 3, he correspondng heorecal resuls of he convergence and he sably are also gven. In Secon 4, he heorecal resuls are denfed by some numercal examples. In he followng ex, we frs recall he basc defnons of fraconal calculus [6]. Usually, a Dα denoes he apuo fraconal dervave of order α as a Dα y ( I n α D n y ( = { Γ (n α ( τ n α d n y (τ dτ, a dτn = >a, f n <α<n, { d n { d n y (, f α=n, (

2 Journal of Appled Mahemacs where D n s he classcal dfferenal operaor of order n, y( s n mes connuously dfferenable, and I α denoes he negral operaor of order α as I α y ( = { { ( τ α y (τ dτ, a { I, > a, f α>, f α=, where I s he deny operaor. As s well known, here are some dfferen defnons of fraconal operaor excep he apuo fraconal dervave. From a heorecal pon of vew, he mos naural approach s he Remann-Louvlle defnon as R a Dα y ( =Dn I n a y ( d n = Γ (n α d n ( τ n α y (τ dτ, a >a,n <α n. The relaonshp beween he apuo defnon and he Remann-Louvlle defnon can be gven by he followng Lemma. Lemma (see [5, 6]. Le α>and n= α. Assume ha f( s n mes connuously dfferenable, D α f and R D α f exs; hen ( (3 R D α f ( = D α n f ( + f k ( Γ (k α+ ( k α. (4 k=. ubc Splne ollocaon Mehod for FDEs onsder he nal value problem Dα y ( =f(,y(, I=[, T], <α, y ( =y, where f:[,t] R Rs a gven connuous mappng and sasfes he Lpschz condon f (, y f(, z L y z, I = [, T], y,z R. (6 I s well known ha he IVP (5sequvalenoheproblem [4] y ( =y( + ( τ α f(τ,y(τdτ, y ( =y. Le = h ( =,,...,N, h = T/N,N > be he grd ponsof he unformparonof [, T] no he subnervals (5 (7 I =[, ], =,,...,N.OneachsubnervalI,hecubc -splne funcon S(, h canberepresenedas where S (, h S (, h =S (, h, (8 =(+ ( h h S ( + ( ( h 3 S ( +(+ h ( S ( ( ( h, h 3 S ( and S(, h = S ( =y,s ( =hy (. Le ξ=( /h [, ], ξ= ξ=( /h.wecan wre (9as S (, h =S ( + ξh, h (9 = ξ (ξ + S ( +ξ ξs ( +ξ (ξ+s ( ξ ξs (, ( where S ( =S(,h y(, S ( =hs (,h hy (, =,,...,N.Obvously,S(, h [, T], ands(, h y(, for all [,T]. Remark. In hs algorhm, he nal value y ( mus be provded, bu does no need o be gven n problem. Tha s, f we approxmae y (, hen he approxmaon of y ( s gven by usng he low-order mehods. In hs paper, we obaned approxmaon of y ( by usng one-order BDF mehod. Tha s, Then y(h y(h = y + y ( where h s a ny sepsze. h α Γ (α+ f(,y. ( y(h y (, ( h By usng he collocaon condons n each subnerval I and (5, we have Dα S(, h = +φh =f( +φh,s( + φh, h, φ {φ,φ }, (3

3 Journal of Appled Mahemacs 3 where φ,φ are he collocaon pons, and < φ < φ, s a consan. And accordng o (, we have ( S( +φ,h = (φ (φ + φ φ S ( ( S( +φ,h φ (φ + φ φ S ( +( φ (φ + φ φ φ (φ + φ φ ( S ( S ( n= [ 6φ (n, φ j S ( n +(φ (n, φ j φ (n, φ j 3φ (n, φ j S ( n +6φ (n, φ j S ( +(φ (n, φ j 3φ (n, φ j S ( n ] =Γ(4 α h α f( +φ j h, S ( +φ j h, j =,, (6 =(φ,φ ( S ( S ( +D(φ,φ ( S ( S (, (4 where φ j = φ j,=,. Usng he defnon of he apuo fraconal dervave n heform ( andusng(3, we have Γ ( α [ [ +φ j h n n= n S (τ, h dτ ( +φ j h τ α S (τ, h dτ ( +φ j h τ α ] ] =f( +φ j h, S ( +φ j h, h, j =,. Through calculaon, we can ge 6φ α j (α 3+φS ( +φ α j ( 3 + α + 3φ S ( (5 where φ (n, φ j =( n+φ j α ( α(3 α ( n+φ j α (3 α, φ (n, φ j = ( n+φ j α ( α(3 α ( n+φ j α (3 α +( n+φ j α (3 α, φ (n, φ j =φ (n, φ j +( n+φ j α ( α(3 α +( n+φ j α (3 α +( n+φ j 3 α ( n+φ j 3 α. (7 +6φ α j (α 3+φS ( +φ α j ( φ j +4φ j α+6φ j +6 5α+α S ( In order o oban he correspondng eraon formula, we denoe 6φ α (α 3+φ φ α ( 3 + α + 3φ A =( 6φ α (α 3+φ φ α ( 3 + α + 3φ, 6φ α (α 3+φ φ α ( φ +4φα+6φ +6 5α+α B =( 6φ α (α 3+φ φ α ( φ +4φα+6φ +6 5α+α, H ( = ( ( n= n= [ 6φ (n, φ S ( n +(φ (n, φ φ (n, φ 3φ (n, φ S ( n +6φ (n, φ S ( n +(φ (n, φ 3φ (n, φ S ( n ] [ 6φ (n, φ S ( n +(φ (n, φ φ (n, φ 3φ (n, φ S ( n +6φ (n, φ S ( n +(φ (n, φ 3φ (n, φ S ( n ]. (8

4 4 Journal of Appled Mahemacs Obvously, follows from he nequales φ <φ hahemarx A s nverble, and S = A BS +Γ(4 α h α A f A H (, (9 where f =(f +φ,f +φ T S =(S (,S ( T, =(f( +φ,s( +φ,h,f( +φ,s( +φ,h T. ( Le A= A B, B = A, H( = A H (. ( Then, follows ha S =AS +h α Γ (4 α Bf +H(. ( For (, we can oban her numercal soluons by usng he Newon erave mehod. In addon, applyng he collocaon condons n each subnerval I and usng (7, we have S(, h = +φh =y + +φh ( +φh τ α f(τ,s (τ, hdτ, φ {φ,φ }, (3 where φ,φ are he collocaon pons, < φ <φ, and S (, h =(+ ( h h S ( + ( ( h 3 S ( +(+ h ( S ( ( ( h, h 3 S ( (4 where S(, h = S ( = y, S ( = hy (, S ( = S(,h y(, S ( =hs (,h hy (, =,,...,N. Inhspaper,forconvenence,hecubcsplnecollocaon mehod based on he drec dscrezaon wh (3 scalled drec splne collocaon mehod (DSM for shor; and he cubc splne collocaon mehod based on he fraconal order negral equaons (7and(7 s called ndrec splne collocaon mehod (ISM for shor. 3. Theorecal Resuls Lemma 3 (see [5]. Suppose ha z n, n =,,...,N, and sasfy z n h α n c (n j α z j +c, n=,,...,n, (5 = where <α, c, c >s ndependen of h>.then z n c E α (c (nh α, n=,,...,n, (6 where E α s a Mag-Leffler funcon. Lemma 4. Assume ha he funcon g( [, T]. Then, for all φ [, ], [, T h],onehas +h (s τ α g (τ dτ +h = g ( + φh (s τ α dτ + o (h, (7 where <α, <s Ts a consan and sasfes +h=s or s ( + h h, (, ], o( s nfnesmal of hgher order. Proof. Usng he Taylor formula, we have g (τ = g ( + φh + g ( + φh (τ φh + o (h, τ [, + h]. Applyng he negral mean value heorem, we can oban +h (s τ α g (τ dτ +h = (s τ α (g(+φh+g ( + φh (τ φh+o(h dτ +h = g ( + φh (s τ α dτ +g +h ( + φh (s τ α (τ φhdτ+o(h +h = g ( + φh (s τ α dτ +g +h ( + φh (ξ φh (s τ α dτ + o (h +h = g ( + φh (s τ α dτ +g ( + φh (ξ φh [ α ((s α (s h α ] + o (h, (8 (9

5 Journal of Appled Mahemacs 5 where ξ [,+h].snceg( [, T], g ( s bounded on [, T].Wehave +h (s τ α g (τ dτ +h = g ( + φh (s τ α dτ +O(h α [(s α (s h α ] +o(h. onsder he followng. ( If s=+h,hen +h (s τ α g (τ dτ +h =g(+φh (s τ α dτ +O(h α [(+h α ]+o(h +h =g(+φh (s τ α dτ +O(h α hα +o(h +h =g(+φh (s τ α dτ +O(h +α +o(h +h =g(+φh (s τ α dτ + o (h. ( If h s ( + h,hen (3 (3 (s τ α (s (+h α α h α, τ [, + h], α [(s α (s h α ] Thus, +h = (s τ α +h dτ α h α dτ = α h α. (3 α [(s α (s h α ]=o(. (33 By means of (3, Lemma4s esablshed obvously. Theorem 5 (he local runcaon error. If he analycal soluon of he problem (5 y( 4 [, T], hen he local runcaon error of DSM s O(h 4. Proof. Applyng he Taylor expanson and he defnon of S(, h,wehave y ( =(+ ( h h y( + ( ( h 3 hy ( +(+ h ( y( h ( ( h 3 hy ( +O(( 4 = y ( +O(( 4, y ( = y ( +O(( 3, [, ]. (34 From he defnon of he apuo fraconal dervave, we oban Γ ( α [ [ Tha s, +φ j h n= n n (y (τ +O((τ 3 dτ ( +φ j h τ α y (τ dτ ( +φ j h τ α ] ] =f( +φ j h, y( +φ j h + O(φ j h 4, j =,. (35 +φjh Γ ( α y (τ dτ ( +φ j h τ α =f( +φ j h, y( +φ j h + O(φ j h 4 Γ ( α [ n= [ where j=,.moreover, y( ( hy ( n n y( =A( hy ( +hα Γ (4 α B y (τ dτ ( +φ j h τ α 3 +φ j h O((τ dτ + ] ( +φ j h τ α, ] f( +φ h, y( +φ h + O(φ h 4 ( f( +φ h, y( +φ h + O(φ h 4 (36

6 6 Journal of Appled Mahemacs h α A { n= ( { { n= n n n n y (τ dτ ( +φ h τ α y (τ dτ ( +φ h τ α 3 +φh Γ ( α O((τ dτ ( +φ h τ α } + ( 3 +φh Γ ( α O((τ dτ. ( +φ h τ α } ( } (37 Supposng S(, h = y(, for all [, ],and subsung y( no (3, we have y( y( ( hy ( =A( hy ( +hα Γ (4 α B f( +φ h, S ( +φ h, h ( f( +φ h, S ( +φ h, h + H ( +T, H ( = h α A ( ( = h α A ( n= n= n= n= n n n n n n n n S (τ, h dτ ( +φ h τ α S (τ, h dτ ( +φ h τ α y (τ dτ ( +φ h τ α (38 y (τ dτ, ( +φ h τ α S (, h =(+ ( h h y( + ( ( h 3 hy ( +(+ h ( y( h ( ( h 3 hy (, where T s he local runcaon error. (39 ombnng (38, (37whheLpschzcondonyelds T O((φ h 4 hα LΓ (4 α B ( O((φ h 4 +h α A ( Γ ( α (φ h α O(h3, Γ ( α (φ h α O(h3 T h 4, (4 where s an approprae posve consan. And he norm s he -norm of he marx ( n hs paper. ToproveheconvergenceofDSMforheproblem(5, we frs gve he followng lemmas. Lemma 6. If he analycal soluon of he problem (5 y( 4 [, T] and he marxes (φ,φ and D(φ,φ n (4 sasfy ha D (φ,φ exss, and D (φ,φ (φ,φ <, (4 hen he numercal soluons of ISM for problem (5 S(, h sasfy ha lm h S(, h = y(. Proof. For (3, we have S(, h = +φ j h =y + +φjh ( +φ j h τ α f(τ,s (τ, hdτ, S( +φ j h, h S (, h =y, j =,, =y + +φjh [ ( +φ j h τ α f(τ,s (τ, hdτ n= n ( +φ j h τ α f(τ,s (τ, hdτ] n j =,. (4 I follows from (7 ha he analycal soluon y( of he problem (5 sasfes y( +φ j h =y(+ +φjh [ ( +φ j h τ α f(τ,y(τdτ

7 Journal of Appled Mahemacs 7 n= Moreover, (44 and(45 yeld S( +φ j h, h y ( +φ j h = +φjh { ( +φ j h τ α n= n ( +φ j h τ α f(τ,y(τdτ], n [f(τ,s (τ, h f(τ,y(τ] dτ n ( +φ j h τ α n j =,. (43 [f(τ,s (τ, h f(τ,y(τ] dτ} n= n ( +φ j h τ α n f(τ,s (τ, h f(τ,y(τ dτ} L +φjh { ( +φ j h τ α S (τ, h y(τ dτ + n= = hα L { φj n= n ( +φ j h τ α S (τ, h y(τ dτ} n (φ j τ α S( + τh, h y ( +τh dτ ( n+φ j τ α S( n + τh, h y ( n +τh dτ}, j =,. = hα φ j { (φ j ξ α [f ( +ξh,s( + ξh, h n= ( + φ j n ξ α f ( +ξh,y( + ξh ] dξ [f( n +ξh,s( n + ξh, h f ( n +ξh,y( n + ξh] dξ}, j =,. I follows from he defnon of S(, h ha ( y( +hu (u + u u y( +hv =(u V (V + V V ( y( hy ( +( u (u+ u u V (V + V V ( y( + hy ( + =(u, V ( y hy +D(u, V ( y + hy + +O(h 4, u,v [, ], (45 (46 By usng he Lpschz condon, we have S( +φ j h, h y ( +φ j h +φjh = { ( +φ j h τ α [f(τ,s (τ, h f(τ,y(τ] dτ (44 where u= u,v = V.Obvously,for(4and(46, he marxes (u, V and D(u, V are connuous on [, ] [, ]; hus, we have ( S( +hu,h y( +hu S( +hv,h y( +hv =( u (u + u V V (V + V V S( ( S ( y( hy ( n= n ( +φ j h τ α n [f(τ,s (τ, h f(τ,y(τ] dτ} +φjh { ( +φ j h τ α f(τ,s (τ, h f(τ,y(τ dτ +( u (u+ u u V (V + V V =(u, V ( S( y( S ( hy ( S( ( + y( + S ( + hy ( + +D(u, V ( S( + y( + S ( + hy ( + +O(h4, u, V [, ]. (47

8 8 Journal of Appled Mahemacs Therefore, here exs λ >,λ >such ha λ = sup u<v (u, V, λ = sup u<v D (u, V, λ=λ +λ. Obvously, φ j (φ j ξ α dξ = α (φ j ξ α φ j = α (φ j α α, φ j [,]. Applyng he negral mean heorem, we have ( n+φ j ξ α dξ =( n+φ j θ α ( θ ( n α ( + φ α j θ n ( n α ( + φ j n α. (48 (49 (5 I follows from he nequales < φ <φ and n ha ( n + φ j ξ α dξ ( n α ( + φ α j n ( n α ( + φ j α ( n α α, j =,. ombnng (45, (48, and (49 wh(5 and defnng ε =( S( S( y(, hy ( ε S( +φ,h y( +φ =(, S( +φ,h y( +φ X = max { n + ε n }, where +φj = +φ j h, j =,,wecanoban ε hα L[ (λ ε +λ ε + α + h 4 ((+ hα Γ (α+ + α n= ( n α (λ ε n +λ ε n+ ] (5 (5 h α L[ Γ (α+ λx + α n= ( n α λx n ]+ h 4, (53 where, are some posve consans. If he marx D(φ,φ s nverble, hen follows from (47hawehave ( Hence, S ( + y( + S ( + hy ( + = D (φ,φ (φ,φ ( S ( y( S ( hy ( +D (φ,φ ( S( +hφ,h y( +hφ S( +hφ,h y( +hφ +O(h 4. ε + D (φ,φ (φ,φ ε + D (φ,φ ε + 3 h 4 D (φ,φ (φ,φ X + D (φ,φ ε + 3 h 4, (54 (55 where 3 s an approprae posve consan. Take ε l+ = max n + ε n =X, l.thenx l = max n l+ ε n = ε l+, X =X l,and X l = ε l+ D (φ,φ (φ,φ ε l + D (φ,φ ε l + 3 h 4 D (φ,φ (φ,φ X l + D (φ,φ ε l + 3 h 4. Snce D (φ,φ (φ,φ <,wehave (56 ( D (φ,φ (φ,φ X l D (φ,φ ε l + 3 h 4, X l ε l + 4 h 4, (57

9 Journal of Appled Mahemacs 9 where = D (φ,φ /( D (φ,φ (φ,φ, 4 s an approprae posve consan. Accordng o (53, we oban X l h α L[ Γ (α+ λx l + l α n= ( h α Lλ Γ (α+ X l h α L( (l n α λx n ]+ 5 h 4, l α n= (l n α λx n + 5 h 4, (58 where 5 > s an approprae consan. Obvously, here exs (, and h >such ha h α Lλ/Γ(α+ < for h<h.thus, X l h α Lλ ( ( l α n= Le = α λ/( Γ(α;hen (l n α X n + 6 h 4. (59 ε + X l h α l ( (l n α X n + 6 h 4, (6 n= where 6 >s an approprae consan. Applyng Lemma 3, we ge By means of (4, (47, we oban sup S (, h y( T max { sup (u, V N ε u<v + sup D (u, V ε + }+ 7 h 4 u<v (λ +λ E α ( T α 6 h h 4 [ 7 +(λ +λ 6 E α ( T α ] h 4, where 7 s a posve consan. onsequenly, we have Remark 7. ( If φ =φ,hen s nverble, and (63 lm h S (, h =y(. (64 φ (φ + φ φ D(φ,φ =( (65 φ (φ + φ φ ε + X l 6 h 4 E α ((lh α 6 h 4 E α ( T α. (6 By usng he convergence of he Mag-Leffler funcon E α (z [6], we have + φ D φ (φ,φ =( ( φ +φ 3 + φ φ ( φ +φ + φ φ ( φ +φ. ( φ φ ( φ +φ X l, ε + (h. (6 ( onsder (φ φ +φ +φ ( +φ ( +φ D φ (φ,φ (φ,φ =( φ φ 3φ +φ φ 3φ +φ φ φ ( + φ ( +φ φ φ φ φ φ +3 φ φ φ. (67 When D (φ,φ (φ,φ <, herangeofvalues φ,φ canbeobanedbyusngalgorhm 8, whchcanbe shown n Fgures and,respecvely. Algorhm 8. For φ For φ <φ f D (φ,φ (φ,φ < record φ,φ ; end end end plo(φ,φ.

10 Journal of Appled Mahemacs φ.94 φ φ φ (a (b Fgure : The range of values φ,φ when D <(a or D <(b. and he perurbed problem φ Dα z ( =f(, z (, I = [, T], <α, z ( =z, (69 where f sasfes he Lpschz condon (6,respecvely.Then here exss he consan h >such ha, for all h<h,when ys( n,h zs( n,h y z, here exss he consan >such ha ys( n,h zs( n,h y z ;here > s a consan, n=,,...,n. Proof. Frsly, we gve he proof of he concluson (. Applyng DSM, we have φ Fgure : The range of values φ,φ when D <. Dα S (, h = +φh =f( +φh,s( +φh,h, Snce S(, h [, T],weoban =,...,N. (7 Lemma 9. If he analycal soluon of he problem (5 y( 4 [, T], he splne funcons S(, h and S(, h defned as (9 and (4 on he same grds are he numercal soluons of he problem (5 obaned by DSM and ISM, respecvely; hen we have he followng conclusons. ( If lm h S(, h = y(, henlm h S(, h = y(, where y( s he analycal soluon of he problem (5. ( Assume ha ys(, h, zs(, h and ys(, h, zs(, h are he numercal soluons obaned by DSM and ISM for he problem Dα y ( =f(,y(, I=[, T], <α, y ( =y, (68 S( + φh, h =y + +φh =y + [ n= ( +φh τ α Dα τs (τ, h dτ n ( +φh τ α Dα τs (τ, h dτ n +φh + ( +φh τ α Dα τs (τ, h dτ]. (7

11 Journal of Appled Mahemacs I follows from Lemma 4 ha S( + φh, h =y =y + [ n= ( Dα τ S (τ, h τ= n +φ n ( +φh τ α dτ + o (h n + Dα τ S(τ, h τ= +φh +φh ( +φh τ α dτ + o (h ] + [ Dα τ S (τ, h τ= n +φ n= n ( +φh τ α dτ n + Dα τ S (τ, h τ= +φh =y + [ n= +φh + ( +φh τ α f (τ, S (τ, h dτ + o (h ]+o( n ( +φh τ α f (τ, S (τ, h dτ n +φh + ( +φh τ α f (τ, S (τ, h dτ] + o (, (74 where here exss a consan 8 >,γ>such ha o( = 8 h γ. Based on he defnons of S(, h, S(, h,followsha S( +φ j h, h S( +φ j h, h +φjh { ( +φ j h τ α [f(τ, S (τ, h f(τ,s (τ, h] dτ +φh ( +φh τ α dτ] + o (, where o( as h. By means of (7and(7, we oban S( + φh, h =y + [ f( n +φ,s( n +φ,h n= Applyng Lemma4,we ge S( + φh, h =y + [ n ( +φh τ α dτ n +f( +φh,s( + φh, h +φh ( +φh τ α dτ] + o (. n= n ( ( +φh τ α n f (τ, S (τ, h dτ + o (h (7 ( h γ n= n ( +φ j h τ α n [f(τ, S (τ, h f(τ,s (τ, h] dτ} +φjh { ( +φ j h τ α f (τ, S (τ, h f(τ,s (τ, h dτ n= n ( +φ j h τ α n f (τ, S (τ, h f(τ,s (τ, h dτ} + 8 h γ L +φjh { ( +φ j h τ α n= S (τ, h S (τ, h dτ n ( +φ j h τ α n S (τ, h S (τ, h dτ} + 8 h γ, j =,. (75

12 Journal of Appled Mahemacs Based on he smlar proof o ha of Lemma6, follows from Lemma 3 and he nequaly D (φ,φ (φ,φ < ha ε 9 h γ E α ( T α, =,,...,N, (76 where S( S ( ε =(, (77 S( S ( and s defned n Lemma 6, 9 >s a consan. By means of he convergence of he Mag-Leffler funcon E α (z [6], we ge From Lemma 6,wehave ε, h ; lm S (, h = lm S (, h. (78 h h lm S (, h =y(, h (79 lm S (, h = lm S (, h =y(. h h Now, we gve he proof of he concluson ( of Lemma 6. Accordng o he concluson (, we ge ys ( n, h zs ( n,h = ys ( n,h ys( n,h+ys( n,h Theorem (convergence of DSM. If he analycal soluon of he problem (5 y( 4 [, T] and he marxes (φ,φ and D(φ,φ of (4 sasfy ha D (φ,φ exss, and D (φ,φ (φ,φ <, (83 hen DSM s convergen, and lm h S(, h = y(. Proof. Ths resul follows drecly from Lemmas 6 and 9. Now, we consder he sably of DSM. Theorem (sably of DSM. If he analycal soluon of he problem (5 y( 4 [, T] and he marxes (φ,φ and D(φ,φ n (4 sasfy ha D (φ,φ exss and D (φ,φ (φ,φ <, (84 hen ISM s sable; ha s f ISM s, appled o solve he problem (68 and he perurbed problem (69, respecvely, hen, for all <h<h (h >s a consan, one has ( where ε y z, =,,...,N, (85 zs( n,h+zs( n, h zs ( n,h ys ( n,h ys( n,h + ys( n,h zs( n,h (8 ys( ε =( zs ( ys( zs (, (86 + zs( n,h zs( n,h max N ye (,h + max N ze (,h + y z, n=,,...,n, where ye(, h = ys(, h ys(, h, ze(, h = zs(, h zs(, h. Accordng o he proof of he concluson (, here exss h >such ha, for all h<h, Then, max N ye (,h + max N ze (,h y z. (8 ys ( n, h zs ( n,h (+ y z, n=,,...,n. (8 Obvously, ake =+ ; hence, he concluson ( s rgh. ( > s a consan, {ys (,ys ( } are he numercal soluons of he problem (68,and{zS (,zs ( } are he numercal soluons of he perurbed problem (69,and sup ys (, h zs (, h y z, (87 T where >s a consan, ys(, h s he numercal soluon of he problem (68,andzS(, h s he numercal soluon of he problem (69. Moreover, DSM s sable. Proof. Accordng o Lemma 9, f ISM s sable, hendsm ssable.inhefollowngex,wegveheproofofsablyof ISM.

13 Journal of Appled Mahemacs 3 By usng ISM o solve he problem (68andheproblem (69, we have [f(τ,ys (τ, h f(τ,zs (τ, h] dτ} ys (, h = +φ j h =y + +φjh ( +φ j h τ α f(τ,ys (τ, h dτ, j =,, ys (, h =y, zs (, h = +φ j h =z + +φjh ( +φ j h τ α f(τ,zs (τ, h dτ, j =,, zs (, h =z. (88 =(y z + hα φ j { (φ j ξ α [f( +ξh,ys( + ξh, h f ( +ξh,zs( + ξh, h] dξ n= ( + φ j n ξ α [f( n +ξh,ys( n + ξh, h f ( n +ξh,zs( n + ξh, h] dξ}, The prevous equaons can be wren as ys( +φ j h, h =y + +φjh [ ( +φ j h τ α f(τ,ys (τ, hdτ n= n ( +φ j h τ α f(τ,ys (τ, hdτ], n Applyng he Lpschz condon, we ge ys( +φ j h, h zs( +φ j h, h y z j =,. (9 zs( +φ j h, h =z + +φjh [ ( +φ j h τ α f(τ,zs (τ, hdτ Thus, n= ys( +φ j h, h zs( +φ j h, h =(y z + +φ j h { ( +φ j h τ α j =,, (89 n ( +φ j h τ α f(τ,zs (τ, hdτ], n [f(τ,ys (τ, h f(τ,zs (τ, h] dτ n= n ( +φ j h τ α n j =,. (9 +φjh + { ( +φ j h τ α [f(τ,ys (τ, h f(τ,zs (τ, h] dτ y z n= n ( +φ j h τ α n + +φjh { ( +φ j h τ α [f(τ,ys (τ, h f(τ,zs (τ, h] dτ} f(τ,ys (τ, h f(τ,zs (τ, h dτ n= n ( +φ j h τ α n f(τ,ys (τ, h f(τ,zs (τ, h dτ} y z + L +φjh { ( +φ j h τ α

14 4 Journal of Appled Mahemacs y( Numercal soluon by DSM Exac soluon Fgure 3: The error beween he numercal soluons and he exac soluon of Example ; α =.75. ys (τ, h zs (τ, h dτ n= n ( +φ j h τ α n ys( ε =( zs ( ys( zs (, X = max { n + ε n }, ε ys( +φ,h zs( +φ,h =(, ys( +φ,h zs( +φ,h where +φj = +φ j h, j =,. I follows from (9, (93, (49, and (5ha ε y z +h α L[ y z + (λ ε +λ ε + α α n= ( n α (λ ε n +λ ε n+ ] +h α L[ Γ (α+ (λx + n= α ( n α (λx n ] (93 (94 = y z ys (τ, h zs (τ, h dτ} y z +h α L[ Γ (α+ λx + + hα L { φj (φ j τ α ys( + τh, h zs( + τh, h dτ n= ( n + φ j τ α ys( n + τh, h zs( n + τh, h dτ}, α n= ( n α λx n ]. Take ε l+ =max n + { ε n } = X ;hence,x =X l, l. Accordng o he fac ha D(φ,φ s nverble and D (φ,φ (φ,φ <,wehave X l = ε l+ D (φ,φ (φ,φ ε l (9 + D (φ,φ ε l where j=,. Noe ha he defnons of ys(, h, zs(, h. Le λ = sup u<v (u, V, λ = max D (u, V, u<v λ=λ +λ, D (φ,φ (φ,φ X l + D (φ,φ ε l, ( D (φ,φ (φ,φ X l D (φ,φ ε l. (95 Moreover, X l ε l, (96

15 Journal of Appled Mahemacs 5.3 3E 7.5.5E 7. y(.5. Error E 7.5E 7 E 7.5 5E Numercal soluon by DSM Exac soluon Absolue error (a (b Fgure 4: The error beween he numercal soluons and he exac soluon of he problem (9; h =.5, α =.. where = D (φ,φ /( D (φ,φ (φ,φ. By means of (94, we have X l y z + h α L[ Γ (α+ λx l (97 Applyng Lemma3 yelds X l ( y z E α ((lh α ( y z E α ( T α. ( ( h α Lλ Γ (α+ X l + l α n= (l n α λx n ], Usng he convergence of Mag-Leffler funcon E α (z and denong = ( E α ( T α, ( y z (98 we oban + Lh α ( l α n= (l n α λx n. ε + X =X l y z. (3 Obvously, here exs (, and h h α Lλ/Γ(α + <as h<h.then > such ha Moreover, from he defnons of ys(, h and zs(, h,we have X l ( ( y z + h α L l α n= Le = α Lλ/( Γ(α.Weoban X l ( l (l n α X n. (99 y z + hα (l n α X n. ( n= ( ys( +hu,h zs( +hu,h ys( +hv,h zs( +hv,h ( =( u (u + u ys V V (V + V V ( +( u (u+ u u V (V + zs ( ys ( zs ( ( ys V V ( + zs( + ys ( + zs( +

16 6 Journal of Appled Mahemacs Table : The absolue errors of DSM (φ =.85, φ =.99 and he mehods repored n [, ]; α =.5. Error by DSM Error by [] Errorby[] Table : The absolue errors of DSM (φ =.9, φ =.98 and he mehods repored n [, ]; α =.75. Error by DSM Error by [] Errorby[] =(u, V ( ys( ys ( zs ( ys ( +D(u, V ( By means of (3, we ge zs ( + zs( + ys ( + zs( + sup ys (, h zs (, h T, u,v [, ]. max { sup (u, V N ε u<v + sup D (u, V ε + } u<v (λ +λ y z y z, (4 (5 where =(λ +λ ; ha s, ISM s sable. Accordng o Lemma 9,DSMssable. 4. Illusrave Examples In order o demonsrae our heorecal resuls, we apply DSM o he problem (5 and presen some numercal examples n hs secon. Le N error L =h S(,h y(, error L order L order L = N =(h = S(,h y( = log / error L (h/ error L (h, / = log / error L (h/ error L (h, (6,

17 Journal of Appled Mahemacs 7 Table 3: The error beween he numercal soluons and he exac soluon of he problem (9; α =.5, φ =.88, φ =.98. h error L Order L error L Order L / / / / / Table 4: The error beween he numercal soluons and he exac soluon of he problem (9; α =.75, φ =.95, φ =.99. h error L Order L error L Order L / / / / / where error (h s he error funcon whch depends on he sepsze h. Example. onsder he nal value problem [, ] Dα y ( = y( + + α Γ (3 α, The exac soluon s y( =. I=[, ], <α<, y ( =. (7 For dfferen values of α (,,henumercalsoluons for problem (7 are obaned by usng DSM, fas wavele collocaon mehod (FWM for shor repored n [], and he mehod repored n []. When α =.5,hemesepsze h = /6; he absolue errors of DSM, FWM, and he mehod repored n []are shown n Table.When α =.75, h = /6; he absolue errors of hese mehods are shown n Table. The numercal soluons and he exac soluon are shown n Fgure 3. From he numercal resuls, he resuls obaned by DSM are beer han by FWM and he mehod repored n [] n erms of accuracy f he exac soluon s suffcenly smooh. Therefore, DSM s a vald mehod n solvng fraconal dfferenal equaon. Example. onsder he nal value problem [] 43 Dα y ( = Γ (9 α 8 α 3 Γ (5+α/ Γ (5 α/ 4 α/ Γ (+α +(3 α/ 4 3 y( 3/, The exac soluon s y( = α/ + (9/4 α.obvously, y( 4 [, ]. In order o verfy our heorecal resuls, we modfy problem (8 as follows: 43 Dα y ( = Γ (9 α 8 α 3 Γ (7+α/ Γ (7 α/ 6 α/ Γ (5+α 4 +( 3 +α/ 4 3 y( 3/, y ( =, <α. The exac soluon s y( = α/ + (9/4 4+α. I=[, ], (9 Tables 3 and 4 ls he errors and he error orders of DSM wh α=.5and α =.75 and show ha he errors beween he numercal soluons and he exac soluon are very small, respecvely. Fgures 4 and 5 llusrae hgh accuracy of DSM wh φ =.9, φ =.98 and show ha he error s also very small. All of he numercal resuls show ha DSM for solvng nonlnear FDEs s convergen and he mehod s robus. Example 3. onsder he nal value problem Dα y ( +y ( =f(, I = [, T], <α, y ( =θ ( andheperurbedproblem I=[, ], Dα z ( +z ( =f(, I = [, T], < α, z ( =θ. ( y ( =, <α. (8

18 8 Journal of Appled Mahemacs.5 E 5. y(.5. Error.5E 5 E 5.5 5E Numercal soluon by DSM Exac soluon (a Absolue error (b Fgure 5: The error beween he numercal soluons and he exac soluon of he problem (9; α =.9, h =.5. y( θ =. θ =. (a Error Absolue error (b Fgure 6: The numercal resuls of he problems (and(; h =.5, T =., α =.8, θ =., θ =., φ =.9, φ =.98. We use he cubc splne collocaon mehod o solve he problems ( and(, respecvely. Selecng f( = sn(, T =, θ =., θ =., weobanhenumercal resuls gven n Fgure 6. Whenf( = sn(, T =, θ =., θ =., he numercal resuls are gven n Fgure 7. When f( = e, T =, θ =., θ =., henumercal resuls are shown n Fgure 8. From hese fgures, we can see ha he absolue errors of he numercal soluons of he problems ( and( decrease and fnally end o as ncreases. Thus, we can draw he concluson ha he cubc splne collocaon mehod for nonlnear FDEs s sable. The numercal resuls verfy our heorecal resuls. 5. oncluson Inhspaper,hecubcsplnecollocaonmehodwhwo parameers s successfully appled o he IVPs of general FDEs. The resul of he local runcaon error of hs mehod sgven.andheconvergenceandsablyresulsofhecubc

19 Journal of Appled Mahemacs 9 4 y( θ = θ = (a Error Absolue error (b Fgure 7: The numercal resuls of he problems (and(; h =.5, α =.8, θ =., θ =., φ =.9, φ =.98. y(, z( θ = θ = (a Error Absolue error (b Fgure 8: The numercal resuls of he problems (and(; h =.5, α =.8, θ =., θ =., φ =.9, φ =.98. splne collocaon mehod for he fraconal order negral equaons whch s equvalen o he IVPs of general FDEs are obaned. By usng he relaonshp beween he numercal soluons from he cubc splne mehod for he IVPs of general FDEs and he numercal soluons obaned from he cubc splne mehod for he correspondng equvalen IVPs of fraconal order negral equaons, we also oban some resuls of he convergence and he sably of he mehod for he IVPs of general FDEs. Some numercal examples successfully verfy our heorecal resuls and show ha he gven mehod s effcen. Acknowledgmens Ths work s suppored by projecs from NSF of hna (no. 73, no. 63, and no. 6444, Program for hangjang Scholars and Innovave Research Team n Unversy of hna (no. IRT79, he Ad Program

20 Journal of Appled Mahemacs for Scence and Technology Innovave Research Team n Hgher Educaonal Insuons of Hunan Provnce of hna, Huzhou Scence and Technology Program (no. 3, and NSF of Huzhou Unversy (no. YB5. References [] D. Delbosco and L. Rodno, Exsence and unqueness for a nonlnear fraconal dfferenal equaon, Journal of Mahemacal Analyss and Applcaons, vol.4,no.,pp.69 65, 996. [] J. H. He, Some applcaons of nonlnear fraconal dfferenal equaons and herapproxmaons, Bullen of Scence, Technology,vol.5,no.,pp.86 9,999. [3] V. V. Uchakn and R. T. Sbaov, Fraconal heory for ranspor n dsordered semconducors, ommuncaons n Nonlnear Scence and Numercal Smulaon, vol.3,no.4,pp.75 77, 8. [4] V. E. Tarasov and G. M. Zaslavsky, Fraconal dynamcs of sysems wh long-range neracon, ommuncaons n Nonlnear Scence and Numercal Smulaon, vol.,no.8,pp , 6. [5] K. Dehelm, The Analyss of Fraconal Dfferenal Equaons, vol. 4 of Lecure Noes n Mahemacs, Sprnger, Berln, Germany,. [6] I. Podlubny, Fraconal Dfferenal Equaons,vol.98ofMahemacs n Scence and Engneerng, Academc Press, New York, NY, USA, 999. [7] h. Lubch, Runge-Kua heory for Volerra and Abel negral equaons of he second knd, Mahemacs of ompuaon, vol. 4, no. 63, pp. 87, 983. [8] h. Lubch, Fraconal lnear mulsep mehods for Abel- Volerra negral equaons of he second knd, Mahemacs of ompuaon,vol.45,no.7,pp ,985. [9] h. Lubch, Dscrezed fraconal calculus, SIAM Journal on Mahemacal Analyss,vol.7,no.3,pp.74 79,986. [] K. Dehelm, An algorhm for he numercal soluon of dfferenal equaons of fraconal order, Elecronc Transacons on Numercal Analyss,vol.5,pp. 6,997. [] K. Dehelm and G. Walz, Numercal soluon of fraconal order dfferenal equaons by exrapolaon, Numercal Algorhms,vol.6,no.3-4,pp.3 53,997. [] K.Dehelm,N.J.Ford,andA.D.Freed, Apredcor-correcor approach for he numercal soluon of fraconal dfferenal equaons, Nonlnear Dynamcs,vol.9,no. 4,pp.3,. [3] K. Dehelm and N. J. Ford, Numercal soluon of he Bagley- Torvk equaon, BIT,vol.4, no.3,pp ,. [4] F. Lu, V. Anh, and I. Turner, Numercal soluon of he space fraconal Fokker-Planck equaon, Journalofompuaonal and Appled Mahemacs, vol. 66, pp. 9 9, 4. [5] F.Lu,P.Zhuang,V.Anh,I.Turner,andK.Burrage, Sably and convergence of he dfference mehods for he space-me fraconal advecon-dffuson equaon, Appled Mahemacs and ompuaon,vol.9,no.,pp.,7. [6] A. Pedas and E. Tamme, Splne collocaon mehods for lnear mul-erm fraconal dfferenal equaons, Journal of ompuaonal and Appled Mahemacs, vol.36,no.,pp ,. [7] A. Pedas and E. Tamme, Splne collocaon mehod for negrodfferenal equaons wh weakly sngular kernels, Journal of ompuaonal and Appled Mahemacs,vol.97,no.,pp.53 69, 6. [8] A. Pedas and E. Tamme, On he convergence of splne collocaon mehods for solvng fraconal dfferenal equaons, Journal of ompuaonal and Appled Mahemacs,vol.35,no., pp ,. [9]. L, A. hen, and J. Ye, Numercal approaches o fraconal calculus and fraconal ordnary dfferenal equaon, Journal of ompuaonal Physcs,vol.3,no.9,pp ,. [] J. T. Kemppanen and K. M. Ruosalanen, On he splne collocaon mehod for he sngle layer equaon relaed o mefraconal dffuson, Numercal Algorhms, vol.57,no.3,pp ,. [] X. L, Numercal soluon of fraconal dfferenal equaons usng cubc B-splne wavele collocaon mehod, ommuncaons n Nonlnear Scence and Numercal Smulaon,vol.7,no., pp ,. [] F. Dubos and S. Mengué, Mxed collocaon for fraconal dfferenal equaons, Numercal Algorhms, vol.34,no. 4, pp. 33 3, 3. [3] E. A. Rawashdeh, Numercal soluon of semdfferenal equaons by collocaon mehod, Appled Mahemacs and ompuaon,vol.74,no.,pp ,6. [4] M. Welbeer, Effcen Numercal Mehods for Fraconal Dfferenal Equaons and her Analycal Background, Paperfleger, 6. [5] J. Dxon and S. McKee, Weakly sngular dscree Gronwall nequales, Zeschrf für Angewande Mahemak und Mechank, vol. 66, no., pp , 986.

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